Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Characteristics of unskilled, skilled and highly skilled mental calculators Hope, John Alfred 1984

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1984_A2 H66.pdf [ 12.95MB ]
Metadata
JSON: 831-1.0096641.json
JSON-LD: 831-1.0096641-ld.json
RDF/XML (Pretty): 831-1.0096641-rdf.xml
RDF/JSON: 831-1.0096641-rdf.json
Turtle: 831-1.0096641-turtle.txt
N-Triples: 831-1.0096641-rdf-ntriples.txt
Original Record: 831-1.0096641-source.json
Full Text
831-1.0096641-fulltext.txt
Citation
831-1.0096641.ris

Full Text

CHARACTERISTICS OF UNSKILLED, SKILLED AND HIGHLY SKILLED MENTAL OVLOJIATORS By JOHN (JACK) ALFRED HOPE B. Sc. The University of B r i t i s h Columbia, 1965. M. A. The University o f B r i t i s h Columbia, 1972. A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF EDUCATION in THE FACULTY OF GRADUATE STUDIES Department of Mathematics and Science Education Faculty of Education We accept t h i s d i s s e r t a t i o n as conforming t o the required  standard  THE UNIVERSITY OF BRITISH COLUMBIA September, 1 9 8 4 © J o h n A l f r e d Hope, 1984  In p r e s e n t i n g requirements  this thesis f o r an  of  British  it  freely available  agree t h a t for  that  Library  s h a l l make  for reference  and  study.  I  for extensive copying of  h i s or  be  her  g r a n t e d by  s h a l l not  the  be  of  this  of  Mathematics and  The U n i v e r s i t y o f B r i t i s h 1956 Main Mall V a n c o u v e r , Canada V6T 1Y3 Date  October 1,  1984  this  Science Education  Columbia  thesis my  It is thesis  a l l o w e d w i t h o u t my  permission.  Department  further  head o f  representatives.  copying or p u b l i c a t i o n  f i n a n c i a l gain  University  the  s c h o l a r l y p u r p o s e s may  understood  the  the  I agree that  permission by  f u l f i l m e n t of  advanced degree at  Columbia,  department or for  in partial  written  ii ABSTRACT This  study  p r o c e s s e s and  was  concerned  with  the  procedures which characterized  i d e n t i f i c a t i o n of  the  unskilled, skilled,  and  h i g h l y s k i l l e d mental c a l c u l a t i o n performance of high-school students. F i f t e e n s k i l l e d and f i f t e e n u n s k i l l e d m e n t a l c a l c u l a t o r s w e r e s e l e c t e d f r o m 280  senior  performances  on  1 3 - y e a r - o l d was  secondary mathematics students as a r e s u l t o f a  mental m u l t i p l i c a t i o n  test.  One  highly  l a t e r added t o t h e sample. T h e s e t h r e e  their skilled  s k i l l groups  were compared on a number o f d i m e n s i o n s i n c l u d i n g t h e s t r a t e g i e s u s e d to determine the mental p r o d u c t s o f m u l t i - d i g i t f a c t o r s , a b i l i t y r e c a l l numerical equivalents, The  and s h o r t - t e r m memory c a p a c i t y .  study i d e n t i f i e d twelve mental c a l c u l a t i o n s t r a t e g i e s used  by t h e s e s t u d e n t s . The m a j o r i t y  o f u n s k i l l e d s u b j e c t s made  use  to written  of  to  strategies  more s u i t e d  t a s k s . The more p r o f i c i e n t s u b j e c t s upon p r o p e r t i e s s u g g e s t e d by t h e  than mental  extensive  calculation  t e n d e d t o employ s t r a t e g i e s b a s e d  factors.  T h e r e were s t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e s  between  the  mean p e r f o r m a n c e o f e a c h g r o u p on a m u l t i p l i c a t i o n b a s i c f a c t r e c a l l t e s t . A s t a t i s t i c a l l y s i g n i f i c a n t b u t weak l i n e a r r e l a t i o n s h i p between r e c a l l and m e n t a l m u l t i p l i c a t i o n p e r f o r m a n c e e x i s t e d . The  proficient  subjects  equivalents  during  retrieved the  s i g n i f i c a n t l y more l a r g e  numerical  s o l u t i o n o f a mental c a l c u l a t i o n than d i d the  unskilled  s u b j e c t s . The h i g h l y - s k i l l e d s u b j e c t c o u l d r e c a l l q u i c k l y t h e  majority  o f 2 - d i g i t s q u a r e s and many 3 - d i g i t s q u a r e s . S t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e s between t h e p e r f o r m a n c e s o f the  s k i l l e d and  u n s k i l l e d g r o u p s on  four measures o f  short-term  iii memory c a p a c i t y weak  linear  were f o u n d . F u r t h e r  relationship  between c a p a c i t y  performance  existed.  Certain  calculative  task  the interim  and  analysis  aspects  indicated  that only  and mental  of  the  calculations  calculation  initially were  a  stated  found  t o be  p a r t i c u l a r l y s u s c e p t i b l e t o f o r g e t t i n g by mental c a l c u l a t o r s . T h e s e s u g g e s t i o n s f o r f u r t h e r r e s e a r c h were made: s t u d y i n g t h e characteristics o f subjects m e n t a l sums, d i f f e r e n c e s , memory demands determining  imposed  which  who d i f f e r i n t h e a b i l i t y t o d e t e r m i n e and quotients;  evaluating  the  by d i f f e r e n t mental c a l c u l a t i o n  strategies  c a n be  calculation a b i l i t i e s of lesser skilled  used  t o improve  subjects.  short-term strategies; the mental  iv s  TABLE OF CONTENTS  PAGE LIST OF TABLES  xi  LIST OF FIGURES  xii  CHAPTER I. THE PROBLEM Introduction  1 to the Problem  1  Statement o f the Problem  4  Questions about Calculative Strategies  4  Questions about R e t r i e v a l o f Numerical Equivalents  5  Questions about Short-term Memory Processes  5  D e f i n i t i o n o f Terms...  6  Discussion o f the Problem  7  S i g n i f i c a n c e o f the Study  10  Mental C a l c u l a t i o n : A P r a c t i c a l L i f e - S k i l l  11  Mental c a l c u l a t i o n s and exact solutions  13  Estimation as a form o f mental c a l c u l a t i o n  15  P o t e n t i a l Contributions  to Theory and P r a c t i c e . . . .  I I . REVIEW OF RELATED LITERATURE Introduction  17 22  to the Review  22  The Human Information-Processing System  24  S t r u c t u r a l Resources and Mental Functioning  24  Estimating  storage capacity  27  Estimating  a t t e n t i o n a l capacity  30  V  Strategies and Mental Functioning The C h a r a c t e r i s t i c s o f Expert Mental Calculators  32 36  The C a l c u l a t i v e Acxxanplishments o f Expert Mental Calculators  38  The C a l c u l a t i v e Strategies o f Expert Mental Calculators  42  Mental analogue o f the penc i1-and-paper algorithm  44  Distribution  45  Factoring  48  Expert Mental Calculators' Memory for Numerical Equivalents Expert Mental Calculators' Short-Term Memory Capacity  52 54  Implications For Studies of Non-Expert Mental Calculation Experimental Studies o f STM and Mental C a l c u l a t i o n . . .  56 58  Ordering and Mental Calculation  61  Transforming and Mental Calculation  65  Implications Tasks  66  for Studies o f Mental C a l c u l a t i o n  I I I . METHODS AND PROCEDURES  68  Outline o f the Study  68  S e l e c t i o n o f the Subjects  71  Test Instruments and Procedures  76  The Screening Instrument: CALl  76  Item s e l e c t i o n  76  Test administration  85  Scoring  85  vi  The Probing Instxument: CAL2  86  Item s e l e c t i o n  86  Test administration  87  Scoring  90  The Challenge Test: CAL3  91  Item s e l e c t i o n  91  Test administration  91  Scoring  91  The Test o f Written Paper-and-Pencil Computational S k i l l s : WPP  92  Item s e l e c t i o n  92  Test administration  92  Scoring  93  The Test o f R e c a l l o f Basic Facts: BFR  93  Item s e l e c t i o n  93  Test administration  93  Scoring  94  Forward D i g i t Span: FDS  94  Item s e l e c t i o n  94  Test administration  94  Scoring  94  Backward D i g i t Span: BDS  95  Item s e l e c t i o n  95  Test administration.  95  Scoring  95  Delayed D i g i t Span: DDS  95  vii Item s e l e c t i o n  95  Test administration  96  Scoring  96  Letter Span: LS  96  Item s e l e c t i o n  96  Test administration  97  Scoring  97  C l a s s i f i c a t i o n o f Methods o f Solution Pencil-and-paper  mental analogue  97 100  1. No p a r t i a l product retrieved (P&PO)  100  2. One p a r t i a l retrieved (P&Pl)  100  3. Two p a r t i a l s retrieved (P&P2)  102  4. Stacking  102  Distribution  103  1. Additive  103  2. F r a c t i o n a l  104  3. Subtractive  104  4. Quadratic  105  Factoring  106  1. General  106  2. Half-and-double  107  3. A l i q u o t parts  107  4. Exponential  107  R e t r i e v a l o f a Numerical Equivalent R e l i a b i l i t y o f the Instruments and Procedures  108 109  viii R e l i a b i l i t y o f CALl  109  R e l i a b i l i t y o f CAL2 and CAL3  110  R e l i a b i l i t y of WPP and BFR  I l l  R e l i a b i l i t y o f FDS, BDS, DDS, and LS  112  The R e l i a b i l i t y o f the Introspective Reports  113  R e l i a b i l i t y of the C l a s s i f i c a t i o n Scheme  116  The Design of the Study and Methods of Analysis  116  The Logic o f an Extreme Groups Design  116  The S t a t i s t i c a l Analysis of Extreme Group Data....  119  The Adequacy of the Sample Size  123  IV. PRESENTATION OF THE FINDINGS P r o f i c i e n c y i n Mental C a l c u l a t i o n  126  Performance on CALl, the Screening Test  126  Performance on CAL2, the Probing Test  129  Performance on CAL3, the Challenge Test  132  Choice o f Method of Solution and C a l c u l a t i v e Strategy  135  The U n s k i l l e d Subjects* Choices of C a l c u l a t i v e Methods  145  The use of penci 1-and-paper mental analogue  145  The use of d i s t r i b u t i o n  149  The use of factoring  151  The S k i l l e d Subjects' Choices of C a l c u l a t i v e Methods  152  The use o f penci 1-and-paper mental analogue  152  The use o f d i s t r i b u t i o n The use of factoring  155 160  ix The Highly S k i l l e d Subject's Choices o f C a l c u l a t i v e Methods R e t r i e v a l o f Numerical Equivalents R e t r i e v a l o f the Basic M u l t i p l i c a t i o n Facts  164 167 168  Accuracy o f basic f a c t r e c a l l  168  Time t o access a basic f a c t  170  Reconstruction o f basic f a c t s  172  The R e t r i e v a l o f Large Numerical Equivalents  174  Blocking  174  The 12's Facts  179  The Highly S k i l l e d Subject's Memory for Numerical Equivalents  180  Short-Term Memory Processes i n Mental C a l c u l a t i o n  187  Types o f Forgetting During Mental C a l c u l a t i o n  187  Forgetting the i n i t i a l factors  188  Forgetting a completed c a l c u l a t i o n  189  Losing track o f the d i r e c t i o n o f the calculation  192  Forgetting the order o f a s e r i e s  195  Carrying errors  196  Misalignment during addition  197  A Comparison o f Mental and Written Computation  199  Memory Capacity and Mental M u l t i p l i c a t i o n Performance  201  V. SUMMARY AND CONCLUSIONS  208  Introduction  208  Summary and Discussion o f the Findings  209  Questions about Mental Calculation Strategies  209  X  The s t r a t e g i e s  210  The task and choice o f strategy  213  Adapting t o the task  214  E f f i c i e n c y o f a c a l c u l a t i v e strategy  218  1. Eliminating the c a r r y operation  219  2. L e f t - t o - r i g h t c a l c u l a t i o n  221  3. .Retaining a single r e s u l t  223  Questions about R e t r i e v a l o f Numerical Equivalents  224  Basic f a c t r e c a l l  225  R e c a l l o f large numerical equivalents  226  Questions about Short-Term Memory Processes  227  STM capacity  227  Written versus mental c a l c u l a t i o n performance....  227  Forgetting o f information  228  Memory devices  228  Conclusions about STM processes  230  Implications f o r Instruction  237  Concluding Remarks  241  REFERENCES  "  244  REFERENCE NOTES  253  APPENDIX A: Subject Consent Form  254  APPENDIX B: The Instruments  256  APPENDIX C: Portion o f a CAL2 Interview with a S k i l l e d Subject  264  x i  L I S T OF TABLES TABLE I.  II.  III.  IV.  V.  VI.  VII.  VIII.  IX.  X. XI.  XII.  PAGE A N CK/ERVTEW O F E A C H P H A S E O F T H E STUDY I N C L U D I N G P A R T I C I P A T I N G S A M P L E S A N D T H E INSTRUMENTS AND PROCEDURES U S E D  72  SAMPLING D I S T R I B U T I O N OF SUBJECTS P A R T I C I P A T I N G I N THE SCREENING PHASE  74  D I S T R I B U T I O N OF ITEMS SELECTED FOR THE P I L O T T E S T I N G PHASE  79  GENERAL METHODS A N D S P E C I F I C S T R A T E G I E S U S E D TO S O L V E MENTAL M U L T I P L I C A T I O N T A S K S  101  D I F F I C U L T Y I N D I C E S POR A L L 2 0 C A L l I T E M S A D M I N I S T E R E D TO 2 8 0 GRADE 1 1 A N D 1 2 MATHEMATICS STUDENTS  128  FREQUENCY A N D PERCENTAGE O F GENERAL METHODS A N D S P E C I F I C S T R A T E G I E S U S E D B Y S K I L L E D AND U N S K I L L E D S U B J E C T S TO S O L V E C A L 2 MENTAL M U L T I P L I C A T I O N T A S K S  136  FREQUENCY A N D PERCENTAGE O F GENERAL METHODS A N D S P E C I F I C STRATEGIES USED B Y 1 1 S K I L L E D SUBJECTS AND 1 HIGHLY S K I L L E D S U B J E C T TO S O L V E C A L 3 MENTAL C A L C U L A T I O N TASKS  137  PRODUCTS DETERMINED B Y R E T R I E V I N G RATHER THAN OUJCULATING  177  NUMBER O F S U B J E C T S A B L E TO R E C A L L NUMERICAL ECJUIVALENTS S E L E C T E D FROM T H E " 1 2 * 8 T A B L E "  181  SQUARES R E C A L L E D B Y ONE H I G H L Y S K I L L E D S U B J E C T  184  A COMPARISON O F WRITTEN A N D MENTAL COMPUTATIONAL PERFORMANCE  200  MEASURES O F S T M C A P A C I T Y : GROUP S T A T I S T I C S , S I G N I F I C A N C E T E S T S F O R T H E D I F F E R E N C E S BETWEEN T H E MEANS, T H E C O R R E L A T I O N S A N D T H E I R S I G N I F I C A N C E BETWEEN C A L l MENTAL M U L T I P L I C A T I O N PERFORMANCE AND C A P A C I T Y  203  xii LIST OF FIGURES FIGURE 1. 2. 3.  4.  5.  PAGE FREQUENCY DISTRIBUTION OF CALl MENTAL MULTIPLICATION SCORES FOR 280 GRADE 11 AND 12 MATHEMATICS STUDENTS  127  COMPARATIVE PERFORMANCE OF EACH SKILL GROUP ON CAL2 SCORED UNDER DIFFERING TIME LIMITS  131  COMPARATIVE PERFORMANCE OF 11 SKILLED SUBJECTS, THE MOST SKILLED SUBJECT AND 1 HIGHLY SKILLED SUBJECT ON CAL3 SCORED UNDER DIFFERING TIME LIMITS  134  SCATTER PLOT OF FREQUENCY OF SELECTION OF THE DIGIT-BYDIGIT, RIGHT-TO-LEFT CALCULATIVE STRATEGY (P&PO) TO SOLVE CAL2 TASKS AND SCORES ON CALl  140  FREOUENCY DISTRIBUTION OF NUMBER OF STRATEGIES USED BY SKILLED AND UNSKILLED SUBJECTS TO SOLVE 30 CAL2 MENTAL MULTIPLICATION TASKS  143  6.  SCATTER PLOT OF NUMBER OF STRATEGIES USED BY SUBJECTS TO SOLVE 30 CAL2 MENTAL MULTIPLICATION TASKS AND SCORES ON CALl 144  7.  SCATTER PLOT OF TOTAL TIME NEEDED TO RECALL 100 BASIC FACTS ( ACCESS TIME) AND SCORES ON CALl  171  8.  SCATTER PLOT OF FORWARD DIGIT SPAN AND SCORE ON CALl  204  9.  SCATTER PLOT OF BACKWARD DIGIT SPAN AND SCORE ON CALl.... 205  10.  SCATTER PLOT OF DELAYED DIGIT SPAN AND SCORE ON CALl  206  11.  SCATTER PLOT OF LETTER SPAN AND SCORE ON CALl  207  xiii AC^OWT.FJXSMEMS  The author wishes to express sincere expressions o f gratitude to the many people who have made t h i s study possible. He i s e s p e c i a l l y indebted to the chairman o f h i s d i s s e r t a t i o n committee, Dr. James M. S h e r r i l l , for h i s advice, encouragement, and friendship during the writing o f the t h e s i s . Gratitude i s expressed to the members o f h i s committee, Dr. Doug Edge, Dr. Doug Owens, Dr. Ron Jar man, and Dr. G a i l S p i t l e r , for their cooperation and assistance. S p e c i a l thanks are extended to the Saskatoon P u b l i c and Separate School Boards, and t o the p r i n c i p a l s , teachers, and students for their cooperation and willingness to p a r t i c i p a t e i n t h i s study. An expression o f gratitude i s also extended  to the U n i v e r s i t y  of Saskatchewan who provided the author with a sabbatical leave to complete the d i s s e r t a t i o n . The assistance, advice, and encouragement provided by the author's many u n i v e r s i t y colleagues were g r e a t l y appreciated. F i n a l l y , the author wants to wholeheartedly thank h i s family for their encouragement to pursue t h i s work. He i s g r a t e f u l t o h i s parents, Cedric and Louise, for their support during h i s formative years. The author i s e s p e c i a l l y indebted to h i s wife, Charlene, f o r her patience, understanding, and encouragement, and to h i s c h i l d r e n , Shaun and Kim, for t h e i r cooperation and understanding while he completed t h i s study.  1  CHAPTER I THE PROBLEM Introduction t o the Problem In the everyday world o f the consumer and worker, the need for a mental studies  calculation reviewed  will  often a r i s e . A series o f recent B r i t i s h  by the Committee o f Inquiry into  Mathematics i n Schools  (Cockcroft, 1982) concluded  the Teaching o f that many adults  cannot meet t h i s need because they are unable to perform even the most elementary  of  mental  calculations.  The  reviewers  claimed  that:  There are indeed many adults i n B r i t a i n who have the greatest d i f f i c u l t y with even such apparently simple matters as adding up money, checking their change i n shops or working out the cost o f f i v e gallons o f p e t r o l . Yet these adults are not j u s t the u n i n t e l l i g e n t or the uneducated. They come from many walks o f l i f e and some are very highly educated indeed, but they are hopeless a t arithmetic...(Cockcroft, p. 5) The  difficulties  o f adults i n performing mental  calculations  have been reported by the National Assessment o f Educational Progress (NAEP, 1977, 1983a, 1983b).  Asked  to multiply  90 and 70 " i n the  head," 45% o f 17-year-olds were unable to do so (NAEP, 1983b, p. 2). As w e l l , 55% o f the 17-year-olds were unable to c a l c u l a t e mentally 4 x 625  (NAEP,  exercises,  1983a, almost  p.  32). On  orally  presented  40% o f 17-year-olds were unable  mental  division  to determine the  solution t o either o f the items, 480/16 or 3500/35, within a 10 second time period (NAEP, 1983a, p. 11). Over 60% o f the 17-year-olds could not s e l e c t the c o r r e c t estimate to 3.04 x 5.3: the choices were 1.6, 16, 160, and 1600 (NAEP, 1983b, p. 3). As i n other forms o f human reasoning, the a b i l i t y t o determine  2 a mental c a l c u l a t i o n varies greatly among i n d i v i d u a l s . Some people are so  adept at  calculating  mentally  "lightning c a l c u l a t o r s . " A.C. Dase  (Scripture,  account of  1891)  that  they have been described  as  Aitken (Hunter, 1962), Bidder (1856) and  were a  few  such individuals.  Aitken's mental decimalization  of  1/851  The  following  illustrates  the  magnitude of some expert's powers: The instant observation was that 851 i s 23 times 37. I use t h i s f a c t as follows. 1/37 i s 0.027027027027... and so on repeated. This I divide mentally by 23. ( 23 into 0.027 i s 0.001 with remainder 4 ). In a f l a s h I can see that 23 into 4027 i s 175 with remainder 2, and into 2027 i s 88 with remainder 3, and into 3027 i s 131 with remainder 14, and even into 14,027 i s 609 with remainder 20. And so on l i k e that. Also before I ever s t a r t t h i s , I know how far i t i s necessary to go i n t h i s manner before reaching the end of the recurring period: for 1/37 recurs at three places, 1/23 recurs at twenty-two places, the lowest common multiple of 3 and 22 i s 66, whence I know that there i s a recurring period of 66 places. (Hunter, 1962, p. 245) Such a demonstration of recondite knowledge and  reasoning must appear  p a r t i c u l a r l y freakish i n t h i s age of dime-store c a l c u l a t o r s . Although  certainly  mental  calculators,  expert skilled.  not  some  the  individuals  performances of do  seem  to  In an unpublished study conducted by the writer  1), several young adults  Why  are  calculation 1979), Hitch  25 x 25, and 25 x  Researchers  (1977, 1978), and  i n d i v i d u a l s make use  quite  (Hope, Note  48.  there such apparent i n d i v i d u a l differences  performance?  be  the  could determine the mental products of such  c a l c u l a t i v e tasks as 9 x 742,  how  approaching  such  Howe and Ceci  as  Hunter  i n mental  (1962,  1978,  (1979) have argued that  of t h e i r memory during a mental c a l c u l a t i o n  w i l l be an important factor i n determining t h e i r l e v e l of p r o f i c i e n c y . According to Hunter, a mental c a l c u l a t i o n makes demands on memory that are  of  these  three  distinguishable  kinds:  memory for  calculative  3 method,  memory  for  numerical  equivalents,  memory  for  interrupted  working. The  first  two  demands are  met  mainly  by  calling  long-term nemory to provide both c a l c u l a t i v e strategies and useful  numerical  equivalents.  Calculative  method or  on  one's  a set  strategy  of  is  a  "schematic procedure which decomposes your working into a pre-arranged sequence of steps" (Hunter, 1978, the  basic  constituent  parts  equivalents  comraonly  retrieved  calculation  will  be  the  p. 340).  of  the by  calculation.  the  basic  Numerical equivalents are  facts  average of  The  person  addition,  numerical to  solve  a  subtraction,  m u l t i p l i c a t i o n and d i v i s i o n . The "working  t h i r d demand i s met rremory"  (Hitch,  1977,  performing a c a l c u l a t i o n , a step of the route. written  by  some form of temporary storage or 1978;  Howe  &  subject must f i n d  Ceci,  a way  calculations  working nemory i s v i r t u a l l y  because the  subject can  In  to record each  c a l c u l a t i o n as well as to keep track of the  This demand on  1979).  calculative  eliminated  for  employ ciphers to keep  track of the current and past states of the c a l c u l a t i o n . However, for mental c a l c u l a t i o n s ,  the demand must be met  by  the c a l c u l a t o r ' s  own  i n t e r n a l working memory resources. Thus, i n d i v i d u a l differences memory demands could l i k e l y be  i n one  or more aspects of  these  the reason for the great v a r i a t i o n i n  mental c a l c u l a t i o n performance. What remains to be done by researchers i s to p a r t i c u l a r i z e these i n d i v i d u a l differences i n nemory demands. Do i n d i v i d u a l s who or  different  different  d i f f e r i n mental c a l c u l a t i o n performance use  calculative  numerical  strategies?  equivalents  Do  during  they r e t r i e v e the  solution  the same  the of  same or a  mental  4 c a l c u l a t i o n task? Do they possess the same or d i f f e r e n t c a p a c i t i e s t o store and process temporary c a l c u l a t i v e the  present  study was to study  calculation  performance  information? The purpose o f  individuals  i n order  who d i f f e r e d  i n mental  to provide some answers to these  questions.  Statement o f the Problem The which  study attempted to i d e n t i f y the processes and procedures  characterized  calculation involving  unskilled,  performance  multi-digit  during  factors.  skilled,  and highly  the solution  skilled  mental  of c a l c u l a t i o n  S p e c i f i c a l l y , the following  tasks  research  questions were used t o guide the investigation: Questions About C a l c u l a t i v e  Strategies  1. Can i n d i v i d u a l s who d i f f e r i n mental c a l c u l a t i o n performance be characterized  by the types o f c a l c u l a t i v e strategies used to solve  a task? In p a r t i c u l a r , 1-1.  Which strategies are most frequently applied  by each  of the s k i l l groups? 1-2.  Which s k i l l  group most frequently changes from one  strategy to another i n response to a change i n the c a l c u l a t i v e task? 1-3.  Is  there  a  relationship  between  c a l c u l a t i o n task and the type o f strategy selected  the  type  of  t o solve the  task? 1-4. Can any general c h a r a c t e r i s t i c s  be i d e n t i f i e d  appear to d i s t i n g u i s h e f f i c i e n t from i n e f f i c i e n t  that  strategies?  5  Questions About R e t r i e v a l o f Numerical Equivalents 2. Can i n d i v i d u a l s who d i f f e r i n mental c a l c u l a t i o n performance be  characterized  by the types o f numerical equivalents retrieved t o  solve a mental c a l c u l a t i o n task? In p a r t i c u l a r , 2-1.  Do  quickly  the s k i l l  and  groups  accurately  differ  recall  i n their  the  basic  a b i l i t y to facts  of  multiplication? 2- 2. number  Do  of  skilled large  mental  numerical  calculators  retrieve  equivalents  to  a greater  solve  mental  m u l t i p l i c a t i o n tasks than do unskilled mental c a l c u l a t o r s ?  Questions About Short-Term Memory Processes 3. Can i n d i v i d u a l s who d i f f e r i n mental c a l c u l a t i o n performance be characterized  by the e f f i c i e n c y o f their short-term memory systems?  In p a r t i c u l a r , 3- 1.  Are there  differences  among  the s k i l l  groups on  measures o f short-term memory capacity? 3-2.  Do s k i l l e d mental calculators employ d i f f e r e n t memory  devices than do u n s k i l l e d mental c a l c u l a t o r s forgetting o f the i n i t i a l or interim 3-3.  to minimize the  calculations?  What i s the e f f e c t on performance when mental rather  than written methods must be used to solve computational tasks? 3-4.  Which stages o f a mental c a l c u l a t i o n are p a r t i c u l a r l y  susceptible  to forgetting.  6 D e f i n i t i o n o f Terms Mental c a l c u l a t i o n refers to the cognitive to solve  processes  required  numerical c a l c u l a t i o n tasks without benefit o f any external  memory-aids,  including  pencil  and paper.  For the purposes o f the  study, mental m u l t i p l i c a t i o n was the process that was investigated. S k i l l e d , u n s k i l l e d , and highly s k i l l e d mental c a l c u l a t o r s refer to those i n d i v i d u a l s who exhibited of  extremes o f performance i n a t e s t  mental m u l t i p l i c a t i o n . The tests  these subjects are described Calculative decxDmpose Several  a  task  strategies  calculators  and procedures used  to s e l e c t  i n Chapter I I I .  strategy  refers  to the type o f procedure used to  into  series  o f more  a  that  have been  may  have  identified  been  through  tractable used  by  calculations. expert  mental  a thorough review o f the  l i t e r a t u r e . These are discussed i n Chapter I I .  The strategies used by  the subjects who p a r t i c i p a t e d i n the study are presented i n Chapters III and IV. Retrieval  o f a numerical equivalent  information that a subject  describes  the numerical  r e t r i e v e s , but does not c a l c u l a t e , during  the process o f solving a c a l c u l a t i o n task. Short-term memory (STM) refers to a hypothetical function  system whose  i s to process information held for temporary periods.  This  memory system has been referred to as working memory, primary memory, immediate memory, and memory for interrupted working. Long-term memory r e f e r s to a hypothetical system whose function is  to store  memory.  information  i n a more permanent  form  than  short-term  This system has been referred to as secondary memory.  7 Processing capacity r e f e r s to the supposedly l i m i t e d c a p a b i l i t y of  the short-term  memory  system  to process  and r e t a i n  incoming  information. Researchers have used the terms s t r u c t u r a l resources and short-term memory capacity to describe processing capacity. Information-processing  r e f e r s to the models o f memory which  attempt to analyze the flow o f information within the human organism. The  d i g i t a l computer  has been used as an analogy to describe  approach and d e s c r i p t i o n s o f the processing registers,  loops,  rtiemory  stores,  and  o f information  retrieval  process  information  has  been  an  important  through  routines  accompany such models. The l i m i t e d capacity o f short-term  this  often  memory to  feature  of  most  information-processing theories. • Discussion o f the Problem Hunter's  (1962,  1978,  1979)  analyses  of  expert  mental  c a l c u l a t o r s has l e d him to the conclusion that the type o f c a l c u l a t i v e strategy a subject applies to a mental c a l c u l a t i o n w i l l be a major determiner mental  o f performance. He has argued  c a l c u l a t o r s are able  strategy which minimizes their individual  differences  to select  that  the more p r o f i c i e n t  and apply  an  "efficient"  reliance on temporary memory. Thus,  i n mental  calculation  performance  should  r e f l e c t , according to Hunter's argument, differences i n the choice o f strategy:  the s k i l l e d  mental  calculator s e l e c t i n g more  efficient  strategies than the u n s k i l l e d mental c a l c u l a t o r . Evidence i n support o f Hunter's argument has been sparse. H i s conclusions have been based almost e n t i r e l y upon h i s i n v e s t i g a t i o n s o f A.C. Aitken's extraordinary c a l c u l a t i v e performances.  The subject was  8  so a t y p i c a l that any generalizations performance w i l l  be,  about lesser s k i l l e d c a l c u l a t i v e  at best, speculative.  attempt to specify these strategies. in  general terms such as  Moreover, Hunter made no  Ait-ken's methods were described  "unconventional"  (Hunter, 1978)  or  "novel"  (Hunter, 1962). To  help guide the present investigation,  strategies  was  developed  by  the  researcher  a tentative  through  list  examining  the  accounts of expert mental c a l c u l a t i o n provided by Hunter (1962, 1978),  Scripture  (1944), Bidder  (1891),  (1856), Gardner  l i s t of strategies was mental c a l c u l a t o r s , be  used  Mitchell  (1907),  (1977), and  Ball Smith  (1956),  of  1977,  Jakobsson  (1983). Since t h i s  based upon a highly unrepresentative sample of  i t was  to discriminate  not clear which strategies, between the  unskilled,  i f any,  skilled,  and  could highly  s k i l l e d mental c a l c u l a t o r s selected to p a r t i c i p a t e i n the study. Other researchers such as Levine (1982) and Maier examined the  (1977) have  c a l c u l a t i v e strategies of less a t y p i c a l subjects.  have reported that u n s k i l l e d mental calculators could be  They  characterized  by t h e i r use of a mental analogue of the conventional pencil-and-paper algorithm.  Levine's evidence should be considered as only suggestive  because estimation and study.  Maier, on  not mental m u l t i p l i c a t i o n was  the other hand, has provided no empirical  to support h i s claims. solution  used  the focus of  by  Thus, no  subjects  who  study has have  her  evidence  i d e n t i f i e d the methods of  exhibited  differing  levels  of  mental c a l c u l a t i o n performance. Hunter calculators  can  has  presented  retrieve  large  convincing  evidence  numerical  equivalents  that to  aid  expert in  a  mental c a l c u l a t i o n : for example, A. C. Aitken, the expert c a l c u l a t o r  9  studied by Hunter, could r e t r i e v e i n s t a n t l y such equivalents as 123 45 = 5535 without  appearing  (1962, p. 249).  x  to perform any intermediate c a l c u l a t i o n s  R e t r i e v a l of large numerical  equivalents by  expert  c a l c u l a t o r s has been reported by Jakobsson (1944), S c r i p t u r e (1891), Gardner (1977), and B a l l (1956). Such evidence can be considered only suggestive because what i s true for experts might not n e c e s s a r i l y be true for s k i l l e d but non-expert subjects. The great contrast i n performance between c a l c u l a t i o n s done by pencil-and-paper  and  mental  methods  has  led  some  researchers  to  conclude that p r o f i c i e n t mental c a l c u l a t i o n makes great demands on the l i m i t e d resources of short-term memory. Evidence provided by Howe and Ceci (1979), Hitch (1977, 1978, demonstrated  that  mental  1980), and Merkel and  calculation  does  make  H a l l (1982) has  great  demands  on  short-term memory. Assuming that STM processes are a factor i n mental c a l c u l a t i o n performance, calculation  it could  follows reflect  that the  individual differing  differences  processing  are  skilled  should possess greater STM  mental  c a p a c i t i e s that  each i n d i v i d u a l brings to bear upon the c a l c u l a t i v e task. subjects who  in  Thus, those  c a p a c i t i e s than  those  subjects who  are u n s k i l l e d mental c a l c u l a t o r s .  No  have  investigated  the  multiplication  r e l a t i o n s h i p between  performance and processing capacity.  mental  researchers  10  Significance of the Study  In h i s 1958 described  a  short  future  calculations  were  story The  society  solved  by  in a  Feeling of  which  even  computer.  A  scientific  world by  calculation.  low-grade  Technician,  managing  Aub,  the  startled  to reconstruct  s c i e n t i s t s who  Asimov  elementary  had  become  a virtually  so  forgotten  this  futuristic  the ancient  methods of  Such a discovery d i d not go unnoticed  o f p o l i t i c i a n s and  most  People  dependent upon machines that c a l c u l a t i o n was science.  Power, Isaac  by a s e l e c t group  quickly grasped the s i g n i f i c a n c e of  Aub's discoveries: the p o t e n t i a l l i b e r a t i o n of human thought from the i n t e l l e c t u a l s e r v i l i t y imposed by the machine. Shuman, the society,  surmised  c h i e f Programmer that  the  of  this  possessor of  advanced  technological  such c a l c u l a t i v e knowledge  would be provided with great i n t e l l e c t u a l and p o l i t i c a l power: Nine times seven, thought Shuman with deep s a t i s f a c t i o n , i s sixty-three, and I don't need a computer to t e l l me so. The computer i s i n my head. And  i t was amazing the f e e l i n g of power that gave him. Unlike  not  yet  Asimov's future society, our  progressed  c a l c u l a t i o n are  schools how  to  hand-held position  and  & Sherrill,  still  the  threatened  NACOME (NCTM, 1975) (Robitaille  to  point  enjoyed  by  traditional  methods  has of  Many studies including  the B r i t i s h Columbia Mathematics Assessments 1977;  However,  calculators  where  with extinction.  Robitaille,  spend a considerable  compute.  contemporary society  the  seems the  1981)  have reported  amount of time teaching  nearly  destined  traditional  computation c u r r e n t l y taught i n schools.  universal to  methods  of  children  availability  challenge  the  that  of  favoured  pencil-and-paper  11  The c a l c u l a t o r  far surpasses any other previous invention i n  reducing the memory requirements needed by the user. o f a product such as 123 x 456, neither  numerical  The c a l c u l a t i o n  for example, requires the r e c a l l o f  equivalents nor  calculative  method.  Other  than  knowing which keys to push, the order i n which to push the keys, and r e c a l l i n g which e n t r i e s have been made, the user has been freed from the more taxing memory demands normally associated with c a l c u l a t i o n . Since the c a l c u l a t o r holds an obvious advantage over other more conventional important:  techniques  has  mental  obsolete or can  a  of  calculation,  calculation  as  these  an  questions  become  educational goal become  study o f a possible anachronism  be  justified  on  either p r a c t i c a l or t h e o r e t i c a l grounds? The remainder o f t h i s chapter presents  several  arguments  to  answer  these  important  questions.  Mental C a l c u l a t i o n : A P r a c t i c a l L i f e - S k i l l A  curriculum can  be  evaluated i n any  number of ways.  One  popular method o f assessment has been to examine the s o c i a l u t i l i t y of a  curriculum's  content.  According  to  this  viewpoint,  a  good  curriculum should be based upon those s k i l l s needed to solve problems encountered  i n everyday  activity.  Commonly such  skills  have been  i d e n t i f i e d through the study o f people as they proceed through their regular 1959;  activities.  Sauble, 1955;  Several studies  (Brown, 1957;  Wandt & Brown, 1957)  Flournoy,  have attempted  1957,  to i d e n t i f y  the routine uses o f c a l c u l a t i o n , including those requiring a mental calculation. In one study note  their  (Wandt & Brown, 1957), i n d i v i d u a l s were asked to  calculative  activities  over  a  24  hour  period.  The  12 researchers' intention was to determine those a c t i v i t i e s not d i r e c t l y connected  with  calculation  on-the-job  was  placed  mental-approximate, approximate.  performance.  i n one  of  Each  four  reported  categories:  penc i1-and-paper-exact,  and  use  of  a  mental-exact,  penci1-and-paper  The study reported that approximately 75 percent of the  c a l c u l a t i o n s were done using mental procedures while the remaining 25 percent  required  penci1-and-paper  methods.  Even  i n an  era where  c a l c u l a t o r s were v i r t u a l l y non-existent, most people had more need of mental than written methods to solve tasks requiring a c a l c u l a t i o n . The  study a l s o  found  that mental-exact c a l c u l a t i o n s were much more  numerous than mental-approximate methods. that  "considerable  emphasis  mental-approximate  should be  mathematics  at  Wandt and Brown concluded  placed  both  on  mental-exact  elementary  and  and  secondary  l e v e l s " (p. 153). Certainly studies  of  curriculum  routine  curriculum.  behaviour  are  used  to  exercise judge  Many writers (Niss, 1981; Tyler, 1970)  deriving  content from  rapidly  outmoded  mathematical  developers must  studies of contemporary  curriculum.  skills  whose  The utility  caution  the  merit  can  lead  identification  of  a  remain  of  a  have argued that  life  will  when  to a  set  timeless  is  of a  p a r t i c u l a r l y d i f f i c u l t , i f not impossible, task. There  i s no  doubt  that  the ubiquitous hand-held  calculator  would a f f e c t the f i n d i n g s o f a r e p l i c a t i o n o f Wandt and Browns' e a r l y study.  Today's  citizens  would  likely  not  use  mental  calculation  methods as frequently as Wandt and Browns' subjects d i d 25 years ago. Nevertheless,  mental  calculation  can  still  have  great  practical  u t i l i t y despite the continuous p r o l i f e r a t i o n o f c a l c u l a t o r s . I t w i l l  13 remain a convenient t o o l used both to determine an exact s o l u t i o n for a  narrow range  of  numerical  tasks and  to estimate the  results  of  c a l c u l a t i o n s produced by non-mental methods. Mental c a l c u l a t i o n s and exact solutions. Teachers spend a great deal of time consider  i n teaching c h i l d r e n how  to c a l c u l a t e .  student competence i n computation  That teachers  to be the most important  goal of arithmetic has been well-documented. Major surveys including P r i o r i t i e s i n School Mathematics (NCTM, 1981), the National Advisory Ctommittee  on  Columbia  Mathematics  Robitaille,  Mathematical  1981)  Education  Assessments  have concluded  (NCTM, 1975), and (Robitaille  &  the  British  Sherrill,  1977;  that most teachers have spent  the  majority of i n s t r u c t i o n a l time i n arithmetic i n the p u r s u i t of t h i s goal. To  teachers,  computational  facility  has  meant  usually  p r o f i c i e n c y with the conventional pencil-and-paper algorithms.  These  methods of "ciphering" have been taught with the b e l i e f that they are l i f e - s k i l l s e s s e n t i a l for functioning i n the world of the adult.  Yet  the s k i l l s taught i n school are not necessarily the s k i l l s that people use outside of school. Wandt everyday  and  calculation  pencil-and-paper programs. different  Brown  most everyday  (1977)  those  found  that  the  great  tasks were solved using mental  techniques  Maier from  (1957)  claimed  taught  calculative  so  emphasised that  in  majority  of  rather than the  school  adults apply  arithmetic  methods quite  i n school mathematics classes to solve  tasks.  Referring to these unconventional  and often untaught procedures as "folk math," he wrote:  14 Seme o f the general differences between school math and folk math are c l e a r . One i s that school math i s l a r g e l y paper-and-pencil mathematics. Folk mathematicians r e l y more on mental computations and estimations and on algorithms that lend themselves to mental use. When computations become too d i f f i c u l t or complicated to perform mentally, more and more folk mathematicians are turning to calculators and computers. In folk math, paper and p e n c i l are a l a s t resort. Yet they are the mainstay o f school math. (p. 86) Are calculator  Maier's  observations  correct?  and the expected advances  I f they  are,  the modern  i n c a l c u l a t i v e technology w i l l  pose more o f a threat to penci 1-and-paper than to mental methods of calculation. likely  I f a c a l c u l a t i o n task seems complex,  to reach for a calculator  the user i s more  than a p e n c i l and paper.  I f the  c a l c u l a t i o n task i s reasonable, what can be more convenient than a mental c a l c u l a t i o n . may  not always be  As Maier has suggested, "Other computation tools available,  but folk  mathematicians always  carry  their brains with them" (p. 89). Some recent evidence indicates a renewed i n t e r e s t i n elevating the r o l e o f mental arithmetic report Mathematics Counts  i n mathematics  (Cockcroft, 1982)  programs.  The  British  outlined the mathematics  required i n higher education, eriployment, and adult l i f e ,  generally.  The report suggested that mental c a l c u l a t i o n should be given a f a r more prominent p o s i t i o n i n mathematics programs than has been accorded i n the recent past. The report concluded that the "decline o f mental and o r a l work within mathematics classrooms represents a f a i l u r e to recognize the c e n t r a l place which working 'done i n the head' occupies throughout mathematics"  (p. 75).  Admittedly, mental c a l c u l a t i o n most  types o f c a l c u l a t i v e  capabilities  of  the  tasks,  calculator.  i s a limited  especially Few  t o o l f o r solving  when compared  individuals w i l l  use  to the mental  15 methods to solve computations  such as 123 x 456, for example, when a  c a l c u l a t o r i s a v a i l a b l e . On the other hand, i t does not seem to be an unreasonable goal f o r mathematics educators to expect that most people will  be able to use  some form of mental c a l c u l a t i o n to solve tasks  such as 9 x 300, $1.99  + $5.99, 480/16, 1000 - 501, 2 x 555, and even  25 x 48. Exact mental c a l c u l a t i o n w i l l remain as a p r a c t i c a l method o f c a l c u l a t i o n : the use of t h i s type of c a l c u l a t i o n w i l l be  determined  both by the nature o f the c a l c u l a t i v e task and by the c a p a b i l i t i e s of the user. The challenge for teachers w i l l be to help c h i l d r e n develop the necessary c a p a b i l i t i e s . Estimation as a  form of mental c a l c u l a t i o n . Estimation as a  method of determining the reasonableness of a proposed computation of  s o l u t i o n to a  has become increasingly important, e s p e c i a l l y i n the age  c a l c u l a t o r s . The  NCTM's An Agenda for Action (1980) recommended  that: Teachers should incorporate estimation a c t i v i t i e s i n t o a l l areas of the program on a regular and sustaining basis, i n p a r t i c u l a r encouraging the use of estimating s k i l l s to pose and s e l e c t a l t e r n a t i v e s and to assess what a reasonable answer may be. (p. 7) Trafton (1978), Denmark and Kepner (1980), and Levin (1981) have made similar recommendations. Despite (NAEP, 1977, Reys, 1981, Bourke, 1976)  the 1979,  claimed  importance  1983a; Carpenter,  Robitaille & Sherrill, have demonstrated  of  estimation, many studies  Corbitt,  1977;  Kepner, Lindquist  R o b i t a i l l e , 1981;  &  Keeves &  that both children and adults are not  very good a t estimating solutions to c a l c u l a t i o n tasks.  For example,  only 54% of 17-year-olds and 64% o f young adults could estimate a sum to  the  nearest m i l l i o n  dollars  given four hypothetical government  16 expenditures  (Carpenter,  more s u r p r i s i n g was could provide 1979,  p.  the  Coburn, Reys & Wilson, finding  a reasonable  39).  This  latter  that only  estimate result  since the choices were 1, 2, 19, and Why determining  do  children  and  to the  of  p.  the  30). Even  17-year-olds  item 12/13  seems p a r t i c u l a r l y  + 7/8  (NAEP,  disheartening  21!  adults  q u a n t i t a t i v e estimates?  have concluded  37%  1978,  have  such  apparent  Some authors  difficulty  (Skvarcius,  1973)  that estimation simply has not been taught i n schools.  I f t h i s conclusion i s correct, the disappointing findings reported by many studies merely r e f l e c t a lack of i n s t r u c t i o n a l opportunity. Other studies have a t t r i b u t e d the d i f f i c u l t i e s i n estimating to the  failure of  One  study  the  subject to employ useful estimation s t r a t e g i e s .  (Reys, Rybolt, Bestgen & Wyatt, 1982), which attempted  to  i d e n t i f y and describe the processes used by good estimators, reported that  good  estimators  could  be  characterized by:  (Note:  underline  emphasis i s mine) ...the quick and e f f i c i e n t use of mental computation to produce accurate numerical information with which to formulate estimates. A l l estimators exhibited well-developed s k i l l with multiples of 10 or a limited number of d i g i t s , and many others were fluent i n mentally computing with larger numbers, more d i g i t s , and even d i f f e r e n t types of numbers (eg., f r a c t i o n s ) . On some problems, subjects resorted to mental computation rather than using an estimation technique. For these problems and these students, i t was more e f f i c i e n t for the person to compute mentally rather than estimate, (p. 197-198) The  close  established purporting  r e l a t i o n s h i p between by to  Reys teach  et  estimation  a l . implies  estimation  and  that no  skills  can  mental  calculation  instructional afford  to  program  ignore  the  teaching o f mental c a l c u l a t i o n . Estimation, after a l l , can be thought o f as no more than a form of "less precise" mental c a l c u l a t i o n .  17  P o t e n t i a l Contributions t o Theory and Practice It  has been argued  that  despite  the advances i n e l e c t r o n i c  c a l c u l a t i v e aids, mental c a l c u l a t i o n w i l l continue to be a worthwhile goal i n mathematics education.  I f , as Begle and Gibb  (1980) have  suggested, the purposes o f research i n mathematics education are t o "find  out how and why something works and then to see what works i n  practice" first  (p. 8), the present study could be considered a necessary  step  in  improving  the  proficiency  of  unskilled  mental  calculators. The  study may demonstrate that the s k i l l e d and u n s k i l l e d mental  calculators would  differ  suggest  calculators  i n their  that  could  choices of strategies.  the performance be  improved  o f the lesser  i f teachers  Such a  finding  skilled  mental  provide  them  with  i n s t r u c t i o n on how and when to use the strategies o f the more s k i l l e d mental c a l c u l a t o r s .  Before such i n s t r u c t i o n  i s planned, however, i t  would be wise to compare the s k i l l e d and u n s k i l l e d subjects'  methods  for clues about the e f f i c i e n c y o f p a r t i c u l a r techniques. For example, some methods o f the s k i l l e d amount o f information  that  subjects might be e f f i c i e n t must be dealt  reduced. An examination o f the unskilled efficient difficult  techniques t o carry out,  calculations. during  could  reveal  because the  with a t one time can be subjects'  mental  apparently  operations  which  e s p e c i a l l y during p a r t i c u l a r l y taxing  Knowing what the learner  a mental c a l c u l a t i o n  should do or should  has obvious implications  less are  mental not do  f o r designing  effective instruction. The  possible  finding  that  the s k i l l  groups d i f f e r  i n some  fundamental arithmetic processes such as the a b i l i t y to r e c a l l quickly  18  and  accurately basic m u l t i p l i c a t i o n facts would be a d d i t i o n a l u s e f u l  information and Ford that  f o r p r a c t i t i o n e r s . Several researchers  including Resnick  (1981), Case (1978), and Gagne (1983a, 1983b) are convinced  the information-processing  demands o f complex  tasks  can be  reduced i f the component behaviours have been developed t o the point of automaticity. ensures  that  As Gagne has argued, automatization  the scarce  cognitive  resources  of basic  of attention  skills can be  devoted to the more i n t r i c a t e and complex parts o f the c a l c u l a t i v e task  (1983a, p. 15). Thus, many s k i l l e d subjects' mental c a l c u l a t i o n  performances might be improved by ensuring  that their r e c a l l o f basic  facts becomes highly automatized. On a l e s s o p t i m i s t i c note, s k i l l e d mental c a l c u l a t i o n could be found  to be associated  those possessed  with short-term  memory resources  by the average young adult.  f a r beyond  Such a f i n d i n g would  suggest that p r o f i c i e n t mental c a l c u l a t i o n performance requires some basic  and e s s e n t i a l l y unalterable  mechanistic  processes which some  individuals can never possess. I f that i s so, the study could  help  educators  from  identify  i n d i v i d u a l s who would  not l i k e l y  benefit  i n s t r u c t i o n i n mental c a l c u l a t i o n . External memory-aids might always be a necessity for these subjects. A d i r e c t p r a c t i c a l consequence i s not the only measure o f the significance o f a study. to mathematics education  A study can make a s i g n i f i c a n t contribution i f , through the attempt to explain how people  learn and use a s p e c i f i c s k i l l such as mental c a l c u l a t i o n , i n s i g h t i s gained i n t o how people use memory and cognition, i n general. This type o f information  could be important because many researchers  A l l p o r t (1980a, 1980b), Baddeley (1981), Claxton  including  (1980), H i t c h (1980),  19  Neisser has  (1978), and Nilsson  been  made  towards  (1979) have argued  understanding  how  that l i t t l e  memory  progress  contributes to  cognition i n everyday settings. As  an important  step  towards reform,  researchers have been  urged to move from the study o f memory as i t r e l a t e s to conventional "laboratory  tasks"  to  "real-world" cognition. speak,  the study  o f memory  as  i t relates  to  The view that memory research should, so t o  "come out o f the l a b and see the r e a l  world,"  has been  expressed by Neisser (Claxton, 1980): A psychology that cannot interpret ordinary experience i s ignoring almost the whole range o f i t s natural subject matter. I t may hope to emerge from the l a b some day with a new array o f important ideas, but that outcome i s u n l i k e l y unless i t i s already working with p r i n c i p l e s whose a p p l i c a b i l i t y to natural situations can be forseen. (p. 20) Other  researchers  (1980) argued typical  that  laboratory  ecological v a l i d i t y "  have  expressed  similar  sentiments.  Hitch  there i s a "need to move beyond the bounds o f experimentation  by  considering  the  issue  (p. 157). To Hitch, an e c o l o g i c a l v a l i d  of  study  means that the focus o f the study should be upon tasks that are more representative o f normal everyday cognition. Neisser's (1978) proposed reforms for memory research require a major s h i f t o f thinking for many researchers. He suggested, "We should be c a r e f u l i n what we say about memory i n general u n t i l we learn more about these many memories i n p a r t i c u l a r "  (p. 19). The present study,  which investigated how d i f f e r e n t individuals used memory to cope with the demands o f mental m u l t i p l i c a t i o n , appears  to be both timely and  worthwhile. A  study  o f how  individuals  manage  to cope  with  everyday  20  situations  which  require  some use o f memory can help  researchers  i d e n t i f y those concepts developed i n t r a d i t i o n a l experimental studies which might be fundamentally inadequate to explain the more prosaic uses  o f memory. T h i s  knowledge can be used  theories about memory. For example,  to r e f i n e  particular  Cole, Hood and McDermott (1982)  have argued that because "everyday l i f e contexts f o r thinking d i f f e r in  important ways from the contexts assumed to have been obtained i n  laboratory  tasks"  (p. 373), these differences  could  undermine any  attempts t o generalize from one research setting t o another. Cole conclusions  e t a l . used reached  an interesting  analogy to demonstrate why  i n an experimental setting  do not necessarily  apply to everyday instances o f memory use. They stated: In b r i e f , one i s advised t o think o f our examples o f everyday cognitive tasks as bearing a r e l a t i o n to closed experimental tasks that i s analogous t o the the r e l a t i o n o f a sieve t o a bowl. I f the bowl i s an environment which completely constrains i t s contents, a sieve i s more open space than netting; there i s enough metal netting t o provide the sieve with a recognizable shape o f a bowl. But, l i k e a sieve, and unlike a bowl, our s p e c i f i c a t i o n o f task and behaviour i n everyday l i f e cannot hold water, (p. 372) Empirical investigations o f mental c a l c u l a t i o n processes have been  based on the performance o f subjects on "closed" experimental  tasks. In such a s e t t i n g l i t t l e opportunity e x i s t s for the resourceful person to lessen the memory load by selecting e f f i c i e n t  calculative  strategies and e f f e c t i v e memory techniques. The generalizations about individual  differences  i n calculative  performance  based  on these  l i m i t e d task environments might not apply to the more t r a c t a b l e mental m u l t i p l i c a t i o n tasks used i n the present study. Consequently,  this  study could  identify  some aspects about  memory as i t i s used i n solving mental m u l t i p l i c a t i o n tasks that have  21  not been apparent  i n experimental studies of mental c a l c u l a t i o n .  the very l e a s t , t h i s knowledge could: ...prevent us making hasty statements about what people can't do that they do i n f a c t do i n their normal (extra-experimental) l i v e s . And very l i k e l y i t would stimulate new and s i g n i f i c a n t areas o f research as we watch and puzzle over cognition i n i t s natural habitat. (Claxton, 1980, p. 19)  At  22 CHAPTER I I  REVIEW OF RELATED MTERATL7RE  Introduction t o the Review Since determine  one o f the objectives o f the present  i f skilled  and  unskilled  mental  study  calculators  was to  could  characterized by their d i f f e r e n t i a l uses o f memory, a thorough  be  review  o f memory research was considered to be necessary. Because the sheer magnitude and d i v e r s i t y o f memory research i s beyond the scope o f t h i s study, t h i s review o f research has been r e s t r i c t e d to those conceptual approaches which appeared  to hold the greatest promise f o r explaining  i n d i v i d u a l differences i n mental c a l c u l a t i o n . Information- processing theories seemed to hold that promise. These  theories place great emphasis on sequentially  ordered  events, actions, and manipulation o f information: the very processes necessary f o r p r o f i c i e n t mental c a l c u l a t i o n . Resnick and Ford  (1981)  have stated the focus o f information-processing theories i s on: ... the structure o f knowledge within the mind and on the mechanisms by which knowledge i s manipulated, transformed, and generated i n the process o f solving the myriad problems humans face both i n adapting t o p r a c t i c a l demands o f t h e i r environments and i n following their more i n t e l l e c t u a l pursuits, (p. 197) The approach has been used recently by researchers t o study a v a r i e t y o f the more complex forms o f human cognition. For example, Pascual-Leone  (1970),  Romberg and C o l l i s  Case  (1972,  1974a,  1974b, 1975, 1978) and,  (1981) have provided a d i f f e r e n t perspective from  which to view Piagetian research. The theory has been used to analyze these  other  areas o f human  functioning: problem solving  (Newell &  23  Simon, 1972), i n t e l l i g e n c e (Hiebert,  Carpenter,  &  Moser,  place-value comprehension mathematics Hitch  (Hunt, 1980), arithmetic problem solving 1982),  differences i n  (Brockmann, 1978), learning d i s a b i l i t i e s i n  (Nason & Redden, 1983). Only a few researchers such as  (1977, 1978), Merkel and H a l l  Silikowitz  individual  (1982), Whimbey, Fischhof and  (1969) and, Dansereau and Gregg  (1966) have attempted to  examine mental c a l c u l a t i o n from an information-processing perspective. This chapter w i l l of  the human  i d e n t i f y and explain the prominent features  information-processing  system.  Since  most  empirical  studies o f mental c a l c u l a t i o n performance have used t h i s t h e o r e t i c a l framework, a review o f the major concepts  and issues should  provide  the reader with the necessary background material. This review proved d i f f i c u l t because there seemed to be as many theories as there were researchers. Often the important one  researcher  another  seemed to have been e a s i l y dismissed  researcher.  There  numerous  analysis!  theories  or ignored by  are so many v a r i a t i o n s that Tulving and  Madigan (Shallice, 1979, p. 258) suggested, these  concepts o f  could  be  somewhat f a c e t i o u s l y , that  discriminated  In spite o f t h i s d i f f i c u l t y ,  only  by  factor  information processing  does  provide a u s e f u l framework within which to examine the topic o f the study: the c h a r a c t e r i s t i c s o f subjects d i f f e r i n g i n mental c a l c u l a t i o n performance.  24  The Human Information-Processing System Although  there  are many  different  theories  o f information  processing, they are a l l based on an analogy with a d i g i t a l computer. J u s t as a computer needs several types o f resources t o perform tasks, the human performer  needs them as w e l l .  (1980) and Newell and Simon  Researchers  (1972) have argued  such  as Hunt  that s t r u c t u r a l and  strategy resources are o f p a r t i c u l a r importance to any problem solving machine, be i t computer figured  prominently  performance,  or human. Since these  i n many  two resources have  explanations o f p r o f i c i e n t  cognitive  including mental c a l c u l a t i o n , each w i l l be discussed i n  d e t a i l under separate headings. S t r u c t u r a l Resources and Mental Functioning S t r u c t u r a l resources refer to those "mechanical c a p a c i t i e s f o r storing, r e t r i e v i n g and transforming information" (Hunt, 1980). Just as the hardware c h a r a c t e r i s t i c s o f a d i g i t a l computer s e t l i m i t s on the machine's a b i l i t y t o reason, s t r u c t u r a l resources "set l i m i t s on the effectiveness o f s p e c i f i c information-processing functions"  (Hunt,  p. 471). But  what are these human s t r u c t u r a l resources? E s s e n t i a l l y ,  they involve the processes postulated to be endemic to the short-term or  working-memory system. This memory system, or memory structure as  i t has sometimes been c a l l e d , was developed as a construct to explain why  human  beings  have  such  apparent  difficulty  in  retaining  information over short periods o f time. William distinguish  James  (1956)  was  one o f the f i r s t  theorists  between consciousness and the more permanent  to  long-term  25  memories that  a l l human beings  possess.  He used  the term primary  memory to describe the " j u s t past" (p. 646). The term secondary memory was reserved  to describe memory proper:  state o f mind a f t e r  the "knowledge o f a former  i t has already dropped from consciousness"  (p.  648). Modern t h e o r i s t s postulate that there  i s something  different  about the way humans remember information over shorter periods o f time as opposed to the way they renember information over longer periods o f time.  Such  systems with memory,  differences differing  short-term  are believed processing  store,  to r e f l e c t  different  mechanisms. The terms  primary  memory,  immediate  memory  short-term  memory, and  working memory have been used by d i f f e r e n t t h e o r i s t s to describe the hypothetical  system that processes  ephemeral information. Long-term  memory and secondary memory have been used  to  refer  to the more  i n d e l i b l e forms o f memory. Broadbent  (1958) was one o f the f i r s t researchers  to use the  computer metaphor to explain human memory processes. He proposed that humans, l i k e any other  p h y s i c a l system, are limited i n the rate a t  which they can process information. His c e n t r a l thesis was that t h i s , .quite t h e o r e t i c a l , upper l i m i t on information-processing capacity, i n bits-per-second, nature  was  o f conscious  transmission  flow  the d i r e c t attention  (Allport,  p h y s i c a l basis and  acted  as  f o r the s e l e c t i v e a  1980b, p. 114). This  "bottleneck"  to  idea o f a h i g h l y  specialized memory system with a l i m i t e d capacity t o store information has become a d i s t i n g u i s h i n g c h a r a c t e r i s t i c o f a " s t r u c t u r a l model o f memory" ( Zechmeister & Nyberg, 1982, p. 49). Researchers such as Waugh and Norman (1965), and Atkinson and  26 Shiffrin  (1968, 1971) soon extended and elaborated upon Broadbent's  earlier  model.  In Atkinson  and  Shiffrins'  three-store  model,  short-term memory was given, "a p o s i t i o n o f p i v o t a l importance" (1971, p.  82).  The  considered  authors  explained  that  "the short-term  store i s  a working memory: a system, i n which decisions are made,  problems are solved and information flow i s d i r e c t e d " (p. 83). The  working  memory system  model was considered held  information.  i n Atkinson  and S h i f f r i n s '  early  to be more than a simple store f o r temporarily  The authors  argued  that  c o n t r o l processes,  or  cognitive a b i l i t i e s that directed the flow o f information, were a l s o required  f o r short-term  control  processes  memory retention. The authors  such  as  rehearsal  and  coding  believed that had  "enormous  consequences for performance" (p. 82). Undoubtedly,  these  information-processing these  models  research,  have  had  but most  great  such as A l l p o r t  failures  (1980a, 1980b), Claxton  (1982), and Craik and Lockhart  on  theorists have abandoned  e a r l y multi-store models. The experimental  researchers  influence  c i t e d by  (1980), Crowder  (1972) and the apparent i n a b i l i t y to  explain complex but everyday human cognition (Neisser, 1978; Nilsson, 1979;  Hitch,  1980) have  considerably  weakened  support  f o r these  models. Despite  the shortcomings o f the multi-store models o f memory,  the concept o f a limited-capacity working memory i s s t i l l regarded as sound by many memory researchers. After a l l ,  "any c r e d i b l e theory o f  memory must account f o r the f a c t that people can consciously only  a  limited  Butterfield,  amount  o f material  a t once"  1979, p. 272). For t h i s reason,  process  (Lachman,  Lachman &  the various  successors  27  developed  by Anderson  (1980),  Newell  and Simon  (1972),  (1981), Nason and Redden (1983), and Baddeley and Hitch  Baddeley  (1974) have  retained the notion that every human has a l i m i t e d capacity t o r e t a i n and process temporary information. What Experimental  i s the  capacity  estimates  of  the short-term  of short-term  memory  memory  capacity  system?  have  proved  elusive because, f o r one reason, researchers are not always c l e a r what they  mean  by  capacity  limitations.  What  i s clear  i s that  how  researchers p i c t u r e short-term memory w i l l influence t h e i r attempts to ascertain i t s capacity. Estimating been  storage  capacity.  Short-term  memory capacity has  referred t o as the quantity o f information a memory system can  hold. Researchers who have held t h i s view o f capacity p i c t u r e working memory  as a type  o f container  d i g i t a l computer, although  similar  t o the storage  a more folksy image o f a "bucket" would  a l s o be appropriate. Several writers have appeared Wortman and Loftus does short-term discussing stated, takes  t o hold t h i s view.  (1981) asked t h i s question: "How much information  memory hold?"  the l i m i t e d  (p. 182). Resnick and Ford  processing  (1981), i n  c a p a b i l i t e s of working memory,  "no one knows exactly how many  to ' f i l l  buffer o f a  'pieces' of information i t  up' working memory..." (p. 31). S i m i l a r l y ,  Atkinson,  Atkinson, and H i l g a r d (1983) suggested: Given t h i s f i x e d capacity, i t i s tempting t o think o f short-term memory as a s o r t o f mental box with roughly seven s l o t s . Each item entering short-term memory goes i n t o i t s own s l o t . So long as the number o f items does not exceed the number o f s l o t s , we can r e c a l l items p e r f e c t l y , (p. 225) For container,  those  researchers who have likened short-term memory t o a  i t n a t u r a l l y follows that capacity would be measured by  28  somehow " f i l l i n g degradation storage  the memory container  with  information" u n t i l  some  i n r e c a l l occurs. The most popular method o f estimating  capacity has been d i g i t  span. I t has been used  to assess  "higher mental processes" since about 1885 when the f i r s t experiments on  memory span were recorded  by Ebbinghaus (Dempster, 1981). Memory  span i s ccimmonly measured by requiring a subject to r e c a l l s e r i e s o f d i g i t s that have been presented o r a l l y , usually a t a rate o f about one digit-per-second. The maximum number o f d i g i t s a subject can r e c a l l i n perfect order i s taken as the subject's "iirmediate memory span." Many standardized  tests  including the various  editions o f the Wechsler  Adult I n t e l l i g e n c e Scale have incorporated d i g i t span as a subtest. What  i s the span  of  the average  person?  Unfortunately,  estimates vary. Because there has been no consensus regarding the type o f the information used to f i l l d i f f e r e n t techniques for  short-term  memory to i t s capacity,  using d i f f e r e n t tasks y i e l d d i f f e r e n t  estimates  d i f f e r e n t ages o f subjects. An estimate o f capacity i s influenced by a subject's store o f  long-term knowledge. One subject could view the d i g i t s 1, 4, 2, 8, 5, 7  as a  s e r i e s o f s i x random d i g i t s  while  another  subject  recognize the s e r i e s as the recurring period for the decimal 1/7.  could  fraction  For the former subject, there are s i x units o f information; f o r  the l a t t e r subject, there i s but one u n i t . Miller "chunking"  (1956) would say that each subject employed d i f f e r e n t  procedures.  Miller's  storage capacity has been c i t e d  estimate  o f the average  person's  i n the famous form: "Seven, plus or  minus two." This has meant that the average person can hold  seven  "chunks" or units o f information f o r temporary periods o f time. But,  29  to  use M i l l e r ' s  words,  "we  are not very  definite  about  what  constitutes a chunk o f information" (p. 93). This ambiguity about the s i z e o f a chunk has caused great d i f f i c u l t i e s i n determining  estimates  o f the capacity o f working memory. Zechmeister  and Nyberg  clearer definition, organized pattern"  according  (1982),  i n an attempt  to provide  a  have defined a chunk as a "unit o f information to a  rule  or corresponding  (p. 43). This d e f i n i t i o n s t i l l  to some  familiar  suffers from some ambiguity  since the f a m i l i a r pattern w i l l vary from subject to subject depending upon the type  o f information  that each  subject can r e t r i e v e from  long-term memory. The influence o f long-term memory explains some o f the v a r i a b l e estimates  obtained  f o r short-term  memory capacity. Depending on the  measure used, the short-term store seems capable o f maintaining from 2 to 20 words (Lachman, Lachman & B u t t e r f i e l d , 1979, p. 268). This f a c t that  long-term  memory  processes  capacity o f the short-term  can a f f e c t  the estimate  store has not been very  o f the  satisfying for  those theorists who have postulated the r e l a t i v e independence o f each memory system. That chunking can g r e a t l y influence estimates was well demonstrated by a subject p a r t i c i p a t i n g Ericsson,  Chase, and Falcon  (1980) reported  o f memory span  i n a recent  study.  the performance o f one  subject who used chunking t o enlarge h i s memory span. H i s knowledge and  interest  i n the sport o f long-distance  group d i g i t s according his  memory  span  running  t o a d i s c e r n i b l e "running  increased  from  a average  enabled  him to  time." Consequently  o f seven  digits  to a  phenomenal 79 d i g i t s ! This subject was a t y p i c a l i n other ways as w e l l :  30 he was w i l l i n g t o spend a t o t a l o f some hundreds o f hours p r a c t i s i n g . Not many subjects p a r t i c i p a t i n g i n studies o f xjremory have demonstrated such a willingness consider  to a t t a i n p r o f i c i e n c y i n what most people would  to be a questionable  skill.  Interestingly h i s span f o r  l e t t e r s remained a t seven. Other influence  factors  estimates  besides  a  subject's  o f working-memory capacity.  1974b, 1978) and Pascual-Leone  Case  knowledge  (1972, 1974a,  (1970) have argued that capacity can  change as c h i l d r e n grow older. Obviously a numerical value  background  i t i s d i f f i c u l t to determine  f o r a l i m i t when the l i m i t seems to be constantly  changing. Dempster age  (1978) has another point o f view. He has argued that  differences do not a f f e c t span but method o f presentation and  stage o f p r a c t i c e do. Thus, the age o f the subject, the nature o f the tasks,  and  the subject's  familiarity  with  the material  can be  confounding factors that have t o be taken into account when estimating the capacity o f working memory. What storage  can be  capacity?  concluded  Presently,  about  estimating  no more than  short-term  a fluid  estimate  memory of a  subject's capacity can be obtained. Perhaps Anderson's comment says i t best: Thus, while i t i s clear that the capacity o f short-term memory i s l i m i t e d , how to measure that capacity i s not a t a l l c l e a r . Cognitive psychology i s s t i l l working a t developing adequate measures o f the capacity o f short-term memory. (1980, p. 167) Estimating a t t e n t i o n a l capacity. The container analogy has been a  useful metaphor  f o r viewing  short-term  memory capacity. However,  other views lead t o d i f f e r e n t methods for establishing the capacity o f  31 short-term memory. One certain  processing  view sees capacity as the a b i l i t y to  operations  while  attending  to  new  perform  information  (Wingfield, 1979, p. 43). According to researchers who hold t h i s view, there are p h y s i c a l l i m i t s to attending to or processing information. The is  natural analogy not  a passive  Cognitive  that presents i t s e l f for t h i s notion of capacity container, but  performance  is  rather  limited  a  "limited  because,  as  power  supply."  Allport  (1980b)  explained, " Once the supply i s f u l l y loaded, any more watts consumed by one part o f the system means l e s s for a l l the r e s t , regardless of what they want i t f o r " (p. 116). The  analogy  at  people have experienced  first  glance  seems reasonable  enough. Most  s i t u a t i o n s where performance on some task  has  suffered because of some concurrently competing task. As Norman and Bobrow (1975) have explained, "The processing resources for any system are  limited,  resources,  and  when  several  processes  eventually there w i l l be  compete  for  the  same  a d e t e r i o r a t i o n of performance"  (p. 44). As there i s a lack o f agreement i n measuring storage capacity, there  is a  similar  capacity. A l l p o r t  lack o f agreement about estimating a t t e n t i o n a l  (1980b) has been e s p e c i a l l y c r i t i c a l of the notion  o f a t t e n t i o n a l capacity and  i t s r o l e i n s t r u c t u r a l models of memory.  Commenting on the work of Kahneman, who has devoted much a t t e n t i o n and e f f o r t on the study o f "attention and e f f o r t , " A l l p o r t quipped: ...how can we measure the non-specific demand on general capacity, made by a particular task, and from which predictions o f r e s i d u a l capacity, or performance on some other simultaneous task or performance could be derived? Kahneman c a l l s t h i s a "basic problem for experimental psychology." Well, i t i s c e r t a i n l y a basic problem for h i s theory, (p. 117)  32  Despite  the d i f f i c u l t i e s  work ing-memory processes  i n measuring and even defining the  required for higher mental functioning, most  researchers s t i l l agree that working memory has a l i m i t e d capacity t o hold and process digit  span  information. However, the older notion o f equating  with  discarded. contribution  short-term  memory  Researchers  who  to higher  mental  methods o f estimating Merkel & H a l l ,  capacity  study  short-term  functioning now  capacity  seems  (see,  to have  memory  been  and i t s  incorporate  several  f o r example, Martin,  1978 and  1982). This e c l e c t i c approach to estimating capacity  was used i n the present  study.  Strategies and Mental Functioning Strategy i s another important resource needed by both human and e l e c t r o n i c problem solvers. Put simply, the computer uses programs to solve  problems;  differences  in  humans  use  reasoning  strategies.  could  reflect  Accordingly,  individual  differences  either  s t r u c t u r a l resources, or i n the choice o f strategy, or i n both.  in  It is  no exaggeration  t o say that many information-processing  studies have  attempted  explain  by  studying  from a  subject's  to  differences  differences  i n structural  resources  in  functioning  i n isolation  choice o f strategy.  As some researchers  (Hunt, 1980; Newell & Simon,  1972)  out, t h i s  has not been  have  pointed  paradigm  particularly  f r u i t f u l for explaining d i f f e r e n c e s i n complex human reasoning. Newell and Simon considered  in  performance.  (1972) have argued that s t r a t e g i e s must be  evaluating They  warned,  individual "A  few,  differences and  only  in a  cognitive  few,  gross  c h a r a c t e r i s t i c s o f the human IPS (information processing system) are invariant  over  task  and problem  solver"  (p. 788).  To argue  that  33  differences i n s t r u c t u r a l resources  rather than choice o f strategy i s  the most important factor i n explaining differences i n human reasoning i s to argue that the hardware c h a r a c t e r i s t i c s o f a computer are more important to i t s functioning than i t s software. To d r i v e t h i s point home, consider  the  analogy  to basketball playing  offered by  Hunt  (1980): I f you t r i e d to p r e d i c t a basketball player's scoring p o t e n t i a l from i s o l a t e d p h y s i c a l c h a r a c t e r i s t i c s , you would have only l i m i t e d success. Extreme weakness or lack of stature would be associated with very poor performance, but once the person moved into the "above normal" f i e l d , c o r r e l a t i o n s with p h y s i o l o g i c a l measures break down. The reason i s that there are two quite d i f f e r e n t ways of scoring points i n basketball. Some players score by muscling t h e i r way underneath the basket, then j limping up and slamming the b a l l down i n t o the goal. For players who use t h i s strategy, height and weight are good predictors of success, while hand-eye coordination and depth perception are not. The other strategy for scoring i s to step backwards, away from your opponent, and toss a high, arcing shot up into the goal, over the heads and hands of the opposition. Players who use t h i s strategy need not be p a r t i c u l a r l y large or strong, but must be quick and have excellent depth perception, (p. 456) The  point Hunt makes i s t h i s :  performance and  information-processing  individual's choice o f how  the  r e l a t i o n s h i p between  task  c a p a b i l i t i e s depends upon the  the task i s to be done.  Reasoning i s seen  as an orchestration of s t r u c t u r a l c a p a c i t i e s and s t r a t e g i e s .  The  o f a strategy w i l l be determined both by the nature o f the task and  use by  the s t r u c t u r a l resources possessed by a subject. But what i s meant by a "strategy"? The answer depends upon both the s i t u a t i o n and the i n v e s t i g a t o r : d i f f e r e n t s i t u a t i o n s and d i f f e r e n t investigators c a l l  for d i f f e r e n t meanings. Some discussion of  meanings seems necessary since, as A l l p o r t (1980b) has  these  remarked,  "so  much of the d i f f i c u l t y of psychology has to do with getting c l e a r what i t i s we are talking about" (p. 147).  34  Gagne (1977, 1984) psychologists who  has been one o f many  information-processing  has used the term "cognitive strategy" to describe  the very g l o b a l ways used by i n d i v i d u a l s to focus t h e i r knowledge and skills  on  problem  situations  that  have  not  been  previously  encountered. Strategies are "ways of 'using one's head'" (Gagne, 1977, p.  167). What are some s p e c i f i c ways o f "using one's head"? No  clearly  defined and agreed upon l i s t of strategies has been developed because, according  to Gagne  (1977), "the  i d e n t i f i e d and described"  strategies themselves have not been  (p. 37). Despite t h i s obvious shortcoming, a  p a r t i a l l i s t o f important cognitive strategies has been i d e n t i f i e d . Gagne (1977), himself, perceiving,  coding  for  has  i d e n t i f i e d attending  long-term  storage,  and s e l e c t i v e  retrieval,  and  problem  solving as important i n t e r n a l processes associated with learning remembering  (p. 167).  and  Dempster (1981) i d e n t i f i e d rehearsal, grouping,  chunking, and r e t r i e v a l as important s t r a t e g i c variables (p. 64). Howe and Ceci (1979, p. 71) suggested several cognitive strategies a person can acquire processes, other  including l a b e l l i n g , elaborating, various kinds of using  planned  describe  mediators, organizing  activities.  types o f  To  general  these  thinking  items, rehearsal and  coding various  t h e o r i s t s , cognitive s t r a t e g i e s skills  required  by  individuals  whenever a need to process information a r i s e s . Because  cognitive  s t r a t e g i e s are  mental c a l c u l a t i o n strategy researchers  as an  instance  would  i n other  be  regarded  by  a  many memory  of a cognitive strategy. Instead,  mental process would be considered procedure:  not  supposedly content-free,  such a  knowledge about a learned r u l e or  words, what Gagne  (1977, 1984)  would c a l l  an  °  35 "intellectual s k i l l " and what Anderson (1980) would term knowledge." Thus generalizations characteristics  of  about  content-free  the  cognitive  "procedural  information-processing strategies  may  not  necessarily apply to content-specific mental calculation strategies. Not a l l researchers  seem to make a rigid distinction between  strategies as general thinking processes and strategies as specific learned rules or algorithms. "cognitive strategy"  strategy"  Hitch, for example, has used the terms  (1977,  (1978, p. 319)  p.  337)  and  "information-processing  to refer to the variety of mental addition  methods he has attempted to investigate. Hitch  seems  to  have  taken  the  position  that  cognitive  strategies cannot be divorced from the specific rules used to attack a task  because different rules  may  foster  or  inhibit  the  use  of  different cognitive strategies. In the case of mental calculation, some calculative strategies (learned rules) may be more efficient than others because they provide the user with more opportunity  to apply  cognitive strategies such as chunking, rehearsal, and the like. Thus, to take Hitch's viewpoint, "good" calculative strategies like "good" cognitive strategies minimize the deleterious effects of short-term forgetting. Whether or not generalizations about the information characteristics of calculation  cognitive  strategies can  strategies i s beyond the  However, i t does seem possible to wed that  the  risks  Consequently,  for  of the  invalid purposes  applied  to mental  scope of the present study. the two notions of strategy so  generalization of  be  processing  the  can  present  be study,  nrinimized. a  mental  calculation strategy i s assumed to involve the recall of a  learned  36 procedure  as  well  as  the  attendant cognitive  strategies  used  to  minimize forgetting during the application o f the procedure. In  summary,  calculation,  as  i t has  been  i n other  forms  argued of  that performance  human  information  i n mental processing,  depends upon both the choice o f an e f f i c i e n t strategy and structural  resources.  differences mental  A  number  o f questions regarding  adequate  individual  i n mental c a l c u l a t i o n suggest themselves. Can  calculation  strategies  be  of  these  seem  associated with p r o f i c i e n t  mental  strategies,  i f any,  calculation  performance?  to be Do  identified?  proficient  mental  Which  different  calculators  possess  s t r u c t u r a l resources that are s u b s t a n t i a l l y d i f f e r e n t from the norm? The  remainder o f the chapter w i l l  be used  to explore these  questions. Two sets o f studies which have investigated some aspect o f mental c a l c u l a t i o n calculators;  will  be  reviewed:  (1) studies on expert mental  (2) experimental studies on the r o l e o f STM  i n mental  calculation. The C h a r a c t e r i s t i c s o f Expert Mental Calculators One f r u i t f u l method used to study any form o f human reasoning is  to  experts. insight Although  examine  the  highly  proficient  behaviours  demonstrated  by  Often an analysis o f expert behaviour has provided valuable into how they  a trait  have  had  such as memory contributes to reasoning. little  practical  significance,  expert behaviour i n chess (Chase & Simon, 1973), Go  studies  of  (Reitman, 1976),  and the abacus (Hatano, Miyake, & Binks , 1977; Hatano & Osawa, 1983), have  increased  the  complex reasoning.  knowledge  about  information  processing  during  37 A v a l u a b l e adjunct t o the a n a l y s i s o f h i g h l y s k i l l e d is  to  study  strategies  unskilled  and  or  processes  novice needed  behaviour  as  well.  for proficient  behaviour Often  behaviour  in  s p e c i f i e d a r e a o f c o g n i t i o n under s t u d y c a n be i d e n t i f i e d by t h e b e h a v i o u r o f b o t h t h e e x p e r t and n o n - e x p e r t .  the the  comparing  R e s e a r c h e r s may  be  a b l e t o use t h i s knowledge a b o u t e x p e r t s ' s t a t e g i e s and p r o c e s s e s t o d e v e l o p i n s t r u c t i o n d e s i g n e d t o h e l p l e s s s k i l l e d i n d i v i d u a l s become a t l e a s t more p r o f i c i e n t ,  i f n o t e x p e r t , i n t h e s k i l l under  study.  C a s e (1975, 1978) and Shulman (1976) a r e two r e s e a r c h e r s who h a v e b e e n strong advocates  o f the i n s t r u c t i o n a l i m p l i c a t i o n s o f novice-expert  comparisons. The  a n a l y s i s o f expert behaviour  processes  essential to  highly  general.  Theoretical extensions  reasoning are then p o s s i b l e .  c a n r e v e a l some u n d e r l y i n g  efficient to other  similar  forms  in  o f human  Scripture realized this potential for  t h e o r y g e n e r a l i z a t i o n a s e a r l y a s 1891. behaviour,  c o g n i t i v e behaviour,  he  R e f e r r i n g t o the study  expert  calculative  optimistically  thought  "we  perhaps  g a i n l i g h t o n t h e n o r m a l p r o c e s s e s o f t h e human mind by  of can a  c o n s i d e r a t i o n o f s u c h e x c e p t i o n a l c a s e s " (p. 1 ) . V e r y few r e s e a r c h e r s h a v e examined e x p e r t m e n t a l  calculation;  e v e n fewer r e s e a r c h e r s h a v e a n a l y z e d t h i s f o r m o f p r o f i c i e n t t h i n k i n g from  an  information-processing viewpoint.  calculation  have  observation but  been on  historical  secondary  accounts  Most s t u d i e s o f based  not  s o u r c e s whose r e l i a b i l i t y ,  on at  mental direct times,  a p p e a r s t o be q u e s t i o n a b l e . Few r e s e a r c h e r s have examined t h e n a t u r e o f mental c a l c u l a t i o n ; most r e s e a r c h e r s have been c o n t e n t t o r e p o r t accomplishments  r a t h e r t h a n i d e n t i f y i n g and d e s c r i b i n g t e c h n i q u e s o f  38 solution. Fortunately, which have been h i s own useful  there  are studies o f expert mental c a l c u l a t o r s  i n c i s i v e and revealing. Bidder's  (1856) analysis o f  expert behaviour proved to be an extremely i n s i g h t f u l source.  Scripture's including  Perhaps  (1891).  the  Later  most  comprehensive  early  paper  and was  accounts o f expert c a l c u l a t i v e behaviour  M i t c h e l l (1907), Gardner (1977), and B a l l (1956) have been  based upon t h i s e a r l i e r work. Although the studies o f Hunter  (1962,  1977, 1978, 1979) were r e s t r i c t e d to the analysis o f the introspective reports provided by A.C. Aitken, a very p r o f i c i e n t mental c a l c u l a t o r , Hunter's insights about the r o l e o f memory i n mental c a l c u l a t i o n have been  invaluable.  A  recent  book by Smith  (1983), which  provides  a  valuable overview o f expert mental c a l c u l a t i o n , seems to be the most comprehensive work on the subject. The C a l c u l a t i v e AccxatTplishments o f Expert Mental Calculators The c a l c u l a t i v e performances o f expert mental c a l c u l a t o r s were often,  to use  a  now  hackneyed  awesome. A few examples  but,  nevertheless,  should s u f f i c e  to i l l u s t r a t e  accurate  term,  the extent o f  these experts' powers. Dase,  a  19th  c a l c u l a t i n g powers.  century  German  prodigy,  According to B a l l  possessed  incredible  (1956), Dase once c a l c u l a t e d  the correct product o f 79 532 853 and 93 758 479 i n 54 seconds. In answer to a request to f i n d the product o f two numbers, each o f twenty d i g i t s , he took 6 minutes; t o f i n d the product o f two numbers each o f f o r t y d i g i t s , he took 40 minutes  (p. 476).  There were reports that  Dase once extracted the square root o f a number of a hundred d i g i t s i n 52 minutes (p. 477) 1 B a l l believed that Base's feats:  39  ... far surpass a l l other records o f the kind, the only c a l c u l a t i o n s comparable to them being Buxton's squaring o f a number o f t h i r t y - n i n e d i g i t s , and W a l l i s ' extraction o f the square root o f a number o f f i f t y - t h r e e d i g i t s , (p. 477) Zerah demonstrated years  Colburn was some f a c i l i t y  o l d , he  impressed  regarded  as  a  backward  child  until  with m u l t i p l i c a t i o n . When Colburn was h i s father's  friends  with  the  he 8  following  exhibition: ... he undertook and succeeded i n r a i s i n g the number 8 to the sixteenth power, 281,474,976,710,656. He was then t r i e d as to other numbers, consisting o f one figure, a l l o f which he raised as high as the tenth power, with so much f a c i l i t y that the person appointed to take down the r e s u l t s was obliged to enjoin him not to be too rapid. (Scripture, 1891, p. 14) A.  C.  Aitken  was  both  an  expert mental  calculator  and  talented mathematician. An example o f h i s c a l c u l a t i v e a b i l i t i e s provided by a mathematical colleague who  a was  related a story about t h e i r  j o i n t venture to an e x h i b i t i o n o f desk c a l c u l a t o r s (Gardner, 1977): The salesman-type demonstrator said something l i k e "We'll now multiply 23,586 by 71,283." Aitken said r i g h t o f f "And get..." (whatever i t was). The salesman was too intent on s e l l i n g even to notice, but h i s manager, who was watching, did. When he saw Aitken was r i g h t , he nearly threw a f i t (and so d i d I ) . (p. 75) The most recent account o f extraordinary c a l c u l a t i v e power has been attributed to Shankuntal Devi. According to the 1982 Guiness Book of  Records,  she  demonstrated  the  multiplication  of  two  13-digit  numbers: namely, 7 686 369 774 870 and 2 465 099 745 779. Each number was constructed a t random by a computer. Her correct answer o f 18 947 668 177 995 426 462 773 730 took only 28 seconds (Smith, 1983, p. 97) ! These "mechanical"  talents or  could  be  viewed  "unthinking" exercises.  disdainfully  as  That c a l c u l a t o r s  merely such  as  Aitken were not unthinking automatons and were very f l e x i b l e i n t h e i r  40  methods i s highlighted i n the following p a r t i a l account o f h i s attempt to determine  the square  root o f 851  (Hunter, 1962).  The  reader i s  reminded that the following reasoning was e n t i r e l y "mental." I a t once perceive that 841 i s 29 squared. So 29 i s a good f i r s t approximation. At once I have noted the remainder 10 and halved i t (by my rule) and noted mentally that 5/29 i s 0.172.... At once, I r i s k 29.172 as an answer which i s almost c e r t a i n l y c o r r e c t to f i v e s i g n i f i c a n t d i g i t s . But already I am o f f on another track, because 29.172 i s nearly 29 1/6, that i s , 700/24. And almost before having formulated the procedure i n a r a t i o n a l manner, I have divided 851 by 700/24, that i s , m u l t i p l i e d i t by 24/700. So I f e e l (rather than see or hear) 20,424/700. But then some experience t e l l s me that 700 times 29 1/6 i s 20,416.66666.... Averaging a t speed 20,424 and 20,416.66666... getting 20,420.33333, d i v i d i n g by 700, and placing the decimal point i n the proper place - and a l l o f t h i s i n one continuous follow-through l i k e a good g o l f stroke - I have 29.17190476 (p. 251) He continued on i n t h i s manner u n t i l after less than 15 seconds, had  carried  the  calculation  " s a t i s f i e d to go no further"  to  8  decimal  places  where  he  he was  (p. 251). This solution was done by  no  robot: t h i s was ingenuity, s k i l l , and i n t e l l i g e n c e i n action. Some Buxton,  who  calculative Mitchell  prodigies  (1907)  were  reported  delightfully  "was  very  stupid  boyhood" (p. 63), must have been an interesting character.  eccentric. even  from  He could  give from memory an itemized account o f a l l the free beer he had been given from the age o f 12 on. How  such a claim can be validated i s not  clear. Using apparently clumsy and laborious c a l c u l a t i v e methods, he was  reported to have mentally squared a number of 39 d i g i t s over  a  period o f 2 1/2 months (Mitchell, p. 63). At l e a s t h i s perseverance and i n d e f a t i g a b i l i t y , i f not h i s good judgement, have to be admired. Scripture Buxton's  (1891)  calculative  related talents  an  incident  bordered  on  which  suggests  that  the  obsessive  and  41  monomaniacal. performance  His v i s i t  to London, where he was  taken  to see  a  o f King Richard I I I , i l l u s t r a t e s h i s exaggerated zeal for  numbers and c a l c u l a t i o n s : During the dance he f i x e d h i s attention upon the number o f steps; he attended to Mr. Gar r i c k only to count the words that he uttered. At the conclusion o f the play they asked him how he l i k e d i t . He r e p l i e d "such an actor went i n and out so many times and spoke so many words; another so many, etc.... (p. 5) Reverend behaviour  H. W.  Adams provided another i l l u s t r a t i o n o f curious  demonstrated  by  an  expert  mental  c a l c u l a t o r . Safford's  a n t i c s as he calculated the product o f 365 365 365 365 365 365 and 365 365 365 365 365 365 were described i n the following manner: He flew around the room l i k e a top, pulled h i s pantaloons over the top o f h i s boots, b i t h i s hand, r o l l e d h i s eyes i n their sockets, sometimes smiling sometimes talking, and then seeming to be i n agony, u n t i l , i n not more than one minute, said he, 133,491,850,208,566,925,016,658,299,941,583,225! (Scripture, 1891, p. 30) J.R.  Newmann wryly commented:  "An e l e c t r o n i c computer might do the  job a l i t t l e faster but i t wouldn't be as much fun to watch." I t would be a mistake to surmise that eccentric behaviour  was  the norm. Some mental c a l c u l a t o r s such as Buxton might be properly l a b e l l e d as "retarded savants": b r i l l i a n t i n handling immense numbers; d u l l i n everything e l s e ,  including mathematics.  Generally speaking,  most expert c a l c u l a t o r s were very talented and l i t e r a t e  individuals.  Ampere, Gauss, and W a l l i s were highly p r o f i c i e n t c a l c u l a t o r s who were more w e l l known for t h e i r mathematical and s c i e n t i f i c than  for  engineer.  their  calculative  A.C. Aitken was  prowess.  Bidder  was  accomplishments a  professional  a professional mathematician. As i s often  the case with expertise i n most f i e l d s o f human endeavour, c a l c u l a t i v e prodigies exhibited wide d i f f e r e n c e s as regards heredity, education,  42  general  intelligence,  retention  o f their  powers over  many years,  success i n other f i e l d s , longevity, and the l i k e . Despite  these  c h a r a c t e r i s t i c s that calculators.  differences  i n background,  there  were  appeared t o be common to a l l o f these  some expert  Of p a r t i c u l a r s i g n i f i c a n c e to t h i s study was evidence  which suggested that expert c a l c u l a t o r s could be d i f f e r e n t i a t e d from the non-expert by their memory for c a l c u l a t i v e strategies, memory f o r numerical equivalents,  and enormous s t r u c t u r a l resources.  The C a l c u l a t i v e Strategies o f Expert Mental Calculators The  strategies  used  by expert mental c a l c u l a t o r s  cannot be  e a s i l y determined simply by examining the reports o f their c a l c u l a t i v e performances.  Some  reports  were  vague and rarely  d i d the writer  i d e n t i f y the p a r t i c u l a r method o f solution. As well, since many e a r l y c a l c u l a t o r s earned a l i v i n g they  were  very  reluctant  by staging t o divulge  exhibitions o f t h e i r powers, their  methods  o f reasoning  (Gardner, 1977). A few p r o f i c i e n t c a l c u l a t o r s reasoning. B a l l  seemed unable to explain  their  (1956) wrote o f Buxton: " I t was only i n rare cases  that he was able to explain h i s methods o f work, but enough i s known o f them to enable us to say that they were clumsy" (p. 469). Despite these l i m i t a t i o n s , there seems s u f f i c i e n t evidence t o i d e n t i f y a tentative l i s t o f some experts' methods o f solution. no claim  i s made that the l i s t  While  o f strategies i s complete, the l i s t  does seem extensive. L i k e l y the reader w i l l be intrigued by the unconventionality o f the methods used by these expert mental c a l c u l a t o r s .  Certainly, they  43  contrast sharply with the s t i l l common algorithms solve  written  computations.  Most  used i n school t o  c a l c u l a t o r s claimed  that  these  mental methods were l a r g e l y self-taught. An  attempt  conventional more  has been  made  to l a b e l  each  strategy  mathematical term that appears to denote  salient  features.  A  sample c a l c u l a t i o n w i l l  brief  explanation  be given  f o r each  with  listed  with  a  the method's  an accompanying strategy.  Some  documented use o f the strategy by a mental calculator w i l l be provided wherever possible.  No claim i s made that the categories are mutually  exclusive since many s t r a t e g i e s can be considered other  strategies.  calculative  task  Finally,  i t should  would be solved  as s p e c i a l cases o f  be mentioned  that  often  a  through an a p p l i c a t i o n o f several  c a l c u l a t i v e strategies used s e r i a l l y rather than through the exclusive application o f one strategy. The  l i s t has been ordered to r e f l e c t what the writer  believes  i s each strategy's g e n e r a l i z a b i l i t y . That i s , the strategies which can be  applied  to a wider  e a r l i e r i n the l i s t  range o f c a l c u l a t i v e tasks have been  placed  than the more s p e c i a l i z e d strategies found l a t e r  i n the l i s t . The  list  contains  calculate products. restricted  only  those  There were several reasons why the analysis was  to mental m u l t i p l i c a t i o n strategies. In the f i r s t  most o f the introspective analyses restricted  strategies which were used to  reported  by observers have been  t o m u l t i p l i c a t i o n . Expert c a l c u l a t o r s seemed to f i n d , as  B a l l (1956) reported, "Of the four fundamental processes, subtraction 481).  place,  present  no d i f f i c u l t y  and are o f l i t t l e  addition and i n t e r e s t " (p.  M i t c h e l l (1907) held a s i m i l a r view: "... m u l t i p l i c a t i o n and  44  not addition seems t o be the fundamental and favorite operation i n mental c a l c u l a t i o n " division  (p. 103). As well,  the evidence  tasks seemed to be solved through  multiplication  rather  than  suggests  the i n t e l l i g e n t  the application  of  some  that  use o f  conventional  d i v i s i o n algorithm. Finally  multiplication  i s a process  whereby properties and  a t t r i b u t e s abound. After a l l , the mathematical f i e l d o f number theory is  essentially  multiplication. properties,  no  more  than  the  study  of  the  properties o f  Given that most c a l c u l a t o r s were intrigued by number  perhaps i t should  be no surprise  to f i n d  that expert  c a l c u l a t o r s thrived on challenging m u l t i p l i c a t i o n tasks. Mental  analogue  of  the pencil-and-paper  algorithm.  Expert  mental c a l c u l a t i o n could involve no more than the a p p l i c a t i o n o f a mental analogue o f the conventional paper-and-pencil algorithm taught in  schools.  What  could be more natural than  extension o f a learned procedure  an apparent  to a d i f f e r e n t setting?  (1891) believed that there was evidence which suggested and F u l l e r used such techniques.  simple  Scripture that Buxton  He reported that:  I t i s said o f Buxton that he preserved the several processes of multiplying the multiplicand by each figure o f the lower l i n e i n their r e l a t i v e order, and place as on paper u n t i l the f i n a l product was found, (p. 58) In complex  theory, a t l e a s t , the procedure could be employed to solve calculations  i f the  calculator  possessed  short-term memory resource. Such a calculator  a  prodigious  would be, a  "little  c a l c u l a t o r with a b i g memory" (Mitchell, 1907, p. 117). Any person capable o f mentally squaring a 39-digit number over a period o f 2 1/2 months without losing track o f the c a l c u l a t i o n , as Buxton apparently  45 d i d , could have the necessary resources to use the seemingly awkward mental analogue o f the conventional algorithm for written computation. However, the l i t e r a t u r e has been ambiguous on t h i s point and one i s l e f t t o speculate. Distribution.  Perhaps  the most  well  documented  method o f  s o l u t i o n used by expert c a l c u l a t o r s has been d i s t r i b u t i o n . Often used when  easier  methods  were  not apparent,  the calculator  typically  proceeded by p a r t i t i o n i n g one or both factors into either a s e r i e s o f sums or differences,  carrying  out a s e t o f m u l t i p l i c a t i o n s ,  and  adding or subtracting each o f these p a r t i a l products to obtain a f i n a l solution. Bidder's  (1856) s o l u t i o n  o f the task  89 x 73 t y p i f i e s the  process. Although he solved  the task " i n s t a n t l y , " the speed b e l i e d a  lengthy  He  mental  process.  explained  that  the c a l c u l a t i o n  was  accomplished by using t h i s s e r i e s : 80 x 70, 80 x 3, 9 x 70, 9 by 3 (p. 256). He  claimed  significant digit a  he always  started  with  the left-hand  or most  (p. 260). This l e f t - t o - r i g h t procedure seemed to be  method common to most expert c a l c u l a t o r s  (Smith, 1983, p.  109).  Bidder reported that each o f the p a r t i a l products i n a c a l c u l a t i o n was successively added to produce a running sum rather than postponing the addition o f a l l p a r t i a l products to the end o f the c a l c u l a t i o n . During instance  a public  exhibition  o f h i s powers, Bidder  o f 373 by 279 to i l l u s t r a t e  h i s method.  After  used the quickly  announcing the product o f 104,067, he explained: ...now the way I a r r i v e a t the r e s u l t i s t h i s - I multiply 200 into (sic) 300 = 60,000, then multiplying 200 into 70, gives 14,000, I then add them together, and o b l i t e r a t i n g the previous figures from my mind, carry forward 74,000; I  46  multiply 3 by 200 = 600, and I add that on and carry forward 74,600. I then multiply 300 by 70 = 21,000, which added to 74,600, the previous r e s u l t , gives 95,600, and I o b l i t e r a t e the f i r s t . Then multiplying 70 by 70 = 4,900 and adding that amount, gives 100,500. Then multiplying 70 by 3 = 210, and adding as before, gives 100,710. I then have to multiply 9 i n t o 300 - 2,700, and pursuing the same process brings the r e s u l t to 103,410; then multiplying 9 into 70 = 630, and adding again = 104,040; then multiplying 9 into 3 = 27, and adding as before, gives the product, 104,067. This i s the process I go through i n my mind. (p. 260) Bidder could use d i s t r i b u t i o n to multiply a 12-digit number by another  12-digit number! Not  s u r p r i s i n g l y Bidder explained that the  task "required much time, and was 256).  For  practical  a great s t r a i n upon the mind" (p.  purposes,  he  felt  that  the  use  of  the  d i s t r i b u t i o n procedure was r e s t r i c t e d to c a l c u l a t i n g the products o f , at  most, 3 d i g i t  without reason.  factors.  Apparently he  d i d not  f i x this  limit  He explained:  Each set, or s e r i e s o f 3 figures, constitutes a step i n numbers, 787 i s one s e r i e s , - the second s e r i e s i s 787 thousand, the next s e r i e s 787 m i l l i o n s , the next 787 thousand m i l l i o n s , and the next 787 b i l l i o n s . Therefore, a t the change beyond each t h i r d figure, another idea (emphasis i s mine) must be seized by the mind; and though i t i s but one idea, yet with a l l the t r a i n i n g I have had, when I pass three figures, and jump from 787 to 1,787, I cannot r e a l i z e to myself that i t i s but one idea; - i n f a c t there are two, and t h i s increases the s t r a i n on the r e g i s t e r i n g powers o f the mind. (p. 263) To seemed  put  to be  Bidder's comments able  to extend  through "chunking."  in a  the  more contemporary  limits  An "idea" or chunk was  787  and  787  by 787,  by 1000.  related to a scale whose  chunks" rather  than  For example, to  the task was apprehended by Bidder as 787 This tendency  individual  he  o f h i s short-term memory  radix was 1000: ones, thousands, m i l l i o n s and, so on. c a l c u l a t e 1787  light,  by  to calculate with large "number  digits  can be  said  p r o f i c i e n t c a l c u l a t o r s (Hunter, 1978; Smith, 1983).  to characterize  47  Colburn  (Scripture,  1891)  Jakobsson (1944) were reported o f Bidder's.  and  the two subjects  studied  by  to have used a method s i m i l a r to that  M i t c h e l l (1907) reported that Diamandi and Bidder's son,  who l i k e h i s father was an expert c a l c u l a t o r , used a complex v a r i a t i o n which  involved  cross  m u l t i p l i c a t i o n . Their  c a l c u l a t i o n s progressed  from r i g h t to l e f t instead o f l e f t to r i g h t . Some p r o f i c i e n t c a l c u l a t o r s used the d i s t r i b u t i o n strategy by chunking i n a very f l e x i b l e manner. For example, Aitken's s o l u t i o n o f 123 by 456 involved the p a r t i a l products o f 45 by 123 and 123 by 6 (Hunter, 1962, p. 249). D i s t r i b u t i o n can be used additive  sense.  Subtractive  i n a subtractive as well as i n an  distribution  was  used  by  one  of  Jakobsson's (1944) subjects to solve these c a l c u l a t i v e tasks: I f the m u l t i p l i e r i s a number i n the v i c i n i t y o f some round number, one proceeds i n the following manner: 127 x 359 = 127 x (360 - 1) = 45720 - 127 = 45593; 546 x 784 = 784 x (550 - 4) = 431200 - 3136 = 428064. (p. 187) As  Jakobsson suggested, the subtractive form o f d i s t r i b u t i o n  was frequently applied when one or more o f the factors were perceived by  the subject as being  close to a "round number": i n other words,  some multiple o f a power o f 10.  Ball  (1956) stated that Inaudi made  use o f the strategy to solve 27 x 729 by thinking 27 x 730 - 27 (p. 482). The  ability  t o use the annexation algorithm  (determining  the  product when one factor i s a mutiple o f a power o f 10 by "annexing zeroes") seems to be e s s e n t i a l for using d i s t r i b u t i o n to solve mental c a l c u l a t i o n tasks. Bidder's  s o l u t i o n o f 373 x 279 described  earlier  involved several such a p p l i c a t i o n s o f the algorithm. Often the expert  48  c a l c u l a t o r used an  abridgement to annex zeroes. For  Aitken's solution o f 123 expressed as "123 does not  x  456,  the  an  application of  the  of  squares: a  c a l c u l a t e the product o f 51 and that i s , 51 x 49 = (50 + 1) Jakobsson's subject 78 x 84 = 81  2  - 3  was  thinker  249).  used by  some expert c a l c u l a t o r s  well-known algebraic  2 difference  x 450  x 45 = 5535." Hunter (1962) conmented: "This  Another d i s t r i b u t i o n strategy  the  p a r t i a l product 123  'burden the mind with zeros'" (p.  involved  example, during  equivalence for  2 - b  =(a  +  b ) x ( a - b ) . T o  49, one merely has to think 2500 - 1:  (50 - 1) = 50  - 1  2  = 2500 - 1.  Ingemar,  (1944), solved many tasks i n t h i s manner including = 6561  2  - 9  = 6552 (p.  186).  Notice that t h i s v a r i a t i o n of d i s t r i b u t i o n i s u s e f u l i n those instances where the  two  factors are  of a similar magnitude and  the  mean value o f the two factors i s a convenient reference number such as a quickly r e c a l l e d square. Since most expert c a l c u l a t o r s could r e a d i l y r e c a l l or Likely  calculate  squares, t h i s strategy  this a b i l i t y  to  recall  or  was  calculate  particularly useful.  squares quickly  was  natural outcome of the expert's i n t e r e s t i n c a l c u l a t i n g square (Smith, 1983,  p.  and then applying  many  Mental multiplying by factoring one or both factors the a s s o c i a t i v e law i s a useful and  strategy.  (1977), M i t c h e l l that  roots  123).  Factoring.  calculation  a  Evidence  provided  (1907), Scripture  expert  calculators  by  Ball  (1891), and  favoured  ingenious mental (1956),  Gardner  Smith (1983) suggests  factoring  over  most  other  strategies. Because o f the c a l c u l a t o r ' s almost instant apprehension o f the be  factors needed for the c a l c u l a t i o n , many c a l c u l a t i o n tasks could solved  quickly  through  factoring.  Even  a  person  possessing  a  49  hand-held calculator could not hope to approach an expert's rate o f working. The readiness with which a presented number can lead to factor apprehension has been w e l l documented. When Colburn ( B a l l , 1956)  was  "asked for the factors o f 247,483 he r e p l i e d 941 and 263; asked for the factors o f 171,395 he gave 5, 7, 59 and 83; asked for the factors o f 36,083 he said there were none" (p. 471). in-cheek, B a l l wrote  Possibly with tongue-  that Colburn, "however, found i t d i f f i c u l t  to  answer questions about the f a c t o r s o f numbers higher than 1,000,000" (p. 471).  On another occasion when asked for the product o f 21 734 by  543, Colburn immediately r e p l i e d : "11 801 562." He explained that he had arrived a t t h i s solution by multiplying 65 202 by 181 ( B a l l ,  1956,  p. 472). Aitken  possessed  an  aptitude  for factoring  which  must have  equalled i f not excelled Colburn's. That he could use factoring to a i d i n very complex c a l c u l a t i o n s was evidenced by the incredible response he gave when he was asked to multiply 987 654 321 and 123 456  789:  I saw i n a f l a s h that 987,654,321 by 81 i s 80,000,000,001; and so I m u l t i p l i e d 123,456,789 by t h i s , a simple matter, and divided the answer by 81. Answer: 121,932,631,112,635,269. The whole thing could hardly have taken more than h a l f a minute, (p. 251) Hunter a t t r i b u t e s was 246).  believed  that  "immediate,  Aitken's  apprehension  simultaneous and  of  often autonomous"  (p.  Aitken seemed not to have to set himself to apprehend numbers;  "rather he must set himself to prevent such apprehending" That  numerical  this  apprehension was  often  (p. 246).  autonomous i s i l l u s t r a t e d  i n the  following anecdote offered by Aitken (Hunter, 1962): I f I go for a walk and i f a motor car passes and i t has the r e g i s t r a t i o n number 731, I cannot but observe that i t i s 17  50 times 43. But as f a r as possible, I shut that o f f because i t i n t e r f e r e s with thought about other matters. And a f t e r one or two numbers l i k e that have been factorized, I am conditioned against i t for the r e s t o f my walk. (p. 247) There  i s evidence  Jakobsson  (1944),  calculators  had  a  that  Bidder  (1856),  Gauss, Safford similar  the  two  subjects studied  (Mitchell, 1907),  aptitude  for factoring  and and  by  many other i t s use  in  calculation. Aliquot parts and i t e r a t i o n seemed to be two popular factoring strategies  known  by  factoring strategy  expert  product  original  factors.  procedures,  an  calculators.  involves a s e r i e s o f steps.  m u l t i p l i e d by a multiple, resulting  mental  call  i s divided As  so  "a" to obtain  often  happens  can  be  admittedly awkward verbal d e s c r i p t i o n . i n the following manner to multiply 48  x  (100/4) =  aliquot  parts  F i r s t , one factor i s  i t "a," o f the remaining factor. The by  illustration  The  4800/4 = 1200.  the product o f  i n explaining  more  illuminating  the  mathematical than  this  The c a l c u l a t i o n could proceed  25 by 48, for example: 48 x 25 =  This  technique i s most useful when  c a l c u l a t i n g products with factors such as 25, 50 and 125 because the reference numbers are 100  (25 x4), 100  (50 x 2), and 1000  (125 x 8),  respectively. Some calculation.  calculators  used  other  For example, Gullan  a reference point. following manner:  reference  numbers  to  aid  in  (Jakobsson, 1944) often used 1001 as  She determined the product o f 143 and 674 i n the 143 x 674 = (674 x 1001) / 7 = 674 674 / 7 = 96 382  (p. 188). She extended the technique to include products such as 125 575  i n the manner described  by Jackobsson: 125 x 575 =  by  (575 / 8) x  51  1000.  Realizing  she added 875  (7 x 125)  product 71 875 The  that 575  (p.  use  divided by 8 i s 71 with a remainder of  to 71 000  (71 x 1000)  7,  and obtained the c o r r e c t  187).  of  i t e r a t i v e techniques was  used frequently  by  some  experts to calculate with powers o f two or three. The solution to 27 x 32,  for example, can be found by doubling the factor 27, 5 times: 54,  108,  216,  432,  864.  For  those  "successive  triples"  quickly,  calculating  products  where  t r i p l e d successively as 96, 288 The  number of  the  the  calculate the product of 27 and  calculators technique  factors 32,  and  are  who can  can be  powers  extended of  To  for example, the factor 32 can  be  864.  i t e r a t i o n s needed to determine the  fancied  seem to deter  himself  reasoning  for  would be 486,  to  be  a  squaring 1458,  by 3, 3, 2, 3, 3"  (p.  2916,  some c a l c u l a t o r s .  capable c a l c u l a t o r ,  162:  to  three:  larger factors makes t h i s technique cumbersome to use. f a c t d i d not  determine  "...to  square 162,  of  However, t h i s  Mitchell offered  product  (1907),  the  again,  8748, 26244, i . e . multiplying  who  following the  stages  successively  92).  Buxton's attempt to multiply by 378  seems p a r t i c u l a r l y awkward  (Mitchell, p. 63). He m u l t i p l i e d the other factor a by 5, 20, and 3 to obtain  300a, m u l t i p l i e d  a by  5 and  15  to obtain  75a,  and,  finally,  m u l t i p l i e d a by 3 to obtained the required product of 300a + 75a +  3a.  Despite Buxton's enormous c a l c u l a t i v e powers, the obvious use of  the  annexation algorithm i n t h i s c a l c u l a t i o n escaped  him.  M i t c h e l l surmised that Buxton often used i t e r a t i v e techniques which involved slave Tom  counting multiples o f the multiplicand. The  F u l l e r was  believed  illiterate  to have used counting techniques which  52  were similar to Buxton's (Mitchell, p. 63). Such evidence suggests the possibility details,  that,  these  resources  and,  inefficient  in  order  to  remember  calculators  may  thus,  able  were  the  numerous  have possessed to  get  by  calculative  enormous  with  structural  these  seemingly  techniques. This p o s s i b i l i t y w i l l be discussed l a t e r i n  the chapter. Expert Mental Calculators' Memory for Numerical Equivalents Hunter  (1978) concluded  that expert  calculators  acquired  a  store of numerical equivalents i n order to save a l o t o f c a l c u l a t i v e effort.  These experts, unlike the average adult, have a "vastly large  fund of numerical equivalents upon which they can draw with speed and accuracy" efficient  (p. 340). I f h i s conclusion i s v a l i d , what could be a more strategy  than  the  simple  retrieval  of  a  "fact"  from  long-term memory. Jakobsson he  called  an  (1944) concluded that experts acquired and used what  "extended  multiplication  table."  His  studies of  two  precocious c a l c u l a t o r s l e d him to surmise that: ...they make use of 'extended m u l t i p l i c a t i o n Tables' ( s i c ) , that i s to say, m u l t i p l i c a t i o n Tables up to 30 by 30. These products as w e l l as several products by higher numbers which had become f i x e d i n their memory, they knew by heart, (p. 186) Scripture the  squares  Scripture  of  (1891) believed that Mondeux knew "almost by heart a l l the  suspected  numbers less  that Colburn,  tables which reached rhetorically:  entire  "Did any  to products  Buxton,  than 100..."  and  as high as 100  of the prodigies possess  answered: "Considering their  Dase also x 100.  (p. 48). possessed He  asked  such a table?"  enormous powers o f memory i t would  He be  53  almost ^ e x p l a i n a b l e i f they d i d not" (p. 46). Gardner (1977) concluded that p r o f i c i e n t c a l c u l a t o r s possessed a l i b r a r y o f numerical data which could be accessed a t w i l l .  He s a i d :  Aitken"s s k u l l housed an enormous memory bank of data. This i s t y p i c a l o f the lightning c a l c u l a t o r s ; I doubt that there has ever been one who d i d not know the m u l t i p l i c a t i o n table through 100, and some a u t h o r i t i e s have suspected that Bidder and others knew i t t o 1,000 but would not admit i t . (p. 73) However,  other  authorities  such  as M i t c h e l l  have  taken an  opposing viewpoint. He argued:, There i s no warrant, then, f o r supposing that any o f the prodigies except Gauss and Dase used a m u l t i p l i c a t i o n table larger than that o f ordinary mortals; and even i n these two cases there i s no direct evidence, only a bare p o s s i b i l i t y ; . . . (p. 110) He thus concluded: We may therefore dismiss the theory of enlarged m u l t i p l i c a t i o n tables, a t l e a s t u n t i l i t s advocates have brought forward further and more d e f i n i t e evidence than any that has y e t been produced, (p. I l l ) One weak. as  o f h i s arguments to support h i s skepticism seemed rather  For some reason, he believed that the only way information such  multiplication  deliberate study.  products  can be committed  to memory i s through  As he explained:  ...but i n dealing with the enlarged m u l t i p l i c a t i o n table theory, we must insist that the only legitimate interpretation o f the theory i s that such a table i s d e l i b e r a t e l y committed to memory (emphasis i s Mitchell's) by the c a l c u l a t o r . . . . (p. 106) He seemed unwilling  to accept  the f a c t that people who are highly  interested i n a subject, as c a l c u l a t o r s obviously were, can memorize a lot  o f information without  a conscious  and deliberate  attempt to  commit anything to memory. Undoubtedly, Hunter (1978) would have disagreed with M i t c h e l l ' s  54 viewpoint.  In attempting  to explain  the a c q u i s i t i o n  o f Aitken's  formidable store o f knowledge, he stated: "We might suppose that the acquisition  of  memorization"  this  store  involves  a  great  deal  of  deliberate  (p. 343). But he argued that the supposition would be  wrong as "there i s r e l a t i v e l y l i t t l e deliberate memorization bleakly undertaken f o r i t s own sake: i f there were, the l i k e l y outcome would be boredom and discouragement" (p. 343). Bidder  believed  multiplication extremely  table.  rapid  that  he  Rather,  calculation  he  did  not  thought  possess that  an  enlarged  h i s a b i l i t y for  gave observers the mistaken impression  that he possessed such a l i b r a r y o f numerical facts  (p. 256).  He  stated, "I acquired the whole m u l t i p l i c a t i o n table up to 10 times 10; beyond which I never went; i t was a l l that I required" (p. 258). As stated e a r l i e r , Gardner  (1977) d i d not believe Bidder's claim.  Ball  (1956) has expressed a similar skepticism o f Bidder's s e l f d e n i a l . Smith  (1983) argued  that  since most educated people possess  part o f an extended m u l t i p l i c a t i o n table, the important question to determine i s the extent o f the expert c a l c u l a t o r ' s table. He believed that there was a dearth o f evidence to support what he thought were the extravagant claims about the s i z e o f the expert's memory store. According t o Smith, Wim K l e i n  i s the only calculator known to have  used a m u l t i p l i c a t i o n table o f 100 by 100 (p. 60). Expert Mental Calculators' Short-Term Memory Capacity Information-processing  t h e o r i s t s .such as Hunt  (1980), Newell  and Simon (1972), Resnick and Ford (1981), Romberg and C o l l i s (1981), and Anderson  (1980) have a l l expressed views about the r o l e played by  55  working memory i n l i m i t i n g  the performance on any s e r i e s o f mental  operations. Normally the l i m i t e d capacity o f working memory does not inhibit  performance  continuously written  i n tasks  where  available for visual  calculation  there  the problem  information  inspection. For example,  is little  demand  is  during  on short-term  memory  processes. A l l stages o f the c a l c u l a t i o n are recorded and, thus, are a v a i l a b l e for continual examination. Mental c a l c u l a t i o n  can be another  matter: the luxury o f an  external memory-aid i s not a v a i l a b l e . Consequently a c a l c u l a t i o n can involve a great deal o f processing by short-term memory. Therefore, i t could  be  argued  that  expert  mental  calculation  would  processing c a p a b i l i t i e s f a r exceeding the limited resources by  the average person. Is there  expert c a l c u l a t o r s possessed  any evidence  which  require possessed  indicates that  such memory resources?  A l f r e d Binet studied many o f the great 19th century c a l c u l a t o r s and  attempted  to estimate  s l i g h t l y d i f f e r e n t techniques the  conclusion  inescapable.  that  Binet  these  their  memory spans.  Although  he used  for measuring span than are used today, calculators  s e t the l i m i t  had  o f Inaudi's  (Mitchell, 1907, p. 71), f o r example.  large memory  spans  seems  span  a t 42  This figure stands  i n stark  contrast to the average adult d i g i t span o f about 7. M i t c h e l l reported that one u n i d e n t i f i e d c a l c u l a t o r was said to be able t o repeat 150 f i g u r e s i n order after a single hearing! also able  to repeat  He was  the s e r i e s backwards and name the 30th or 50th  figure from either end (p. 86). One could imagine that t h i s subject would have surpassed with ease the c e i l i n g o f instruments measure such notions o f capacity as backward d i g i t  designed to  span and probed  56  serial recall. Perhaps the most r e l i a b l e capacity was conducted by Hunter was  assessment o f a great c a l c u l a t o r ' s (1977).  The subject, A.C. Aitken,  reported to have had a span o f 13 with auditory presentation o f  random d i g i t s  and a span o f 15 with v i s u a l presentation o f random  d i g i t s (p. 156). Of  course,  i t would  be  tempting  to  attribute  a  casual  r e l a t i o n s h i p between span and p r o f i c i e n t c a l c u l a t i v e thinking. which would be reciprocal  the cause and which  relationship  in  which  the e f f e c t ? increasingly  But,  There could be a proficient  mental  c a l c u l a t i o n r e s u l t s i n increasingly higher estimates of ?span which, i n turn, begets increasingly p r o f i c i e n t mental c a l c u l a t i o n . Suggestive evidence about the existence o f such a r e c i p r o c a l relationship  has  been  provided  by  studying and  grand Osawa  experts  in  "abacus-derived mental c a l c u l a t i o n . "  Hatano  (1983)  found  that these abacus experts had d i g i t  spans which far exceeded  their  memory for l e t t e r s and common f r u i t names. They concluded that: We reasonably assume that i t was through extensive p r a c t i c e i n abacus and mental computation that their d i g i t span was extended to double the "norm," because o f the domains p e c i f i c i t y o f the extension, (p. 102)  Implications for Studies o f Non-Expert Mental C a l c u l a t i o n An attempt has been made to review those studies which  have  investigated some aspect o f expert mental c a l c u l a t i o n . In p a r t i c u l a r , it  was  hoped  characterize  to  identify  skilled  and  those  unskilled  factors  which  could  performance. There  be  used  i s at  to  least  suggestive evidence that i n d i v i d u a l differences i n mental c a l c u l a t i o n performance could r e f l e c t several differences i n these uses o f memory:  57  the choices o f c a l c u l a t i v e s t r a t e g i e s , the r e c a l l o f useful numerical equivalents,  the a b i l i t y  to process information  i n the short-term  memory system. However, since expert mental c a l c u l a t o r s are a very s e l e c t and unrepresentative group  o f people, any conclusions about  individual  differences i n mental c a l c u l a t i o n performance must remain tentative. These experts could be as d i f f e r e n t from s k i l l e d mental c a l c u l a t o r s as skilled  calculators  are from  unskilled  subjects.  Skilled  mental  c a l c u l a t o r s could gain t h e i r advantage over the u n s k i l l e d through some u n i d e n t i f i e d process. Little researchers unskilled examined  i s known about have  mental  attempted  m u l t i p l i c a t i o n tasks. Maier rely  to  calculators.  the d i f f i c u l t i e s  calculators  unskilled  mental  identify  the strategies  In p a r t i c u l a r ,  that  some  multiplication. used  no researchers  subjects  have  with  Few by have  mental  (1977) has claimed that u n s k i l l e d mental  on procedures  similar  to pencil-and-paper mental  reproductions. A procedure which, he has argued, does not lend  itself  to mental use. However, no evidence was provided to support h i s claim. Levine's study (1982) has provided some i n d i r e c t evidence that u n s k i l l e d mental c a l c u l a t o r s may favour the pencil-and-paper mental analogue to solve mental arithmetic computations. She found that by f a r the most popular strategy used by the lowest-scoring subjects on a t e s t measuring the a b i l i t y to estimate products and quotients was the pencil-and-paper mental analogue. In f a c t , 10% o f the college  she reported that almost  student sample p a r t i c i p a t i n g  i n her study made  exclusive use o f t h i s estimation strategy. The l i t e r a t u r e was also inconclusive on whether or not s k i l l e d  58  and  unskilled  mental  c a l c u l a t o r s could  be characterized  by  their  a b i l i t y to r e c a l l u s e f u l numerical equivalents during the s o l u t i o n o f a mental m u l t i p l i c a t i o n  task.  In f a c t ,  there was some disagreement  among researchers about whether experts possessed a larger than normal storehouse mental  o f u s e f u l "number f a c t s . " Whether non-expert but s k i l l e d  c a l c u l a t o r s do possess  an  extensive  library  of  numerical  equivalents remains to be demonstrated. Despite the obvious f a c t that some mental c a l c u l a t i o n s require a  great  deal o f short-term  memory processing,  i t i s not c l e a r i f  s k i l l e d and u n s k i l l e d mental c a l c u l a t o r s can be characterized by the efficiency  of  their  short-term  c a l c u l a t o r s often possessed can  only  speculate  memory  stores.  extraordinary  Although  structural  expert  resources, one  that s k i l l e d mental c a l c u l a t o r s have greater STM  c a p a c i t i e s than u n s k i l l e d subjects.  Experimental Studies o f STM and Mental C a l c u l a t i o n Many researchers Ford,  1981) have  (Hunt 1980; Lindsay & Norman, 1977; Resnick &  argued  that  any task  requiring a great  deal o f  information processing w i l l suffer degraded performance because o f the l i m i t e d processing c a p a b i l i t i e s a v a i l a b l e to a subject. Hitch (1977, 1978),  Merkel  and H a l l  (1969),  and Dansereau  mental  calculation  (1982),  Whimbey,  and Gregg  requires  a  Fischhof,  and S i l i k o w i t z  (1966) have a l l demonstrated great  deal  of  short-term  that  memory  processing. Hitch's studies (1977, 1978) have been among the best attempts to  examine  studied  how  mental  calculation  short-term  under  experimental  f o r g e t t i n g contributed  conditions.  He  to errors o f mental  59  calculation.  The tasks he used were o f low to medium complexity. The  range o f d i f f i c u l t y i s best i l l u s t r a t e d by the examples, 325 + 431 and 6345 x 5. He postulated  that mental c a l c u l a t i o n  proceeds s e r i a l l y and  that the following s e r i e s o f mental operations are involved: r e t r i e v a l of  some information,  usually i n the form o f d i g i t s ;  an arithmetic  combination retrieved from long-term memory; storage o f the "carried" d i g i t , i f applicable; and the storage o f the sum or, i n the case o f a sum greater than 10, the separate  storage o f both the decade d i g i t -  the "carry" - and the concomitant u n i t d i g i t . He evaluated a number o f hypotheses  about  the  role  of  c a l c u l a t i o n performance using  short-term  t h i s elementary  forgetting  in  mental  information-processing  analysis. First,  he hypothesised  that the l a t e r steps i n a c a l c u l a t i o n  would show more errors since information stored i n working memory i s l o s t r a p i d l y during any a c t i v i t y interpolated between presentation and recall  (1977, p. 333).  individual  items  The data  i s increased  suggested that "the forgetting o f  by interpolated a c t i v i t y  during the  i n t e r v a l p r i o r to r e t r i e v a l and combination" (p. 334). He a l s o attempted to determine i f the d i f f i c u l t y o f r e c a l l i n g an item might be increased by the presence o f other stored items. He hypothesised  that  the more  information  being  memory, the greater the amount o f forgetting.  processed  by working  Hitch found that t h i s  "storage load," as he described i t , d i d contribute to l o s s o f r e c a l l . Dansereau and Gregg (1966) attempted to analyze the underlying processes  involved i n mental m u l t i p l i c a t i o n . Because their findings  were based on the study o f a s i n g l e subject, generalizations about the  60  process of mental c a l c u l a t i o n are  l i m i t e d . They found that the  time  taken i n the mental s o l u t i o n of various m u l t i d i g i t products correlated h i g h l y with the "add,"  t o t a l number o f processing  "multiply,"  "carry,"  and  stages,  "hold."  No  corresponding  attempt  was  made  to to  i d e n t i f y the types o f c a l c u l a t i v e strategies used by the p a r t i c i p a t i n g subject. These  few  studies  have  provided evidence suggesting underlying  processes as  been  important  in  that  they  have  that mental c a l c u l a t i o n involves the same  more conventional  short-term  memory tasks.  However, they leave a number of i n t e r e s t i n g questions unanswered about the  contribution of  short-term  memory to  i n d i v i d u a l differences i n  mental c a l c u l a t i o n performance. I f , as Hitch (1978) has implied, there was  some variance  i n performance  even on  these  relatively  simple  tasks, how can such differences be explained? I t seems u n l i k e l y the d i f f e r e n c e s i n performance r e f l e c t choice of  c a l c u l a t i v e strategy:  the  subjects  were asked  to  use  the  same  strategy and a l l subjects reported that they implemented the requested strategy.  I t seems very u n l i k e l y that erroneous addition combinations  retrieved  from  long-term  memory  contributed  performance: a l l subjects were adults and  to  the  variation in  could be expected to have  good r e c a l l o f "basic addition f a c t s . " A p l a u s i b l e explanation reflected  differences  in  STM  i s that the differences i n performance capacity.  Hitch  did  not  report  any  measures o f h i s subjects' c a p a c i t i e s so there i s no way o f t e s t i n g the hypothesis. subsequent assumed  Nevertheless, studies  of  Hitch  (1978)  noted  and  mental c a l c u l a t i o n should  characteristics of  short-term  suggested  incorporate  memory such  as  that other  i t s limited  61  capacity to hold items (p. 322). Only a few researchers have attempted to determine i f there i s a  relationship  calculation  between  performance.  short-term One  memory  difficulty  capacity in  and  mental  investigating  this  r e l a t i o n s h i p has been the s e l e c t i o n o f the instrument used to measure STM capacity. As has been discussed, there has been l i t t l e agreement among researchers as t o the most v a l i d method o f capacity assessment. Perhaps the most reasonable approach for a researcher to take in  the absence o f a concensus would be to i d e n t i f y  processes  required  i n a mental  calculation  and then  first  the key  select  those  measures o f capacity which seem most l i k e l y to measure such processes. This was the approach taken to s e l e c t the STM measures that were used in  the present study. A review o f the research l i t e r a t u r e seemed to  suggest  that ordering and transforming were two STM processes  that  could be necessary for p r o f i c i e n t mental c a l c u l a t i o n .  Ordering and Mental C a l c u l a t i o n Ordering i s one process that seems to be required f o r success in  both mental c a l c u l a t i o n and many short-term memory tasks. Almost  a l l the i n i t i a l and interim information i n a mental c a l c u l a t i o n must be stored i n an ordered manner. Haphazard storage o f information i s to be avoided a t a l l costs. When m u l t i d i g i t factors are presented to a subject, they must be stored i n a way that preserves the place value of  each d i g i t .  In a similar manner, the order o f each stage o f the  c a l c u l a t i v e strategy must be remembered. The subject must constantly consider these questions: what have I done and what have I l e f t to do? Thus, a basic process  underlying  a l l mental c a l c u l a t i o n s could be  62  ordering or what some researchers  (Merkel & H a l l , 1982; Martin, 1978)  have preferred to c a l l memory for order. Perhaps  the most  common  measure  of this  process  has been  forward d i g i t span. The technique has been included as a subtest i n many  psychometric  measures o f i n t e l l e c t u a l  Weschler Intelligence Scale  f o r Children - Revised  and the Weschler Adult I n t e l l i g e n c e Scale Dempster digit  span  (1981) has c i t e d  correlated  function  highly  (Weschler, 1974)  (Weschler, 1955).  studies  with  including the  which  various  found  that  mathematics  achievement  measures, including an r = 0.77 on SAT - Math (p. 65). He "there  i s ample  evidence  that  the factors  forward  underlying  claimed,  span  also  underlie performance i n more complex domains and that memory span taps some basic aspect o f human information processing"  (p. 65).  Memory for order has been used i n a few studies to study tasks which involve some type o f mental c a l c u l a t i o n . Merkel and H a l l (1982) attempted to determine i f memory for order was related "to the amount o f mental manipulation involved i n tasks" simple  form  manipulation." tasks  of  mental  calculation  (p. 428). They chose a very as  a  measure  of  "mental  Each subject was required to solve 12 mental addition  involving 2 or 3 d i g i t addends.  Each task was c l a s s i f i e d as  either involving a "carry" operation  (MMC) or "not involving a carry"  operation  were  (MMNC).  The MMC  tasks  assumed  to require  more  processing and thus were expected to induce a higher memory load than were the MMNC tasks. A that Hitch  basic assumption not made e x p l i c i t by the researchers  each  subject  used  the strategy  (1977) has suggested  that  this  suggested  by their  i s not always a  was  analysis. reasonable  63 assumption because some subjects can reduce the processing demand by choosing and using an e f f i c i e n t strategy. I f a subject can reduce the processing demand of a c a l c u l a t i v e task, STM capacity l i k e l y w i l l not be a s i g n i f i c a n t factor i n determining performance i n that task. Merkel and H a l l  (1982) reported c o r r e l a t i o n s between  d i g i t span on MMC and on MMNC o f 0.34  and 0.03,  forward  respectively (p. 434).  These c o r r e l a t i o n s are not p a r t i c u l a r l y high and neither c o r r e l a t i o n was  s t a t i s t i c a l l y s i g n i f i c a n t . Not s u r p r i s i n g l y , these mental addition  tasks  were  particularly  consequently, ceiling  easy  f o r the c o l l e g e - l e v e l  subjects and,  l i t t l e variance i n performance resulted. Because of t h i s  effect,  the r e l i a b i l i t i e s  for MMC and MMNC were quite low:  MMNC was 0.36 and MMC was 0.52 (p. 432). A stronger r e l a t i o n s h i p may have been detected i f more r e l i a b l e measures were used. Merkel  and H a l l  instead o f adults. on MMC were 0.42  replicated  the study  using  fifth  graders  The c o r r e l a t i o n s o f forward d i g i t span on MMNC and  and 0.28,  r e s p e c t i v e l y . Only the c o r r e l a t i o n between  MMNC and forward d i g i t span was  s t a t i s t i c a l l y significant.  Merkel and H a l l attempted to explain the d i f f e r e n t r e s u l t s for the younger and older subjects by arguing that "short-term or working memory increases or changes i n nature p l a u s i b l e explanation college-level  not  subjects  with  age"  (p. 440).  i d e n t i f i e d by the researchers could  have  used  more  Another  i s that the  efficient  mental  c a l c u l a t i o n strategies than the younger subjects. Since the study d i d not report the types o f s t r a t e g i e s used by the subjects i n solving the mental addition tasks, t h i s explanation cannot be validated. Another measure that has been used to estimate the a b i l i t y of a subject t o process temporary information i s delayed d i g i t span. This  64 measure i s similar to forward d i g i t span i n that a series o f d i g i t s i s presented  to a subject. Instead o f immediately  recalling  the d i g i t  s e r i e s as i s required i n estimating forward d i g i t span, the subjects must f i r s t attend t o a short intervening a c t i v i t y before r e c a l l i n g the s e r i e s . This task would seem to require the use o f similar memory processes as mental c a l c u l a t i o n : both tasks require a subject to store some information for a short period o f time while other information i s being received or processed. Whimbey, Fischhof and S i l i k o w i t z (1969) t r i e d to determine i f a c o r r e l a t i o n a l r e l a t i o n s h i p existed between delayed d i g i t special  form o f mental addition.  solve these types o f tasks: t h i s to 2B and 5D.  span and a  Each adult subject was asked to  "You have 8A, 3B, 2C and 5D and you add  How many o f each category do you now have?" (p.  57). Vvhimbey e t a l . obtained a c o r r e l a t i o n o f 0.77 and, i n a l a t e r replication,  they  reported  a correlation  o f 0.67.  Correcting the  c o r r e l a t i o n o f 0.77 f o r the u n r e l i a b i l i t y o f each measure produced a c o r r e l a t i o n o f 0.95. Like Merkel and H a l l (1982), they a l s o concluded that  some  basic  process  measured  by d i g i t  span  and not general  i n t e l l i g e n c e was responsible for these high c o r r e l a t i o n s . Whether or not a s i m i l a r l y high r e l a t i o n s h i p between delayed digit clear. least  span and mental m u l t i p l i c a t i o n performance would e x i s t i s not In everyday  instances o f mental m u l t i p l i c a t i o n ,  an opportunity  to apply  a wide  variety  there i s a t  of strategies.  In  contrast, the mental c a l c u l a t i o n task incorporated by Whimbey e t a l . provides a subject with  little  opportunity to apply a d i v e r s i t y o f  c a l c u l a t i v e strategies. And, as Hunt (1980) has argued: Performance i n such a s i t u a t i o n w i l l be more determined by mechanistic information-processing functions than by choice  65  o f a problem-solving strategy simply because o f the limited range o f strategies possible, (p. 458) Transforming  and Mental C a l c u l a t i o n  Transforming identified  i s another  as basic  t o tasks  short-term  memory  process  requiring information  often  processing. In  addition to an i n i t i a l storage o f information, many tasks require that the o r i g i n a l  store be transformed  to a form more i n keeping with the  mental operations required to solve the task. been  a sub-test  i n many psychometric  Backward d i g i t span has  measures including the WAIS  (Wechsler, 1955) and the WISC-R (Weschler, 1974). These sub-tests have been used to estimate a subject's capacity to transform An  estimate  of this  short-term  memory  capacity  information.  i s obtained  by  requiring a subject to r e c a l l a s e r i a l l i s t o f d i g i t s i n the reverse order o f presentation. Researchers  such  as Hiebert,  Carpenter  Romberg and C o l l i s (1981), Scardamalia (1974) have  included backward  digit  memory measures used to estimate  and Moser  (1982),  (1977), and Case and Globerson span as part o f a battery o f  "M-capacity": a construct developed  by Pascual-Leone (1970) and which has been postulated to be a measure o f the basic i n t e l l e c t u a l l i m i t a t i o n Romberg backward  digit  and C o l l i s  (1981)  o f children. have  span measure a basic  argued  process  that  estimates  underlying  of  many more  complex information-processing tasks since the task requires a form o f "short-term  memory and an information  operation  or transformation.  The numbers i n the task not only have to be held i n mind, they have to be held i n mind while operating on them i n some way" (p. 6). Similar processing may take place during a mental c a l c u l a t i o n .  66  Those subjects who  tend to use a s e r i e s o f " r i g h t - t o - l e f t " procedures  to solve a c a l c u l a t i v e task have to transform digits  into a  reversed  listing,  solve  the  the o r i g i n a l l y stored  various  sub-tasks  and,  f i n a l l y , output the product again by reversing a s e r i a l l y stored set o f d i g i t s . Thus, backward d i g i t  span could be a good predictor for  those types o f mental c a l c u l a t i o n tasks where a subject must transform a s e r i e s i n t o a reversed arrangement. Although backward d i g i t examined  the  no  studies of  mental c a l c u l a t i o n have  span as an experimental r e l a t i o n s h i p between  v a r i a b l e , a few  the  basic  incorporated studies have  processes  underlying  backward d i g i t span and some form o f mathematical reasoning.  Hiebert  e t a l . investigated, among other things, the r e l a t i o n s h i p s between the processes  that f i r s t - g r a d e c h i l d r e n used to solve verbal addition and  subtraction tasks and  an  information-processing  the information-processing  task was  task: i n t h i s  backward d i g i t span.  case,  They found  that o f the four cognitive v a r i a b l e s included i n the study, backward digit  span was  performance.  the v a r i a b l e most c o n s i s t e n t l y related to arithmetic  They concluded  that:  Viewing children's mathematical behaviour from an information processing perspective i s a r e l a t i v e l y new approach, and the search continues for appropriate measures of processing capacity as w e l l as analysis procedures that specify the processing demands o f i n d i v i d u a l tasks, (p. 97) Implications for Studies of Mental C a l c u l a t i o n Tasks These few  experimental  studies have demonstrated that mental  c a l c u l a t i o n requires a great deal o f STM STM  capacity  is  related  to  performance  c a l c u l a t i o n tasks i s not c l e a r . The  processing. Whether or on  a l l types  of  not  mental  studies that have established a  67  r e l a t i o n s h i p between capacity  and mental c a l c u l a t i o n performance have  used tasks for which a v a r i e t y o f approaches was not possible. Whether a r e l a t i o n s h i p e x i s t s between capacity on  mental  strategies calculative  calculation could tasks  be  tasks  such  available  remains  to be  as  to deal  and performance  multiplication with  where  the demands  more  of the  seen. One o f the purposes o f the  present study was to examine the r e l a t i o n s h i p between several measures o f STM capacity and mental m u l t i p l i c a t i o n performance.  68  CHAPTER III METHODS AND PROCEDURES Outline o f the Study The three phases of the study were the p i l o t t e s t i n g phase, the screening phase, and the interview and assessment phase.  Each phase  can be characterized by d i f f e r i n g purposes and procedures. The  purpose  feasibility  of  of  the p i l o t  some testing  testing phase was  procedures  and,  to evaluate the  in particular,  to help  develop several instruments designed to assess p r o f i c i e n c y i n mental multiplication.  The  information gained from  the p i l o t  testing  was  used to r e f i n e many aspects of the subsequent phases of the study. One of the major goals of the study was  to acquire a d e t a i l e d  analysis of the processes involved i n mental c a l c u l a t i o n . The methods of  solution  used  by  subjects to solve a  c a l c u l a t i o n were o f p a r t i c u l a r i n t e r e s t .  task  requiring  a  mental  However, r a r e l y can methods  o f solution be i d e n t i f i e d through an analysis of either a written or stated solution alone. gather solve  introspective a calculation  A f a r more useful technique of a n a l y s i s i s to reports either  task or  as  immediately  a subject i s attempting after  a solution  has  to  been  completed. The  collection  of  introspective  reports requires that  researcher meet with i n d i v i d u a l subjects i n a private s e t t i n g .  the  As can  be appreciated, such a procedure i s very time-consuming. To make the data  collection  and  analysis  more manageable, a  relatively  small  sample by psychometric standards was selected: 30 high school students  69  and  one  elementary-school student. Thus, the purpose of the  phase was  to s e l e c t the  larger sample of 280  interview and  who  calculation, differ  assessment phase sample from a  senior high school mathematics students.  I f the study was subjects  screening  to be able to i d e n t i f y the c h a r a c t e r i s t i c s of  exhibited  differing  levels  of  proficiency  a means of ensuring that the p a r t i c i p a t i n g  i n c a l c u l a t i v e performance had  to be  found.  in  mental  subjects d i d  Consequently  the  r e s u l t s of a screening t e s t CALl were used to categorize subjects as either  s k i l l e d or u n s k i l l e d mental c a l c u l a t o r s .  fifteen  unskilled  subjects were  subject agreed to p a r t i c i p a t e One  highly  skilled  selected  i n the  by  Fifteen skilled t h i s method  interview and  elementary-school  student  and  and each  assessment phase. was  selected  to  assessment  phase was  to  distinguish  skilled  p a r t i c i p a t e i n t h i s phase of the study as well. The identify  purpose  of  the  characteristics  unskilled instruments thirty-item  mental were  interview  that  appeared  calculators. administered  mental  To to  task,  the  meet  multiplication  subject was  to  test  CAL2  stating asked  solution i n as much d e t a i l as possible. questions i f the  this  purpose,  a l l participating  i n d i v i d u a l l y to each subject. After calculation  and  to  The  a  several  subjects.  was  A  administered  solution  explain  from  to a mental  the  method  of  researcher asked probing  report seemed to lack the d e t a i l needed to c l a s s i f y  the method of s o l u t i o n . These explanations were recorded on audio tape for l a t e r t r a n s c r i p t i o n and complete each task was Two individual  analysis. The  time needed by a subject to  also measured.  tests of arithmetic fundamentals were also administered subjects.  One  test  of  written  to  paper-and-pencil  70  computational  performance  WPP  required  a  subject  to  solve  ten  m u l t i - d i g i t m u l t i p l i c a t i o n tasks using any written algorithm. A t e s t o f basic f a c t r e c a l l BFR was used to assess each subject's a b i l i t y to r e c a l l 100  basic multiplication facts.  The  times taken  to complete  each of these t e s t s were a l s o measured and recorded for a l l subjects. Four memory  measures  processing  incorporated 1955),  in  forward  administered  designed  to  capacity  the  were  Weschler  digit  delayed d i g i t span DDS,  and  Two  a  subject's  administered.  Adult  span FDS  individually.  estimate  Two  Intelligence backward  short-term measures  Scale  digit  (Weschler,  span  other measures, l e t t e r  BDS,  were  span LS  and  were administered to a l l the subjects i n small  groups of 4 to 5 subjects. DDS and LS were not administered to the one highly s k i l l e d subject. Eleven subjects who during  the  interviews  m u l t i p l i c a t i o n t e s t CAL3. identified  through  t e s t procedures o f CAL2. these  demonstrated a high l e v e l of p r o f i c i e n c y  were  given  the  more  challenging  mental  This t e s t contained 15 items that had been  the p i l o t  testing as being very d i f f i c u l t .  The  were similar to those used during the administration  Each subject was asked to explain the method of s o l u t i o n and  explanations  were recorded,  reviewed,  and  l a t e r transcribed.  Each subject's s o l u t i o n time was a l s o measured. A great deal of q u a l i t a t i v e data was gathered during t h i s phase of  the  study.  performed  a  The  p h y s i c a l gestures  calculation  and  their  of  some  accompanying  subjects  as  they  comments  provided  a d d i t i o n a l insights into the process of mental c a l c u l a t i o n . In order to help the reader keep track of the various phases, samples,  instruments,  and  procedures,  an overview  of the study  has  71  been included i n Table I. Selection o f the Subjects With  the  exception  of  one  subject,  a l l subjects  who  p a r t i c i p a t e d i n the screening as w e l l as the interview and assessment phases  of  the  study  were  senior  high-school  students  currently  enrolled i n a grade 11 or 12 mathematics course. A high-school sample seemed  preferable  to an  elementary-school  sample  because  i t was  expected that older students would demonstrate a greater v a r i a t i o n i n mental m u l t i p l i c a t i o n performance than would younger students. expected  greater  variation  i n performance  seemed a t t r a c t i v e  This for a  study i n v e s t i g a t i n g i n d i v i d u a l differences. In the screening phase, a test o f mental m u l t i p l i c a t i o n was administered senior-level  CALl  to 286 students enrolled i n ten d i f f e r e n t classes of  mathematics.  mathematics  classes  mathematics  classes  This  containing containing  sample 139  147  included  students students.  and  five  grade  11  five  grade  12  A l l subjects  were  studying mathematics courses that were part of the academic u n i v e r s i t y entrance program. Alternative General  No subject was enrolled i n either the l e s s rigorous  Mathematics  program  Mathematics program.  or  the more  practically-oriented  A l l grade 11 students  had completed  Algebra 9 and 10, and a l l grade 12 students had completed Algebra 9_, 10, and 20.  The majority of students had a l s o completed Geometry 10.  Thus, a l l the high-school subjects who p a r t i c i p a t e d i n the study had reasonably comparable mathematical opportunities. Two  secondary schools,  Saskatchewan,  agreed  or c o l l e g i a t e s as they  to p a r t i c i p a t e i n the study.  are c a l l e d i n Four  classes  72 TABLE I AN OVERVIEW OF EACH PHASE OF THE STUDY INCLUDING PARTICIPATING SAMPLES, AND THE INSTRUMENTS AND PROCEDURES USED  Instruments and Procedures  Sample  The P i l o t Testing Phase 180 university students  Five p i l o t tests; 80 mental multiplication items; 30-second item-presentation rate.  61 university students  Two trial-forms of a 20-item mental multiplication screening test; 20-second item-presentation rate. The Screening Phase  280 senior high school mathematics students  CALl: 20-item mental multiplication test; 20-second item-presentation rate. Used to select the s k i l l e d and unskilled sample.  The Interview and Assessment Phase 15 s k i l l e d and 15 unskilled students  CAL2: 30-item individually administered mental multiplication test. Used to determine method of solution, accuracy, solution time, type of errors, and memory devices. WPP: 10-item written multiplication test. BFR: 100-item multiplication basic fact r e c a l l test. Used to determine accuracy and speed i n arithmetic fundamentals. FDS: forward d i g i t span. BDS: backward d i g i t span. DDS: delayed d i g i t span. LS: letter span. Used to estimate STM capacity.  11 most s k i l l e d students  CAL3: 15-item d i f f i c u l t mental multiplication test. Used as a supplement to CAL2.  1 highly s k i l l e d 13-year-old g i r l  A l l tests and procedures administered except for DDS and LS. Knowledge of primes and squares also was ascertained.  73  containing 121 students were enrolled i n school A which was under the j u r i s d i c t i o n of the Saskatoon Separate (Catholic) School Board.  Six  classes containing 165 students were enrolled i n school B which was under the j u r i s d i c t i o n of the Saskatoon Public School Board. Each mathematics  school  used  classes.  a  form  School A  of  used  ability  streaming  the nomenclature  A  f o r the and B to  designate the top two streams while school B used the nomenclature AA, A, and B to represent the top three streams. In school A, the A and B streams constituted  approximately 30% and 50%, respectively, o f a l l  the grade 11 and 12 students. The remainder of subjects were either enrolled i n a non-academic mathematics program or they were not taking any mathematics courses. According t o the mathematics department head of school B, the enrollment i n the three top streams varies greatly from year to year. She reported that the present AA, A, and B streams constituted about 10%,  25%, and 50%, respectively,  o f the grade  11 and 12 student  enrollment. As was the case f o r school A, the remainder o f students were  either  enrolled  taking  a  non-academic  i n any mathematics  mathematics  class  or were not  courses. The sampling d i s t r i b u t i o n  that  was used during the screening phase i s presented i n Table I I . The eligible  subjects  who  to p a r t i c i p a t e  participated  i n the screening  i n the subsequent  phase  were  interview and assessment  phase only i f t h e i r performance on CALl met the standards established for t h i s s e l e c t i o n purpose.  These standards w i l l be discussed l a t e r  i n t h i s chapter. The  original  sample  of 286  subjects  was  reduced  to t h i r t y  subjects by applying these s e l e c t i o n standards to the r e s u l t s of the  74 TABLE I I  SAMPLING DISTRIBUTION OF SUBJECTS PARTICIPATING  IN THE SCREENING PHASE  L e v e l o f Mathematics School  A  B  Total  A b i l i t y Stream  Grade 11  Grade 12  Total  A  27  34  61  B  32  28  60  AA  24  27  51  A  28  24  52  B  28  34  62  139  147  286  75 screening mental  test  CALl.  calculators  Fifteen  and  subjects  fifteen  were designated as s k i l l e d  as unskilled  mental  calculators.  Nineteen subjects were chosen from school A and eleven from school B. Originally  twenty  subjects  attending  school  A  were  selected  to  p a r t i c i p a t e i n the f i n a l phase o f the study but one subject found i t d i f f i c u l t to make arrangements to meet with the researcher.  By mutual  agreement, he withdrew from the study. All school.  skilled  subjects were enrolled i n the top stream i n each  Likewise, a l l u n s k i l l e d  subjects  were  from  the lower  mathematics streams. Seven grade 11 and eight grade 12 students formed the  skilled  group  and seven grade 11 and eight grade  formed  the u n s k i l l e d  formed  the s k i l l e d  12 students  group. Thirteen male and two female  group  and f i v e male and ten female  subjects  formed the  u n s k i l l e d group. These  thirty  subjects  met i n d i v i d u a l l y  with  the researcher  where the purpose o f the study was explained and a l l procedures were described.  At  this  time,  an  eligible  subject  was  invited  to  p a r t i c i p a t e i n the interview and assessment phase o f the study. A l l thirty  subjects  volunteered  to p a r t i c i p a t e  i n the study and each  signed the subject consent form included i n Appendix A. One a d d i t i o n a l subject was l a t e r i n v i t e d to p a r t i c i p a t e i n the study  as a  result  of a  fortuitous  approached  the researcher after  explained  that  event.  A  grade  12 student  a screening t e s t administration and  she had a young  friend  exceptionally s k i l l e d mental c a l c u l a t o r .  who  she believed  was an  Fortunately, her o f f e r to  make the necessary arrangements was accepted because t h i s 13-year-old subject  proved  to have  calculative  powers  f a r beyond  any o f the  76 s k i l l e d subjects who  participated  i n the  study. The  analysis  of  her  c a l c u l a t i v e techniques led to a better understanding of the process of mental c a l c u l a t i o n .  In t h i s study, the  term highly  used to describe the performance of t h i s one To  skilled will  subject.  summarize, a group o f 15 unskilled subjects,  s k i l l e d subjects,  and  one  be  a group of  highly s k i l l e d subject p a r t i c i p a t e d  in  15 the  interview and assessment phase of the study.  Test Instruments and Procedures The Screening Instrument: CALl Item s e l e c t i o n .  The  process of item selection for a l l three  mental c a l c u l a t i o n t e s t s deviated somewhat from conventional practice. To  develop a t e s t  researcher  for a study of written  typically  selects  items  by  computational s k i l l s ,  first  restricting  the  the  item  domain to a s e r i e s of 1-digit by 2-digit products, 1 - d i g i t by 3 - d i g i t products and,  so  forth.  considering  which  typically,  1-digit  tasks,  carry  developed, the value d i g i t s procedure  a  place by  ten;  Often the values  works  would  involve  2 - d i g i t tasks, no and,  so  items for the at  items are categorized further  random. well  forth.  because  the  "carried  digit":  c a r r i e s ; 1 - d i g i t by  2-digit  Once such a  t e s t can  In  a  scheme has  been  be created by s e l e c t i n g place  case of written  the  by  researcher  calculation,  makes  assumption that almost a l l subjects w i l l apply the  the  this  reasonable  same strategy  to  solve each computation. Such a scheme could not be used to s e l e c t items for a t e s t of mental c a l c u l a t i o n performance. determine  i f skilled  and  The major purpose of the study was  unskilled  mental c a l c u l a t o r s  differed  to in  77  their  choices  of calculative  selected  to ensure  subjects  to apply  literature  that a  on expert  strategies.  at least  Thus,  items  an opportunity  d i v e r s i t y o f strategies. mental c a l c u l a t i o n  existed  Since  suggested  had to be for the  the research  that  apprehended  number properties were a useful c a l c u l a t i v e a i d , the majority o f items were selected  according  t o the number properties  suggested by the  factors. The  following  scheme was created  by the researcher to ensure  that a large proportion o f items would cue the hypothesized strategies i f some subjects d i d i n f a c t possess a knowledge o f such strategies. All  items selected  f o r the study met one o f the following  conditions  (A few examples are given to c l a r i f y the item s e l e c t i o n procedures): 1. and  The prime f a c t o r i z a t i o n o f both factors included only 2, 3,  5 as prime f a c t o r s .  (Examples: 25 x 32, 12 x 15, 64 x 250, 125 x  125). 2.  The item d i d not meet condition 1 and a t l e a s t one factor  was either a multiple o f 10 or a multiple o f 25.  (Examples: 50 x 39,  8 x 70, 25 x 65, 75 x 201). 3.  The item d i d not meet either condition  1 or 2 and either  factor was one greater or one l e s s than a multiple o f a power o f 10. (Examples: 8 x 99, 11 x 79, 49 x 51). 4.  The  item  d i d not meet either  condition  1,  2, or 3.  the best  chance o f  (Examples: 87 x 23, 8 x 4211, 73 x 83, 13 x 13). Condition eliciting  a  1 items were thought  variety  of  calculative  to have  strategies  and,  therefore,  two-thirds o f the items were o f t h i s type. The condition 4 items, on the other hand, seemed l e a s t l i k e l y to e l i c i t a v a r i e t y o f strategies  78  and, The  consequently, only 9 items were chosen to meet t h i s  condition.  numbers of items chosen to meet a l l four conditions are included  i n Table I I I . In order considered  t o reduce the s i z e of the item pool,  an item was  for s e l e c t i o n only i f the factors were of the  forms: 1 - d i g i t  by 2 - d i g i t ,  1-digit by 3-digit, 1 - d i g i t  following  by 4 - d i g i t ,  2 - d i g i t by 2 - d i g i t , 2 - d i g i t by 3 - d i g i t , 3 - d i g i t by 3 - d i g i t . The number of The  item forms included  i n each category i s summarized i n Table I I I .  majority o f the items were of the 2 - d i g i t by 2 - d i g i t form while  the other forms were used  sparingly.  There were a number o f reasons why the great majority of items selected  for t e s t i n g  contained  2-digit  by 2 - d i g i t  factors.  First,  2 - d i g i t by 2 - d i g i t  factors were expected to provide a subject with  more opportunities  to use a v a r i e t y  of methods f o r determining  a  mental product than 1-digit by 2 - d i g i t factors. Second, finding the mental products o f either  2 - d i g i t by 3 - d i g i t or 3 - d i g i t by 3 - d i g i t  factors was expected to be a much more d i f f i c u l t task than finding the mental product o f 2 - d i g i t by 2 - d i g i t factors. In order t o ensure that less skilled  subjects d i d not become overly  d i f f i c u l t items were used  discouraged, these more  sparingly.  Five p i l o t tests were constructed from the 80 item pool.  Four  tests contained 15 items and one t e s t contained 20 items. Some s p e c i a l precautions had to be taken to ensure that a l l subjects  used  administration  only  mental  calculation  procedures.  The  group  of any instrument designed t o evaluate p r o f i c i e n c y i n  mental c a l c u l a t i o n poses s p e c i a l problems: each subject needs a p e n c i l to record a s o l u t i o n but a p e n c i l cannot be used as a c a l c u l a t i v e a i d .  79  TABLE I I I  DISTRIBUTION OF ITEMS SELECTED FOR THE PILOT TESTING PHASE  Conditions Forms  1  2  1 - d i g i t by 2 - d i g i t f a c t o r s  2  1  1 - d i g i t by 3 - d i g i t f a c t o r s  2  0  1  1-d i g i t by 4 - d i g i t f a c t o r s  0  0  0  1  1  2-d i g i t by 2 - d i g i t f a c t o r s  39  3  6  4  52  2-d i g i t by 3 - d i g i t f a c t o r s  9  2  1  3-d i g i t by 3 - d i g i t f a c t o r s  1  Total  53  0  6  3  4  4  Total  0  7  4  7  0  12  0  0  1  12  9  80  80  Persuasion was the method used to prevent a subject from using a  pencil  little the  to c a l c u l a t e .  A l l subjects were informed that  there was  to be gained from using a p e n c i l i n t h e i r c a l c u l a t i o n s  results  o f the test would have no e f f e c t on t h e i r  since  coursework  grades. The researcher emphasized that a p e n c i l was to be used only to record  the solution.  They were  instructed  to record  neither  the  factors nor any intermediate c a l c u l a t i o n s . As  a check  to determine  i f the test  procedures  had been  followed, the answer sheets were c a r e f u l l y examined for any evidence indicating that some stage o f the c a l c u l a t i o n had been recorded. As a result of this before  scrutiny,  any analysis  the r e s u l t s o f ten subjects were removed  o f the p i l o t  testing  data was conducted.  Of  course, there was no way to determine with c e r t a i n t y that a subject d i d not make some s u r r e p t i t i o u s record o f a c a l c u l a t i o n . There was an a d d i t i o n a l unconventional testing requirement: the time given to complete each item had to be controlled t o some degree. For  the purposes  of this  study,  skilled  mental c a l c u l a t o r s  were  defined as being more capable and faster than l e s s s k i l l e d subjects i n completing calculation  mental as  calculations.  an  additional  The  decision  criterion  of  to  use  mental  speed  of  calculation  performance was based upon both experimental precedent and p r a c t i c a l considerations. time  to  Reys, Rybolt, Bestgen, and Wyatt (1982) used response  select  good  computational  estimators.  Levine  (1982)  recommended that the e f f e c t o f time constraints on estimation a b i l i t y be  examined i n future  calculation  i s often  studies.  Furthermore, since  a p r a c t i c a l necessity,  a "quick" mental  the incorporation  solution time constraint seemed both appropriate and desirable.  of a  81  In  a  group  situation,  however,  it  is  very  difficult  to  determine the s o l u t i o n times o f a large number of subjects who would complete each task l i k e l y a t d i f f e r i n g times. A p r a c t i c a l method to estimate  solution times  i n a large group setting  i s to c o n t r o l the  item-presentation rate so that the time for each item i s the same for all  subjects.  only  the  I f the item-presentation rate i s s u f f i c i e n t l y  most  skilled  subjects  should  be  able  to  rapid,  complete  each  calculation. These  methods  of  presenting  the  items  and  controlling  item-presentation rate were considered: audiotapes, videotapes, 35 slides  (see  Reys,  microcomputer these  controlled  possible modes  presentation was are  Rybolt,  obvious.  As  medium before  monitor.  of  used.  Bestgen  item  &  After  the  researcher  1982,  careful  presentation,  The convenience  well,  Wyatt,  a  p.  185),  the mm a  consideration of  recorded  audiotape  and s i m p l i c i t y of the medium had  used  t h i s presentation  (Hope, Note 1) and the p a r t i c i p a t i n g subjects appeared  to have l i t t l e d i f f i c u l t y i n remembering the factors despite the very short duration of the spoken word. The  five  pilot  tests were recorded  on  i n i t i a l item-presentation rate o f 30 seconds.  item order for each t e s t was  the end  of  the  arrangement was  t e s t contained expected  university  five pilot students  an  expressed  so that the beginning  equally d i f f i c u l t  items.  and  T h i s item subjects to  items.  tapes were administered who  using  Nine times f i f t y - o n e . "  to encourage the less s k i l l e d  attempt the l a t e r presented The  scrambled  tape  Each item was  using the term "times": for example, "Item 1. The  audio  were  currently  to 180  enrolled  in  undergraduate mathematics  82  education c l a s s e s . There were several reasons why u n i v e r s i t y rather than  high  school-students  were  used  bo p i l o t  the t e s t  items. An  important consideration was subject a c c e s s i b i l i t y . U n i v e r s i t y students were f a r easier f o r the researcher to access  than were high-school  students. A further and more important consideration was that the item r e s u l t s f o r each sample were expected  to be very s i m i l a r  since the  u n i v e r s i t y and high-school students d i d have comparable mathematical backgrounds. When each sample's r e s u l t s on a number o f common items were compared, the performance patterns o f each sample were found to be very s i m i l a r . Thus, the u n i v e r s i t y subject pool seemed to be an adequate sample for p i l o t - t e s t i n g Each Prior  test  purposes.  administration was supervised  by the researcher.  to each administration, the general nature and purpose o f the  study were explained. withdraw  from  identifying  A l l students were offered the opportunity to  the t e s t i n g .  their  answer  They were a l s o given  sheet  explain the testing procedures  or not.  The t o t a l  the choice o f time  needed to  and administer the t e s t t o each c l a s s  was about 15 minutes. No more than one p i l o t t e s t was administered to a c l a s s but one t e s t was administered to two classes. Each item s o l u t i o n was scored as either c o r r e c t or incorrect. A  difficulty  index  (p-value)  and  a  discrimination  index  ( p o i n t - b i s e r i a l r) were calculated for every item used i n the p i l o t study.  The conventional p r a c t i c e o f selecting items with d i f f i c u l t i e s  between 0.3 and 0.7 (Allen & Yen, 1979, p. 121) seemed inappropriate for  a  test  performance.  designed  to d i s t i n g u i s h  between  two  extremes  of  Thus, i t was decided to s e l e c t items that were either  very d i f f i c u l t or very easy.  S p e c i f i c a l l y , an item was considered for  83  inclusion i f i t was either d i f f i c u l t  (0.03 < p < 0.30)  0.85).  and  The  expected  inclusion to  of  provide  very  good  non-extreme performances  easy  very  discrimination  or easy  difficult between  (p >  items  was  extreme  and  (Allen & Yen, 1979, p. 121). The i n c l u s i o n o f  a r e l a t i v e l y large number o f very easy items (p > 0.85) was to ensure that only the most u n s k i l l e d mental c a l c u l a t o r s would be unable to answer a t l e a s t one or two items c o r r e c t l y . Two  potential  t r i a l - f o r m s o f CALl were constructed from the  pool o f 80 items that had been administered i n the preliminary stages of  the  pilot  study. Each  trial-form  contained 10  difficult  designated as Part A and 10 easy items designated as Part B.  items Part A  had d i f f e r e n t items but the items o f Part B were i d e n t i c a l for each trial-form. The items 25 x 25, 25 x 48, and 16 x 16 had d i f f i c u l t y of  0.42,  0.41,  and  0.34,  respectively.  Although  these  indices  difficulty  indices were not within the range stipulated for i n c l u s i o n , the items were  included  in  a  trial-form  because  each  item  contained many  a t t r i b u t e s that were expected to a i d i n a mental c a l c u l a t i o n . The  final  administer Sixty-one The  each  procedure of  university  these  in  seconds  was  used  development  trial-forms  students agreed  t e s t procedures were similar  stages o f the p i l o t  the  to  of  another  to complete  to those used  CALl pilot  was  to  sample.  both  trial-forms.  i n the  preliminary  t e s t i n g . However, an item-presentation rate o f 20 instead  of  30  seconds.  This  change  in  item-presentation rate was expected to further increase the d i f f i c u l t y of each t r i a l - f o r m . The analysis o f the t e s t r e s u l t s indicated that each t r i a l - f o r m  84  was equally d i f f i c u l t  (10.95 and 10.56) and the variance was about the  sane (3.96  for each t e s t . The most r e l i a b l e t r i a l - f o r m  and 5.06)  retained as CALl. The  was  f i n a l t r i a l - f o r m of CALl selected to be used i n  the screening phase i s included i n Appendix B. The r e s u l t s o f these two t r i a l - f o r m s were a l s o used to evaluate the  selection  criteria  that  had  been  established  p a r t i c i p a t i o n i n the interview and assessment phase. attempt was set  both  for  subject  E s s e n t i a l l y , an  made to set a basal l e v e l for s k i l l e d performance and to  a  basal  and  a  ceiling  level  for  unskilled  performance.  S p e c i f i c a l l y , the screening c r i t e r i a were: 1.  Any subject who answered 7 or more of the 10 d i f f i c u l t CALl  items c o r r e c t l y and who  a l s o answered at l e a s t 9 of the 10 easy items  c o r r e c t l y was defined as a s k i l l e d mental c a l c u l a t o r . 2. easy CALl the  Any subject who  answered at l e a s t 2 but l e s s than 10 of the  items c o r r e c t l y and who  difficult  items  correctly  also answered no more than one  was  defined  as  an  unskilled  of  mental  calculator. These external standards w i l l provide any researcher planning a r e p l i c a t i o n with more s u i t a b l e information than provided by  relative  standards such as "the s k i l l e d are those subjects whose scores were i n the top 10%."  The  seemed reasonable for  unskilled  participating  basal l e v e l established for the s k i l l e d  for such an exploratory study.  performance was  chosen  to  ensure  subjects  The basal l e v e l set that a l l subjects  i n the l a t e r interview and assessment phase would have  a t l e a s t some minimal p r o f i c i e n c y i n mental c a l c u l a t i o n . Fewer than two the  pilot  study  percent of the u n i v e r s i t y subjects sampled i n  phase met  the  proposed  standards  established for  85  s k i l l e d mental c a l c u l a t i o n performance. When the screening  standards  were applied to the r e s u l t s o f one t r i a l - f o r m o f CALl, only 1 o f the 61  university  subjects  demonstrated  skilled  performance  performances o f 6 u n i v e r s i t y subjects were designated After  the standards  trial-form  of  were  CALl,  no  applied subject  to the r e s u l t s was  found  who  while  the  as u n s k i l l e d . of  met  the the  second skilled  performance standards while 17 subjects were designated as u n s k i l l e d mental c a l c u l a t o r s . Test administration. administration  o f CALl  The screening phase, which involved the  to high-school  students,  began i n school A  during mid-October and was completed i n school B by l a t e  November.  A l l testing took place during a regularly scheduled mathematics c l a s s . The researcher supervised every t e s t administration. All  subjects and teachers were most cooperative and no major  problems were encountered. such  as  an  unscheduled  There were the expected minor d i f f i c u l t i e s fire  drill  i n the middle  of  one  test  administration. Despite  the researcher's  emphasis,  a few i n d i v i d u a l s  either  misunderstood or chose to ignore the instructions not to use a p e n c i l as a c a l c u l a t i v e a i d . apparent both  This deviation from the t e s t i n s t r u c t i o n s was  from observing  a subject's behaviour  during  the test  administration and from the evident p e n c i l marks on a subject's answer sheet.  S i x subjects  ignored  this  procedural  requirement  and  their  r e s u l t s were removed from any analyses. Scoring. incorrect.  A l l answers  were  scored  either  as  c o r r e c t or  For each subject, a t o t a l score, a score on Part A, and a  score for Part B were recorded.  86  When the s e l e c t i o n standards were applied to these r e s u l t s , 16 of  the 280  skilled  subjects  performance.  classified  (approximately 6%)  met  Ninety-four subjects  the standards set for  (approximately 34%) were  as u n s k i l l e d mental c a l c u l a t o r s . Because  the s t a t i s t i c a l  techniques used i n the study assume large differences i n performance between the two extreme groups, the subjects demonstrating the most extreme performances were selected. Each extreme group contained 15 subjects. CALl was also administered to the highly s k i l l e d subject.  She  e a s i l y met the c r i t e r i a established for s k i l l e d performance. On the advice o f a teacher, some u n s k i l l e d subjects were not asked  to p a r t i c i p a t e  i n the interview and assessment phase  o f the  study. Subjects who demonstrated comparable performance were chosen as replacements. The Probing Instrument: CAL2 Item s e l e c t i o n .  CAL2 was the instrument used i n the interview  and assessment phase to determine each p a r t i c i p a t i n g subject's method of  solution.  The 30 m u l t i p l i c a t i o n tasks were a l l selected from the  80 item pool used i n the p i l o t testing phase. The item s e l e c t i o n process f o r CAL2 as i n the construction o f CALl was guided more by the expected strategies that an item might e l i c i t than by i t s psychometric properties.  However, an attempt was  made to s e l e c t a cross-section o f items that d i d vary i n d i f f i c u l t y . Some easy items were included so that the u n s k i l l e d mental c a l c u l a t o r s would be encouraged to attempt the more d i f f i c u l t items. items  were needed  t o challenge  Difficult  and maintain the i n t e r e s t  o f the  87  skilled  mental  calculators.  About  d i f f i c u l t y indices between 0.3 0.7,  and  the  remaining 30%  50%  and  had  of  0.7,  the  20% had  items  selected  had  indices greater  indices l e s s than 0.3.  selected for CAL2 had also been included i n CALl.  Five  than items  Appendix B contains  a l i s t i n g o f a l l 30 items included i n CAL2. Test  administration.  Unlike  large groups o f subjects, CAL2 was subject  met  explaining that  a  with  the  calculation  stated o r a l l y . As  task  administered  administered to i n d i v i d u a l s .  researcher  the purpose of the  CALl which was  in  a  t e s t , the  requiring  private  setting.  researcher t o l d  the  Each After  subject  mental m u l t i p l i c a t i o n would  mental method that seemed "natural."  I f i t was  Each subject  was  any  natural to talk out  loud or to use some a i d such as fingers, the subject was so.  be  soon as both factors were heard, the subject would  be expected to make an attempt to determine the solution by using  do  to  instructed to  encouraged to ignore the presence of  the  researcher as a c a l c u l a t i o n attempt was made. All  subjects  were t o l d that accuracy was  more important than  speed, but they were instructed to complete the c a l c u l a t i o n as quickly as possible.  For the f i r s t two  f u l l view o f the subject was for  each item.  t e s t administrations,  a stop watch i n  used to assess a subject's s o l u t i o n time  However, t h i s method of timing was  i t proved to be very d i s t r a c t i n g for subjects.  abandoned because  Instead, the solution  times were calculated from the audio record made of each Each subject  was  t o l d that the researcher would not  whether any answer was  c o r r e c t or not.  to  skilled  prevent  discouraged.  the  less  interview.  Such information  mental c a l c u l a t o r s  was  indicate withheld  from becoming  However, some exceptions to t h i s procedural r u l e had  too to  88  be made: a few that  a  unskilled  solution was  subjects were encouraged by  c o r r e c t and  the  appeared more w i l l i n g  knowledge to  attempt  subsequent tasks. Subjects were often asked to make more than one fact,  some subjects made up  attempt at a  solution.  In  to four attempts.  Any  subject who  requested another attempt was always allowed to do so but  only the f i r s t attempt was used to determine the accuracy and speed of a  solution.  A  subject's  subsequent  attempts  were  used  by  researcher to help i d e n t i f y the method of s o l u t i o n used i n the  the  initial  attempt and to provide a d d i t i o n a l information about the d i f f i c u l t i e s a subject was experiencing during a mental c a l c u l a t i o n . Whether or not a c o r r e c t s o l u t i o n was  stated, each subject  asked to explain the c a l c u l a t i v e method that was  applied.  was  Usually a  subject's method o f s o l u t i o n was evident from the i n t r o s p e c t i v e report that was provided.  I f the method could not be c l e a r l y i d e n t i f i e d , the  researcher would ask the subject to provide more d e t a i l .  For example,  the question, "Did you multiply the 25 and 8 by doing 8 x 5 , 4, and  carry the  so on or d i d you j u s t think 200?," was posed to some subjects  a f t e r an attempt to solve the task 25 x 48. At times, some students needed a p e n c i l and paper to describe a method of s o l u t i o n but such external aids were NOT the attempt to solve a given task. that e i t h e r the method of  a v a i l a b l e during  When the researcher was  s o l u t i o n had  been c l e a r l y  convinced  identified  or  further probing would be f u t i l e , he progressed to the next item. The order o f item presentation during the interview corresponds to  the item order l i s t e d i n Appendix B.  but one  This order was used with a l l  subject. This u n s k i l l e d subject was  unwilling to attempt  any  89  item  that seemed d i f f i c u l t ,  interview. administered  especially  i n the e a r l y stages  o f the  In order  to o f f e r encouragement, the e a s i e s t items were  first.  As a r e s u l t o f t h i s deviation from the usual  testing procedures,  she appeared to relax and made an attempt to solve  a l l items with the exception o f 25 x 480. The  amount o f time needed to complete the administration o f  CAL2 varied g r e a t l y from subject to subject.  T y p i c a l l y , with  subjects, about one hour was needed to administer each subject's method o f s o l u t i o n .  skilled  CAL2 and discuss  However, the slow responses and  numerous attempts a t r e - c a l c u l a t i o n g r e a t l y increased the time needed to complete the interviews with the unskilled subjects. of  one u n s k i l l e d  subject,  f o r example,  took about  The interview three  one-hour  sessions t o complete. An attempt was made to ensure that consistent standards o f procedure  were maintained  phase: s p e c i f i c  tasks,  throughout  task  the interview  sequences,  and assessment  i n s t r u c t i o n s , and questions  were common to a l l interviews. Nevertheless,  the researcher  d i d not  hesitate to deviate from a planned interview i f an opportunity to gain a d d i t i o n a l i n s i g h t i n t o the process o f mental c a l c u l a t i o n arose. Some subjects  were  presented  with  unique  tasks  and questions:  only  6  subjects were given the mental c a l c u l a t i o n 123 x 456; only one subject was asked t o multiply mentally from such  8 x 25. Often the information  spontaneous decisions helped  gained  improve the q u a l i t y o f the  subsequent interviews. Appendix C contains a p a r t i a l CAL2 interview with one s k i l l e d subject.  This  procedures used  interview i n this  provides study  further  clarification  o f the  to determine a subject's methods o f  90  solution. Scoring. incorrect.  Each item o f CAL2 was scored as either c o r r e c t or  I f a subject had requested  that the factors be repeated  a f t e r the c a l c u l a t i o n had begun, the item was scored as i n c o r r e c t . But if  the subject had requested  confirmation o f the task by  responding  with a question such as, "Was i t 12 x 15?," the item was scored on the basis o f the subject's completed solution. If without  a  subject  immediately  any prompting  correct.  an  erroneous  response  from the interviewer, the item was scored as  One subject's  scoring procedure.  corrected  response  to 8 x 4211 should  clarify  this  He said, "32 611...No, wait...It's 33 688 rather."  A maximum CAL2 p r o f i c i e n c y score o f 30 was possible. The time taken by a subject t o complete a s o l u t i o n was measured by using a stopwatch. Only the f i r s t attempt a t a task was timed. This timing  information  was gathered  records made f o r each subject. researcher  had stated  during  Timing was i n i t i a t e d as soon as the  the question  subject had stated a s o l u t i o n .  the analysis o f the audio  and terminated  as soon as the  Each s o l u t i o n time was estimated t o  the nearest second. Often protracted  a subject's statement  s o l u t i o n time was increased because o f a  o f the s o l u t i o n .  For example, one subject  responded, "6000,..600, and..78." This statement took 4 seconds. The solution  time  was  increased  corrected a stated s o l u t i o n .  f o r those  subjects  who  immediately  This increase i n s o l u t i o n time i s w e l l  i l l u s t r a t e d by the following subject's response to the task 32 x 500. He s a i d , "1600,... No, wait, 2600..., No, wait. 15 000,.. 16 000!"  32 x 500 i s 1500,  91  The Challenge Test: CAL3 Item s e l e c t i o n . was  designed  to  CAL3 was a mental m u l t i p l i c a t i o n t e s t which  challenge  p a r t i c i p a t i n g i n the study.  the  most  skilled  mental  calculators  The purpose o f the t e s t was t o determine  i f these subjects would use other strategies not i d e n t i f i e d during the administration o f CAL2. had  been  difficult  identified  The challenge test included 15 items which  during  (p < 0.10).  the p i l o t  A  list  testing  phase as being  o f the items  included  very  i n CAL3 i s  presented i n Appendix B. Test administration.  The procedures  used t o administer CAL3  were i d e n t i c a l to those used for the administration o f CAL2. was given to only one subject a t a time. read  to a subject and immediately  The t e s t  Each c a l c u l a t i v e task was  after a solution was stated, the  subject was asked t o provide an explanation o f the method o f solution. A l l responses were recorded on audio tape f o r l a t e r a n a l y s i s . The highly  t e s t was administered  skilled  subjects  subject. CAL3 was  because  participation. supplementary  to 11 s k i l l e d  they  Since  were  the  information  only about  strategies known by s k i l l e d  subjects and the one  not administered not  enthusiastic  purpose the  of  types  CAL3 of  to four  skilled  about  further  was  mental  to  provide  calculation  c a l c u l a t o r s and no comparisons between  s k i l l e d and u n s k i l l e d performers were to be made, 11 subjects seemed an adequate sample s i z e . CAL3 was not administered t o any u n s k i l l e d subject. Scoring. or incorrect.  Each item was scored and recorded as either c o r r e c t The s o l u t i o n time f o r each c o r r e c t response was a l s o  estimated from the audio records.  92  The Test of Written Paper-and-Pencil Computational S k i l l s : Item s e l e c t i o n . r e s u l t s o f WPP  The  purpose of WPP  was  two-fold.  WPP First,  were to be used to eliminate any subjects who  the  could not  c o r r e c t l y apply a written algorithm. For the purposes of the  present  study, there seemed to be l i t t l e point i n studying such dysfunctional subjects. No subjects were eliminated from the study because of their poor performance on A written  more  WPP.  important purpose of WPP  c a l c u l a t i o n performances.  Any  was  to compare mental  differences  and  i n performance  resulting from the absence of the permanent memory-store served by the written page were assumed to r e f l e c t  the  a d d i t i o n a l burden on  the  l i m i t e d resources o f the short-term memory system. To ensure that mental and written c a l c u l a t i o n performance could be compared, the same d i f f i c u l t items included i n CALl were included i n WPP.  These 10 items are l i s t e d i n Appendix B.  Test administration. subject. WPP subject  items, WPP  WPP  was administered  could have been administered  took no  (the A-iterns)  more than  was administered  2-3  i n d i v i d u a l l y to each  as a group t e s t . Since each  minutes to complete  the  10  written  i n d i v i d u a l l y at the mid-point o f each CAL2  interview. A l l 10 items were l i s t e d on a sheet i n a scrambled presentation order.  Each  questions  subject  was  instructed to solve  these m u l t i p l i c a t i o n  by applying any written method of c a l c u l a t i o n .  They were  asked to record a l l steps i n the c a l c u l a t i o n . Instructions were given to complete the c a l c u l a t i o n as q u i c k l y as possible as the time taken to complete the test would be However, the  researcher  a l s o emphasized that accuracy  recorded.  should  not  be  93 s a c r i f i c e d for an increase i n speed. A minor problem arose  when a few subjects misunderstood the  directions  and t r i e d  solution.  In these cases, the subject was stopped and the procedures  were c l a r i f i e d . few  skilled  to determine a mental rather  than  a written  Despite t h i s a d d i t i o n a l i n s t r u c t i o n i n procedures,  subjects  continued  to use mental c a l c u l a t i o n  a  t o solve  tasks such as 1 6 x 1 6 . No comment was made about t h i s minor deviation from procedure and the subjects were allowed  to complete  the test  without any further interruptions. Scoring.  Each item i n WPP was scored and recorded  c o r r e c t or i n c o r r e c t .  The maximum possible score was 1 0 .  as either The time  needed to complete the test was recorded for each subject. The Test o f R e c a l l o f Basic Facts; BFR Item s e l e c t i o n . quickly  This instrument was designed  and accurately a subject could  multiplication.  Since  these  numerical  recall  to determine how  the basic  equivalents  facts of  are the basic  "building blocks" o f most c a l c u l a t i o n s , an assessment o f r e c a l l seemed v i t a l to any analysis o f i n d i v i d u a l differences i n mental c a l c u l a t i o n performance. BFR included a l l 1 0 0 m u l t i p l i c a t i o n d i g i t factors.  items formed from s i n g l e  The order o f these items was scrambled t o ensure that  the s o l u t i o n o f one item could not be used to a i d i n the r e c a l l o f a successive item. Test  administration.  BFR was administered  the administration o f WPP was completed. listing  iitimediately after  A subject was given a sheet  the 1 0 0 items and instructed to state the s o l u t i o n to each  94  item as quickly as p o s s i b l e .  The subject was reminded that, although  the time taken t o complete the test would be measured, accuracy was considered t o be more important than speed. The interviewer recorded a subject's i n c o r r e c t reponses as the testing proceeded. Scoring. The  Each item was scored as either c o r r e c t or incorrect.  maximum possible score on BFR was 100.  The t o t a l access  time,  defined as the time needed to complete BFR, was a l s o estimated to the nearest second and recorded for each subject. Forward D i g i t Span: FDS Item s e l e c t i o n . Scale  A subtest o f the Wechsler Adult > Intelligence  (Wechsler, 1955) was used to estimate  This subtest contains two l i s t s digits.  digit  span FDS.  ( T r i a l I and II) o f seven s e r i e s o f  Each s e r i e s o f d i g i t s v a r i e s i n length from 3 to 9 d i g i t s . Test administration.  subjects. be  forward  read  FDS was administered  i n d i v i d u a l l y to a l l  The subjects were instructed that a s e r i e s o f d i g i t s would to them.  As soon as a s e r i e s was read and the researcher  paused, a subject was expected to r e c a l l the d i g i t s i n the same order as the presentation.  Each s e r i e s was read a t a rate o f one d i g i t - p e r -  second. Beginning researcher  with  continued  the shortest t o present  series l i s t e d  successively longer  in Trial  s e r i e s u n t i l an  error was made by a subject. Thereafter, the researcher s e r i e s o f equivalent length from T r i a l I I .  I, the  presented  This process  a  continued  u n t i l e i t h e r a l i s t was completed or a subject f a i l e d on both t r i a l s o f a given s e r i e s . Scoring.  A subject's score on FDS was the number o f d i g i t s i n  95  the  l a r g e s t s e r i e s repeated  max imum possible score  without  error i n T r i a l  I or I I .  The  f o r FDS i s 9. But i n the event a subject  "reached the c e i l i n g " o f either FDS or BDS, longer s e r i e s were to be administered.  These longer s e r i e s were needed for only one subject's  r e - t e s t o f FDS.  Backward D i g i t Span; BDS Item s e l e c t i o n . used  to assess  trials  The subtest o f the WAIS (Wechsler, 1955) was  backward d i g i t  span BDS. This  o f seven s e r i e s o f d i g i t s whose length  subtest  contains two  ranges from 2 to 8  digits. Test administration. used  to administer  The test procedures were similar to those  FDS with  the exception  instructed t o "say the s e r i e s backwards."  that  each  subject was  To ensure that a l l subjects  understood t h i s change i n procedure, a practice s e r i e s was given. Scoring. determining  A  score  the number  for each  of digits  subject  was  i n the longest  backwards without error i n T r i a l I or I I .  calculated series  by  repeated  The maximum p o s s i b l e score  for BDS was 8.  Delayed D i g i t Span; DDS Item s e l e c t i o n .  The items for DDS used i n the present  study  were constructed using the procedures outlined i n a study conducted by Vvhimbey and Lieblum constructed. length  (1967).  S i x t r i a l s o f 5 s e r i e s o f d i g i t s were  The d i g i t s i n each s e r i e s were selected a t random. The  o f each  s e r i e s varied  from  4 to 8 d i g i t s .  The order o f  presentation o f each s e r i e s was scrambled for each t r i a l . included i n DDS are presented  i n Appendix B.  A l l series  96  Test administration. s i z e from 3 to 6 students. that  a l l subjects  DDS was administered to groups ranging i n Small group sizes were necessary  followed  the  somewhat  to ensure  "unconventional"  testing  procedures. Each recall.  subject  was  each  sheet  On  given was  a a  sheet  to  record  section which  listed  word-letter p a i r s : bear-Q, bird-L, crab-V, f i s h - H presentation o f each s e r i e s o f d i g i t s was statement o f one o f these s i x words.  the  f  attempts the  at  following  frog-R, f l e a - J .  The  followed immediately by  a  A l l subjects were instructed to  locate the word-letter section after hearing any one of these words, f i n d the word, and write the corresponding place on the record sheet before attempting  letter  i n the  designated  to write the d i g i t s e r i e s .  A p r a c t i c e t r i a l was given to ensure that a l l subjects understood the directions. Each  s e r i e s of  digits  was  read  at a  rate o f  2  digits-per-  second. The  researcher  l i s t e n e d through an earphone to an e l e c t r o n i c  metronome to help pace the reading o f each s e r i e s . Scoring. was  reproduced  determined by  In scoring DDS, in  the  alloting  order  a s e r i e s was considered c o r r e c t i f i t presented.  subject's  score  was  4 points for each 4 - d i g i t s e r i e s c o r r e c t , 5  points for each 5 - d i g i t s e r i e s correct, and procedure was  A  so f o r t h .  i d e n t i c a l to that used by Vvhimbey and  The maximum possible score for DDS was  This scoring  Lieblum  (1967).  180.  Letter Span: LS Item s e l e c t i o n .  Another measure commonly used to assess  capacity i s l e t t e r span LS.  The  STM  items for LS were constructed using  97  procedures similar to those trials  containing  9  used by Brown and  s e r i e s of  letters  selecting consonants from B to M.  Kirsner  (1980).  were developed  by  Four  randomly  The absence of vowels ensured that  a s e r i e s contained no recognizable English words which could be used by a subject to foster retention. Each s e r i e s length varied from 3 to 11 l e t t e r s .  The order o f each s e r i e s i n a t r i a l was  Test administration.  LS was  administered  subjects i n the same t e s t i n g session as DDS.  scrambled.  to small groups of  Each subject was given a  sheet designed to record a given s e r i e s . The in  subjects were instructed to attempt to r e c a l l each s e r i e s  the same order  as the presentation. Each l e t t e r  read at a rate of one letter-per-second.  One  i n a s e r i e s was  t r i a l l i s t was  completed  to ensure that a l l subjects understood the i n s t r u c t i o n s . Scoring.  Each s e r i e s was considered c o r r e c t i f i t was  i n the order presented. for  recalled  A score was determined by assigning 3 points  a 3 - l e t t e r s e r i e s , 4 points for a 4-letter s e r i e s , and so f o r t h .  The maximum possible score for LS was  252.  C l a s s i f i c a t i o n of Methods of Solution Since one the  calculative  mental  o f the major purposes of the study was s t r a t e g i e s used  multiplication  tasks,  a  by  high-school  strategy  to i d e n t i f y  students  classification  needed. I t became evident a f t e r a somewhat f r u i t l e s s  to  solve  scheme  search  of  was the  l i t e r a t u r e that the researcher would have to develop a scheme. The  analyses  of  experts*  Chapter II d i d guide the research  calculative  methods  outlined  i n the preliminary stages o f  in the  study. Nevertheless, the continual intertwining o f data c o l l e c t i o n and  98 analysis helped  the researcher develop and r e f i n e the c l a s s i f i c a t i o n  scheme used to answer the research questions. An example w i l l i l l u s t r a t e how analysis o f the accumulated data led  to a refinement  early  stages  o f the strategy c l a s s i f i c a t i o n  o f the interview  scheme. In the  and assessment phase  the u n s k i l l e d  subjects almost always used the penci1-and-paper analogue while the s k i l l e d subjects r a r e l y used t h i s method o f s o l u t i o n . These patterns o f approach t o the task skilled  continued  subject. Unlike the other  u n t i l the l a s t  interview with a  s k i l l e d subjects, she made almost  exclusive use o f t h i s method f o r even p a r t i c u l a r l y d i f f i c u l t  tasks.  This "exception to the r u l e " forced the researcher to elaborate on the penc i1-and-paper  mental  analogue.  As  a  result,  four  specific  paper-and-penci1 mental s t r a t e g i e s were i d e n t i f i e d and included i n the c l a s s i f i c a t i o n scheme. To f a c i l i t a t e the c l a s s i f i c a t i o n o f the c a l c u l a t i v e procedures used by a subject to solve the CAL2 and CAL3 mental c a l c u l a t i o n tasks, a l l audio  records o f the interviews were transcribed.  300  pages resulted from  typed  recorded  solutions  for  two  this  transcription  subjects  were  Approximately  process.  destroyed  A few by  the  tape-recorder: s p e c i f i c a l l y , an u n s k i l l e d subject's reported solutions for 5 items and a s k i l l e d subject's reported solutions f o r 3 items. In these  cases,  the written  notes  made  during  the CAL2  and CAL3  interviews were substituted f o r the missing audio records. If  the interviewer  thought that the subject had not provided  sufficient details for classifying was questioned remember  f o r more d e t a i l s .  the s p e c i f i c  the applied strategy, the subject Although the subjects might not  numbers that were involved i n a computation,  99  they usually had no problem remembering and describing the method of solution. As i s often the case with any classification scheme, a certain amount of  ambiguity  had  to be  tolerated.  In  particular,  some  strategies were d i f f i c u l t to classify under a single heading because the  subject  seemed to combine several strategies.  Completing  a  calculative task by combining strategies was especially true of the skilled subjects during where  a  combination  CAL3  the  of  interviews.  strategies was  For those situations  used,  the  strategy  was  classified according to what the interviewer believed was the primary or dominant strategy. so  that a  If a subject appeared to have arranged the task  factoring strategy could  be  applied, for example,  but  completed a stage of the calculation using another strategy such as additive  distribution,  the  solution  attempt  was  classified  as  factoring. The annexation algorithm —calculating with factors which are multiples  of  a  power of  10  by  annexing  zeroes—  is a  form of  factoring. But annexation played a secondary rather than a primary role in solving the types of calculative tasks included in this study. For example, even though many subjects began to calculate 25 x 480 by reasoning, "take off the zero and add i t later," another strategy was needed to solve the d i f f i c u l t portion of the computational task, in this case, 25 x 48. There were no items included in CAL2 and CAL3 which could  be  solved  through  the  exclusive  application of  the  annexation algorithm. The analysis of the introspective reports revealed that four general methods of solution were used by the subjects to solve the  100  CAL2 and CAL3 mental m u l t i p l i c a t i o n tasks. Eleven s p e c i f i c c a l c u l a t i v e s t r a t e g i e s were also i d e n t i f i e d . The r e l a t i o n s h i p between the general methods and the s p e c i f i c s t r a t e g i e s reported by some or a l l subjects i s summarized i n Table IV. Pencil-and-paper mental analogue mental  calculation  involves  the  (P&P).  This general method o f  application  of  the  conventional  pencil-and-paper algorithm to solve mental c a l c u l a t i o n tasks.  These  four main v a r i a t i o n s were i d e n t i f i e d : (1) no p a r t i a l product r e t r i e v e d (P&PO);  (2) one  p a r t i a l product retrieved  products retrieved P&PO, P&P1,  and  (P&P2);  (P&P1);  (4) stacking. The  P&P2, respectively,  during a c a l c u l a t i o n a subject who  (3) two  partial  d i g i t s 0, 1, and 2 i n  represent the number of  used the pencil-and-paper  times mental  analogue retrieved a numerical equivalent larger than a basic f a c t to determine a p a r t i a l product. 1. No p a r t i a l product r e t r i e v e d strategy w i l l make no attempt mental  medium.  Regardless  calculation w i l l  proceed  (P&PO).  A subject using t h i s  to adapt pencil-and-paper methods to a of  the  type  of  in a digit-by-digit,  task  presented,  the  r i g h t - t o - l e f t manner.  Each p a r t i a l product w i l l be calculated and no numerical equivalents larger than basic facts w i l l be retrieved during the c a l c u l a t i o n . One u n s k i l l e d subject's unsuccessful attempt to solve 25 x 480 provides  a  good  illustration  of  an  application  of  t h i s strategy:  Let's see. 480 on the top and 25 on the bottom. 5 x 0 , 5x8 i s 40, carry 4, and 4 i s 24. I have to r e a l i z e that the second number i s one over. 2 x 0, 2 x 8 i s 16, carry 1; 2 x 4 i s 8 and 1 i s 9 so 960, 9600. So 9600 and 2400 i s 0, 0, 19 thousand and....860. 2.  One  p a r t i a l retrieved  (P&P1). Rather  than completing  the  c a l c u l a t i o n of each p a r t i a l product by proceeding d i g i t by d i g i t ,  a  101  TABLE IV GENERAL METHODS AND SPECIFIC STRATEGIES USED TO SOLVE  MENTAL MULTIPLICATION TASKS  General Method  S p e c i f i c Strategy  Penc il-and-Paper Mental Analogue  P&PO: No p a r t i a l product r e t r i e v e d P&P1: One p a r t i a l product r e t r i e v e d P&P2: Two p a r t i a l products r e t r i e v e d Stacking  Distribution  Additive Fractional Subtractive Quadratic  Factoring  General Half-and^double Aliquot parts Exponential  Retrieval o f a Numerical E q u i v a l e n t  102  subject who uses P&P1 w i l l p a r t i a l product.  r e t r i e v e a numerical  1200."  f o r one  The following report involved an a p p l i c a t i o n o f P&P1  to c a l c u l a t e 25 x 48: "5 x 48 i s . . . 5 x 8 = 2 x 48 i s 96.  equivalent  I know that.  40, carry 4, 24, 240. And  I think o f 96 brought over one. So i t ' s  Notice that the p a r t i a l product  240 was calculated d i g i t by  d i g i t but the second p a r t i a l product 96 was retrieved as a numerical equivalent. 3.  Two p a r t i a l s retrieved (P&P2).  pencil-and-paper  In t h i s v a r i a t i o n o f the  mental analogue, the c a l c u l a t i o n w i l l be arranged so  that two p a r t i a l products  can be retrieved as numerical  rather than c a l c u l a t e d d i g i t by d i g i t .  equivalents  One subject's explanation o f  12, x 250 i l l u s t r a t e s an a p p l i c a t i o n o f t h i s strategy.  She explained,  "2 x 250 i s 500 and 1 x 250 i s 250. Move over one, 3000." 4. the  Stacking.  This strategy can be applied to questions where  factors are o f a 1 - d i g i t by x - d i g i t  partial  product  will  carrying, each product  be completed  form.  digit-by-digit  Essentially,  each  but, instead o f  w i l l be v i s u a l i z e d i n a stacked arrangement.  One reported s o l u t i o n o f 8 x 999 i s a good example:  "I thought 8 x 9  i s 72, 72, and 72, r i g h t across." The subject who provided t h i s report said that she never " c a r r i e d " during the c a l c u l a t i o n . She determined the t o t a l sum by adding from l e f t to right. To describe her thought processes to the researcher, she drew the diagram below: 72 72 72 7992  103 Distribution. transforming  one  differences.  The  following four 1.  This  or  method  of  more factors  calculation  calculation  into a  proceeds  is  s e r i e s of  by  initiated  either  applying  Additive.  one  of  This strategy  sum.  The  following  solution  of  8  so i t ' s 33 600.  the  As the c a l c u l a t i o n  x  4211  to produce involved  application o f additive d i s t r i b u t i o n : "8 x 4000 i s 32 000, 1600,  these  i s based on the p r i n c i p l e of  progresses, each p a r t i a l product w i l l be added successively running  sums or  strategies:  d i s t r i b u t i v e law of m u l t i p l i c a t i o n over addition.  a  by  8 x 11 i s 88, so the answer i s 33  an  8 x 200  is  688."  unlike the use of P&PO where the c a l c u l a t i o n proceeds d i g i t digit,  additive  multiples  of  distribution  powers  of  involves  ten.  Thus,  calculating knowledge  frequently  of  the  series o f Instead,  some c a l c u l a t i o n s a factor w i l l not be addends whose terms involve the  computation w i l l  equivalents can be retrieved. by thinking, i s 240,"  "15  be The  multiples  arranged  so  with  annexation  algorithm i s e s s e n t i a l to the successful application o f t h i s For  by  strategy.  transformed into a of a power o f  that  s k i l l e d subject who  ten.  known numerical solved 15 x 16  squared i s 225 which i s a f a c t I know and another 15  i l l u s t r a t e s such an a p p l i c a t i o n : i . e . , 15 x 16 = 15 x  (15 +  1). There were some reported difficult  to d i s t i n g u i s h  cases, the researcher had the  classification  solutions  from additive  these  two  distribution.  P&P3 were  In  these  few  to r e l y on additional information to guide  decision.  The  language  describe a c a l c u l a t i o n procedure was between  where P&P2 and  strategies.  used  by  a  subject  to  often h e l p f u l i n d i s t i n g u i s h i n g  Terms  characteristic  of  written  104  m u l t i p l i c a t i o n procedures such as "12 on top, and 15 on the bottom," "carry  the 1," and "move over one" were often  describe  the pencil-and-paper strategies.  used by subjects to  Such  terms were  usually  absent i n reported uses o f d i s t r i b u t i o n . Generally speaking, the c a l c u l a t i o n s f o r pencil-and-paper and distribution  strategies  proceeded  pencil-and-paper c a l c u l a t i o n s  in  proceeded  opposite  from the l e a s t  "directions": to the most  s i g n i f i c a n t d i g i t s (right t o l e f t ) and d i s t r i b u t i o n proceeded from the most to the l e a s t s i g n i f i c a n t d i g i t s ( l e f t to r i g h t ) . This  difference  i n d i r e c t i o n often was used to d i s t i n g u i s h P&P2 and P&P3 from additive distribution. Another d i s t i n g u i s h i n g c h a r a c t e r i s t i c o f additive d i s t r i b u t i o n was that  most c a l c u l a t i o n s  usually  involved  an a p p l i c a t i o n  o f the  annexation algorithm which was expressed by the user i n terms such as "9 x 7 i s 63 and add 3 zeroes." An application o f pencil-and-paper strategies usually  incorporated  some abridgement such as using 12 to  represent 120. 2.  Fractional.  For those factors which have a u n i t  equal to 5, f r a c t i o n a l d i s t r i b u t i o n can be applied. x 48 can be calculated  digit  A task such as 15  as, "10 x 48, 480 and 1/2 o f 480 i s 240; so  i t ' s 720." Larger f a c t o r s such as 125 x 125 can be reasoned as, "100 x 125, 12 500 and 1/4 o f 12 500 i s 3125. Although calculation,  factoring  t h i s strategy  i s needed  So i t ' s 15 625."  to complete  a portion  o f the  was c l a s s i f i e d as a type o f d i s t r i b u t i o n  rather than as a type o f factoring because the c a l c u l a t i o n proceeds by p a r t i t i o n i n g each factor i n i t i a l l y into a sum rather than a product. 3.  Subtractive.  This strategy  i s based upon the d i s t r i b u t i v e  105  p r i n c i p l e o f m u l t i p l i c a t i o n over subtraction. can  proceed,  the subject  must be able  Before the c a l c u l a t i o n  to express  a  factor as a  difference between two numbers which the subject thinks w i l l make the c a l c u l a t i o n more t r a c t a b l e .  For example, 8 x 999 was solved by the  majority o f s k i l l e d subjects who reasoned 8000 - 8 = 7992: i . e . , 8 x 999 = 8 x (1000 - 1 ) . A  few  incorporating  subjects a  can  retrieval  apply  of  a  subtractive  large  numerical  distribution  by  equivalent.  For  example, one s k i l l e d subject solved 15 x 16 by thinking, "16 squared i s 256 and then minus 16, so 240": i . e . , 15 x 16 = (16 - 1) x 16. 4. some  Quadratic.  subjects  Certain properties of quadratics can be used by  as a  calculative  aid.  There  were  three  forms o f  quadratic d i s t r i b u t i o n i d e n t i f i e d i n the study. If  the square  o f the mean value  o f the two factors i n a  m u l t i p l i c a t i o n task can be r e t r i e v e d by a subject, the c a l c u l a t i o n can be completed by using  the algebraic i d e n t i t y for the d i f f e r e n c e o f 2  squares:  i . e . , (x - y) (x + y) = x  2 - y .  For 49 x 51, one subject  reasoned, "50 x 50 minus 1" and, f o r 23 x 27, he thought, "25 squared minus 2 squared, so i t ' s 625 - 4 = 621." The remaining  two v a r i a t i o n s o f quadratic d i s t r i b u t i o n were not  used by any subject i n the CAL2 interviews.  However, they were used  to solve some CAL3 items. One strategy involved binomial expansion: 2 2 2 i . e . , (x + y) = x + 2xy + y^. To determine a square, one subject reasoned, "48 x 48 = (50 - 2 ) = 2500 - 200 + 4." 2  This same subject  used quadratic d i s t r b u t i o n to c a l c u l a t e 125 x 125 by reasoning, 25)  2  (100 +  = 10 000 + 5000 +625. The a p p l i c a t i o n o f another v a r i a t i o n o f quadratic d i s t r i b u t i o n  106  i s l i m i t e d to those squares whose u n i t d i g i t i s 5. can be calculated by reasoning, 7 x 8 = 56 and add  For example, 75  2  (annex) 25, so i t ' s 2  5625. Such a strategy i s based on the knowledge that (lOx + 5) lOOx (x + 1)  +25.  Factoring. i n that one  =  This method o f solution d i f f e r s from d i s t r i b u t i o n  or more factors i n the presented c a l c u l a t i v e task can  transformed into a s e r i e s o f products or quotients rather s e r i e s o f sums or d i f f e r e n c e s .  be  than into a  Several types of factoring  strategies  were i d e n t i f i e d . 1.  General.  factor one  To solve 25 x 48, one  reasoned, "5 x 48  applied  additive  i s 240  distribution  =  5  x  200  +  5  x  40,  the  associative  to  i s 1200." This subject  determine  the  general  factoring  rather  for 5x5 also  intermediate  i . e . , 5 x 48 = 5 x 40 + 5 x 8 and 5 x respectively.  Despite  the  a d d i t i o n a l use o f a d d i t i v e d i s t r i b u t i o n , the reasoning was as  law  subject factored 25 i n t o  and 5 x 240  c a l c u l a t i o n s 5 x 48 and 5 x 240: 240  the user must be able to  or more factors before applying  multiplication. and  To apply t h i s strategy,  than d i s t r i b u t i o n because  subject's classified  the  subject  transformed the computation i n i t i a l l y into a s e r i e s of products rather than sums.  Additive  a n c i l l i a r y rather  d i s t r i b u t i o n can  be  thought o f as playing  than primary c a l c u l a t i v e r o l e during an  an  application  o f factoring. The in  a  remaining v a r i a t i o n s of factoring could a l l be considered,  mathematical  However, since  sense,  as  special  these v a r i a t i o n s  did  cases  of  general  seem to d i f f e r  with  factoring. regard  to  their range o f applications, any use of these v a r i a t i o n s were reported as separate strategies.  107  2.  Half-and-double.  I f one factor i n a m u l t i p l i c a t i o n task i s  a multiple o f 2, a s p e c i a l form o f factoring can be employed.  The  calculation  and  proceeds  usually  by  "doubling" the remaining factor.  " h a l f i n g " the  multiple  of  This process of h a l f i n g and  2  doubling  continues to a point where the subject can complete the c a l c u l a t i o n by applying another strategy. The  following examples i l l u s t r a t e  how  two  subjects  who  both  used the half-and-double factoring strategy d i f f e r e d i n their method o f terminating t h i s process. One x  16,  "I d i d doubling  Doubling 16 gives 32.  and And  subject reported that to c a l c u l a t e 12  h a l f i n g again.  I d i d 1/2  6 x 30 i s 180,  so 192."  of 12  Another subject  explained, "I j u s t decided to multiply by 2, or 12 x 2, four 3.  A l i q u o t parts.  or more factors as subject  using  Instead of transforming  i s the  case with  a l i q u o t parts  other  times."  a factor i n t o two  factoring s t r a t e g i e s , a factor into a  quotient.  For example, although she erred i n c a l c u l a t i n g 25 x 480, one  subject's  explanation i l l u s t r a t e s how  transforms one  i s 6.  a l i q u o t parts was applied.  She  "When I multiply by 25 I think o f money, l i k e quarters.  explained,  So I divided  480 by 4, 120 and then changed i t to 1200." This strategy i s applied frequently to -those computations where one  factor " f " i s a factor o f a power of 10  "p" and  the  remaining  factor i s a multiple o f the quotient "p/f". Items such as 25 x 48 12 x 250,  for example, can be solved through an a p p l i c a t i o n o f a l i q u o t  parts: i . e . , 25 x 48 = 100/4  x 48 = 100 x 48/4  = 100 x 12 = 1200  12 x 250 = 12 x 1000/4 = 12/4  x 1000 = 3 x 1000  =  4. calculate  and  Exponential. the  products  This of  form of  powers  3000.  factoring can  through  the  and  be  used  to  a p p l i c a t i o n of  an  108  exponential r u l e .  To solve 32 x 32, one subject reasoned: "I solved  i t by thinking powers o f 2.  32 i s 2 to the f i f t h , so squared i s 2 to  the tenth which I j u s t know i s 1024." There were some reports  that could have been c l a s s i f i e d as  applications o f e i t h e r half-and-double  or exponential f a c t o r i n g . For  example, 32 x 64 was reasoned by one subject as "32 x 32 i s 1024, which  i s a power  o f 2, so double  i t , 2048." Further  questioning  revealed that the subject knew 1024 was a power o f 2 but he d i d not know that 1024 was equal t o 2  1 0  some knowledge o f exponential  . I f the subject seemed t o incorporate arit_imetic  into the c a l c u l a t i o n , the  strategy was c l a s s i f i e d as exponential factoring. Retrieval products, by  of a  numerical  equivalent.  To  determine  some  a subject appears to do no c a l c u l a t i o n and solves the task  retrieving  the product  from memory.  For example, many  subjects stated immediately "625" when presented  with  skilled  the task 25 x  25. A subject's method o f s o l u t i o n was c l a s s i f i e d as r e t r i e v a l only i f both the subject and researcher were convinced had  taken place.  that no c a l c u l a t i o n  For example, the response, "I d i d 5 x 25, then I  kind o f remembered that 25 x 25 i s 625," was c l a s s i f i e d as general factoring  rather  than r e t r i e v a l because the subject admitted  that some c a l c u l a t i o n had been necessary.  later  A quick response (less than  3 seconds) was considered as a d d i t i o n a l evidence that r e t r i e v a l rather than c a l c u l a t i o n had taken place.  109  R e l i a b i l i t y o f the Instruments and the Procedures All the  instruments and procedures used i n the various phases o f  study  were  Consequently,  a  designed variety  for differing of  methods  purposes  was  needed  and to  functions. assess  the  " r e l i a b i l i t y " o f these various measures. One problem reliability All  inherent i n extreme group designs i s c a l c u l a t i n g  coefficients  that can be interpreted with some meaning.  subjects who p a r t i c i p a t e d  i n the interview and assessment phase  were not selected a t random from the population. Rather the s e l e c t i o n of the sample was based upon the performance standards established f o r t h i s purpose. Any r e l i a b i l i t y c o e f f i c i e n t based on the combined scores of the s k i l l e d and u n s k i l l e d subjects would apply only to t h i s type o f extreme sample. An attempt  was made, however, to incorporate some  method o f assessing the r e l i a b i l i t y o f each measure.  R e l i a b i l i t y o f CALl CALl served two major functions i n the study. F i r s t , was  used  to s e l e c t  performance.  subjects who  differed  i n mental  Second, CALl was the instrument used  this test  calculation  to determine the  strength o f r e l a t i o n s h i p between mental m u l t i p l i c a t i o n performance and various other measures used i n the study. Therefore, i t was important to calculate some index o f the instrument's r e l i a b i l i t y . Two estimates o f the r e l i a b i l i t y o f CALl were c a l c u l a t e d .  To  estimate the i n t e r n a l consistency, the Kuder-Richardson 20 formula was applied and a c o e f f i c i e n t o f 0.80 was obtained. This suggested  that  CALl was a r e l a t i v e l y homogenous test. Furthermore,  as CALl was used to select students who d i f f e r e d  110 i n mental c a l c u l a t i o n performance, i t seemed important an  to determine i f  i n d i v i d u a l ' s score remained uniform over a period o f time and t o  ensure, a t the very l e a s t , that the r e l a t i v e standings o f the subjects were maintained. CALl would not be a p a r t i c u l a r y r e l i a b l e instrument  screening  i f a subsequent re-administration o f the t e s t indicated  that some u n s k i l l e d subjects should be r e - c l a s s i f i e d as s k i l l e d and, conversely, some s k i l l e d subjects were i d e n t i f i e d as u n s k i l l e d . To determine the s t a b i l i t y o f a subject's score on CALl over a period o f time, a large sample o f the subjects who p a r t i c i p a t e d i n the initial  screening  re-administered weeks a f t e r seemed  phase  to four  the f i r s t  sufficiently  "carry-over e f f e c t "  was  calculating  105 students  administration.  to imnimize  CALl  was  about  nine  Such a period o f time  the p o t e n t i a l problem  of a  (Allen & Yen, 1979, p. 77) which could reduce an  estimate o f the r e l i a b i l i t y . by  Specifically,  classes totalling  test  long  re-tested.  An index of r e l i a b i l i t y was determined  the c o r r e l a t i o n  between the 105 p a i r s o f t e s t and  r e - t e s t scores. The t e s t - r e t e s t r e l i a b i l i t y was calculated to be 0.79 which seemed t o i n d i c a t e that a subject's r e l a t i v e standing remained reasonably stable over a period o f time. R e l i a b i l i t y of CAL2 and CAL3 The procedures  function  of  that  subject  multiplication  a  both  CAL2 used  and to  CAL3 solve  was a  i n Chapter  IV,  the  of  mental  two t e s t s ,  though  determined  and  statistical  analyses. Thus, r e l i a b i l i t y c o e f f i c i e n t s f o r the CAL2 and  CAL3 performance scores were not necessary.  were  identify  variety  tasks. A subject's scores on these reported  to  not  used  The r e l i a b i l i t y  i n any  o f the  Ill strategy c l a s s i f i c a t i o n scheme which was based on the tasks used i n CAL2 and CAL3 was o f p a r t i c u l a r estimate  this  form o f r e l i a b i l i t y  interest. will  The procedures  used t o  be discussed l a t e r  i n the  chapter.  R e l i a b i l i t y o f WPP and BFR Both the 10-item t e s t o f written m u l t i p l i c a t i o n s k i l l s WPP and the 100-item tasks  that  school  t e s t o f basic f a c t m u l t i p l i c a t i o n r e c a l l BFR contained were expected  students. This  to be easy p a r t i c u l a r l y  assumption  proved  f o r senior-high  to be c o r r e c t :  the mean  performances on WPP and BFR for the e n t i r e 30-subject sample were 8.7 (s  = 1.28) and 98.3 (s = 2.21), respectively. Expressed another way,  X  X  the average p-values for the items i n WPP and BFR were 0.87 and 0.98, respectively. /Any estimate o f r e l i a b l i t y based on item variance such as KR20 would have y i e l d e d an extremely low c o e f f i c i e n t since the item variance  was minimal.  An accumulation o f scores a t the f l o o r or  c e i l i n g o f a t e s t w i l l always lower the r e l i a b i l i t y o f a t e s t (Allen & Yen, 1979, p. 214). Each  subject's performances  on basic f a c t r e c a l l and written  computation s k i l l s were l a t e r assessed by re-administering WPP and BFR several weeks a f t e r the i n i t i a l testing had taken place. As might be expected with the highly overlearned s k i l l s assessed by these t e s t s , little  change i n performances  was observed. The mean score on the  r e - t e s t o f WPP changed by 0.30 (3%) and the mean score on the r e - t e s t of BFR changed by only 0.37 (0.37%). Furthermore, over the two t e s t i n g sessions, 80% o f the subjects' WPP scores d i d not vary by more than a score o f 1. Likewise, 70% o f the subjects' BFR scores d i d not change  112  by more than 1. These r e s u l t s seemed to indicate that a l l subjects' performances  on  these  selected  arithmetic  fundamentals  were  quite  consistent. In t h i s sense, WPP and BFR could be considered r e l i a b l e . Time needed  to complete  the basic  fact  recall  test  found to remain r e l a t i v e l y constant over separate testing  BFR  was  sessions.  The mean change i n s o l u t i o n time was about 8 seconds: t h i s represented an  average  solution trials.  change  of  about  times never changed  8%.  Eighty  percent o f  the  subjects'  by more than 10 seconds over the  two  Thus, access time or the time required to complete the 100  items o f BFR seemed to remain r e l a t i v e l y stable over time.  R e l i a b i l i t y o f FDS, BDS, DDS, and LS Each subject's forward d i g i t span and backward d i g i t span were assessed by using the appropriate Weschler Adult Intelligence Scale (Weschler, reliability young the  1955)  subtests.  The  test  c o e f f e c i e n t o f r = 0.71  manual reported  a digit  span  for the standardized sample  adults. Unfortunately, t h i s reported r e l i a b i l i t y  of  was based on  combined rather than the separate scores o f FDS and BDS. However,  Dempster (1981, p. 65) suggested that most psychometric tests o f d i g i t span are moderately r e l i a b l e (about r = 0.66). Each subject was retested cn FDS and BDS i n order to determine the  s t a b i l i t y o f a subject's score over a period o f several weeks. The  mean changes over the two assessments of FDS and BDS were c a l c u l a t e d to be 0.6 and 0.8, r e s p e c t i v e l y : representing average changes o f about 9% and 15% for FDS and BDS the  scores, respectively. Furthermore, 90% o f  subjects' FDS scores and 80% o f their BDS scores d i d not deviate  by more than 1 from one t e s t i n g session to the other.  113  Some  skilled  subjects  increased  their  scores  on  these  two  memory measures by changing strategies over the two t e s t i n g sessions. One  subject commented, "Last time (the f i r s t testing session) I t r i e d  to remember 2 - d i g i t s e r i e s and t h i s time (the second t e s t i n g session) I t r i e d to remember groups of three. I t seemed to work better." This apparent change i n "chunking" strategy by some subjects and not others will  change some subjects'  scores  i n an unpredictable  type  of  could  reduce  measurement  error  the  manner. This  r e l i a b i l i t y of  these  measures. The other measures used to provide an estimate of STM were delayed DDS  d i g i t span DDS  and  letter  span LS. The  capacity  construction of  was based on Whimbey and Leiblum's (1967) recommendations. DDS  usually more r e l i a b l e used  to  estimate  than either FDS  capacity.  The  or BDS  because more t r i a l s are  researchers  reported  a  split-half  r e l i a b i l i t y , corrected for the f u l l length of the test, of r = The measure of l e t t e r span LS was by Brown and Kirsner 0.92  (1980) who  is  0.88.  developed from the suggestions made reported a s p l i t - h a l f r e l i a b i l i t y of  based upon a sample o f c o l l e g e - l e v e l students.  The R e l i a b i l i t y of the Introspective Reports The  accuracy o f a subject's introspective report i s not e a s i l y  v e r i f i e d . The l i k e l i h o o d e x i s t s that the verbal reports bear no close r e l a t i o n s h i p to the a c t u a l cognitive processes complete  a  task.  There  are  precautions  that  used by a subject can  be  to  undertaken,  however, to ensure that introspective reports are a c r e d i b l e source of data. C a r e f u l consideration must be given to the procedures used to  114  gather  the verbal data. How  the introspective reports are generated  w i l l determine to a large degree the types of information that can be reported r e l i a b i l y . Two present  study  were  data c o l l e c t i o n procedures considered for the concurrent  and  retrospective  verbalization  (Ericsson & Simon, 1980). Concurrent  verbalization  or  "thinking-aloud"  refers  to  information verbalized a t the time a subject i s attending to a task. An  advantage of t h i s procedure over other  forms of introspection i s  that the information under the conscious attention of the subject can be  traced d i r e c t l y  and,  hence, the  researcher  i s provided  with  an  i n d i r e c t assessment o f the i n t e r n a l stages of the cognitive process under study. There were several reasons why reporting was  t h i s type o f i n t r o s p e c t i v e  not used i n the present study to determine a subject's  method of s o l u t i o n . F i r s t , as Ericsson and Simon (1980) have pointed out, one consequence of thinking-aloud task performance study,  a  i s a s l i g h t decrease i n the speed of  (p. 226). Although for the purposes of the  reduction  in  usual  calculation  speed  would  not  have  present been  an  i n t o l e r a b l e outcome, a comparison o f each s k i l l group's a b i l i t y  to  c a l c u l a t e q u i c k l y seemed d e s i r a b l e . There was a more important reason that concurrent introspection was  not used i n the present  identify  the processes  cope with  the  and  demands of  study.  The  intent of the study was  to  procedures that people normally  used  a  required  mental c a l c u l a t i o n .  To  have  subjects to engage i n overt v e r b a l i z a t i o n during a c a l c u l a t i v e  to  task  when such an a c t i v i t y was not part of their normal c a l c u l a t i v e routine seemed  an  unnecessary  and  undesirable  obtrusion  that  could  have  115  affected the course of a mental c a l c u l a t i o n . Retrospective rather than concurrent v e r b a l i z a t i o n seemed to be the preferred method of gathering the required information about each subject's c a l c u l a t i v e methods. Obviously t h i s form o f reporting cannot p r o t r a c t the time needed for an already completed c a l c u l a t i o n . Whether post hoc  explanations of one  subsequent  calculation  retrospective  c a l c u l a t i o n can a f f e c t the course o f a  is  subject  verbalization  verbalization  to  interfere  seemed with  a  to less  debate. likely  subject's  Nevertheless,  than  concurrent  normal approach  to  a  mental c a l c u l a t i o n . An obvious weakness of retrospective reporting i s the p o t e n t i a l loss  of  calculative  temporarily-held STM  detail  the  method of  calculation  task.  calculation  and  A  the  each  the  i n the  forgetting  study  s o l u t i o n immediately a f t e r  short start  subject  delay of  the  between  the  verbal report  previously heeded information would s t i l l Simon, p.  with  of  information. Since unattended information stored i n  i s r a p i d l y forgotten,  report on  associated  was  asked  completing  completion ensured  reside i n STM  of  to the the  that some (Ericsson &  226).  Some l o s s o f information was  p e r f e c t l y acceptable  abundance o f c a l c u l a t i v e d e t a i l often was  not necessary  because an  to c l a s s i f y a  subject's strategy. For example, the f a c t that a subject could r e c a l l only that "some number was o f the carry was  c a r r i e d " but could not remember the value  s u f f i c i e n t l y d e t a i l e d for the purpose of strategy  classification. Attempts were made to corroborate  each subject's  report:  by  ensuring that both the researcher and subject understood the meanings  116  o f the various terms involved i n the discussion and explanations; using probing  questions  to e l i c i t a d d i t i o n a l information not  by  reported  i n i t i a l l y by the subject. Although the accuracy o f a subject's report can  never be  fully  demonstrated, the  approaches used  i n the  seemed to ensure that a l l subjects gave as complete and as  study  accurate  accounts of t h e i r thought processes as possible.  R e l i a b i l i t y of the C l a s s i f i c a t i o n Scheme. Once  all  introspective  reports  had  been  compiled  and  c l a s s i f i e d , the r e l i a b i l i t y with which the reports had been c l a s s i f i e d needed to be estimated. Since c l a s s i f i c a t i o n schemes can be ambiguous, an  attempt  introspective  was  made  reports  to  ensure  would  not  that vary  the  classification  markedly  from  of  the  researcher  to  researcher. In order to ensure that a subject's reported method of s o l u t i o n was  accurately c l a s s i f i e d ,  another mathematics educator was  with a written d e s c r i p t i o n of the c l a s s i f i c a t i o n c l a s s i f y a 10 percent sample of the 900  scheme and  provided asked to  introspective reports.  There  was mutual agreement on 95 percent of these reports.  The Design of the Study and Methods of Analysis The l o g i c of an Extreme Groups Design Comparing the performances o f extreme groups on a v a r i e t y o f measures i s a research methodology which has been used often i n the exploratory phase i n the i n v e s t i g a t i o n of a psychological construct. Such investigations are random sample o f  initiated  subjects  by  evaluating  v i a some measure of  a the  relatively  large  construct  under  117  study. High and low performing subgroups are selected by using the d i s t r i b u t i o n o f scores which r e s u l t s . These two extreme subgroups are either assigned to a treatment condition or evaluated on a v a r i e t y o f dimensions that  the researcher  suspects contribute  to the observed  differences i n performance (Feldt, 1970, p. 133). By  comparing  each  group's  performance  on  these  other  dimensions, the researcher hopes to i d e n t i f y some v a r i a b l e s that can be  incorporated  differences measures,  exist the  into  subsequent  between  existence  the  of  a  experimental  extreme monotonic  groups  studies.  on  If  no  these secondary  relationship  between  the  c r i t e r i o n and secondary performance measures seems u n l i k e l y .  Through  this  can  comparison  process,  possible  sources  of  variation  be  eliminated from further t h e o r e t i c a l consideration. Differences between the extreme groups must be interpreted with more caution. Although the extreme groups were selected to d i f f e r on one dimension, a researcher has no way o f knowing whether or not the groups d i f f e r there  on a multitude o f other dimensions, as w e l l .  is difficulty  i n proving  that any one p a r t i c u l a r  Hence,  difference  between the groups on a secondary measure o f performance i s the source o f the observed difference i n the performances o f each group on the c r i t e r i o n measure. The need for prudent analysis and interpretation i n comparative group research has been expressed by Cole and Means (1981): .. .we do want to underscore the need for extreme caution. The researcher r i s k s error when: (1) he goes beyond the behaviour observed under a p a r t i c u l a r set o f circumstances to hypothesize about general cognitive processes; (2) he t r i e s to i n t e r p r e t differences i n performance manifested by groups that vary i n unspecified but c e r t a i n l y numerous ways.... (p. 11).  118  Since t h i s study employed a comparative or extreme groups design, some ways o f nrinimizing  spurious  inferences  about group  differences i n  mental c a l c u l a t i o n performance had to be found. One way o f attempting to eliminate a l t e r n a t i v e explanations o f group differences i s to s e l e c t the comparison groups so that they are matched i n terms o f c h a r a c t e r i s t i c s that are not under study but whose presence could also produce the same behavioural outcomes (Cole and Means, p. 39).  In t h i s study, both the s k i l l e d and u n s k i l l e d groups  were reasonably matched  for age  and  l e v e l o f mathematics  studied.  Admittedly, t h i s matching does not necessarily solve the problem o f guarding against other relevant and unaccounted for variables such as i n t e l l i g e n c e , q u a l i t y o f mathematics  instruction, or sex.  However,  t h i s matching process was considered adequate for an i n i t i a l into  the  nature  of  individual  differences  i n mental  inquiry  calculation  performance. Another approach used to improve findings  of  the  present  study  was  performance within the selected groups.  the i n t e r p r e t a b i l i t y o f the  to  consider  the  patterns  of  In p a r t i c u l a r , an attempt was  made to determine the d i s t i n g u i s h i n g c h a r a c t e r i s t i c s o f high and low scoring members o f the same s k i l l group. strategies  used  by  low  and  high  scoring  For example, unskilled  the types o f subjects  were  i d e n t i f i e d and compared. The  identification  and  explanation o f deviant behaviour  was  a l s o another method o f a n a l y s i s employed i n t h i s study. The researcher attempted to examine the behaviour o f those subjects who  were more  s i m i l a r on one dimension to the members o f the other s k i l l group than they were to the members o f their own  s k i l l group.  For example, i t  119  seemed important to understand why of  some subjects who  used the methods  solution c h a r a c t e r i s t i c of the u n s k i l l e d calculators were able  to  a t t a i n a l e v e l o f s k i l l e d performance. Examining different  performance patterns o f  calculation  tasks  employed i n the study.  One  such  unskilled  subjects,  processes  as  also  group over  another  a  method  number of  of  inquiry  purpose of the d i f f i c u l t tasks included  the challenge t e s t CAL3 was the  was  a  in  to make the s k i l l e d subjects perform l i k e  the  l o g i c being that the connection between  methods of  solution  and  calculation  performance  could become more apparent.  The S t a t i s t i c a l Analysis of Extreme Group Data Estimating the magnitude of a hypothesised l i n e a r r e l a t i o n s h i p among the  variables  selected  extreme group researchers. direct  application  of  The  the  for  study poses s p e c i a l problems  nature of analytic  the  design precludes  techniques  so  common  for the in  experimental c o r r e l a t i o n a l research. All met  subjects i n the  present study were selected  some standard o f mental c a l c u l a t i o n performance.  procedure  in  effect  d i s t r i b u t i o n from any  removed  the  "middle  part"  of  because they  This s e l e c t i o n the  population  subsequent s t a t i s t i c a l analyses. The  removal o f  t h i s p a r t i c u l a r range of scores invalidates any conclusions based upon conventional c o r r e l a t i o n a l or regression  analyses.  the  some variable  skilled  combined,  and  any  exaggerate the  unskilled  calculated s i z e of  subjects on correlation  the  s e l e c t i o n v a r i a b l e X (see,  I f the r e s u l t s of Y  c o e f f i c i e n t would,  r e l a t i o n s h i p between variable for example, A l l e n & Yen  were  simply  typically, Y and  the  (1979, p. 36)  for  120 discussion). Thus, l i n e a r r e l a t i o n s h i p s detected by the indiscriminate use  of  correlational  and  regression analysis w i l l  be  more  likely  a r t i f a c t u a l than r e a l . A number o f researchers 1940;  Feldt, 1970;  presented gathered  (Pearson, 1903;  A l f & Abrahams, 1975;  Peters & Van  Voorhis,  Abrahams & A l f , 1978)  have  a v a r i e t y o f s t a t i s t i c a l techniques to analyze information from "widespread c l a s s e s . " The v a l i d use of each  technique  rests on several assumptions. F i r s t , i t must be assumed that both high and low scoring groups have been selected from a r e l a t i v e l y large sample (preferably N >  100)  based upon the performance scores on a v a r i a b l e X. In the case of the present study, X represents a subject's performance on the screening t e s t CALl. Second, estimates o f performance on a variable Y are assumed to have been obtained for only those subjects with high and low scores on X. In the present study, the various instruments used i n the interview and assessment phase and described i n t h i s chapter were used to obtain scores on the v a r i a b l e Y. An important techniques  further assumption that underlies the s t a t i s t i c a l  for analysing extreme group data i s that the v a r i a b l e s X  and Y are normally d i s t r i b u t e d . I f a l l of these assumptions have been met  or  at least  reasonably  approximated, there are several methods  that can be used to estimate the r e l a t i o n s h i p between X and Y. Perhaps the score  means  of  significance o f this:  any  simplest  the  two  s t a t i s t i c a l method i s to compare the  extreme  groups  using  a  t  the d i f f e r e n c e between the means. The  relationship  i n the  test  for  reasoning  population between X and  Y will  Y  the is be  121  reflected  i n the s i g n i f i c a n t difference  between the Y d i s t r i b u t i o n  means for the "high" and "low" subgroups (Feldt, 1970, p. 133). In the present study, t t e s t s were used to compare the means o f the s k i l l e d and u n s k i l l e d relationship  groups  i n an attempt  to determine the presence o f a  between CALl mental m u l t i p l i c a t i o n performance and the  various secondary measures o f performance used i n the interview and assessment phase. Using a t t e s t to compare the extreme groups* performances can provide  useful  but" somewhat  incomplete  information  about  the  relationships under study. McNemar (1960) has argued that a comparison o f the means o f two extreme  subgroups can provide information about  the presence only and not the strength o f a relationship. Without some estimate  o f the degree  iirportance  of  Furthermore,  a  trivial t  test,  o f association  between two variables,  relationships  can  as McNemar  pointed  easily  be  the  exaggerated.  out, i s " p a r t i c u l a r l y  f a l l a c i o u s i n case the underlying r e l a t i o n s h i p happens, unbeknownst, to be nonlinear" (p. 298). Fortunately, estimate  there  the strength  are  of a  statistical  linear  methods  relationship  available  existing  to  i n the  " i n t a c t " population when s t a t i s t i c s about only the extreme subgroups' performances  are available.  To estimate the full-range  correlation  between CALl mental m u l t i p l i c a t i o n performance and performance on the other cognitive tasks used i n the study, the "covariance information s t a t i s t i c " was used ( A l f & Abrahams, 1975; Abrahams & A l f , 1978; Garg, 1983). The c a l c u l a t i o n s needed to estimate R, the c o r r e l a t i o n i n the i n t a c t population, are:  122  R = r ' (S^/sy  ( 1 - ( r ' ) + (r«) ( S ^ ) ) " 2  2  2  1 / 2  where: r' i s the c o r r e l a t i o n between X and Y i n the sample consisting o f the combined high and S  x  low groups.  i s the standard deviation of X i n the population or random sample.  s  x  i s the unbiased standard deviation of X i n the sample consisting o f combined high and low groups.  In the present study, S the scores o f the 280 and s  x  refers to the standard deviation  x  subjects who  compeleted the CALl screening t e s t  r e f e r s to the standard deviation o f the combined CALl scores o f  the 15 s k i l l e d and 15 u n s k i l l e d mental c a l c u l a t o r s . The value of was  of  S^s^^  3.25/6.72 = 0.484. Garg  (1983)  empirically  the  designed  efficacy of  a  Monte  several  Carlo  study  to  investigate  strategies used to estimate  the  degree of r e l a t i o n s h i p i n an extreme group setting. He concluded that the covariance information strategy was  superior  strategies  correlation,  in  square error  terms of for  estimation  a l l values of  the  of  to the other proposed power, and  mean  population c o e f f i c i e n t and  the  proportion of subjects i n each " t a i l " o f the d i s t r i b u t i o n (p. 370).  As  well,  to  the  calculate  covariance than  the  information  strategies  strategy  developed  by  is  less  cumbersome  Peters and  Van  Voorhis  (1940), and F e l d t (1970). To t e s t the s i g n i f i c a n c e of the c o r r e l a t i o n r ' (and, s i g n i f i c a n c e o f R), 1975,  p.  565):  the following  thus,  the  t t e s t can be used (Alf & Abrahams,  123  t = r ' (N where N and N  Q  Q  Q  - 2)  1 / 2  (1 - ( r ' ) ) " 2  1 / 2  i s the number o f p a i r s of observations i n the combined upper  lower groups. The  s i g n i f i c a n c e o f the t test i s determined using  - 2 degrees o f freedom.  The Adequacy of the Sample S i z e The c o l l e c t i o n and analysis of i n d i v i d u a l introspective reports formed  an  integral  part  of  the  present  study.  This  procedural  requirement had to be kept i n mind when the s e l e c t i o n of a sample was being  considered.  The  s i z e of the sample had  to be large enough to  ensure that the proposed q u a n t i t a t i v e analyses would be meaningful, yet small enough to ensure that each interview would y i e l d detailed  qualitative  information.  horns of a dilemma: was to  ensure  the  The  researcher  adequately  i s placed  on  the  the chosen sample s i z e of 30 subjects adequate  validity  of  both  the  proposed  quantitative  and  q u a l i t a t i v e analyses? Viewed statistical  from  analyses,  the the  perspective sample  of  size  the  standard  be  considered  can  tenets  of  small.  If  sample s t a t i s t i c s such as means, standard deviations, and c o r r e l a t i o n s are  to generalize to a population,  an adequately  large sample  must be chosen. Generally speaking,  the larger the sample s i z e ,  more  sample  confidence  one  has  that  the  statistics  are  size the  reliable  estimates of the population parameters. Compared to the very  large sample s i z e s used by many  survey  studies such as the National Assessment of Educational Progress, s e l e c t i o n o f 30 experimental  subjects seems t r i f l i n g .  On  the  the other hand, i f the  studies of information-processing reviewed i n Chapter II  124  and which formed the t h e o r e t i c a l foundations o f the present study are examined, a sample s i z e o f 30 subjects appears to be about the norm (for example, see Hitch, 1978, N = 30; Whimbey, Fischhof & S i l i k o w i t z , 1969,  N = 24; Merkel & H a l l , 1982, N = 30; Dansereau & Gregg, 1966, N  = 1). White  (1980)  has  measures o f association,  suggested  that  f o r inferences  involving  the optimum sample s i z e can be answered only  when the researcher comes t o g r i p with the question: "What degree o f association  between the two variables w i l l have implications  educationally  interesting  or s i g n i f i c a n t ? "  (p. 49). The exploratory  nature o f the study means that almost any s t a t i s t i c a l l y relationship variables  between mental c a l c u l a t i o n  incorporated  What  would  population statistical  have  t o be  i n order  help  identify  i n mental c a l c u l a t i o n  the degree  o f association  for a particular  significant  performance and each o f the  i n the study could  sources o f i n d i v i d u a l difference  that are  potential  performance.  i n the parent  sample c o r r e l a t i o n  to reach  significance?  Using the c o r r e l a t i o n a l formulas described i n t h i s chapter, the correlation  between  a  variable  Y  and  CALl  mental  calculation  performance would have to be a t l e a s t 0.19 for a sample s i z e o f 30 (df = 28) and 0.12 for an sample s i z e o f 62 (df = 60) to reach s t a t i s t i c a l significance  a t the p = 0.05 l e v e l . Thus, a r e l a t i v e l y weak l i n e a r  r e l a t i o n s h i p could be detected using a sample s i z e o f 30. Considering the f a c t that some data c o l l e c t i o n techniques and analyses would have had  t o be eliminated i f the sample s i z e had been doubled, the apparent  gain i n s t a t i s t i c a l power r e s u l t i n g from such an increased sample s i z e seems poor compensation.  125  Viewed from a case  study perspective, the sample s i z e chosen  for the present study would be seen as excessively large. The d e t a i l e d analyses often associated with t y p i c a l "N = 1" case studies cannot be attained  when the  sample  size  becomes excessively large: the more  subjects, the more d i f f u s e the q u a l i t a t i v e data. But behaviour  i f the  researcher  had  chosen to focus the study on  of a very small number o f students, the g e n e r a l i t y of the  quantitative data would have been, at best, g r e a t l y reduced worst,  the  completely  individual  illusory.  differences  inter-subject  in  variablility  In  a  mental is  an  study  proposing  calculation, essential  or,  at  to investigate  information  requirement.  about  To  have  focussed on the behaviour o f a single case would have been, to say the l e a s t , a reductio ad absurdum. Considering the purposes o f the present study, a sample s i z e of 30 subjects seemed to be an adequate compromise: large enough to place some confidence that an  i n the s t a t i s t i c a l  analyses; small enough to ensure  adequate amount of q u a l i t a t i v e data  could be  amassed.  choice of the p a r t i c u l a r sample s i z e provided the researcher with opportunity  to  ally  the  analytic  techniques  of  case  study  The an and  s t a t i s t i c a l research. This a l l i a n c e seemed to exploit the strengths o f each rather than compound t h e i r weaknesses.  126  CHAPTER IV PRESENTATION OF THE FINDINGS P r o f i c i e n c y i n Mental C a l c u l a t i o n  Performance on CALl, the Screening Test The mean performance of the 280 the 20-item CALl screening t e s t was  grade 11 and 12 students for  11.1  (S  = 3.25).  x  Scores ranged  from a low of 2 to a high of 20. The mean performance of the s k i l l e d group on CALl was  18.6  u n s k i l l e d group was scores obtained 1983a;  5.7  (s  x  (s  = 1.24)  while the mean performance of the  = 1.54). Compared with the usually high  i n written computation  Robitaille  and  Sherrill,  1977)  (see for example: NAEP, by  this  age  group,  1977, mental  c a l c u l a t i o n i s a s i g n i f i c a n t l y more d i f f i c u l t task. The  d i s t r i b u t i o n of  scores on  CALl  for the  280  subjects i s  i l l u s t r a t e d i n Figure 1. The formula used to c a l c u l a t e the c o r r e l a t i o n from widespread classes assumes normality i n the t o t a l d i s t r i b u t i o n of the population t r a i t . An examination  of the d i s t r i b u t i o n reveals the  assumption of normality seemed tenable. The d i f f i c u l t y indices f o r each item used i n CALl are l i s t e d i n Table V. reported  Comparisons are d i f f i c u l t to make because few studies have any  findings regarding  mental m u l t i p l i c a t i o n  performance.  The most recent National Assessment of Educational Progress d i d report the r e s u l t s of one were not given.  released item, 90 x 70, but the t e s t conditions  F i f t y - f i v e percent  solved t h i s mental c a l c u l a t i o n  of the 17-year-old  (1983a, p. 32)  NAEP sample  and 92 percent o f the  sample p a r t i c i p a t i n g i n the screening phase of t h i s study solved the  127  o  c 3 cr  F i g u r e 1.  Frequency d i s t r i b u t i o n o f CALl mental m u l t i p l i c a t i o n s c o r e s f o r 280 g r a d e 11 a n d 12 m a t h e m a t i c s s t u d e n t s . X = 11.11  S  x  =  3.25  128  TABLE V  DIFFICULTY INDICES FOR ALL 20 CALl ITEMS ADMINISTERED TO 280  GRADE 11 AND 12 MATHEMATICS STUDENTS  Easy Items Item  D i f f i c u l t Items  p-value  .  Item  p-value  7 x 51  0.95  32 X 64  0.08  30 x 200  0.95  27 X 32  0.09  8 x 70  0.94  16 X 72  0.12  12 x 12  0.94  15 X 48  0.20  60 x 40  0.93  25 X 65  0.22  70 x 90  0.92  16 X 45  0.23  7 x 511  0.86  15 X 64  0.26  12 x 500  0.81  24 X 24  0.27  8 x 99  0.79  16 X 16  0.36  2 x 592  0.79  25 X 48  0.40  129  same item. Performance on CAL2, the Probing Test The  CAL2  subjects skilled  performance  differed and  levels  substantially.  of The  u n s k i l l e d groups were 24.3  5.08),  respectively. On  scores  ranged  from a  this  high  30-item  of  29  to a  the  skilled  mean  unskilled  performances  (s  = 3.92)  test,  the  low  and  and  9.00  skilled  of 17.  for  The  (s  the =  subjects' unskilled  subjects' scores ranged from a high of 19 to a low of 1. The h i g h l y s k i l l e d subject obtained a p e r f e c t score of 30 on CAL2. A  slight  overlap  i n the d i s t r i b u t i o n of each group's  scores  existed. The most p r o f i c i e n t subject i n the unskilled group obtained a score that was  equivalent to two s k i l l e d subjects' scores and greater  than one s k i l l e d subject's score. This overlap i n the d i s t r i b u t i o n of scores d i d not e x i s t for CALl, the screening instrument used to s e l e c t the two groups.  Such a discrepant finding could be used to question  the v a l i d i t y o f CALl as a screening instrument. the  total  distribution  statistical  technique  of  used  the  population  to estimate  Besides normality i n  i n both  a correlation  v a r i a b l e s , the i n the  intact  population from the s t a t i s t i c s of "widespread classes" assumes sharp truncation of the t a i l s of the d i s t r i b u t i o n 1940, p.  presented and  Voorhis,  386)  This procedures  (Peters & Van  discrepancy  can  be  accounted  used to administer each instrument. so that a subject had  record a solution.  administration o f CAL2.  No  for  by  the  differing  Each item i n CALl  was  no more than 20 seconds to complete  such time l i m i t was  imposed during  the  Thus, the apparent improvement i n performance  130  by  this  one u n s k i l l e d  subject could  a v a i l a b l e for a s o l u t i o n . since Thus,  the average i f time  determination  This seems to be a reasonable  s o l u t i o n time  to  be due to the increased  calculate  for this  a  solution  subject was  time  explanation 34  seconds.  i s considered  i n the  o f mental c a l c u l a t i o n p r o f i c i e n c y , no overlap  i n the  d i s t r i b u t i o n o f CAL2 scores f o r each group existed. As a general r u l e , the u n s k i l l e d group could be characterized as being slow to c a l c u l a t e a s o l u t i o n . This finding i s not s u r p r i s i n g since time to c a l c u l a t e a s o l u t i o n formed part o f the c r i t e r i a f o r s e l e c t i o n o f subjects. Nevertheless,  the contrast between the s o l u t i o n  times o f the two groups was considerable.  Over  30 percent  o f the  c o r r e c t responses o f the u n s k i l l e d group were completed i n 30 seconds or  longer.  For some items,  complete a c a l c u l a t i o n .  a great  deal o f time was  needed  to  For example, one unskilled subject solved 49  x 51 i n 93 seconds while another solved 17 x 99 i n 124 seconds.  In  contrast, two s k i l l e d subjects solved 49 x 51 and 17 x 99 i n 5 and 6 seconds, respectively. Figure  2  demonstrates  g r a p h i c a l l y the vast  differences i n  c a l c u l a t i v e performance and s o l u t i o n times between the various groups. The data points f o r the graph were obtained  skill  by c a l c u l a t i n g  each subject's score on CAL2 f o r a number o f d i f f e r e n t time l i m i t s . For  example, i f the imposed  previously solution After  correct  response  time l i m i t was 30 seconds, a subject's was  regarded as c o r r e c t only  time f o r that item was l e s s than'or equal  each item was scored  according  i f the  to 30 seconds.  to t h i s timing c r i t e r i o n , the  mean performance f o r each group was calculated.  Thus, the v e r t i c a l  axis represents the mean score o f each group and the h o r i z o n t a l a x i s  131  10  20 Time  30 limit  40 in  50  6  seconds  Figure 2. Comparative performance o f each s k i l l group on CAL2 scored under d i f f e r i n g time l i m i t s .  o  Unskilled  A  Skilled  0  Highly s k i l l e d  0  132  represents the imposed time l i m i t . An examination o f the response curve f o r the u n s k i l l e d reveals  that  performance  l i m i t s were imposed.  dropped  gradually  as more demanding time  Greatly degraded performance resulted when the  time l i m i t s were l e s s than 30 seconds.  A t a 20 second time l i m i t the  mean performance had dropped t o only 4 (13%). level  group  This low performance  compares favourably with the r e s u l t s o f CALl where the mean  performance f o r t h i s group was 5.67 (28%). As  can be  performance  level  appreciably u n t i l  seen of  by the  a limit  examining skilled  the graph  group  i n Figure  d i d not  seem  2, the  to  suffer  o f l e s s than 20 seconds was reached. The  performance l e v e l o f the s k i l l e d subjects with a 6 second l i m i t  still  exceeded that o f the u n s k i l l e d group's regardless o f the imposed time limit. The  response  curve o f  the highly  skilled  subject  portrays  dramatically her a b i l i t y to respond quickly and accurately to a mental calculation  task.  An examination o f the curve i n Figure 2 reveals  that her perfect score on CAL2 remained i n t a c t u n t i l a 6 second l i m i t was imposed.  Even with a l i m i t  o f 4 seconds, her score o f 26 was  greater than the mean score o f the s k i l l e d group f o r every value o f the  imposed time l i m i t .  Performance on CAL3, the Challenge Test As was expected, the 15-item challenge test CAL3 proved to be more  difficult  than  subjects selected  CAL2.  The mean  score  to take the t e s t was 9.27  score f o r these same subjects was 25.91 (86%).  for those 11 (62%).  skilled  The CAL2 mean  The standard deviation  133 calculated for t h i s sample was 3.07 for CAL3 and 3.05 for CAL2. The c a l c u l a t i o n items chosen for CAL3 had an appreciable e f f e c t on  this  sample's  solutions  times.  The  c o r r e c t l y answered items was 26.9 seconds. demonstrates how time l i m i t . and  3  median  solution  time for  An examination o f Figure 3  the mean score on CAL3 was affected by reducing the  A v i s u a l comparison of the response curves i n Figures 2  demonstrates  that  performance  on  CAL3 diminished  much more  r a p i d l y with a reduced time l i m i t than performance on CAL2. The response curve o f the highly s k i l l e d subject indicates how rapid her solution times were r e l a t i v e to the other s k i l l e d subjects. Seven items including 75 x 75, 32 x 64, 18 x 72, 48 x 48, 125 x 125, 64 x 250, and 64 x 64 were a l l answered i n 2 seconds or l e s s . comparison purposes, the response curve for the most s k i l l e d has been included  i n Figure 3.  For  subject  I t i s probably no exaggeration to  state that the highly s k i l l e d subject's a b i l i t y to c a l c u l a t e mentally was  as superior to those subjects who were i d e n t i f i e d as s k i l l e d as  their a b i l i t y was  superior to those subjects who were i d e n t i f i e d  as  unskilled. Evidence  has  been  provided  to demonstrate  that  c a l c u l a t i o n performance l e v e l s o f each o f the groups who in  the  interview  and  assessment  phase d i f f e r e d  the  mental  participated  substantially.  The  remainder o f t h i s chapter w i l l be used to present evidence that the groups d i f f e r e d on a number o f other dimensions as w e l l . Each group  could  be  characterized  by  differences  i n the following:  skill (1)  choice o f methods o f solution used to solve a mental c a l c u l a t i o n ; (2) r e t r i e v a l o f numerical equivalents useful for a mental c a l c u l a t i o n ; (3) short-term memory capacity.  Though the l o g i c o f a comparative  134  15  1 2 _  9  _  6  _  3 _  20 Time Figure 3.  A  30 limit  40 in  50  6 0  seconds  Comparative performance o f 11 s k i l l e d subjects, the most s k i l l e d subject, and 1 h i g h l y s k i l l e d subject on CAL3 scored under d i f f e r i n g time l i m i t s . Skilled  •  Most s k i l l e d subject  0 Highly  skilled  135  groups design precludes  any  attempt to a t t r i b u t e the differences i n  mental c a l c u l a t i o n performance  to any  variable investigated i n  the  study, any proposed explanation of mental c a l c u l a t i o n performance w i l l have to take some of these v a r i a b l e s into account. Choice of Method of Solution and C a l c u l a t i v e Strategy  It  became  completed  that  evident  skilled  after  and  a  number  of  interviews  had  been  u n s k i l l e d mental c a l c u l a t o r s could  characterized by the methods chosen to solve c a l c u l a t i o n tasks. frequency and  proportion of general methods and  be The  s p e c i f i c strategies  reported by the u n s k i l l e d and s k i l l e d groups during the administration o f CAL2 are summarized i n Table VI.  The  analysis of the s t r a t e g i e s  and methods used by the 11 s k i l l e d subjects and the one highly s k i l l e d subject  during  the  administration  of  the  challenge  test  CAL3 i s  contained i n Table VII. Invalid  approaches  to  a  calculative  task  added  to  d i f f i c u l t y of the c l a s s i f i c a t i o n of the methods of s o l u t i o n and calculative  strategies. For  instance,  to  solve  15  x  15,  subjects reasoned, "10 x 10 = 100 and 5 x 5 = 25, so 125." calculative  approach  will  yield  incorrect  understanding of additive d i s t r i b u t i o n  a  faulty  "strategy"  scheme. These p a r t i a l  was  incorporated  into  the  several  partial  In order  ensure that a l l subjects' c a l c u l a t i v e approaches could be this  the  Though t h i s  solutions,  seemed evident.  the  to  tabulated,  classification  forms o f d i s t r i b u t i o n have been designated  in  Tables VI and VII as incomplete d i s t r i b u t i o n . In response to the computation 16 x 16, one u n s k i l l e d subject said,  "244.  It  just  came  into  my  head."  Her  approach  was  not  136  TABLE VI FREQUENCY AND PERCENTAGE OF GENERAL METHODS AND SPECIFIC STRATEGIES USED BY SKILLED AND UNSKILLED SUBJECTS TO SOLVE CAL2 MENTAL MULTIPLICATION TASKS  Groups General methods and specific strategies  Unskilled Frequency Percentage  Pencil-and-paper mental analogue  387  No p a r t i a l retrieved One p a r t i a l retrieved Two p a r t i a l s retrieved Stacking  Distribution  22  71  28  6  52  12  51  11  15  3  18  4  0  0  4  1  12 36 0 4 0 13  7  General Half-and-double Aliquot parts Exponential  101  320  53  Additive Fractional Subtractive Quadratic Incomplete  Factoring  86  Skilled Frequency Percentage  244 8 0 1 0 3  2 2 3 2 0  54 187 15 32 8 2  61 0 1 0 0  42 3 7 2 0  14 15 15 30 1  3 3 7 0  Retrieval of a numerical e q u i v a l e n t  2  0  44  10  Guess  1  0  0  0  450  100  450  100  Totals  137  TABLE VII FREQUENCY AND PERCENTAGE OF GENERAL METHODS AND SPECIFIC STRATEGIES USED BY 11 SKILLED SUBJECTS AND 1 HIGHLY SKILLED SUBJECT TO SOLVE CAL3 MENTAL CALCULATION TASKS  Groups General methods and specific strategies  Pencil-and-paper mental analogue  Skilled Frequency Percentage  34  No p a r t i a l retrieved One p a r t i a l retrieved Two p a r t i a l s retrieved Stacking  Distribution  Factoring  Totals  a  3  0  0  16  10  0  0  15  9  0  0  0  0  0  0  .  2  0  102  0  62 68 2 24 7 1  29  General Half-and-double Aliquot Parts Exponential  Retrieval of a numerical e q u i v a l e n t  21  3  Additive Fractional Subtractive Quadratic Incomplete  Highly S k i l l e d Frequency Percentage  4 41 1 15 4 1  2 0 1 1 0  18 13 4 6 6  27  6 8 2 4 4  40 5 0 1 0  0  0  5  33  165  101  15  100  Due t o rounding, some t o t a l s do not equal 100%.  13 0 7 7 0  33 0 7 0  138  c l a s s i f i e d as r e t r i e v a l but instead was tabulated as a guess. For an explanation  of  the  classification  scheme used  i n this  study, the  reader i s referred to Chapter I I I . To determine i f the choices of general methods of s o l u t i o n were different  for s k i l l e d  and  unskilled  independent samples was used. was  groups, a chi-square t e s t for  S p e c i f i c a l l y , t h i s method o f analysis  used to t e s t the n u l l hypothesis that the proportion o f general  methods of s o l u t i o n chosen to solve mental m u l t i p l i c a t i o n tasks d i d not d i f f e r between s k i l l e d and unskilled groups. For the purposes o f this  analysis,  distribution"  the  and  number  of  invalid  responses,  "incomplete  "guess," were not included. Since these responses  accounted for only 2% o f the t o t a l responses, t h i s loss of information seemed i n s i g n i f i c a n t . Since the magnitude of the calculated chi-square was 393 3, p  <  0.001), the n u l l hypothesis  differ  i n their  concluded  that  choices skilled  that the two groups d i d not  o f methods was and  (df =  unskilled  rejected.  mental  Thus,  calculators  i t was differed  s i g n i f i c a n t l y i n the methods o f solution they chose and used to solve mental m u l t i p l i c a t i o n tasks. One  consequence  calculation  i s that  of  interim  temporary information  storage produced  during  a  mental  i n the course of  computation w i l l undergo rapid forgetting i f not u t i l i z e d immediately. This suggests that those mental c a l c u l a t i o n strategies which require a large number o f c a l c u l a t i o n stages should be susceptible to a great deal would  of be  forgetting expected.  hypothesised  that  and, To  consequently, r e l a t i v e l y carry  the  argument  poor  further,  performance i t can  be  the d i g i t - b y - d i g i t , pencil-and-paper strategy P&PO  139  with i t s greater number of steps compared to other s t r a t e g i e s should be more c l o s e l y associated with poor performance. An analysis o f the data seems to support t h i s hypothesis: unskilled  subjects  made much more  frequent  use  of  P&PO  the  than  the  s k i l l e d subjects. The mean number o f uses of P&PO by the s k i l l e d unskilled  groups  was  1.9  (s  =  2.61)  and  21.3  (s  =  v  and  8.46),  respectively. The difference between the means was highly s i g n i f i c a n t (t  2 8  = -8.49; p < 0.001). The  t t e s t for the d i f f e r e n c e between means assumes that the  two underlying populations have equal variances. The large difference between  the  variances  groups suggested However,  if  of  the  scores  of  that t h i s assumption may  unequal  population  the  skilled  and  not have been  variances  are  unskilled reasonable.  assumed  and  the  s i g n i f i c a n c e o f the d i f f e r e n c e between the means i s determined using the  method  outlined by  Snedecor  and  Cochran  (1967, p.  difference between the means remains s i g n i f i c a n t  115),  at the p  <  the 0.001  level. The possible existence o f a negative c o r r e l a t i o n between mental m u l t i p l i c a t i o n performance and the frequency o f the s e l e c t i o n of P&PO was  determined i n a s e r i e s o f steps. As a f i r s t step i n determining  the strength of t h i s hypothesized was  constructed.  Figure  4  l i n e a r relationship, a scatter p l o t  is a  scatter p l o t  where  the  x-axis  represents the number of times a subject selected P&PO to solve the CAL2 items and the y-axis represents that subject's score on CALl, the screening  test.  An  examination  of  the  scatter p l o t  i n Figure  '4  indicates that a moderately strong l i n e a r r e l a t i o n s h i p existed between the a p p l i c a t i o n of P&PO and performance on CALl.  140  20 kA A  *  A  •A 15  10  10 Number to Figure 4.  of  20  15 times  solve  P&PO CAL2  2 5 selected  tasks  Scatter p l o t o f frequency o f s e l e c t i o n o f the d i g i t - b y d i g i t , r i g h t - t o - l e f t c a l c u l a t i v e strategy (P&PO) to solve CAL2 tasks and score on CALl. •  Unskilled  A  Skilled  30  141  The two groups were selected on the basis o f t h e i r performance on CALl while the methods o f s o l u t i o n were determined as each subject solved  an item on CAL2.  Thus,  to measure the association between  mental c a l c u l a t i o n performance and choice o f method o f s o l u t i o n , the proportion o f times a subject chose P&PO to solve the CAL2 items must be assumed to be an accurate estimate o f the proportion o f times the strategy would be used to solve the CALl items. not  This assumption d i d  seem to be unreasonable, e s p e c i a l l y i n the case o f the u n s k i l l e d  subjects who r a r e l y deviated from the use o f P&PO. Alf III  and Abrahams covariance information s t a t i s t i c (see Chapter 1  f o r discussion)  apparent l i n e a r  was used  r e l a t i o n s h i p . The following  c a l c u l a t e R: r" = -0.86, SJs for  the entire  significant ( t  t o determine  2 Q  intact  the magnitude  o f the  information was used to  = 3.25/6.72 = 0.484. The estimated R  population  was  -0.63 which  was  highly  = -8.92; p < 0.001). Thus, a moderately strong l i n e a r 2  relationship variance knowing  existed  since  approximately 40%  i n mental m u l t i p l i c a t i o n the degree  digit-by-digit,  to which  right-to-left  performance  a subject  (R  =  0.40) o f the  can be explained by  seems to depend  form o f the pencil-and  upon the  paper  mental  analogue to solve mental c a l c u l a t i o n tasks. Useful information about how frequently each s k i l l group used particular  mental  multiplication  calculation  tasks  strategies  i s provided  by  to solve  Table  the CAL2 mental  VI. What  cannot  be  determined from t h i s tabulation i s how frequently i n d i v i d u a l subjects within each s k i l l group changed strategies i n response to a change i n the  calculative  determination.  task.  Other  analyses were  required  to make  this  142 As a f i r s t step i n the analysis, the number o f strategies used to solve the 30 CAL2 items was calculated  for each subject. The mean  number o f strategies used by the s k i l l e d and unskilled groups was 6.4 (s  = 1.80) and 2.8 ( s = 1.32), respectively. The difference between v  X  X  the means was highly s i g n i f i c a n t  = 6.25; p < 0.001).  Figure 5 represents how the number o f strategies chosen by each subject was d i s t r i b u t e d over the two s k i l l groups. An examination o f the d i s t r i b u t i o n demonstrates how the s k i l l e d subjects were much more l i k e l y to change strategies t o correspond to a change i n the nature o f a  calculative  task.  This  finding  suggested  that  a possible  linear  r e l a t i o n s h i p between the number o f strategies a subject used to solve a  series  of calculative  tasks  and mental c a l c u l a t i o n  performance  existed. Further  evidence  of a  linear  examining the scatter p l o t included the  number o f strategies  selected  relationship  can be found by  i n Figure 6. The x-axis represents by a subject to solve the 30 CAL2  items and the y-axis represents that subject's score on CALl. A v i s u a l inspection o f the scatter p l o t indicates a l i n e a r trend. The strength of  the apparent l i n e a r r e l a t i o n s h i p was determined by using A l f and  Abrahams' covariance  information  statistic.  The c o r r e l a t i o n  r ' was  0.76 and the estimated R for the i n t a c t population was estimated to be 0.49  (t  2 8  = 6.19; p < 0.001). Thus, a s t a t i s t i c a l l y  moderate l i n e a r relationship An subjects  analysis suggested  existed.  o f the introspective that  s i g n i f i c a n t and  there  were  reports  also  provided  important  by the  qualitative  differences between the two groups that could not be revealed through a quantitative  analysis alone.  A d e t a i l e d discussion o f each group's  143  Figure 5.  Frequency d i s t r i b u t i o n o f number o f s t r a t e g i e s used by s k i l l e d and u n s k i l l e d subjects t o solve 30 CAL2 mental m u l t i p l i c a t i o n tasks. •  Unskilled  A  Skilled  144  20  1 6  < O  12  c o <D w  o  8  o  CO  I  I  2 Number to Figure 6.  l  i  4 6 of s t r a t e g i e s  solve  C A12  i  8 used  10  tasks  Scatter p l o t o f number o f s t r a t e g i e s used by subjects to solve 30 CAL2 mental m u l t i p l i c a t i o n tasks and scores on CALl.  •£ Unskilled  A  Skilled  145  nethods o f s o l u t i o n , accomr.>anied by many i l l u s t r a t i v e examples, w i l l be  presented. The  following nomenclature w i l l be used i n t h i s study to refer  to subjects p a r t i c i p a t i n g i n the interview and assessment phase. The letters  U  and S w i l l  respectively. on  refer  to an u n s k i l l e d  and s k i l l e d  subject,  The number r e f e r s to the rank o f that subject's score  CAL2 without  regard  f o r the t i e d  scores that existed: 1 i s the  highest rank and 15 i s the lowest rank.  The single highly s k i l l e d  subject w i l l be referred t o as HS.  The U n s k i l l e d Subjects' Choices o f C a l c u l a t i v e Methods Use subjects  of  pencil-and-paper  strongly  favoured  mental  The  unskilled  the use o f the pencil-and-paper  analogue to solve a c a l c u l a t i v e task. o f the attempted tasks.  analogue.  mental  This method was applied to 86%  Dividing the u n s k i l l e d group i n t o high and  low scoring subjects based on CAL2 performance revealed that the low scoring  subjects  relied  mental m u l t i p l i c a t i o n  almost e n t i r e l y  tasks.  on P&PO and P&P1 to solve  These two strategies were applied by  these l e a s t s k i l l e d subjects t o 94% o f the c a l c u l a t i o n tasks used i n the interview and assessment phase. The  subjects  who  relied  on P&PO and P&P1 referred to each  c a l c u l a t i v e stage often i n s p a t i a l terms more suited to a written than a mental medium.  Subject Ul3's attempted s o l u t i o n o f 4 x 625 contains  numerous such references.  She reasoned, "625 i s on the top, 4 on the  bottom, 4 x 5 i s 20, put down the 0, carry the 2. 4 x 2 , 8, and 2 i s 10.  Carry 1 and then...5 x 6 i s 30...30 and 1 i s . . . Put down the 0...  Can I s t a r t again?"  146  L i t t l e attempt was made by the u n s k i l l e d subjects who favoured t h i s method to examine the c a l c u l a t i v e task for any number properties or relationships which could be used as an a i d i n c a l c u l a t i o n .  Even  the most transparent properties were not heeded. For example, 11 o f the 30 items i n CAL2 included factors which contained 1 as a d i g i t i n the units or tens place value p o s i t i o n : 12 x 81, 17 x 99, 12 x 15, and so f o r t h . Thus, the subjects who  applied the pencil-and-paper method  were not expected to c a l c u l a t e each p a r t i a l product d i g i t by d i g i t to solve these types o f items.  Rather, i t was thought that the p a r t i a l s  would be determined by either reasoning, "1 x a = a" or by using an equivalent  process  Surprisingly, 61%  such  as  thinking,  "bring  down  the  a."  of the u n s k i l l e d group's attempts to solve these 11  items d i d not incorporate t h i s elementary p r i n c i p l e to expedite the c a l c u l a t i o n . I t was found that 32% and 85% of the high and low scoring u n s k i l l e d subjects' attempts, respectively, d i d not use the i d e n t i t y property o f m u l t i p l i c a t i o n as an c a l c u l a t i v e a i d for these 11 items. Some subjects who  employed  P&PO d i d arrange the factors  seemingly take advantage o f t h i s obvious property.  to  However, instead  o f r e t r i e v i n g the p a r t i a l product, they went through a d i g i t - b y - d i g i t c a l c u l a t i v e process. Subject U13's attempted solution o f 49 x 51 i s an example  of  such  calculative  bottom, 49 on the top. Put down the  behaviour.  She  thought,  I t seems easier that way.  1x9,  "51  on  the  1x4,  49.  0..."  Subject U2  was  asked  process to c a l c u l a t e 13 x 13. through? To be sure.  why  she went through a She responded,  "Why  digit-by-digit  would I multiply  Because the l a s t time I was doing a problem l i k e  t h i s I just brought the number down." Her reference to "the l a s t time  147  I brought a number down" was to the item 16 x 16 where she calculated the f i r s t p a r t i a l as 96 and used t h i s value instead o f 16 (160) as the second p a r t i a l product. for  Using the f i r s t p a r t i a l as a substitute value  the second p a r t i a l product was a r e l a t i v e l y common error made by  unskilled  subjects. The various  types  o f mental c a l c u l a t i o n errors  w i l l be discussed l a t e r i n t h i s chapter. Those subjects who depended upon P&PO t o solve most c a l c u l a t i o n tasks sometimes introduced c a l c u l a t i o n s which, though correct, can be considered explained "with  redundant  and unnecessary.  For example,  subject  that her s o l u t i o n o f 50 x 64 included a p a r t i a l  a l l O's on the top."  Further  probing  revealed  U10  product  that she had  v i s u a l i z e d the c a l c u l a t i o n arranged as i l l u s t r a t e d below: x  64 50 00 3200  Perhaps the most graphic example o f over-dependence on the use of  P&PO was subject  Ul3's  attempted  s o l u t i o n o f 20 x 30.  She  explained, "30 i s on the top, 20 i s on the bottom. 0 x 0 i s O , 0 x 3 i s 0. add  Put down the 0.  And 2 x 0 i s 0 and 2 x 3 i s 6.  them together and you'd g e t ....6....600?"  And then you  Her c a l c u l a t i o n took  34 seconds to complete. Most u n s k i l l e d  subjects  used a d i g i t - b y - d i g i t ,  right-to-left  c a l c u l a t i v e process during the a d d i t i v e phase o f a mental c a l c u l a t i o n . As U l explained, "I do i t (addition) from r i g h t to l e f t and then read the answer out backwards." This tendency o f u n s k i l l e d subjects to fragment a c a l c u l a t i o n could have been one o f the reasons that even the most obvious sums  148  were calculated rather than r e t r i e v e d . x  250,  was  heard  to say,  "So  five  Subject U l l , i n c a l c u l a t i n g 12 hundred  and  two  hundred" before beginning the process o f addition.  thousand  But instead o f  r e t r i e v i n g a sum o f 3000, she immediately began to say, "OK. and 2, 5, 0, 0,... asked l a t e r  five  5, 0, 0,  would be ....0, 5, 7, 2....2, 7, 5, 0?".  She  was  i n the interview to add 2500 and 500 and again used a  d i g i t - b y - d i g i t , r i g h t - t o - l e f t additive process. The tendency o f u n s k i l l e d subjects to view c a l c u l a t i o n i n terms o f an imaginary s p a t i a l arrangement similar to the format used during written computation became very evident during the additive phase.  U9  would use expressions such as, "Move over one, 4, that's under the 6, and 6, that's under the 1..." the p o s i t i o n o f each d i g i t  Subject U3 commented that to remember  he had  to repeat constantly,  "something  l i k e 4 over the 6, 2 over the 9, e t c . " Some u n s k i l l e d  subjects appeared to become disenchanted with  t h i s type o f protracted a d d i t i v e process and began, more  flexible  methods which  property o f the addends.  were  based  Subject U2,  on  instead, to use  some recognized  number  a t the midpoint of the CAL2  interview, attempted to solve 17 x 99. She said: I got 990 and 693 but I had problems adding. I'd do 9 and 9 i s 18, carry the 1,... but I'd forget what I'd f i n i s h e d . F i n a l l y for 990 and 693, I took 10 o f f the 693, and got 1000 and then 1683. At f i r s t I thought t h i s addition method might be harder but i t was e a s i e r . F i r s t time I've done t h i s method before. The use o f P&PO and P&P1 was accompanied often by motions which indicated that the subject was attempting to "write" each stage o f the calculation.  Fingers were used to form c a l c u l a t i o n s either i n the a i r  or on a desk. One subject used a p e n c i l poised just above the table to  149  record  her c a l c u l a t i v e  subjects solve  were observed  the majority  stages. using  Eleven  of  an imaginary  o f mental  calculation  the f i f t e e n  "writing tasks.  unskilled  instrument"  Similar p h y s i c a l  gestures were observed during the screening phase o f the study. CALl was administered  to  When  to one c l a s s o f 34 students, 11 students were  observed using f i n g e r s as an apparent c a l c u l a t i v e a i d . All  subjects who used  "writing" believed the p r a c t i c e helped  them cope with the demands o f the c a l c u l a t i o n . her  f i n g e r s helped  Subject U2 explained  her "remember" a completed p a r t i a l product.  For  example, to solve 12 x 15, she "did 120 and then as soon as I d i d 60 I forgot 120.  The second time I put an emphasis on the 60 by pressing  my finger down harder."  Subject U9 explained that fingers helped him  "see the problem" and improved h i s concentration. The  forgetting o f the p o s i t i o n of each d i g i t  of a  partial  product during the additive phase o f a mental c a l c u l a t i o n was a great source o f error f o r u n s k i l l e d subjects. o f fingers helped  her remember  these  Subject U6 believed the use  positions.  She explained she  aligned the fingers o f each hand t o represent the place-value p o s i t i o n o f each d i g i t i n the two p a r t i a l s and then t r i e d to v i s u a l i z e a d i g i t on each finger. The use of d i s t r i b u t i o n .  The u n s k i l l e d group made l i t t l e use  o f t h i s strategy: 12% o f t h i s group's attempted solutions involved an a p p l i c a t i o n of t h i s method. almost 80%  Two subjects, U4 and 07, accounted for  of the number o f reported applications o f d i s t r i b u t i o n .  Nine subjects never used the method once. The one u n s k i l l e d subject who d i d favour additive d i s t r i b u t i o n over a l l other  methods was subject U7.  He was an unusual subject.  150  Despite h i s sound reasoning, he often erred i n the fundamentals.  For  example, to solve 9 x 742, he attempted, "9 x 742, 9 x 7 = 72, so 7200 and 9 x 4 i s 28.  So 280 and 7200 i s 7480.  9 x 2 i s 18, so 7498."  In  the l a t t e r stages o f the interview, he was asked to r e c a l l 9 x 7 .  He  r e p l i e d , "Let's see. two questions."  9 x 9 i s 81, 72, 63.  Frequently,  the larger basic f a c t s .  Well, there go the l a s t  he had to c a l c u l a t e rather than r e c a l l  He commented he "didn't bother to remember  larger numbers" because he r e l i e d on h i s c a l c u l a t o r instead. There were some subjects who seemed to be attempting d i s t r i b u t i o n but who This  error  failed  frequently  distribution  t o only  additive  t o complete the necessary c a l c u l a t i o n s .  occurred  when  a  subject  applied  the decades and units o f each  c:  additive  factor.  example, subject U15 calculated 25 x 25 as 20 x 20 and 5 x 5 .  For  He and  subject U7 accounted for over 70% o f the u n s k i l l e d subjects' attempts to apply t h i s form o f "incomplete d i s t r i b u t i o n . " A comment made by U15 suggests that t h i s i n v a l i d procedure was used  i n an apparent  attempts  to apply  attempt  to break  pencil-and-paper  away  from  the  methods to t h i s  unsuccessful  mental medium.  A f t e r applying P&PO to the f i r s t 11 items i n CAL2, he t r i e d to solve 25 x 25 by reasoning,  "400 and 25."  He commented, "I don't f e e l too  confident because I don't know i f my method works or not. t r i e d i t before."  Apparently,  he never d i d determine that the method  was i n v a l i d since he l a t e r applied the method t o s i x other Despite unskilled calculative  their  subjects  considerable could  I haven't  not  a i d . The researcher  mathematical  correctly noticed  apply  that  items.  experience, annexation  neither  subject  two as  a  used  annexation to help determine the s o l u t i o n to the CAL2 item, 32 x 500.  151 Ull  was  given  the  task  50  x  700  to  test  her  understanding.  explained, "you multiply the 5 and the 7 and add 5 zeroes." she  thought  that  the  number  of  annexed  zeroes  was  She  Apparently  determined  by  counting the number o f d i g i t s rather than the "zeroes" i n each factor. As a further t e s t of her understanding,  she was given 300 x 800.  She  responded, "24 and s i x zeroes." Subject U15  could annex the c o r r e c t number of zeroes but used  the following f a u l t y l o g i c to complete the c a l c u l a t i o n of 30 x 40.  He  explained, "That's easy, 120. I j u s t timesed the two numbers (3 and 4) and  there's two  Obviously  these  zeroes two  there. So  subjects  i t ' s (the product) i n the 100's."  would  have d i f f i c u l t y  using  d i s t r i b u t i o n since the c o r r e c t a p p l i c a t i o n of the annexation  additive algorithm  i s a necessary requirement. Sub t r a c t i v e distribution  distribution  applied  by  the  was  the  unskilled  r e s t r i c t e d to only one subject.  only  other  subjects  and  type  of  i t s use  was  She used the strategy to c a l c u l a t e 8  x 99 and 17 x 99 successfully but her attempts to solve 8 x 999 and  15  x 48 were unsuccessful. The unskilled  use of f a c t o r i n g . group was  very  The use of the factoring method by the  infrequent.  Only  two  percent  attempted solutions involved some factoring strategy.  of  their  Eleven of the  15 u n s k i l l e d subjects made no attempts a t f a c t o r i n g . Although their f i r s t attempts to factor were unsuccessful, the use  of  the  method was  strategy by  two  unskilled  subjects  suggested  that  the  a recent discovery i n s p i r e d by an apparent d i s s a t i s f a c t i o n  with pencil-and-paper  methods. Subject  U9  reported  that he  obtained  4000 as the s o l u t i o n to 12 x 250 by trying a " d i f f e r e n t " method.  His  152  introspective report indicated that t h i s method was a l i q u o t parts. He explained, "I pulled a f a s t one here! times does i t take t o get to 1000? 3000.  I said 250, how many  I t ' s 4 and 3 sets o f 4 i n 12.  Did I say 4 (4000) the f i r s t time?"  So  Subject Ul2's f i r s t answer  i n c a l c u l a t i n g 25 x 120 was 1500 because she "divided 8 into 120 and added two zeroes."  Asked why she chose to divide 120 by 8,. she said  "because.. .oh, I should have divided by 4. into 120 i s 4..., 30. So 3000."  Four 25's are 100. So 4  When asked i f she had ever used t h i s  method before, she r e p l i e d , "No, I j u s t discovered i t today because o f a l l these questions."  The S k i l l e d Subjects' Choices o f C a l c u l a t i v e Methods The  use  of  the pencil-and-paper  mental  analogue.  The  pencil-and-paper mental analogue was used infrequently by the s k i l l e d group.  Approximately 22% o f the s k i l l e d group's introspective reports  were c l a s s i f i e d under t h i s method. method  f a r more  frequently  than  Five s k i l l e d subjects • used any other  subjects.  These  this five  subjects accounted f o r about 85% o f the method's use by the s k i l l e d group. the  A t the other extreme were 6 s k i l l e d subjects who never used  method t o solve any items i n CAL2. Some s k i l l e d subjects d i d r e l y on t h i s method to solve mental  multiplication  questions but, unlike  the unskilled  subjects,  they  tended to avoid the use o f the d i g i t - b y - d i g i t P&PO strategy. While 71% of the  the u n s k i l l e d group's responses were c l a s s i f i e d as P&PO, only 6% o f s k i l l e d group's responses were so c l a s s i f i e d . Subject S3 was the most p r o f i c i e n t subject to make extensive  use o f the pencil-and-paper method. However, she used P&P2 and P&P3  153  rather  than P&PO. She  relied  on her a b i l i t y  to r e t r i e v e numerical  equivalents which seemed to minimize the number of c a l c u l a t i v e steps needed to solve a computation.  For example, instead o f c a l c u l a t i n g  each p a r t i a l product for 15 x 16 d i g i t by d i g i t , she thought, "15 x 16.  80 and 16.  Move one over, 160.  So 240."  These r e t r i e v a l s o f numerical equivalents had  the e f f e c t o f  decreasing the solution time for many c a l c u l a t i o n tasks. The solution to 15 x 16 took S3 less than 4 seconds. In contrast, the successful solutions o f subjects U3 and U4 who used P&PO took 45 and 52 seconds, respectively. S3's l e f t - t o - r i g h t method o f adding the p a r t i a l products also  deviated  from  the  right-to-left  procedures commonly  used  by  u n s k i l l e d subjects. S3 used a pencil-and-paper strategy which was termed stacking (see Chapter III for example and discussion). She used t h i s strategy to  solve  calculation  tasks  of  the  form  1-digit  by  x-digit:  s p e c i f i c a l l y , 8 x 99, 9 x 74, 8 x 625 and 8 x 999. She commented that she  liked  the  strategy  because  "I don't  have to carry  with  this  method." An interesting pattern emerged when each CAL2 item was according  to the  number  of  applied by the s k i l l e d group.  times  ranked  a pencil-and-paper strategy  was  Pencil-and-paper strategies were used  most frequently to solve these items: 8 x 4211, 9 x 742, 8 x 612, 32 x 500, 9 x 74, 50 x 64, 32 x 500, 25 x 65.  The format o f four o f these  items i s 1-digit by x - d i g i t and, with the exception o f 25 x 65, the f a c t o r s o f the remaining items, 50 x 64 and 32 x 500, can be e a s i l y transformed algorithm.  to  this  format  by  an  application  of  the  annexation  154 Mental products involving a 1-digit factor d i d seem to e l i c i t the  most  frequent applications o f the pencil-and-paper  there were a few important  exceptions.  Only one s k i l l e d  applied the method to solve 4 x 625 and 8 x 625. applied  this  method  f a r more  method but  frequently than  subject  Subject S3 who  the other  skilled  subjects was asked why she d i d not use the pencil-and-paper strategy to  solve 4 x 625.  She r e p l i e d , "usually something t e l l s me whether  i t ' s easier to go r i g h t to l e f t right  (apply d i s t r i b u t i o n ) .  make the decision."  (apply pencil-and-paper) or go l e f t to  My knowledge o f 4 x 25 = 100 helped me  Subject S13 made a similar comment:  "I started  to do i t with the 625 on top o f the 8 and then I l o s t my numbers. I thought 8 x 25 i s 200 and 8 x 600 i s 4800. together.  Then I added these two  I saw i t was a 25 type o f problem."  Generally  speaking,  skilled  pencil-and-paper method either for  So  subjects  seemed  to  use  the  f o r very "easy" c a l c u l a t i v e tasks or  tasks which have no apparent properties. The relationship between  the absence o f d i s c e r n i b l e number properties and the application o f the pencil-and-paper method became much more evident during the CAL3 interviews. of the  Two o f the most d i f f i c u l t iterns (p = 0.33) for the sample  s k i l l e d subjects were 87 x 23 and 73 x 83. number  of  subjects  who  applied  For these two items,  the pencil-and-paper  method  increased to 5 from a median o f 2. The researcher asked those subjects who normally avoided the use o f the pencil-and-paper method why the method had been applied to these p a r t i c u l a r items and not to other items. Subject SI explained that he had more d i f f i c u l t y with 87 x 23 than with the other  items  "because I had kind o f a gimmick to solve the other questions." He  155  believed  that f o r such items,  "one method was as good as another."  Subject S9 said he couldn't solve 87 x 23 "because he had to multiply i t out j u s t l i k e on paper."  He chose t h i s method because the "problem  doesn't  that  have  unsuccessful normally  any  numbers  you  can work  with."  After h i s  attempt to solve 87 x 23, subject S l l was asked i f he  avoided  commented,  the  use  of  the  pencil-and-paper  method.  He  "Yes, I only use i t with r e a l l y weird numbers l i k e that."  He also said that he was forced t o use the method to solve 73 x 83 because he "couldn't half.  The  think o f any way of doing i t such as double-and  numbers  (factors)  weren't  close  to  some  convenient  numbers." The almost  unskilled  habitually  discriminating. applied  subjects  but  the  Generally  applied  skilled  the pencil-and-paper  subjects'  use  was  method  much  more  speaking, the pencil-and-paper method was  t o a c a l c u l a t i v e task  by the s k i l l e d  subjects  only  i f no  d i s c e r n i b l e properties were apparent. In those cases where t h i s method was applied,  the s k i l l e d  than c a l c u l a t e the p a r t i a l  subjects made attempts to r e t r i e v e rather products.  The use o f d i s t r i b u t i o n . frequently  by  the  skilled  D i s t r i b u t i o n was the method used most  group.  Over  one-half  of  the  skilled  subjects' attempted solutions involved a d i s t r i b u t i o n strategy.  The  data i n Table VII indicate that d i s t r i b u t i o n was the most frequently reported method i n the CAL3 interviews, as w e l l . The  number o f uses o f a d d i t i v e d i s t r i b u t i o n to solve the CAL2  c a l c u l a t i v e tasks ranged from a high o f 23 to a low o f 1. The median number o f uses was 11.  The a d d i t i v e d i s t r i b u t i o n strategy was always  applied i n a s e r i e s o f stages.  The c a l c u l a t i o n was usually i n i t i a t e d  156  by  determining  the product  of  the most s i g n i f i c a n t d i g i t s  factor and annexing the appropriate number of zeroes. of  the c a l c u l a t i o n progressed  The  of each direction  from the more s i g n i f i c a n t to the  less  s i g n i f i c a n t d i g i t s i n each factor. In a  long  s e r i e s of such c a l c u l a t i o n s ,  would determine a  running  sum  of  the p a r t i a l products  delay the addition u n t i l a l l p a r t i a l s had S8's solution o f 8 x 4211  8 x 200,  subjects  rather  been calculated.  than  Subject  i l l u s t r a t e s both t h i s l e f t - t o - r i g h t sequence  of c a l c u l a t i o n s and progressive addition. 4000 i s 32 000,  the s k i l l e d  1600,  so 33 600.  He thought, "8 x 4211, And  8 x  8 x 11 i s 88 so 33  688." Many s k i l l e d subjects believed that progressive addition helped minimize forgetting. Subject S5 explained, "I add as I go along. works better that way. confused." and  Otherwise you get too many numbers and  It get  Subject S8 said, "I t r y to add the numbers as I go along  forget the others.  those numbers."  I u s u a l l y s t a r t at the l e f t ,  add,  and drop  When asked to explain why he added i n t h i s manner, he  said, "there was too much to remember otherwise." One  subject explained that c o n s i s t e n t l y "working from l e f t  r i g h t " helped  him  to  "keep track" of  completed intermediate c a l c u l a t i o n s .  the completed  and  yet  to  to be  He solved the CAL3 item 24 x 625  by reasoning, "20 x 600 = 12 000, 20 x 25 = 500, 12 500 and f i l e d that away; 4 x 600 = 2400 and 4 x 25 i s 100, so 2500. 000."  To  explain h i s method of track-keeping,  Add to 12 500, so 15 he  took a piece of  paper and diagrammed each stage i n the manner indicated below: visualized calculated  157  He a l s o explained that he " t r i e s to break the problem i n t o 'two simple numbers' to work with."  He drew the following diagram to i l l u s t r a t e  h i s method:  Although  additive  6  25  2  4  distribution  requires  the  a p p l i c a t i o n o f some addition method, the conventional digit-by-digit  procedure,  so common  i n written  generally ignored by most s k i l l e d subjects. was  completed by using  continual  right-to-left,  computation,  Frequently,  the addition  some form o f a l e f t - t o - r i g h t process.  solution o f 480 + 96 i l l u s t r a t e s one such process.  was  He  S9's  explained,  "Bump up the 4 to 5 and 7, 6." Other sums were determined by arranging  one addend so that a  multiple o f a power o f 10 was obtained as a p a r t i a l sum.  Subject S6  added 960 and 64 by thinking, "960 and 40 i s 1000, add the 24 which was  left  over."  For those  sums where an addend was close  to a  multiple o f a power o f 10, some subjects would apply a compensation strategy. 100  S6 reasoned that 480 and 96 equalled 576 because "480 and  i s 580 then minus 4 i s 576." Fractional  which  d i s t r i b u t i o n was a form o f additive d i s t r i b u t i o n  incorporated  factoring t o complete part  o f the c a l c u l a t i o n .  This strategy was frequently applied to those c a l c u l a t i v e tasks where one factor had 5 as a u n i t d i g i t .  For example, the item with the  largest number o f reported a p p l i c a t i o n s o f t h i s strategy was 15 x 48. A l l four subjects who used t h i s strategy simply reasoned "480, 1/2 o f 480,  240, so 720."  158  Subject S14 distribution  accounted for 12 of the 15 instances o f f r a c t i o n a l  reported  during  the  CAL2  interviews.  He  obviously  favoured t h i s technique because, whenever possible, he t r i e d to change the task into a form that made a subsequent a p p l i c a t i o n of f r a c t i o n a l d i s t r i b u t i o n more f e a s i b l e . 24 - 24,"  and then he said, "I went 24 x 10, 240.  I took 1/2 of 240, 120. 24, so  To solve 24 x 24, he f i r s t thought, "25 x  Added 120 and 480,  480.  Then I took o f f a  576." S14  CAL3 item. successful,  was  the only subject who Although h i s  his  attempt  explanation  applied t h i s strategy to solve a to c a l c u l a t e 125  illustrates  strategy to solve more complex items. double that, 2500. 125  so 600.  Doubled that,  x 10."  And  He  then I took 1/4  he  125  tried  reasoned, "125  x 10,  not the 1250,  of that and added that onto should  have  process s i m i l a r to the following: 125 x 125 = (100 +  25) x 125 = (100 x 125) = 12 500 + 3125  was  to adapt  To have applied the strategy c o r r e c t l y , he  used a reasoning  = 15  + 1/4  x  (100 x 125)  = 12 500 + (1/4 x 12  500)  625.  Sub t r a c t i v e d i s t r i b u t i o n was subjects  how  x  used by  to solve items such as 8 x 99,  the majority o f  17 x 99, and  skilled  8 x 999 where  each item included a factor whose magnitude was close to a multiple of a power of 10. For the three items, 8 x 99, 17 x 99, and 8 x 99, 28 of the s k i l l e d subjects' solutions incorporated subtractive d i s t r i b u t i o n . In comparison, only 3 of the u n s k i l l e d subjects' solutions involved t h i s strategy.  The  CAL3 items which prompted the most attempts of  t h i s strategy were 18 x 72 48  (2400 - 96).  (1440 - 144), 48 x 64  (3200 - 128)  and 48 x  159  Subtraction  strategies  which  differed  from  pencil-and-paper  methods always accompanied an a p p l i c a t i o n o f subtractive d i s t r i b u t i o n . The  subtraction was usually  accomplished  i n a s e r i e s o f stages by  renaming the subtrahend as either a sum or difference.  To solve the  computation 8190 - 91, subject S14 thought, "8190 - 90 = 8100, so 1 more o f f i s 8099."  He also subtracted 96 from 2400 by subtracting 100  and then compensated by adding 4. Sixteen during  instances  o f quadratic  the CAL2 and CAL3 interviews.  likely  more  explained,  other  subjects  inefficient  used  reported  Over 60% o f the attempts were  applied to solve 49 x 51 and 89 x 91. squares while  d i s t r i b u t i o n were  Subject SI used difference o f  a mathematically equivalent  procedure.  After  he  solved  49  x  but  51, S5  "50 x 50 i s 2500, and minused 50 because o f the 49 and then  added 49": i . e . , 49 x 51 = (50 - 1)(50 + 1) = 2500 - 50 + 50 -1 = 2500 - 50 + 49. Other  variations  c a l c u l a t i n g squares.  o f quadratic  d i s t r i b u t i o n were applied  to  Subject SI used binomial expansion to solve 48 x  48 by thinking, "50 squared minus 200 and add 4": i . e . , 48 x 48 = (50 - 2 ) = 50 2  2  - 2 x (2 x 50) + 2  2  = 2500 - 200 + 4. The task 125 x  125 was solved by thinking, "100 squared plus 5000 and add 625": i . e . , 125 x 125 =100  2  + 2 x (25 x 100) +25  2  =10 000 + 5000 + 625.  Subject S l l was aware o f the r u l e for squaring  factors ending  i n 5 but he was unable to determine a c o r r e c t solution i n h i s i n i t i a l attempts. i.e.,  In c a l c u l a t i n g 75 x 75, he annexed a 5 instead o f a 25:  he calculated 75 x 75 = 7 x 8 x 100 + 5 = 5605 instead o f  c a l c u l a t i n g 7 x 8 x 100 + 25. For 125 x 125, he calculated 13 x 13 x 100 +  25  instead  o f 12 x 13 x 100 + 25.  As he explained h i s  160 reasoning, he r e a l i z e d and corrected these errors. The  use of f a c t o r i n g .  In contrast to the infrequent use  of  factoring by the u n s k i l l e d subjects, the majority of s k i l l e d subjects used  factoring  involved  to  i n 14%  solve  some  the  skilled  of  calculative  tasks.  group's attempted  Factoring  was  solutions.  The  number of applications reported for each subject varied from a high o f 10 to a low o f 0.  However, only three s k i l l e d subjects d i d not report  any attempts at f a c t o r i n g . General f a c t o r i n g accounted for about 3% of the s k i l l e d group's reported  strategies.  Generally speaking,  i t s use was  restricted  to  those items which contained f a c t o r s that were multiples o f either 2 or 5.  At l e a s t one  items: 8 x 625,  attempt at general factoring was 25 x 48, 25 x 480,  25 x 120,  reported for these  12 x 250,  25 x 32, 15 x  48. The  higher  incidence of general  items as compared to other  factoring reported  for  these  items such as 87 x 23 was due to the f a c t  that each c a l c u l a t i v e task can be reformulated to involve multiples of powers o f 10.  solved 12 x 250  by  using general factoring reasoned "since 12 = 3 x 4 then 4 x 250  is  1000,  so  3  For example, the four subjects who  x  1000  or  3000."  Similarly,  subject S4  applied  the  strategy to solve 25 x 120 by thinking "5 x 120 i s 600 and 5 x 600 i s 3000."  Subject  S8  used general  factoring for those  products  that  could be decomposed into what he described as "easy oombinations." solve 75 x 240, o f 25.  And  To  a CAL3 item, he said, "I broke up the 75 i n t o groups  I broke up the 240  i n t o groups of 4.  When I worked out  the combinations I got the number of 100's." General factoring was a l s o applied to two items, 25 x 25 and  25  161 x 65, where each factor i s a multiple o f 5 but neither factor i s a multiple o f 2. was  completed  distribution.  The f i n a l stage o f the c a l c u l a t i o n , i n these cases, by applying  some  other  strategy  such  as additive  The computation 25 x 25 was reasoned as "5 x 25, 125; 5  x 120 i s 600 and 25, 625."  And 25 x 65 was reasoned as "25 x 65 = 25 x  (64 + 1), so 25 x 64 i s 4 x 25, 100, 'times' 16, so 1600,  and add 25,  1625." Half-and-double i s a f a c t o r i n g strategy which was frequently used to solve c a l c u l a t i v e tasks where a t l e a s t one factor was an even number.  I f the second  factor  is a  multiple  o f 5,  multiple  applications o f half-and-double w i l l eventually produce a factor which i s a multiple o f a power o f 10. of  the reported  interviews  applications  Subject S l l , who accounted for most of this  and f o r a l l the reported  strategy  during  the CAL2  instances  during  the CAL3  interviews, used two consecutive applications o f half-and-doubling to solve 24 x 625.  He explained,  "I doubled and halfed u n t i l 1250 x 12.  Which I thought would be 15 000. h a l f i n g one more time.  However, I checked by doubling and  So 2500 x 6 which i s 15 000."  The most common factoring strategy applied by the s k i l l e d group was  a l i q u o t parts.  One-half o f the number o f attempted applications  o f t h i s strategy involved three items: namely, 25 x 48, 25 x 120, and 25 x 32. 50,  Subject S5 explained  he would apply  that i f a task had a factor o f 25 or  the strategy by d i v i d i n g the remaining factor by  either 4 or 2, respectively, and complete the c a l c u l a t i o n by annexing the appropriate  number o f zeroes.  strategy's range o f usefulness,  When asked to elaborate on t h i s  he r e p l i e d ,  " I f i t ' s a large number  (the second factor i n the given product) then I would t r y to use i t  162  but not with  items such as 50 x 20.  I t ' s e s p e c i a l l y u s e f u l i f one  factor i s an even number but odd f a c t o r s can be confusing." However, some subjects d i d apply a l i q u o t parts to items with odd  factors.  Subject  SI solved  the c a l c u l a t i v e task 25 x 65 i n a  novel manner. He incorporated decimal arithmetic into h i s c a l c u l a t i o n by reasoning over.  "65 divided by 4 i s 16.25  So 1625."  Move the decimal  two places  Subject S5 solved the same item by applying a l i q u o t  parts to c a l c u l a t e 64 x 25 and adding 25 to t h i s r e s u l t . In the CAL3 interviews, there were 7 instances o f a l i q u o t parts reported  and  every  attempt  was  directed  at  solving  64  x  250.  Interestingly, subject S14 solved t h i s item by applying a l i q u o t parts to only part o f the c a l c u l a t i o n .  He explained, "I took 64 x 10, 640.  M u l t i p l i e d by 10, so 6400; t r i p l e d that, 19 200; and then subtracted 1/2 o f 6400." The following a n a l y s i s demonstrates the l o g i c behind h i s method: 250 x 64 = 25 x 10 x 64 = 25 x 640 = (30 - 5) x 640 = 3 x 10 x 640 - 5 x 640 = 3 x 6400 - 6400/2 = 19 200 - 3200 = 16 000. The tendency  of  skilled  subjects  to  incorporate  several  calculative  strategies to solve the d i f f i c u l t CAL3 items added to the problems o f strategy c l a s s i f i c a t i o n . Several subjects commented that they applied a l i q u o t parts only after  reformulating  analogies  the  task  into  more  concrete  were mentioned by several subjects.  terms.  Subject  Money  S13 always  described her applications o f t h i s strategy by making such monetary references.  She commented, "when I multiply by 25, I think o f money,  l i k e quarters.  So I divided by 4 to c a l c u l a t e d o l l a r s . "  She thought  that her knowledge o f t h i s strategy was acquired through a p r a c t i c a l life-experience.  She explained,  "When I was i n Brownies, we always  163  used to s e l l cookies and subtracting,  and  they were $1.25.  multiplying by  So I got used to  quarters."  The  adding,  carments of  some  u n s k i l l e d subjects indicated that they were aware that such analogies could possibly have aided their c a l c u l a t i v e attempts. the pencil-and-paper  A f t e r applying  method to solve 25 x 25, subject U l said, "Maybe  I should have counted quarters." There was  only one  instance of the a p p l i c a t i o n of  factoring recorded during the CAL2 interview.  exponential  Subject SI solved 32 x  32 by reasoning that the product equalled 2 to the tenth which he knew was equivalent to 1024.  Although subject S5 r e a l i z e d that the product  would be a power of 2, he could not r e t r i e v e the numerical equivalent. When f i r s t presented with t h i s task, he said, "I suppose 2 to the 10th is  not  good  enough."  When the  researcher  said  no,  he  correctly  calculated the product by applying a d d i t i v e d i s t r i b u t i o n : i . e . , 32 x 30 + 32 x 2. There  were  six  instances  during the CAL3 interviews. the items 32 x 64  and  of  exponential  factoring reported  Three subjects used the strategy to solve  64 x 64.  reference point. Subject S9, who  In each case, 1024 d i d not know 1024  was  used as a  was 2 to the 10th,  applied an interesting t a c t i c to complete h i s c a l c u l a t i o n of 32 x 64. He  surmised that 64 x 64 would be "a power of 2 ending i n 6."  he  knew that 1024  was  Since  large power o f 2 close to but l e s s than the  square of 64, he "doubled 1024  twice and got 4096."  164  The Highly S k i l l e d Subject's Choices of C a l c u l a t i v e Methods HS r e l i e d  heavily on an apparent a b i l i t y  number properties u s e f u l f o r c a l c u l a t i o n . were  the most  reports  frequently  obtained  during  applied  were combined, she solved over one-half  form o f d i s t r i b u t i o n .  Factoring  and  methods of solution.  the administration  factoring or r e t r i e v a l .  to quickly discern  o f CALl,  retrieval When the  CAL2, and CAL3  o f the items by using either  For the remaining items, she applied some She never  once applied  a pencil-and-paper  strategy to solve any mental products given i n the study. She seemed capable o f applying sophisticated strategies without any  apparent  mental  i n s t a n t l y reasoning  effort:  f o r example, she solved  "69 x 30 - 69."  87  x  23 by  The complexity o f t h i s s o l u t i o n  i s i l l u s t r a t e d by the following a n a l y s i s : 87 x 23 = (29 x 3) x 23 = 29 x  (3 x 23) = 29 x 69 = 69 x (30 - 1) = 69 x 30 - 69 = 2070 - 69 =  2001. The  greatest  challenge  t o her c a l c u l a t i v e a b i l i t i e s was the  item 123 x 456 which was a l s o administered subjects  who  attempted  this  details  were  difficult  to remember  several  times.  This  item  difficulty  to 5 s k i l l e d subjects. A l l  commented and  that  the c a l c u l a t i v e  had  to re-calculate  they  i n retaining the numbers  c a l c u l a t i o n g r e a t l y protracted the s o l u t i o n times.  i n this  The s o l u t i o n times  for the four s k i l l e d subjects who c o r r e c t l y answered t h i s item were 50, 115, 228, and 350 seconds. Each s k i l l e d  subject  reported  using  either d i s t r i b u t i o n or a  pencil-and-paper method. Subject S3, who favoured the pencil-and-paper method,  used  calculated.  progressive A l l skilled  addition  as  subjects  each  partial  reported  that  product there  was were  165 no  recognizable  number  properties other than  the obvious  identity  property ("1" x 456) that could be used t o expedite the c a l c u l a t i o n . HS completed t h i s task i n 50 seconds and offered the following explanation: I f i r s t thought that 456 x 123 i s 152 x 41 x 9. Then I thought 19 x 41 x 8 x 9. Did 779 x 8 by thinking 5600, 560, and so 6160. 6160 and 72 i s 6232. F i n a l l y I m u l t i p l i e d 6232 by 9 by thinking 9 x 62 i s 558, which I know. So 55 800 and 9 x 32. 55800 and 288 i s 56 088. The reader i s reminded that a l l t h i s reasoning was done " i n the head" without any opportunity to refresh the factors by a v i s u a l inspection. HS d i f f e r e d from the majority o f s k i l l e d subjects because she seemed able to sense immediately whether the numbers i n a c a l c u l a t i v e product  could  be  readily  factored.  Her  acute  sense  of  "number  f a c t o r a b i l i t y " became quite evident during an interview designed to assess her knowledge o f prime numbers. Hunter  (1962) assessed Aitken's a b i l i t y to determine whether a  number was prime or composite as a demonstration of the "readiness with which a presented number leads on to numerical properties" (p. 247). This task o f factoring i s , a t l e a s t , more time-consuming i f not more  difficult  number  of  than  mental  calculations  multiplication  that  must  be  because done  to  of the greater ascertain  the  f a c t o r a b i l i t y o f a number. A l i s t o f 27 numbers containing both primes and composites was constructed. The primes were chosen a t random from a table o f prime numbers.  S i x primes were selected because they were used by Hunter.  The composite numbers were selected to ensure that no even composites were included. Many composites with r e l a t i v e l y such as 899 and 667 were selected  large prime  to increase  factors  the d i f f i c u l t y o f  166 factorization. HS was instructed to state immediately i f she thought a given number was either prime or composite. She was to determine a t l e a s t one prime factor i f the number was composite. She c o r r e c t l y i d e n t i f i e d 20 o f 27 numbers as being either  prime or composite. Her greatest  error was stating a composite number as  prime.  The following i s a p a r t i a l account o f the number f a c t o r a b i l i t y assessment.  The reader should note how she was able to use various  number theory concepts such as d i v i s i b i l t y checks to determine i f the given number was composite. The numbers i n the parentheses refer to her solution times; R r e f e r s to the researcher. R: Try 507. HS: That's not prime (1 s ) . R: What are the factors? HS: L e t me see...Oh, 169 and 13 x 13. R: How d i d you know i t wasn't prime? Did you t r y d i f f e r e n t factors? HS: Well, I knew immediately i t was not prime because i t ' s a multiple o f 3. The sum o f the d i g i t s are d i v i s i b l e by 3. R: Try 599. HS: That's prime (1 s ) . R: How about 187? HS: That's not prime. R: How d i d you know? HS: Well, l i k e 1 and 7 are 8, so I knew i t was a multiple o f 11. R: Were you taught t h i s d i v i s i b i l i t y rule? HS: Well, I knew them before I was taught them. R: How d i d you learn them? HS: J u s t playing around with numbers. R: Try 833. HS: That's not prime (2 s ) . R: What are the factors? HS: 7, 219,...no, 119. R: So 7 and 119? HS: Yes, and 49 and 17. R: Try 529. HS: That's the square o f 23 ( I s ) . R: Try 667.  167 HS: That's not prime ( 2 s ) . Do you want the factors? R: Yes, i f possible. HS: Let's s e e . . i t ' s 23 x 29 (4 s ) . R: Try 1063. HS: That's not prime ( 4 s ) . R: What are the factors? HS: Uhmm....(10 s ) . I don't know but i t ' s not prime. R: I t i s prime. HS: Oh, I didn't think i t would be. R: HS: R: HS: R: HS:  Try 301. That's prime (2 s ) . No, i t ' s . . . Oh, yes. I t ' s 43 x 7. Try 179. I t ' s prime (2 s ) .  R: Try  509.  HS: That's prime (1 s ) . R: Try 533. HS: That's prime (2 s ) . R: No. HS: OK. This w i l l be a bad one. Could I have a pen, please?....Wait, d i d you say 553? R: No, 533. HS: Oh, well then, i t ' s 13 and 41. R e t r i e v a l of Numerical Equivalents Two  processes  involved  i n mental c a l c u l a t i o n are the  initial  s e l e c t i o n o f a c a l c u l a t i v e strategy and the subsequent r e t r i e v a l of a s e r i e s of numerical  equivalents.  The  purpose of a strategy i s to  decompose and organize a c a l c u l a t i o n into a s e r i e s of more tractable sub-calculations. continues equivalent that  the  until  This the  process  subject  of  i s able  from memory. Evidence has skill  groups  differed  in  decomposition  and  organization  to r e t r i e v e a needed been provided their  choices  numerical  which indicated of c a l c u l a t i v e  strategies to solve a mental m u l t i p l i c a t i o n task: there were a l s o a number  of  ways that  the  skill  groups d i f f e r e d  i n the  ability  to  168  r e t r i e v e numerical equivalents u s e f u l for mental m u l t i p l i c a t i o n . R e t r i e v a l of the Basic M u l t i p l i c a t i o n Facts The most comrnonly accessed numerical equivalents were the f a c t s of m u l t i p l i c a t i o n . task  when the  distribution  The  most popular  were  applied  numerical equivalents. between s k i l l e d  and  a b i l i t y to solve a mental c a l c u l a t i o n strategies  depended  To  such  upon  subjects  P&PO and  ready  additive  access  in retrieving  c a l c u l a t i v e information, a t e s t of the 100 BFR was  a  as  determine i f there were any  unskilled  basic  to  these  differences this  type  basic m u l t i p l i c a t i o n  of  facts  administered to a l l subjects. Accuracy of  perfect basic  basic  f a c t r e c a l l . Each group exhibited  combined groups was groups  99.06. The were  99.9  The  median score for  the  for the s k i l l e d  and  mean scores on BFR (s  =  0.35)  and  96.7  (s  X  respectively.  to  f a c t r e c a l l . More than 70% of the a l l the p a r t i c i p a t i n g  subjects made l e s s than three errors on BFR.  unskilled  close  Despite the  t h i s difference was  =  3.10),  X  small difference  between the group means,  s t a t i s t i c a l l y s i g n i f i c a n t ( t g = 3.97;  p < 0.001).  2  I f the variances of the two populations are assumed to be unequal, the difference  between the  means remained s i g n i f i c a n t at  the  p  <  0.01  level. The strength  covariance information strategy was  used to determine  of  between  multiplication  the  relationship  recall  and  mental  existing  multiplication  c o r r e l a t i o n r' for the combined groups was the  intact  group R  was  estimated  to  0.58;  equal 0.33  basic  the fact  performance.  The  the c o r r e l a t i o n  for  (t g  3.77;  p  <  0.001). Thus, a s t a t i s t i c a l l y s i g n i f i c a n t but weak relationship  (R  <  2  =  169  0.11)  existed  between basic  fact recall  and mental m u l t i p l i c a t i o n  performance. However, basic f a c t r e c a l l may be a more d i f f i c u l t task  during  a mental c a l c u l a t i o n . In the t e s t o f r e c a l l BFR, the subject had to attend  to only one task: namely, the task o f r e c a l l i n g  calculating) a basic  fact of multiplication.  This  (or possibly  i s not the case  during mental c a l c u l a t i o n where a subject must attend t o a v a r i e t y o f tasks as a r e t r i e v a l process i s under way. calculative stage  task,  The given f a c t o r s of the  the calculated p a r t i a l products, and the current  o f the strategy  must  a l l be remembered while  the subject  attempts t o r e c a l l some f a c t . Because the a b i l i t y to r e c a l l numerical equivalents  requirements  of mental  c a l c u l a t i o n , an attempt was made to assess each subject's  a b i l i t y to  r e c a l l basic  may  be affected  by the extra  f a c t s under the conditions o f c a l c u l a t i o n by examining  the introspective reports. The  number  of  basic  c a l c u l a t i o n was estimated  fact  errors  made  during  a  f o r each CAL2 item. The t o t a l  mental  number of  basic f a c t r e t r i e v a l errors assessed i n t h i s manner was estimated a t 1 for the s k i l l e d group and 18 f o r the u n s k i l l e d group. These  seven  items  accounted  f o r a l l the basic  fact  recall  e r r o r s : 9 x 74, 15 x 16, 8 x 625, 8 x 612, 16 x 16, 9 x 742, 8 x 4211. The  three f a c t s 8 x 6 , 9 x 7 , and 9 x 4 accounted f o r over 70 percent  o f these basic f a c t errors. As might be anticipated, those who made the greatest  subjects  number o f errors on BFR also made the most  r e t r i e v a l errors during a mental c a l c u l a t i o n . These findings suggest that the i n a b i l i t y t o r e t r i e v e a basic  170  f a c t during a mental c a l c u l a t i o n contributed minimally to differences i n mental c a l c u l a t i o n performance.  A f t e r a l l , the combined number o f  e r r o r s i n mental c a l c u l a t i o n made by the s k i l l e d and unskilled groups was  400.  By  far the majority o f errors  c a l c u l a t i o n must be have been due  (95%)  made during mental  to some processes other than the  f a i l u r e to r e c a l l a basic f a c t . Thus, differences i n basic f a c t r e c a l l performance  can be discounted as a major source of v a r i a t i o n i n the  mental c a l c u l a t i o n performance o f young adults. Time  to  access  a  basic  fact.  The  unskilled  and  skilled  subjects appeared to d i f f e r more g r e a t l y on the time needed to access a  basic  complete  fact.  Total  access time,  the 100 items on BFR,  defined  as  the  time needed to  was determined for each subject.  The  mean t o t a l access time for the s k i l l e d and u n s k i l l e d groups was  79.8  seconds (s  =14.74) and 118.9 seconds (s  X  =32.44), respectively. The  X  difference  between  significant ( t  2 Q  these  mean  total  access  times  was  highly  = -4.25; p < 0.001).  Figure 7 contains a scatter p l o t i n which t o t a l access time has been plotted against performance on CALl. plot  reveals  that t o t a l basic  Examination of the scatter  f a c t access time seems to be a weak  predictor o f mental c a l c u l a t i o n performance. "faster"  t o t a l access times tended  Those subjects with the  to have higher scores on CALl;  those with the "slower" times tended to have lower CALl scores. The performance  degree  of  association  on CALl was  between  total  access  time  and  estimated. The c o r r e l a t i o n for the combined  groups r ' was calculated to be -0.61. The c o r r e l a t i o n R for the i n t a c t sample was -0.35 which was s t a t i s t i c a l l y s i g n i f i c a n t ( t g = -4.07, p < 2  0.001).  Thus,  about  12%  of  the  variance  in  mental  calculation  171  2 0 t-  • •  •  AA  •  15 < O  c  o  10  CD  O  o CO  50  10 0 Time  Figure 7.  15 0 in  seconds  S c a t t e r p l o t o f t o t a l time needed to r e c a l l 100 basic m u l t i p l i c a t i o n f a c t s (access time) and score on CALl. •  Unskilled  •  Skilled  2  00  172  performance can be explained by the t o t a l time needed to r e t r i e v e 100 basic m u l t i p l i c a t i o n f a c t s . Reconstruction  of basic f a c t s . Many u n s k i l l e d subjects paused  noticeably for c e r t a i n items such as 8 x 6 during the o f BFR.  The  subject's  researcher  answers  administration  suspected that these pauses indicated that a  were  reconstructed  rather  than  retrieved. Some  subjects admitted they had to c a l c u l a t e rather than r e c a l l some basic facts.  For  example,  subject  U l l made no  errors  on  BFR  but  she  frequently paused for items which included 8 as a factor. These pauses greatly  inflated  the  estimate  questioned a f t e r the BFR  of  her  total  administration,  tables were weak." Rather  than  she  access  time.  When  admitted that her  r e t r i e v i n g 48  for  8x6,  "8's  she  had  reasoned, "6 x 8 i s 48 because 7 x 8 i s 56." There was  some d i r e c t observational evidence c o l l e c t e d during  the administration o f CAL2 which suggested that reconstruction during a  mental  c a l c u l a t i o n contributed  information.  Those subjects  who  to  had  the  forgetting  of  numerical  the greatest access times often  l o s t track of the c a l c u l a t i o n as they attempted to reconstruct rather than r e t r i e v e a basic f a c t . I f these subjects were able to complete the c a l c u l a t i o n , the a d d i t i o n a l reconstructive steps  increased  their  solution times. The r e l a t i o n s h i p between t o t a l access time, reconstruction, and errors  in  mental  calculation  becomes  introspective reports are analysed.  The  more  evident  if  a  few  reports of the subjects  U5,  a f t e r attempting to solve 8 x 625  for  U7, and U l l are p a r t i c u l a r l y i l l u m i n a t i n g . The 67  slowest subject U5,  seconds,  reported  that  he  had  lost  the  numbers. He  gave  the  173 following  report  as  he  calculated  (the  s e r i e s of dots i s used  to  represent a r e l a t i v e measure of time): "8 x 25, 40,....8, 16 and 2 i s 18.  And  8 x  6  i s ....  uhm....48.  I have to go  back.  8 x  625  is....40, so....4000....1 forgot the numbers." The lengthy pause after 8x6  suggests that a reconstructive process had taken place and  process may  have contributed  to the  forgetting of previously  this  stored  calculations. U7, who  made the greatest number of errors on BFR  (10 e r r o r s ) ,  admitted that for the "longer basic f a c t s I have to reason them out." His c a l c u l a t i o n of 8 x 625  i l l u s t r a t e s how  c a l c u l a t i v e d e t a i l s can  be  forgotten during the reconstruction o f a basic f a c t . He t r i e d to count out  multiples of 600  After repeat  28  to determine the p a r t i a l product of 8 x  seconds into the  the  question.  This  600,....4 times i s 1200. be  4800.  were  So  the  8  c a l c u l a t i o n , he  No,  was  the  asked  full  the  report  researcher  he  i t would be 6, 12, 18,....  times 600  i s 4800 and  the  600.  gave: No,  last  to  "Well.  i t would  two  numbers  44?" The  reconstructive  process  needed  hindered h i s c a l c u l a t i o n o f 8 x 612. 6 is....46. digit  So  reasoning  So 460 4600,  i n 60  and  8 times 10  ,4680,  seconds.  and  to  calculate  8 x 6  also  He thought, "8 x 5 i s ....40 and i s 80.... Can't remember the next  16  is  4696."  In contrast,  the  He  completed  typical skilled  this  subject  c o r r e c t l y solved t h i s c a l c u l a t i v e task i n about 5 seconds. o  Even  a  fact  such  as  8  x  4  was  reconstructed  rather  than  retrieved by U7. To c a l c u l a t e 8 x 4211, he reported, "8 x 4000 i s 24 000, No, 8 x 3 i s 24 and 8 must be 32. I t ' s 32 000 Can I have  the  next  digit  please?"  Thus,  after 31  seconds,  some main  174  feature of the  task, i n t h i s case the d i g i t s of a factor, had  been  forgotten. Although  subject  Ull  had  perfect  recall  on  BFR,  she  reconstructed some facts to solve some mental c a l c u l a t i o n tasks. those c a l c u l a t i v e tasks where there was  evidence of  her  was  response  features  of  reconstruct  times the  were  slow  calculation.  8x6,  can't remember my 8's  previously  by  Because  of  unable the  table.  8x6  28:  correct  this  reconstruction, to  time  reported, "8 x 5 i s 40,  28, 29,..so,....2, 9, 4, 0." replaced  she  her incorrect solution o f 8 x 625  seconds to complete. She  was  and  For  retain  she  some  needed  to  took a lengthy 56  and....8 x 6 i s . . . .  is....48 and 8 x 2 i s 16, carry  1,  In t h i s task, the calculated value of  48  substitution  c a l c u l a t i o n was  an  of  incorrect  error  made by  numbers  for  a  many u n s k i l l e d  subjects. U l l gave an 612  for 74 seconds!  carry 1,  8,  pause)....So  9,  and  answer of 5792 a f t e r attempting to calculate 8 x Again, she had 8x6  i t would  be  to reconstruct  8,...no, 6.  So,  "2 x 8,  16,  is....My times tables... .36?.... (very long 6  x  7  =  commented, "I had to reason out 8 x 6 . add  8x6:  i t ' s 48.  42,....56.  No,  57....92."  She  I t ' s 56 because 6 x 7 = 42  and  I must have been thinking o f 8 x  7?"  The R e t r i e v a l of Large Numerical Equivalents Blocking.  Many  described as "blocking" solve a c a l c u l a t i v e  skilled  thinkers  used  what  they  to reduce the number of c a l c u l a t i o n s needed to  task.  before c a l c u l a t i n g 125  calculative  x 125,  For  example, subject S7  explained  that  he organized the c a l c u l a t i v e task into  these three e a s i l y determined blocks: 100  x 125,  25 x 100,  25 x  25.  175  He conpleted the c a l c u l a t i o n by annexing zeroes to obtain the products 12 500  and 2500 and by r e t r i e v i n g the large numerical equivalent  625  from memory. The a b i l i t y to organize a computation i n t o a smaller number of blocks  seemed to be  large numerical  associated with  equivalents.  f a c t s , the computation had blocks  reflecting  this  a  subject's a b i l i t y  to  access  I f a subject could access only basic  to be p a r t i t i o n e d i n t o a larger number of  limited  storage  of  information.  Thus,  the  o  popularity of P&PO amongst the u n s k i l l e d subjects could have been due p a r t i a l l y to the f a c t that each block necessitated the r e t r i e v a l o f only basic f a c t s . On the other hand, i f a subject had access to a store of large numerical equivalents, there was  an increased opportunity to organize  a computation i n t o a smaller number of e a s i l y determined blocks. In f a c t , some subjects used u n i t blocking (organizing a computation i n t o one  block)  equivalent  when  the  computation  that could be  was  recognized  as  a  numerical  retrieved from memory. Forty-six responses  were c l a s s i f i e d as u n i t blocking: that i s , the c a l c u l a t i v e task  had  been solved e n t i r e l y by r e t r i e v i n g one large numerical equivalent. To ensure that the s o l u t i o n was each subject was been attempted.  always asked i f any  retrieved and not calculated, intermediate  c a l c u l a t i o n s had  T y p i c a l l y , a subject would respond to t h i s query with  phrases such as, "I j u s t know i t , " "I memorized i t , " " I t ' s a f a c t , " or as one  subject said, " I t ' s corrorimon knowledge; everyone knows that."  The very short s o l u t i o n times (1 to 4 seconds) which were associated with  these  responses  c a l c u l a t i v e process.  also  suggested  a  retrieval  rather  than  a  176  The blocking  skilled while  subjects  the  accounted  unskilled  for 44  accounted  instances  for two  of  unit  instances.  The  differences were so large that a t e s t f o r s t a t i s t i c a l s i g n i f i c a n c e seemed tasks  unnecessary. by  Thus,  retrieving  skilled  subjects  a  large  numerical  calculative  tasks  solved  solve  more c a l c u l a t i v e  equivalent  than u n s k i l l e d  subjects. The calculating  are displayed  i n Table  by  retrieving  rather  than  VIII. Examination of Table  VIII  indicates that 44 o f the 46 tasks solved by r e t r i e v a l were squares. The  only  square  i n CAL2  not  solved  by  retrieving  a  numerical  equivalent was 32 x 32, the l a r g e s t square included i n CAL2. Although equivalent, organizing  not solved  by  a  single  retrieval  of a  numerical  there were many c a l c u l a t i v e tasks which were solved by the computation  i n t o several blocks and by r e t r i e v i n g a  large numerical equivalent associated with a t l e a s t one o f the blocks. Several  solutions  to the item  process: r e t r i e v i n g 15  2  15  x  16  illustrate  this  blocking  and c a l c u l a t i n g "225 + 15"; r e t r i e v i n g 16  ?  and  c a l c u l a t i n g "256 - 16"; r e t r i e v i n g 5 x 16 and c a l c u l a t i n g "160 + 80." I f the numerical  equivalents o f 1 x 15 and 1 x 16 are excluded, no  u n s k i l l e d subject r e t r i e v e d a numerical equivalent larger than a basic f a c t f o r t h i s same task. No s k i l l e d subject used u n i t blocking to solve a CAL3 item but several computations were blocked so that a large numerical could be retrieved.  equivalent  The most frequently accessed large numerical  2 equivalent was "25  = 625" which was retrieved 8 times. This  numerical  equivalent was r e c a l l e d during these computations: 75 x 75 = 25  x 3 x  3, 24 x 625 = 24 x 25 , 125 x 125 = (100 + 2 5 ) . Several subjects knew 2  2  177  TABLE  VIII  PROJDUCTS D E T E R M I N E D B Y R E T R I E V I N G R A T H E R THAN  CALCUTATING  Groups  Task  Skilled Number o f Subjects  Unskilled Number o f Subjects  25 x 48  1  0  12 x 15  1  0  16 x 16  9  1  25 x 25  8  1  13 x 13  13  0  24 x 24  1  0  15 x 15  11  0  Total  44  2  178 "2  = 1024"  10  and applied t h i s knowledge to solve the CAL3 items 32 x  64 and 64 x 64.  Subject SI applied h i s knowledge that "36  = 1296"  2  to  c a l c u l a t e the s o l u t i o n to the CAL3 item 36 x 72: he said, "I doubled 1296." The large numerical equivalents that were retrieved and applied to a c a l c u l a t i o n frequently involved squares and other powers.  Those  subjects who could r e c a l l these large equivalents were often asked they  could  remember  such  apparent  e s o t e r i c information. 2  why  Several  2  subjects commented that powers of 2 such as 16  and 32  were memorized  because of "working with computers and the binary system." Subject S9, who  was  asked why  he knew immediately that 1024  was  a large power of  2, r e p l i e d : When I was a l i t t l e k i d we'd go on these long t r i p s and used to s i t i n the back of the car and think 1 + 1 i s 2, 2 2 i s 4, and go a l l the way....Just for something to occupy mind. I would recognize that 1024 i s a power of 2 but wouldn't know that i t was 2 to the 10th. 2  1296.  Subject SI was  asked why  he could remember that 36  He explained,  "Probably working with p r o b a b i l i t y .  I + my I  equalled In r o l l i n g  dice you have a chance o f 1/6 of r o l l i n g a 1 so i f you do that 4 times i n a row suggest  i t would be 1/6 that  this  to the 4th or 1 i n 1296."  numerical  information  was  Their answers  memorized  through  the  pursuit o f some a c t i v i t y which interested them rather than through a deliberate intention to commit these f a c t s to memory. There was  some evidence,  however, which suggested  that some  s k i l l e d subjects take pleasure i n "memorizing, merely for the sake of memorization."  Subject  t e s t . He r e p l i e d "I'm to see how  SI was  asked i f he had  ever  taken a memory  not sure but once a f r i e n d and I had a contest  f a r we could each memorize p i . I got as far as 250  places  179  and  then I l o s t i n t e r e s t i n the problem." At a subsequent interview,  he was  asked to r e c i t e t h i s sequence. He  series  correctly  without  any  r e c a l l e d 46 d i g i t s i n the  preparation  for  this  task.  No  other  subject i n the study reported a s i m i l a r i n t e r e s t i n number r e c i t a t i o n . Several subjects, including some u n s k i l l e d subjects, commented that when a c a l c u l a t i v e task required the determination of a square, they  often  thought  "knowing" often because  the  retrieval  that  had  subject  process  they  the  "should  effect  seemed  before  of  to go  know that."  This  protracting the through  a quick  s e l e c t i n g some other  sense  of  s o l u t i o n times and  unsuccesful  method of solution.  There were many instances of t h i s " c a l c u l a t i v e deja vu." When SI was pause punctuated by  given  the  item 24  the phrase,  x 24,  "Hmmm."  felt  I  should  know  that  one."  a noticeable  When questioned  pause, he r e p l i e d , "I had to multiply i t out I  there was  about  the  (using distribution) but  Subject  S l l claimed  that  an  unsuccessful search for 16 x 16 was made before applying d i s t r i b u t i o n . After  the c a l c u l a t i o n was  completed,  he  said, "256  rang  a bell.  I  2 should  have known that 16  believed retrieved "It  would  that the  i s a fact."  squares of 13,  rather than calculated. be  handy to know these  15, One  Even 7 u n s k i l l e d  16,  and  25  subjects  should  have been  u n s k i l l e d subject  explained,  for c a l c u l a t i n g  square roots i n  algebra." The "12's  12's Facts. The u n s k i l l e d subjects would r a r e l y r e t r i e v e a  fact,"  calculation.  the product  of 12 and  The  subjects, on  skilled  another factor, to a i d a mental the other  hand, often solved  c a l c u l a t i v e tasks through these r e t r i e v a l s . For example, 12 x 81  was  solved by several s k i l l e d subjects by thinking, "8 x 12 i s 96; so 960  180  and  12 i s 972."  To determine i f the infrequent r e t r i e v a l o f these  s p e c i f i c m u l t i p l i c a t i o n f a c t s by the u n s k i l l e d subjects was either a matter o f choice  or a matter o f memory, each subject's a b i l i t y to  r e c a l l products from the "12*s m u l t i p l i c a t i o n table" was assessed. Each subject was asked to r e c a l l q u i c k l y the s o l u t i o n to 5 x 12,  6 x 12, 7 x 12, 8 x 12, 9 x 12, 11 x 12, and 12 x 12. A l l these  items were presented randomly.  The easy items 1 x 12, 2 x 12, 3 x 12,  4 x 12, and 10 x 12 were not presented.  I f the subject responded  quickly and claimed that no c a l c u l a t i o n was needed, the s o l u t i o n was c l a s s i f i e d as r e t r i e v a l o f a large numerical equivalent.  The r e s u l t s  are summarized i n Table IX. Examination existed  of  the  table  between the two groups  indicates that i n the a b i l i t y  large d i f f e r e n c e s to r e t r i e v e  these  selected numerical equivalents. These r e s u l t s provide further evidence that the a b i l i t y  to r e t r i e v e numerical  facts of multiplication  equivalents beyond the basic  seems to be more c h a r a c t e r i s t i c o f s k i l l e d  than u n s k i l l e d mental c a l c u l a t o r s . The Highly S k i l l e d Subject's Memory f o r Numerical Equivalents The  highly  skilled  r e t r i e v e large numerical  subject  HS  equivalents.  had a remarkable a b i l i t y  to  She solved 16 o f the 45 CAL2  and CAL3 items by an immediate r e t r i e v a l o f the s o l u t i o n .  For those  items where she d i d not r e c a l l the product immediately, she frequently retrieved  large  calculation. involved  numerical  equivalents  to solve  some part  Nine items were solved i n t h i s manner.  numerical  equivalents  that  no other  o f the  Her c a l c u l a t i o n s  subject  i n the study  r e c a l l e d . Several examples can be used to i l l u s t r a t e her r e t r i e v a l and c a l c u l a t i o n processes:  23 x 27 was solved through r e t r i e v i n g 23  and  181  TABLE IX  NUMBER OF SUBJECTS ABLE TO RECALL NUMERICAL EQUIVAI-OTS  SELECTED FROM THE "12'S TABLE"  Groups Unskilled  Tasks  Number o f subjects ' correctly recalling  Skilled Number o f Subjects correctly recalling  5 x 12  1  14  6 x 12  0  14  7 x 12  0  14  8 x 12  0  14  9 x 12  0  14  11 x 12  0  14  12 x 12  5  15  Total  6  99  182  c a l c u l a t i n g 23  2  c a l c u l a t i n g 18  2  + 92; 18 x 72 was  solved through r e t r i e v i n g 18  x 4;  solved through r e t r i e v i n g 27  27 x 81 was  ? 2  and and  2 c a l c u l a t i n g 27  x 3.  This researcher suspected that the items i n CAL2 and CAL3 d i d not  provide  an  adequate  estimate  of  HS's  memory  for  numerical  equivalents. The following procedures were used to estimate her memory 2 2 for squares: a l l 2 - d i g i t squares from 11 of  to 99 ,  excluding multiples  10, were used to test her r e c a l l ; each item was presented o r a l l y ;  the presentation order was  scrambled  to ensure that no item could be  determined i n advance. She c o r r e c t l y r e c a l l e d a t o t a l of 69 squares. M l  (85%) of the 81 2 - d i g i t  her solutions were stated within 1 or 2 seconds after the  square had been presented o r a l l y . The reasoning such as "I j u s t know it"  or  " I t just  evidence that a  popped  into  my  head"  was  considered  additional  r e t r i e v a l rather than a c a l c u l a t i v e process had taken  place. For the 12 squares that HS claimed were "guesses," her answers were  i n error  correct  by  except  only a  few  for the d i g i t  percent. Frequently, the order:  for example,  instead of 7396. I f given a second attempt, retrieved the c o r r e c t solution i n a few HS for  she  answer stated  was 7936  she either calculated or  seconds.  stated a l l 4 - d i g i t products as a pair of 2 - d i g i t numbers:  example, 7056 was  stated as  "Seventy,  fifty-six."  Speculating  that some subtle memory process could underlie t h i s method of stating the answer, the researcher asked her to explain why expressed fifty-six'  as  pairs.  than  'seven  She  said,  thousand  " I t ' s easier fifty-six'."  the products were to  say  Another  'seventy, potentially  183 p r o l i f i c research area nipped i n the bud! To expedite the process o f t r y i n g squares HS  could r e t r i e v e ,  to estimate how  many more  the following procedures were used  in a  second interview. Starting a t the number 101, she was asked to state i n s t a n t l y that number's square. I f she couldn't r e c a l l t h i s square, she was she  instructed not to bother with any c a l c u l a t i o n and,  was  to proceed  to  the  next  instead,  square  i n the progression. This  because  she began to t i r e o f the  2 procedure was  terminated a t 349  a c t i v i t y . Table X l i s t s the squares that she could r e c a l l instantly. As  can  be  seen  by  examining  this  table,  her  knowledge o f  squares was exceptional. She c o r r e c t l y r e c a l l e d 46 3-digit squares and the  one 4 - d i g i t square she r e c a l l e d was  576! to  Often she would respond with a statement such as "I'm  know that one,"  can't  the square o f 1024 or 1 048  remember  supposed  "I used to know that one," or "I knew i t but I  i t now"  when she could  expressions o f recognition  suggest  not r e c a l l  that  she may  a square.  These  have been able to  r e c a l l even more squares i n the past. Certainly her parents thought that she was not as knowledgeable now as she had been i n the past. For  several squares, she would  stated answer was feasibility  of  sophisticated  an and  incorrect and answer,  she  surmise  immediately  that her  state an a l t e r n a t i v e . To check  the  often  was  surprisingly  applied  rapid.  Her  reasoning which response  i l l u s t r a t e s her rapid appraisal o f a p o t e n t i a l answer. 125, no that's wrong i t ' s 99 225."  Asked why  to  315  2  She said, "90  she thought that 90 125  was incorrect, she explained, "Because i t ' s not a multiple o f 9."  The  reader i s reminded that both the r e t r i e v a l o f two reasonable responses and the reason for r e j e c t i n g one o f these choices took place within 5  TABLE X  SQUARES RECAliLED BY ONE HIGHLY SKILLED SUBJECT  Interval  Recalled squares  101-109  All  111-119  A l l except 115  121-129  121 , 123 ,  131-139  135  2  141-149  144  2  151-159  152  2  161-169  162  2  171-179  171  2  181-189  None r e c a l l e d  191-199  None r e c a l l e d  201-209  201  211-219  None r e c a l l e d  221-229  225 , 228  231-239  None r e c a l l e d  241-249  242  251-259  252 , 256  2  261-269  261 , 264  2  271-279  275  2  281-289  288  2  291-299  294  2  301  301 , 304 , 312 , 315 , 326 , 343  +  2  2  2  2  202 , 204  2  2  124 , 1 2 5 , 126 , 128  2  r  and 118  2  2  2  2  2  2  2  2  185  seconds. Further questioning revealed  that she was able to apply such  d i v i s i b i l i t y rules as "casting out nines" to check the reasonableness of  her c a l c u l a t i o n s . She used reasoning similar to the following to  determine the reasonableness o f the answers 90 125 and 99 225: The sum o f the d i g i t s i n 315 i s d i v i s i b l e by 9 so that the square must be a multiple o f 9. But the sum o f the d i g i t s i n 90 125 i s not d i v i s i b l e by 9 and, therefore, i t cannot be the square o f 315. Consequently, 99 225 seemed the most p l a u s i b l e solution. The remarkably  following  anecdote  is  another  illustration  quick and acute power o f reasoning.  o f her  I remarked  quite  2 i n c o r r e c t l y that her i n i t i a l response o f 4356 to 66 was incorrect. Her response to the re-administration o f t h i s item was revealing. R: T r y 66 squared again. HS: 4356 ( i n s t a n t l y ) . R: OK. That one's wrong, but you're close. HS: 4536 ( i n s t a n t l y ) . R: Yes. Does that happen sometimes, you get the r i g h t d i g i t s but i n the wrong HS: I t ' s 4356. R: No, i t ' s 4536 HS: No, because 4536 i s 504 x 9! I know i t ' s 4356. R: (checks calculation) My apologies, you're correct. At  a subsequent interview, her memory o f t h i s event was refreshed and  she was asked t o explain her reasoning. She said, "4536 couldn't be the  square o f 66 because 9 i s not a factor o f 66 but i t i s a factor o f  both 504 and 9."  The apparent l o g i c o f her reasoning follows:  4536 can be factored into 504 and 9. 504 has a factor o f 9, therefore, 9 x 9 i s a factor o f 4536. On the other hand, 9 i s not a factor o f 66 so 9 x 9 cannot be a factor o f 66 squared. Therefore, 4536 i s not the square o f 66. Both these c a l c u l a t i o n and v e r i f i c a t i o n a c t i v i t i e s took place during a time i n t e r v a l o f a few seconds. During  one o f the interviews,  HS commented  that  she could  186  recall  most  2-digit  squares  "but some o f them  I get caught on,  e s p e c i a l l y prime numbers l i k e 59." When asked what she would do i f she forgot  the square o f 59,  she said,  " I t ' s 59 x 60 - 59."  Thus,  c a l c u l a t i v e s t r a t e g i e s were a l s o used t o check the reasonableness of "retrieved" squares.  The researcher was able to i d e n t i f y some other  ways that she used t o determine the reasonableness o f her answers. These methods are best i l l u s t r a t e d through c i t i n g a few examples. 2 The following discussion relates to the task 62 : HS: 37...let me see, 3844 (5 seconds). R: I noticed a pause there, what were you thinking. HS: I was thinking 3724 but I knew that was wrong so I figured i t out again. R: So you thought 3724 but f e l t i t was wrong. Why would you think i t ' s wrong? HS:  I t ' s too close t o the square o f 61.  Subsequent  discussion  revealed  this  thinking process: " I t might be  3724 but 6 1 i s 3721 which i s too close to 3724 so i t can't be 3724." 2  To complete  the process, she thought 62 x 60 and 62 x 2. Both the  estimation and c a l c u l a t i o n a c t i v i t i e s were completed  i n l e s s than 5  seconds. Often she had to decide between two l i k e l y answers. When asked to r e c a l l the square o f 97, she answered 9049 i n s t a n t l y . she  was t o l d  that  the answer 9049 was incorrect,  responded, "Then i t ' s 9409."  As soon as  she immediately  She said, "I guessed" t o explain t h i s  second answer. Apparently, she suspected that either 9049 or 9409 was correct.  Therefore, once she obtained the information that one answer  was incorrect, she knew that the other a l t e r n a t i v e was c o r r e c t . There unrelated example:  were  times  when  she recalled  a  number  that  to the square. This process can be i l l u s t r a t e d  seemed by t h i s  187 R: HS: R: HS: R: HS: R: HS: R: HS: R: HS:  Try 88 squared. 7744 (5 seconds). How d i d you determine that? I just thought for a moment. Do you mean you j u s t concentrated u n t i l the answer appeared? Yes. Did you do any c a l c u l a t i o n s a t a l l ? No. Were any other numbers thought of before you said 7744? Yes, 7004. Why would you think of t h i s number? I don't know...Well 88 x 8 with a zero i n the middle o f i t .  In other words, she thought 88 x 8 which equals 704.  And 7004 can be  thought o f as "704 with a '0* i n the middle of i t . "  Perhaps such an  involuntary c a l c u l a t i v e response was cued by the d i g i t s contained i n 2 88 . She  commented that  this  response  had  "hurt" the speed of the  calculation. Short-Term Memory Processes i n Mental Calculation Types of Forgetting During Mental C a l c u l a t i o n One of the major types of e r r o r s made by a l l subjects as they performed a mental c a l c u l a t i o n was either  f a i l i n g to reirember some aspect o f  the i n i t i a l c a l c u l a t i v e task or the interim c a l c u l a t i o n s .  The  researcher had hoped to be able to categorize the d i f f e r e n t types of forgetting  and  assess  the  relative  frequency  of  each  error  type.  However, a r e l i a b l e assessment of each type of error and i t s frequency was not possible for several reasons. Although a l l subjects could remember and describe the method o f solution, they often had d i f f i c u l t y r e c a l l i n g the numerical d e t a i l s of a c a l c u l a t i o n after a solution had been stated.  When probed for these  d e t a i l s , some subjects could provide only ambiguous responses such as "I think I got l o s t i n the adding," "I may  have forgotten to carry,"  188 or  " I t was memory again."  Even l e s s d e t a i l was provided  students who became f r u s t r a t e d , confused,  by those  and simply stated, "I don't  know," "I l o s t track," or "I'm l o s t . " A second d i f f i c u l t y i n c l a s s i f i c a t i o n was the tendency for some subjects to make several types o f e r r o r s during a c a l c u l a t i o n . i f a l l the e r r o r s could be i d e n t i f i e d , design  could  not be used  the present  study's  to determine which error type  Even  research  singularly  affected the c a l c u l a t i o n performance. However, could  recall  there were some types with  classification. calculated  the  Many  errors  "out loud."  identified: task;  sufficient  o f forgetting which  detail were  These  f o r reasonably  also  common  identified  errors  direction  series;  o f the c a l c u l a t i o n ;  unambiguous  as a  subject  o f forgetting were  (1) forgetting the f a c t o r s o f the i n i t i a l  (2) forgetting a completed c a l c u l a t i o n ;  subjects  calculative  (3) losing track o f  (4) forgetting the order  of a  (5) forgetting to "carry" or carrying the wrong number;  (6)  misalignment during addition. The conclusions  data provided should  be  on error c l a s s i f i c a t i o n are "soft" and any treated  q u a l i t a t i v e data do provide  with  caution.  Nevertheless,  the  simple confirmation that forgetting some  aspect o f temporarily-held information i s a much more important  source  o f error i n mental than i n written c a l c u l a t i o n tasks. Forgetting reported  the  initial  factors.  that the retention o f the i n i t i a l  The  unskilled  subjects  factors was a d i f f i c u l t  task. The researcher could i d e n t i f y t h i s type o f error when a subject asked either a question o f v e r i f i c a t i o n such as "Is i t 8 x 612?" or a question about the task  such as "What was the question again?" The  189  s k i l l e d and  unskilled  subjects d i f f e r e d i n the  number of times they  asked the researcher to either v e r i f y or repeat the question: 9 times for the s k i l l e d and 64 times for the Subject factors  U6  had  particular  i n a c a l c u l a t i v e task.  this difficulty: .... 612  x 8?  unskilled. difficulties  Her  "....Is i t 612?  in  remembering  response to 8 x 612 (5 seconds into  illustrates  the  calculation)  (20 seconds into the c a l c u l a t i o n ) . . . . 1296?"  seen from t h i s example, t h i s d i f f i c u l t y i n completing the was  not  due  to misunderstanding the  a f t e r several  seconds had passed and  the  As can  be  calculation  task. This d i f f i c u l t y  occurred  substantial c a l c u l a t i v e progress  had been made. Two  c a l c u l a t i o n tasks, i n p a r t i c u l a r , had  of t h i s type of and  forgetting.  Five  unskilled  and  4 skilled  subjects,  7 unskilled and 1 s k i l l e d subjects, asked the researcher to either  repeat or v e r i f y the  factors of 9 x 742  Si's  he  comment,  commented,  after  solved  "I have t r i c k s for  8  x  numbers and  and  4211,  8 x 4211, seemed  some problems but  remember these long ones (1-digit by the  the highest incidence  respectively.  instructive. I find  x-digit factors).  then I have to remember a s t r i n g of  i t hard I  He to  calculate  them as I keep  calculating." Forgetting a completed c a l c u l a t i o n . calculation subjects.  was  another  difficult  Retaining the numbers of a  operation  for  the  unskilled  This type of forgetting occurred frequently as the  attempted to calculate another p a r t i a l product.  subject  After c a l c u l a t i n g for  47 seconds, U4 gave up i n her attempt to solve 23 x 27. She  explained:  I got both numbers (the p a r t i a l s ) but couldn't seem to remember them. I had 161 and, I think, 460 was the other one. I never d i d add. I would calculate the second number and lose the f i r s t and then go back to calculate the f i r s t  190  and  lose the second. This alternation between c a l c u l a t i o n and  protracted  the  calculated partial  17  solution x 99  products.  times  only a f t e r However,  of  some  several the  recalculation  subjects.  U2  greatly  correctly  repeated c a l c u l a t i o n s  solution  took  over  2  of  minutes  the to  complete! In order to terminate a lengthy c a l c u l a t i o n , some subjects used only a p a r t i a l l y r e c a l l e d c a l c u l a t i o n to determine a solution. he said 1525  was  I went to add  the solution to 25 x 65,  I had  forgotten  subject U2 explained, "when  the f i r s t one  o f made up what I thought i t could be... memory and  After  (partial) so I j u s t sort  I took something out of  my  i t wasn't r i g h t . "  Often a subject would complete a c a l c u l a t i o n but  inadvertently  either reduce or increase t h i s c a l c u l a t i o n by a multiple of a power of 10.  For  example, subject S12  applied  d i s t r i b u t i o n to solve 23 x  and  obtained as a p a r t i a l product 4600 instead  had  460  but  i t must have changed to 4600."  they frequently  forgot to annex zeroes.  of 460.  He  said,  27 "I  Many subjects complained  This d i f f i c u l t y i n retaining  the number o f "zeroes" became apparent during the c a l c u l a t i o n of 25 x 480  where seven s k i l l e d  instead o f 12  subjects gave an  initial  a common technique used by a l l subjects to help  r e t a i n a completed c a l c u l a t i o n .  When subject U2  repeated  many  partial  product  so  c a l c u l a t e the second p a r t i a l , she firm." very  SI  1200  000.  Rehearsal was  a  response of  times  calculation  123  before  asked why  she  continuing  to  r e p l i e d , " I t helps make the numbers  made a similar comment a f t e r he  difficult  was  x  456.  He  successfully commented  solved  that  he  the had  191  obtained number  a p a r t i a l product o f 10 448 (23 x 456) and "went over that a couple o f times."  r e p e t i t i o n , he responded,  When asked  to give  reasons for t h i s  "So I wouldn't forget i t .  I t reinforces  it." There was some evidence that a few s k i l l e d subjects used other memory techniques besides  rehearsal  t o help  remember a c a l c u l a t i o n .  Such memory devices became evident only during CAL3 where the s k i l l e d  subjects  the administration o f  commented that  "the numbers seemed  more d i f f i c u l t to remember" than the easier items included i n CAL2. S7 employed a number o f d i f f e r e n t memory t a c t i c s to help r e t a i n his  interim c a l c u l a t i o n s . To solve  32 x  64, he applied  d i s t r i b u t i o n and calculated a p a r t i a l o f 1920. instead o f memorizing (rehearsing) date.  additive  He explained,  "Now  i t I j u s t t r i e d t o remember i t as a  I thought that i t would s t i c k that way better.  F i r s t time I  came across a number that I could remember as a date." A more common memory t a c t i c f o r t h i s subject was to examine a number f o r a recognizable pattern or property:  to r e c a l l the p a r t i a l  product 2880, he said, "I t r i e d t o remember that pattern o f numbers 2, 8, 8"; f o r the calculated p a r t i a l o f 16 800, he s a i d , "I needed a way to r e c a l l that so I thought 8 i s 1/2 o f 16 and stored that as 1000's"; for the p a r t i a l product o f 7280, he remembered, "72 i s a multiple o f 8"; f o r the p a r t i a l 3840, he said, "I t r i e d to remember by saying 4 i s 1/2 o f 8 and a 3 a t the beginning." His successful s o l u t i o n o f the item 123 x 456 i s a fascinating i l l u s t r a t i o n o f an a b i l i t y to coordinate a number o f d i f f e r e n t thought processes explained:  during  a particularly difficult  mental c a l c u l a t i o n .  He  192  100 x 456 i s 45 600. I s e t t±iat to the l e f t o f my "mental vision." 20 x 456 i s 2 x 450, 900 and 2 x 6, 12 so 912. Somehow i t changed to 906 instead o f 912. I d i d i t again to get 9120. Added 9000 and 45 000 so 54 000. And 600 +120 i s 720, so 54 720. I then remembered t h i s by thinking they're (54 and 72) quite frequent i n the times tables so I t r i e d to remember "54 and 72 i n the thousands." I remembered I had blocked 100, 20, and 3 so I had t o multiply the remaining 3. 3 x 400, 1200. 3 x 50, 150 so 1350 and 3 x 6 i s 18, so 1368. I went over i t a couple o f times so I wouldn't forget i t . Added l e f t to r i g h t , so 54 and 1, 55 000; 300 and 700, that s c r o l l s up to another 1000. So 55 600 and, f i n a l l y , 56 088. It's easy to get l o s t i n t h i s c a l c u l a t i o n . Losing  track o f the d i r e c t i o n  o f the c a l c u l a t i o n .  Another  problem inherent i n mental c a l c u l a t i o n appeared to be keeping track o f what c a l c u l a t i o n s had  been  performed and remained to be performed.  There were many instances where subjects l o s t track o f the c a l c u l a t i o n and proceeded to either miss a step, duplicate a c a l c u l a t i v e step, or introduce some unnecessary numbers i n t o the computation. took 66 seconds to complete a c a l c u l a t i o n o f 12 x 16.  Subject U3  After several  r e c a l c u l a t i o n s o f the p a r t i a l s , she s a i d : 16 on the top and 12 on the bottom. 6 x 2 = 12, carry 1, 2 x 1 i s 2, add equals 3.... So 32. Put down the 0. 6x1=6. Add 1 i s 7...so 72. So you have 32 and 72... (begins to add) ...723? What appeared to happen was that she somehow interchanged the " s p a t i a l p o s i t i o n s " o f the factors 12 and 16.  The diagrams below  illustrate  how she l i k e l y viewed the c a l c u l a t i o n .  x first partial  16 12 32  x second p a r t i a l  12 16 72  Several subjects commented that they had "a problem rernembering which number was on the top and which was on the bottom." particularly  evident where, as one  This problem was  subject explained,  "the numbers  193  (factors) were close to being  equal."  One common track-keeping c a l c u l a t i o n o f squares. because,  instead  difficulty  was associated  with the  This type o f forgetting was easy to i d e n t i f y  of calculating  the second  partial  product, the  subject repeated the f i r s t p a r t i a l product and incorporated t h i s value i n t o the addition stage o f the computation. 16 x 16 i l l u s t r a t e s t h i s type o f e r r o r . carry 3.  6 x 1 = 6, add 3 i s 9.  She explained:  So 96.  i s 6 + 0 = 6, 9 + 6 = 15, carry 1,  Subject Ul3's attempt a t "6 x 6 = 36,  Put down the 0.  960 and 96  so 1, 5, 0, 6....1506."  were 26 instances o f t h i s type o f forgetting reported  There  for u n s k i l l e d  subjects. NO s k i l l e d subject made t h i s type o f error. Almost one-third o f the u n s k i l l e d subjects' responses t o the s i x squares included i n CAL2 involved t h i s error. Therefore, a  repeating rather than c a l c u l a t i n g  second p a r t i a l i n the determination  o f a square appeared to be a  s i g n i f i c a n t source o f error f o r the u n s k i l l e d subjects. There were comments made by some subjects which suggested that the decision to repeat repetition  rather  o f the factors  than c a l c u l a t e was influenced by the  i n a square.  Subject  U2 who used the  p a r t i a l product o f 96 and 960 to c a l c u l a t e 16 x 16 said, "I thought the bottom number  (960) should  should be 1 x 6 and l x l .  be the same but now I know that i t  I thought that with the r e p e t i t i o n o f the  16, the top and bottom numbers should be the same." s i m i l a r comment a f t e r c a l c u l a t i n g 25 x 25.  Subject U9 made a  He explained,  " t h i s time I  thought that since 5 x 25 i s 125 and the numbers (factors) are the same, i t ' s going to be another 125." Subject  U3 applied  a variation of this  invalid  strategy to  solve 32 x 32 and offered the following j u s t i f i c a t i o n : "32 and 32, 2 x  194  3 and 3 x 2 then you know i t ' s going t o be the opposite because you're doing the same thing i n a d i f f e r e n t order.  So instead o f 64 i t would  be 46 and then you kind o f s h i f t i t over."  In other words, he added  the p a r t i a l products o f 64 and 460 to determine the incorrect solution o f 504. There  was only one instance o f t h i s  pencil-and-paper computation t e s t WPP.  type o f error  in  the  Subject U l l t r i e d to solve 24  x 24 by substituting 960 f o r 480 as the second p a r t i a l product.  No  other subject made t h i s type o f error during written c a l c u l a t i o n . The f a c t that t h i s type o f forgetting existed i n mental but not i n written c a l c u l a t i o n demonstrates  how the absence o f an external memory-store  can a f f e c t performance and introduce d i f f e r e n t types o f errors. The  tendency  t o lose  track  o f a c a l c u l a t i o n was l i k e l y one  reason why some attempted solutions seemed very unreasonable.  Subject  Ul3's solution o f 9206 f o r the item 17 x 99 was such an example. Subject  Ul4's  attempts 1056,  solutions  were  often  unreasonable:  she made  three  to solve 24 x 24 and her answers f o r these attempts were  4160 and 3460. For the l a s t attempt, she had o r i g i n a l l y stated  "30 thousand and 460" before deciding on 3460. Since the s k i l l e d than  the u n s k i l l e d  subjects tended to use d i f f e r e n t strategies  subjects,  track-keeping errors.  they  committed  different  types o f  For example, a common error made by subjects  who used subtractive d i s t r i b u t i o n was l o s i n g track and using the wrong subtrahend. Usually, one o f the o r i g i n a l factors o f the c a l c u l a t i v e task was used  as the subtrahend:  f o r example, S4 used  99 as the  subtrahend instead o f 17 to c a l c u l a t e 17 x 99. Some s k i l l e d  subjects who attempted  to use factoring made an  195  error by incorporating this  a wrong factor i n the c a l c u l a t i o n .  Frequently,  type o f f a i l u r e occurred when a l i q u o t parts was applied.  For  example, subject S l l calculated 4000 instead o f 3000 as the product o f 25 x 120.  During h i s introspection,  400....No, 300.  So 3000.  I think I know what happened I thought 4  into 12, 3 times but then went 4000. Forgetting  he said, "12 'twenty-fives' are  I t j u s t happened."  the order o f a s e r i e s .  Losing the order o f a s e r i e s  o f d i g i t s was another type o f forgetting type o f error  usually  occurred  either  i n mental c a l c u l a t i o n . This during  the f i n a l  stages o f  c a l c u l a t i n g the product o f a 1 - d i g i t and a x - d i g i t factor or after the sum o f the p a r t i a l s had been determined by using a d i g i t - b y - d i g i t , r i g h t - t o - l e f t addition procedure. Subject U l l ' s solution o f 15 x 15 i l l u s t r a t e s how t h i s problem can  a r i s e during a mental c a l c u l a t i o n .  carry 2. and  She thought, "5 x 5 i s 25,  And 5 x 1 i s 5, and 2 i s 7...so 75.  7, 5 i s . . . . 5 , 6, 7....7, 5, 6...756?"  And 15,...so 1, 5, 0  She also encountered t h i s  d i f f i c u l t y during a c a l c u l a t i o n o f a p a r t i a l product.  Her  initial  c a l c u l a t i v e steps i n solving 23 x 27 were "3 x 7 i s 21, carry the 2, 3 x 2 i s 6,..7, 8, and..86." This d i f f i c u l t y i n retaining the order may be related to her low backward d i g i t span o f 3. Subject U l l seemed to forget the  initially  even the order o f the d i g i t s i n  presented factors a f t e r she began to calculate.  After  attempting to solve 25 x 48 for 20 seconds, she forgot her completed calculations,  began again, and said,  "OK. 54 x 28 i s . . . "  The most  graphic demonstration o f changing the order and values i n a s e r i e s was her c a l c u l a t i o n o f 25 x 32 where she began to c a l c u l a t e 321 x 12!  196 Carrying subject  either  calculation.  errors. forgot This  There were a t l e a s t 51 instances to "carry"  error  where a  or added a wrong value during  was observed  37 and 14  u n s k i l l e d and s k i l l e d subjects, respectively.  times  for  a the  For some subjects such  as U8, mental carrying was a major source o f d i f f i c u l t y . She made a t l e a s t 7 carry errors during the CAL2 interviews. Some mental c a l c u l a t i o n errors occurred interchanged the carry and hold d i g i t s .  because a subject had  For example, as subject U10  attempted to solve 9 x 74, he reasoned, "9 x 4 i s 36, carry 3, put down the 6, and 9 x 7 i s 63.  Add 6 so 69.  So 696." Subject U l l  frequently added only a c a r r i e d 1. During the attempt o f 25 x 48, she said,  "8 x 5 i s 40, carry the 1....8 x 2 i s 16, 17...170..." Again,  these types o f errors were unique to mental c a l c u l a t i o n and were not evident i n the pencil-and-paper t e s t . Those calculation  subjects would  who  used  frequently  their  complete  fingers  the carry  to  operation  counting instead o f r e t r i e v i n g a basic addition f a c t . and  accompany  a  by skip  Subjects U l l  U12 often used t h i s process t o complete the carry c a l c u l a t i o n .  When asked t o c a l c u l a t e 8 x 25, U l l s a i d , "8 x 5 i s 40, carry 4, 8 x 2 i s 16,..17, 18, 19, 20, 21,..210."  She tapped her fingers i n unison  to t h i s skip counting process and, as can be seen from her proposed s o l u t i o n o f 210, she l o s t count o f t h i s additive process. Ul2's second attempt a t solving counting  process.  She said,  8 x 999 involved  "... Should be 72, 72, and 72.  (begins t o tap fingers) 73, 74, 75, 76, 77, 78, 79, 80, 8002."  t h i s same So  80 002, no  When she was asked l a t e r i n the interview to c a l c u l a t e the sum  o f 62 and 7, she i n s t a n t l y responded with the c o r r e c t solution.  Asked  197 why  she used  confused. revealed  her fingers  to count  units,  I have to work i t out sometimes."  "I j u s t get  Further  questioning  that she used a reconstructive rather than r e t r i e v a l process  to solve some basic f a c t s o f addition. said,  she said,  When asked t o solve 9 + 7 , she  "Oh, I didn't need to c a l c u l a t e because you j u s t go one step  lower."  She meant that 9 + 7 could be reconstructed by thinking "17 -  1 = 16." Misalignment digit-by-digit  during  addition  addition.  process  that  During was  the  right-to-left,  typically  used  by the  u n s k i l l e d subjects, "remembering where the numbers went" was a common complaint. This i n a b i l i t y to " a l i g n the columns" had many v a r i a t i o n s . The remembering  most  common  "to move  alignment  one place  difficulty over."  seemed  For example,  to  be not  subject  U2  c o r r e c t l y calculated each p a r t i a l product f o r the item 12 x 250 but gave  750 as a solution.  In her second  c a l c u l a t i o n attempt, she  r e a l i z e d that she had forgotten to "move over one" before adding: i n other words, she added 500 and 250, Other  subjects  had great  instead o f 500 and 2500. difficulties  i n renembering the  p o s i t i o n s o f the d i g i t s i n each addend. These d i f f i c u l t i e s resulted i n solutions that seemed t o r e f l e c t a process which could be described as "sliding  alignments."  For example, subject U9 c o r r e c t l y calculated  the p a r t i a l s o f 325 and 130 for the item 25 x 65. stage,  During the addition  he reasonded, "5, 2 + 3 i s 5, 3 + 3 i s 6, 1, so 1655."  The  diagram below i l l u s t r a t e s how the alignment o f each p a r t i a l had the appearance o f " s l i d i n g " as the c a l c u l a t i o n proceeded:  198  +  This  325 + 130 655  325 + 130  325 130 5  55  same subject appeared  rather than adding d i g i t s .  325 130 1655  +  t o calculate  one sum by annexing  He c o r r e c t l y calculated  240 and 96 (960) for the item 25 x 48 . s  the p a r t i a l s of  In h i s t h i r d attempt a t adding  96 and 240, he determined a solution o f 9260. The researcher drew a diagram and asked the subject i f the c a l c u l a t i o n had been completed i n the manner i l l u s t r a t e d below:  He exclaimed, "You know, I think you're r i g h t ! " This i n a b i l i t y to a l i g n the p o s i t i o n s o f the p a r t i a l s became an insurmountable  barrier  f o r some  subjects.  Ul  commented,  after  spending 78 seconds i n c a l c u l a t i n g 17 x 99, "I can remember 99 and 693 but  I can't get them back together.  I keep putting  the 99 i n the  wrong place." Subject Ul2's attempted s o l u t i o n o f 15 x 48 i s perhaps the best illustration instances  of  recalculation. calculation.  of  this  skip  alignment counting,  difficulty. finger  Her  "writing,"  solution  included  rehearsal,  and  She spent 104 seconds i n attempting t o complete the She said:  48 x 15 i s .... 48 x 15 is....40...I think my brain's done i n . . . . (recalculates). 48 x 15.. .40.. .20, 21, 22, 23, 24, ...240....0,...48 (begins adding) and 24,...16,...240, ..240, 240,..12,.8,...48, 40, 80 and 240 i s (adds again) 12,...24,...24 and 48, 24,..2, 6, 6. Uhh...24 and 48. I can't do t h i s . I know what i t i s but I can't do i t . ...24 and 48...Oh, i t ' s 700.  199  A Comparison of Mental and Written Computation The  analysis  of  the  types  of  forgetting  during  mental  c a l c u l a t i o n suggested that many important features of a c a l c u l a t i o n can be l o s t when a subject has no opportunity either to refresh the f a c t o r s or to record each completed c a l c u l a t i o n . performance  o f written  By comparing  the  and mental computation, an estimate o f the  degree o f forgetting due to the change i n presentation and response mode was determined. To ensure that a v a l i d comparison could be made, 10 items from CALl were a l s o used to create WPP, computational performance.  a t e s t o f written paper-and-penci 1  The timing requirements used for each t e s t  administration, though not i d e n t i c a l , d i d seem comparable. c a l c u l a t i o n t e s t had a 20-second item-presentation o f timing was not used i n WPP. complete  the written  The mental  rate. This method  Instead each subject was instructed to  t e s t q u i c k l y and accurately. The  longest  time  taken by a subject to complete WPP was 183 seconds and the t o t a l time for presenting the 10-item portion o f the mental c a l c u l a t i o n test was 200 seconds. Thus, a l l subjects were given a t l e a s t as much or more time to complete the mental c a l c u l a t i o n tasks as they had been given to  complete  the  written  calculation  tasks.  Any  difference  in  p r o f i c i e n c y should not be due to d i s s i m i l a r testing conditions. Table XI medium. WPP  An examination o f the table reveals that on the written test  there was  unskilled  includes a summary o f the r e s u l t s o f each t e s t i n g  l i t t l e difference i n performances between s k i l l e d  and  subjects. The difference between the group means was  not  statistically  significant  (t  2 g  = 0.48;  p > 0.20). Thus, the superior  mental c a l c u l a t i o n performance o f the s k i l l e d subjects c e r t a i n l y  200  TABLE XI  A COMPARISON OF WRITTEN AND MENTAL COMPUTATIONAL PERFORMANCE  Groups Skilled T e s t i n g medium  X  Unskilled  S  X  X  S X  Written (10 WPP items)  8.9  1.10  8.7  1.16  Mental (10 CALl items)  8.5  1.24  0.1  0.25  201  cannot  be due t o any superior  knowledge  o f conventional  written  computational techniques. Without the benefit o f the external memory store served by the written  page,  however,  the u n s k i l l e d  group's  performance  dropped  dramatically: 14 unskilled subjects d i d not solve any items mentally; one  subject  skilled  solved only one item mentally. On the other hand, the  groups'  performance  was  not s i g n i f i c a n t l y  reduced  by an  apparent change i n the testing medium. Alf  and  Abraham's  covariance  information  statistic  was  calculated to determine the magnitude o f the l i n e a r relationship that existed  between written  and mental m u l t i p l i c a t i o n  computation. The  combined groups' c o r r e l a t i o n r ' was determined t o be 0.15 and R, the c o r r e l a t i o n for the i n t a c t population, was estimated to be 0.07. Since the r e s u l t i n g  t  2  g  equalled  0.80  (p > 0.20),  a linear  relationship  between the written and mental c a l c u l a t i o n performance o f young adults does not e x i s t . Memory Capacity and Mental M u l t i p l i c a t i o n Performance Several measures o f STM capacity were used t o determine i f the two  groups  could  be distinguished  by d i f f e r i n g  STM c a p a c i t i e s . In  particular,  forward d i g i t span FDS, backward d i g i t span BDS, delayed  digit  DDS,  span  and l e t t e r  subject's memory capacity.  span  LS were  used  to estimate each  The r e s u l t s o f each group's performance on  each o f these four tests are presented i n Table XII. An examination o f Table XII shows that the s k i l l e d group had greater mean scores than the u n s k i l l e d group on every STM measure used i n the study. The difference between the two groups' mean scores was  202  statistically  significant  for each  capacity  measure. Thus, the  existence of a linear relationship between STM capacity and mental calculation performance seemed plausible. Scatterplots of CALl mental multiplication performance and each measure of STM capacity were used as an i n i t i a l test of linearity. Figures 8 through 11 contain these plots. An examination of each graph  indicates that a weak linear  relationship existed for each capacity measure used i n the study. As further suggestive evidence that STM capacity was not a major source of  individual differences in mental calculation performance was the  finding that HS had a forward and backward digit  span of 8 and 6,  respectively. The span estimates of most of the skilled subjects and a few of the unskilled subjects were greater than hers. The correlation between CALl performance and each measure of STM capacity was determined using the A l f and Abrahams covariance 1  information statistic. The values and statistical significance of each correlation R are presented in Table XII. Although each correlation was  statistically  significant,  the  relationship  between  mental  calculation performance and STM capacity seems to be weak: no STM measure accounted for more than about 11% of the variance in mental calculation performance.  203  TABLE XII MEASURES OF SIM CAPACITY: GROUP STATISTICS, SIGNIFICANCE TESTS FOR THE DIFFERENCES BETWEEN THE MEANS, THE CORRELATIONS AND THEIR SIGNIFICANCE BETWEEN CALl MENTAL MULTIPLICATION PERFORMANCE AND CAPACITY  Group S t a t i s t i c s Skilled STM Capacity Measures  Correlations  Unskilled  _ X  S  X  FDS  7.8  1.15  6.3  1.22  3.47  BDS  6.2  1.32  4.8  1.26  2.97  DDS  65.0  33.55  38.0  30.32  2.31  LS  83.1  33.54  52.5  25.67  2.81  *  p < 0.05  *** p < 0.001  S  28  fc  ** ** * **  R  48 ***  0.33  3.87  0.30  3.39  0.26  2.90  0.28  3.22  ** ** **  204  20 i-  16  < O  12  c o a> v_ o  8  o CO  l  I  6  7 Forward  Figure 8.  8 digit  span  S c a t t e r p l o t o f forward d i g i t span and score on CALl. #  Unskilled  A  Skilled  205  2 0  A A  r  • A  16k  < O  12  c o _  8  o o cn  i  i  4 5 Backward Figure 9.  i  i  6 digit  7 span  8  S c a t t e r p l o t o f backward d i g i t span and score on CALl. •  Unskilled  A  Skilled  206  2 0  1  i -  A —  —  •  16  < o  12  c o  8 O o  CO  i  5 0 Delayed Figure 10.  digit  10 0 span  S c a t t e r p l o t o f delayed d i g i t span and score on CALl. #  Unskilled  A  Skilled  15 0  207  Figure 11.  Scatter p l o t o f l e t t e r span and score on CALl. #  Unskilled  A  Skilled  208  CHAPTER V SUMMARY AND CONCLUSIONS Introduction E f f e c t : The second of two phenomena which always occurs together i n the same order. The f i r s t , c a l l e d a Cause, i s said to generate the other- which i s no more sensible than i t would be for one who has never seen a dog except i n the p u r s u i t o f a rabbit to declare the rabbit the cause of the dog. Ambrose Bierce, The Devil's Dictionary, 1911.  Comparative not  studies  such as t h i s present one can suggest but  conclude. Since the purpose  attempt was factor study  of  the study was  no  made to i s o l a t e and c o n t r o l variables which could be a  i n mental c a l c u l a t i o n performance. almost  background,  exploratory,  any  outstanding  feature  C l e a r l y , i n a comparative  of  a  subject's  character,  or knowledge can be interpreted as a possible source of  influence: a l l that i s needed i s a l i t t l e  imagination. For example,  the f a c t that Scripture (1891) reported that Colburn, an exceptionally p r o f i c i e n t mental c a l c u l a t o r , had  supernumerary  d i g i t s on each hand  and foot suggests that these bodily c h a r a c t e r i s t i c s were thought to be of  some  importance.  A  similar  extrapolation  was  made  by  the  phrenologist G a l l who examined the young Colburn and: ...without any previous intimation of h i s character, r e a d i l y discovered on the sides of the eyebrows c e r t a i n protuberances and p e c u l i a r i t i e s which indicated the presence of a f a c u l t y for computation (Scripture, p. 17). Consequently,  a t t r i b u t i n g group  differences  to any  variable  investigated  i n the study i s fraught with d i f f i c u l t i e s  and e n t a i l s  great  However,  be  risk.  the  results  of  the  study  can  used  to  209  articulate mental  and c l a r i f y  calculation  questions  f o r which  about  answers  individual can be  differences  sought  by  in  future  e m p i r i c a l l y based studies. Hopefully, researchers who plan to conduct investigations  of  mental  calculation  and  i t s close  relative,  estimation, can benefit from such a r t i c u l a t i o n and c l a r i f i c a t i o n .  Summary and Discussion o f the Findings The  intent  o f the study was to i d e n t i f y  procedures which characterized mental  calculation  the processes and  unskilled, s k i l l e d , and highly  performance during  the s o l u t i o n  skilled  of calculation  tasks involving m u l t i - d i g i t factors. The data c o l l e c t i o n and analyses of the study were guided by the following major research questions: 1. Can individuals who d i f f e r i n mental c a l c u l a t i o n performance be characterized  by the types of c a l c u l a t i v e strategies used to solve  a task? 2. Can individuals who d i f f e r i n mental c a l c u l a t i o n performance be  characterized  by the types of numerical equivalents retrieved to  solve a mental c a l c u l a t i o n task? 3. Can i n d i v i d u a l s who d i f f e r i n mental c a l c u l a t i o n performance be characterized  by the e f f i c i e n c y o f their short-term memory systems?  Questions About Mental Calculation Can unskilled  skilled  mental  .counterparts  strategies affirmative  used  calculators  by their  i n mental  Strategies be distinguished  choices o f c a l c u l a t i o n  computations?  The evidence  from  their  methods or suggests an  answer to t h i s question. The study demonstrated that the  s k i l l e d subjects possessed a large repertoire o f c a l c u l a t i v e plans but  210 the u n s k i l l e d subjects possessed few plans. The by  strategies. Which strategies were most frequently  applied  each o f the s k i l l groups? Although additive d i s t r i b u t i o n was the  favoured  calculative  tool  of  the s k i l l e d  subjects,  the  skilled  subjects knew and applied many o f the non-routine strategies known to have been used by expert c a l c u l a t o r s . These included fractional  distribution,  distribution,  general  subtractive  factoring,  the following:  distribution,  factoring  quadratic  by halfing-and-doubling,  factoring by a l i q u o t parts, exponential factoring. Some  calculations  were  determined  pencil-and-paper mental analogue but only  by  applying  r a r e l y would  the  the s k i l l e d  subjects attempt t o apply the r i g h t - t o - l e f t , d i g i t - b y - d i g i t procedure commonly  associated  with  written  computational  methods. When the  pencil-and-paper mental analogue was used, the s k i l l e d subjects would attempt to apply an abbreviated form o f t h i s method o f s o l u t i o n . Such an  abridgement took  the form o f r e t r i e v i n g rather  than  calculating  each p a r t i a l product. As  was the case f o r expert c a l c u l a t o r s , the s k i l l e d  subjects  tended to use a l e f t - t o - r i g h t sequence o f c a l c u l a t i o n s rather  than the  right-to-left  sequence  so  commonly  associated  computational methods. Progressive addition left-to-right  calculation  of  the p a r t i a l  with  written  always accompanied products.  The  this  skilled  subjects used such methods i n the b e l i e f that t h e i r memory load would be reduced. The using  skilled  an addition  subjects  tended  to complete  process  unlike  the conventional r i g h t - t o - l e f t ,  d i g i t - b y - d i g i t written  addition  algorithm. Rather  the c a l c u l a t i o n by  than  partitioning  211  each addend into a series of i n d i v i d u a l d i g i t s , the would arrange the determined. The  addends so that a convenient p a r t i a l sum  calculation  a  strategies  such  subtractive  method to complete the  the  suitably  would be  subtracting  from  s k i l l e d subjects  as  completed by  chosen number to subtractive  conventional written  the  either  p a r t i a l sum.  distribution calculation,  methods would  which  could  be  adding  or  For  required  a similar  be  those a  departure  undertaken by  the  s k i l l e d subjects. HS, who  the youngest and most highly s k i l l e d mental calculator  participated  in  f a c t o r a b i l i t y . Her number  combined  equivalents  the  study, possessed an  quick apprehension of with  useful  an  for  ability  the  to  multiplication  acute sense of  number  factors of a presented  retrieve  large  contributed  numerical  greatly  to  her  c a l c u l a t i v e powers. Unlike the majority of the other subjects i n study, she  never once used any  form of  the  the  pencil-and-paper mental  analogue to complete a c a l c u l a t i o n . HS rather are,  than  for  Hunter  and  the  other  individual  more s k i l l e d subjects worked with numbers  d i g i t s to a i d a mental c a l c u l a t i o n . Numbers  these p r o f i c i e n t subjects, r i c h i n association  remarked  that  proficient  calculators  such  as  and A.  meaning. C.  Aitken  apprehend a number "as a m u l t i p l i c i t y of numerical properties and, to speak, as b r i s t l i n g with s i g n a l l i n g properties"  (1962, p.  246).  so A  s i m i l a r remark could be made of the most p r o f i c i e n t subjects i n t h i s study and,  i n p a r t i c u l a r , the highly  properties"  appear  to  act  as  retrieval  c a l c u l a t i v e strategy being influenced the  subject.  In  a  sense, the  s k i l l e d subject HS. cues:  the  "Signalling  choice  of  a  by the properties apprehended by  proficient  subjects seemed to  view  212  calculation painting.  in  much  Both  "unskilled  the  same manner  individuals  eye."  Colour,  see  form,  as  an  artist  relationships and  space  might  view  unnoticed  cue  the  by  artist;  a the  number  properties cue the p r o f i c i e n t mental c a l c u l a t o r . The  most s k i l l e d calculators were able to orchestrate  o f strategies to solve a complex mental c a l c u l a t i o n s . HS's 123 x 456 described able  to  carry  activities solution  as  a an  oversimplication this  sense,  provides an  the  solution of  i n Chapter IV, for example, indicated that she  through a  in  a number  m u l t i p l i c i t y of  r e l a t i v e l y short application of  the  of  space  complex and of  general  inter-related  time. C l a s s i f y i n g  factoring  seems  a  this gross  complete mental c a l c u l a t i o n process and,  strategy  classification  scheme  was  used  in  the  in  study  inadequate description of these more complex c a l c u l a t i v e  schemes. M a i e r s contention that many young adults "are enslaved to 1  slow and awkward procedures learned be  i n school"  (1977, p. 92)  seems to  an accurate description of those u n s k i l l e d mental c a l c u l a t o r s  participated  i n the  study. In contrast  to the  the  who  d i v e r s i t y of methods  possessed by the s k i l l e d subjects, u n s k i l l e d mental c a l c u l a t o r s seemed tethered  to the r i g h t - t o - l e f t , d i g i t - b y - d i g i t , pencil-and-paper mental  analogue. Regardless of subject's approach was  the  factors  included  in a  task, an  unskilled  t y p i c a l l y the same, the fragmententation of  the  c a l c u l a t i o n i n t o a series of i n d i v i d u a l d i g i t s as a preparatory step for  the  strategy and  subsequent application of P&PO. Their  reluctance  the  right-to-left, digit-by-digit  to discard even the p h y s i c a l  s p a t i a l terminology associated  actions  with written methods r e f l e c t s  how  213  the  thinking  of  the  particular calculative  unskilled  subjects  was  dominated  by  this  strategy.  Generally speaking, the unskilled mental c a l c u l a t o r s worked not with numbers but with i n d i v i d u a l d i g i t s . Thus, the u n s k i l l e d subjects were content to attend to only the surface d e t a i l s of the task. Even seemingly obvious properties the  that might a i d the c a l c u l a t i o n such  i d e n t i t y p r i n c i p l e of m u l t i p l i c a t i o n were ignored usually by  majority of the u n s k i l l e d  as the  subjects.  During the additive phase of a c a l c u l a t i o n , the r i g h t - t o - l e f t , d i g i t - b y - d i g i t pattern continued as the u n s k i l l e d c a l c u l a t o r s  applied  the  process  algorithm used for written  addition  tasks. This addition  proved to be p a r t i c u l a r l y d i f f i c u l t since the subjects had to remember not  only  the  two  series  of  d i g i t s representing  the  value of  the  calculated p a r t i a l products but the "column p o s i t i o n s " as well. The  task  and  choice  between  the  types  of  selected  to  solve  those  calculators,  of  strategy.  calculative tasks?  If  tasks the  the answer must be stated  been discussed, d i f f e r e n t  Is and  there the  a  type  relationship of  question concerns  strategy unskilled  i n the negative since, as  items seemed to e l i c i t  the  same type  has of  c a l c u l a t i v e response. The  skilled  subjects were much more discriminating  choice of strategy.  prime  approach. The parts,  factors,  general  factoring  would  uses of subtractive d i s t r i b u t i o n ,  factoring  by  their  Certain general patterns of responses were noted.  I f the numbers i n the c a l c u l a t i v e task were replete small  in  halfing-and-doubling,  be  with r e l a t i v e l y the  most  factoring by  likely aliquot  fractional distribution,  exponential factoring, and the various types of quadratic d i s t r i b u t i o n  214  were  most  evident  f o r those  numbers were present.  tasks  where  particular  "signalling"  Factors close to a multiple of a power of 10  such as 999, f o r example, often acted as a " s i g n a l " for subtractive d i s t r i b u t i o n ; factors such as 25 often s i g n a l l e d f a c t o r i n g by a l i q u o t parts. Because additive d i s t r i b u t i o n can be used t o solve a wide range of  calculative  tasks,  this  strategy was much i n evidence.  However,  there were some subjects such as the highly s k i l l e d subject who used additive d i s t r i b u t i o n f o r only the more prosaic 1 - d i g i t factor tasks such as 9 x 742 or those numbers  had  few  pencil-and-paper  tasks such as 87 x 23 where the presented  factors.  The  skilled  subjects  would  use  the  mental analogue to solve a p a r t i c u l a r l y "easy" task  such as 9 x 74 or a task such as 73 x 83 that was described by one s k i l l e d subject as having "weird numbers." The orchestrate  ease  which  a composition  some d i f f i c u l t between  with  some  skilled  of several c a l c u l a t i v e  calculative  tasks  task.  Consequently,  were  plans  able  to  to complete  tended t o obscure the r e l a t i o n s h i p  the s e l e c t i o n o f a c a l c u l a t i v e  calculative  subjects  strategy  and  the type  of  the r e l a t i o n s h i p s outlined by the  researcher should be considered only a rule-of-thumb. Adapting from  one  calculative  to the task. Which s k i l l group most frequently changed  strategy task?  to  another  Although  i n response  a l l subjects  to  were  a  change  i n the  instructed  to use  whatever methods they thought were n a t u r a l l y suited t o the task, large differences existed between the a b i l i t i e s o f the s k i l l e d and u n s k i l l e d subjects to change the approach to correspond  with a change i n the  task. A moderately strong linear r e l a t i o n s h i p between the number of  215  calculation  strategies  calculative  tasks  a  subject  applied  and mental c a l c u l a t i o n  to solve performance  a  series  of  was found to  exist. The  p r o f i c i e n t mental calculators demonstrated  that  they not  only possessed a v a r i e t y o f c a l c u l a t i v e strategies but l i k e the good estimators studied by Reys, Rybolt, Bestgen, and Wyatt (1982) and the capable problem solvers studied by Krutetski (1969) they could r e a d i l y change from one strategy to another i f the task was thought t o warrant the  change.  The  highly  skilled  subject's  almost  habitualized  v e r i f i c a t i o n a c t i v i t i e s she demonstrated during the " f a c t o r a b i l i t y and square r e c a l l i n g " assessments are further testimony to the v a r i e t y o f approaches  which  some subjects can use to c a l c u l a t e mentally. She  often applied an i n i t i a l strategy, checked by applying an a l t e r n a t i v e strategy, and i n some cases used some number theoretic p r i n c i p l e such as a d i v i s i b i l i t y r u l e to provide a further check o f the c a l c u l a t i o n before attempting t o state the solution. The s e l e c t i o n o f a checking procedure often similar  varied  observations  from  about  task  t o task. Hunter  the v e r i f i c a t i o n  (1962) has made  activities  o f expert  mental c a l c u l a t o r s . Do these findings about the f l e x i b l e approaches towards mental multiplication  tasks demonstrated by s k i l l e d  subjects indicate  that  t h i s behaviour i s c h a r a c t e r i s t i c o f p r o f i c i e n c y on a l l types o f mental c a l c u l a t i o n tasks? For instance, do highly s k i l l e d mental c a l c u l a t o r s change  strategies  more frequently  than less  skilled  presented with a series o f mental sums? U n t i l provided,  any conclusions  about  subjects when  further  evidence i s  the adaptive behaviour o f s k i l l e d  c a l c u l a t o r s should be r e s t r i c t e d to the types o f mental m u l t i p l i c a t i o n  216  tasks used i n the present study. Not subjects  a l l skilled  subjects r e a d i l y  strongly favoured  switched  strategies. A few  additive d i s t r i b u t i o n ;  the very  subject S3 favoured the use o f the pencil-and-paper  skilled  mental analogue.  S3's mastery o f t h i s method was such that she was able to solve even the d i f f i c u l t  task 123 x 456. These exceptions  to the r u l e  that there i s l i k e l y no simple r e l a t i o n s h i p between adaptive and  suggest  behaviour  p r o f i c i e n t mental c a l c u l a t i o n . Perhaps, there are many types o f  c a l c u l a t i v e expertise. Hatano and Osawa (1983), f o r example, have argued that there are experts who are p r o f i c i e n t because o f the speed, accuracy, and automaticity  o f their  procedural  skill.  terminology,  S3 might  be described  Using  as a  Hatano  "routine  and Osawa s  expert."  1  Those  c a l c u l a t o r s who demonstrate expertise through f l e x i b l e planning and adaptation experts."  to new c a l c u l a t i v e tasks could be described as "adaptive The most  proficient  skilled  subject  SI and the highly  s k i l l e d subject HS would seem to be representative o f t h i s l a t t e r form of expertise. How t o d i s t i n g u i s h between these types o f experts i s not c l e a r . Despite the ambiguity i n c l a s s i f y i n g p r o f i c i e n t subjects as routine or adaptive, behaviour  researchers  interested i n investigating  expert cognitive  would be well-advised to consider that d i f f e r e n t tasks may  require d i f f e r i n g forms o f expertise. One calculative  cannot  complete  task without  the  discussion  of  adaptation  to a  making some reference t o the behaviour o f  u n s k i l l e d mental c a l c u l a t o r s . Unlike the adaptive stance taken by many skilled  subjects,  the unskilled  subjects exhibited behaviour  which  217  could  be  described  as  "calculative  monomania." Regardless  of  the  change i n the nature of the c a l c u l a t i v e task, most of the u n s k i l l e d subjects  would  attempt  to  take  the  same and  usually  unsuccessful  d i g i t - b y - d i g i t , r i g h t - t o - l e f t approach. Researchers disabilities  who  would  have  likely  subjects' c a l c u l a t i v e  be  studied  various  tempted  behaviour as  to  forms  describe  of the  learning unskilled  "perseverative." Woodward  (1981)  defines perseveration as the "tendency to continue an a c t i v i t y once i t has been started and to be unable to modify or stop the a c t i v i t y even though  i t i s acknowledged  Glennon  (1981)  to have become inappropriate"  describes  the  process  as  "the  (p.  inability  189).  of  the  i n d i v i d u a l to switch with ease from one stimulus s i t u a t i o n to another" (p. 63). Given the evidence does  seem  to  apply  to  provided by the present study,  the  behaviour  of  the  the term  unskilled  mental  calculators. The argued  to  inflexible  behaviour of  the  be  perseverative  tendency  not  a  unskilled at  subjects could a l l , but  simply  be a  Hobson's choice: the type of behaviour one would expect from subjects who  have a l i m i t e d knowledge of a v a r i e t y of c a l c u l a t i v e  Regardless  of how  techniques.  unsuccessful a subject's approach has proved to be,  i f the subject knows only t h i s approach, a s i n g u l a r i t y of approach i s not s u r p r i s i n g . Nevertheless,  there  were  some  aspects  of  the  unskilled  subjects' c a l c u l a t i v e thinking reported i n Chapter IV that seemed to perseverate.  UlO's r e v e l a t i o n that she  calculated a p a r t i a l  "with a l l O's on the top" to solve 50 x 64 and U13's  product  p e r s i s t e n t use of  d i g i t - b y - d i g i t c a l c u l a t i o n s to solve even easy items such as 20 x 30  218  are two instances o f a possible perseverative tendency. The finding  strongest  evidence  that some subjects  of  "rule  bound"  behaviour  was the  always made the same type o f error when  p a r t i c u l a r tasks were presented. For example, a few u n s k i l l e d subjects would substitute the value o f the calculated f i r s t p a r t i a l product for the value o f the second p a r t i a l product whenever a mental product o f a square was required.  Why  this  type o f error  would  mental but not i n written c a l c u l a t i o n i s a question  perseverate  in  that seems worthy  of further a n a l y s i s and investigation. Efficiency characteristics  of  a  calculative  be i d e n t i f i e d  that  strategy.  appear  some  others.  mental  Since  calculation  a  c a l c u l a t i o n strategies moderately  performance  digit-by-digit,  and  strong a  relationship  mental  efficient between  infrequent analogue  evidence  use  to using  further implicate the strategy as being  this  strategy  than mental  of the  existed,  strategy could be i n e f f i c i e n t . The f a c t that HS, the highly mental c a l c u l a t o r , had an aversion  general  suggestive  are more  subject's  pencil-and-paper  any  to d i s t i n g u i s h e f f i c i e n t  from i n e f f i c i e n t strategies? This study provided that  Can  this skilled  seems to  inefficient.  Why would the d i g i t - b y - d i g i t , pencil-and-paper mental analogue be  an  inefficient  mental  c a l c u l a t i o n strategy?  This  strategy has  proven to be p a r t i c u l a r l y useful for written purposes. Because o f the design  o f the study,  efficient  subjects  and e f f i c i e n t  techniques  cannot be disassociated from one another. Thus, one cannot properly conclude that a strategy used by a s k i l l e d subject i s necessarily more efficient studies  than a strategy used by an u n s k i l l e d subject. Only future  which  ensure  that  a l l subjects  use the same strategy can  219 answer t h i s question  about e f f i c i e n c y . However, the findings of the  study  seem to suggest  have  several  eliminate  of  that  these  e f f i c i e n t mental c a l c u l a t i o n strategies  distinguishing  characteristics:  the need f o r the carry operation;  (1)  they  (2) they proceed i n a  l e f t - t o - r i g h t manner; (3) they progressively incorporate each interim calculation into a single result. The  reader  following  should  discussion  characteristics  be  reminded  is  at  directed  of e f f i c i e n t  mental  this  towards  juncture  that  describing  and not written  the the  computational  techniques. 1. Eliminating  the carry operation.  the d i f f i c u l t y o f the "carry operation" & Gregg, 1966; Merkel & H a l l , many errors operation.  Research has demonstrated  (Hitch, 1977, 1978; Dansereau  1982). For example, Hitch found that  i n mental addition were due t o t h i s p a r t i c u l a r mental Moreover, he found  that  "subjects  are j u s t as prone to  forget the absence of carrying as they are to forget i t s presence" (p. 321). Hitch's  findings suggest that a carry operation  requires these  two working memory operations: storing and r e t r i e v i n g the carry d i g i t ; remembering whether carrying was necessary or not. I f the calculatve task involves only a few c a r r i e s , the a d d i t i o n a l burden imposed by a carry operation l i k e l y can be handled by working memory. But with some p a r t i c u l a r l y complex c a l c u l a t i o n s , the burden o f a carry can become excessive and thus c a l c u l a t i o n performance can s u f f e r . An  analysis  calculators The  of  the  strategies  by  proficient  mental  indicates that carrying was conspicuous by i t s absence.  elimination o f the carry operation  ways. F i r s t ,  used  the s k i l l e d  subjects  was accomplished  r a r e l y used  i n several  the d i g i t - b y - d i g i t ,  220  paper-and-pencil mental analogue, a strategy which can involve many carries  both  i n the m u l t i p l i c a t i v e and additive  stages  of the  c a l c u l a t i o n . The carry requirement o f P&PO seems to be one reason why the  skilled  subject  subjects  who never  avoided the use o f t h i s strategy. One s k i l l e d  used t h i s  strategy made t h i s point: he said, "I  don't use the pencil-and-paper method because there's too many c a r r i e s to remember." Additive  distribution,  the " c a l c u l a t i v e drafthorse"  of the  s k i l l e d c a l c u l a t o r s , may be more e f f i c i e n t than P&PO because there i s usually no need to remember a mental carry. Carrying can be eliminated by  arranging  the c a l c u l a t i o n so that  each  factor  has only  one  s i g n i f i c a n t d i g i t . To c a l c u l a t e the product o f 9 and 742, f o r example, the  factors become 9, 700, 40, and 2. No c a r r i e s are required to  calculate  the p a r t i a l  products 9 x 700, 9 x 40, and 9 x 2 . Those  subjects who choose to use the pencil-and-paper mental method, on the other hand, need to use the carry operation  twice.  A d d i t i v e d i s t r i b u t i o n does not eliminate carrying completely i n the additive phase but the need for t h i s operation w i l l be infrequent. Each p a r t i a l product, with the exception  of the product o f the u n i t  d i g i t s , w i l l be a multiple o f 10. Since each o f these p a r t i a l s include "zeroes"  i n the right-most  place value positions, most c a r r i e s can be  eliminated during the c a l c u l a t i o n o f the sum o f two p a r t i a l s . Using 9 x 742 as an example, the c a l c u l a t i o n s w i l l become 6300 + 360 and 6660 + 18. Neither of the other  o f these sums require the carry operation. An analysis strategies used by p r o f i c i e n t subjects reveals a s i m i l a r  absence of the c a r r y operation. Another way that " c a r r i e s " during a mental c a l c u l a t i o n can be  221  elirninated i s by r e t r i e v i n g rather than c a l c u l a t i n g a p a r t i a l product. For example, no carry operations were used to complete the c a l c u l a t i o n of  15 x 16 when one s k i l l e d subject reasoned, "80 and 16, move one  over, 160. And, 160 and 80 i s 240." 2. L e f t - t o - r i g h t c a l c u l a t i o n . Another ccmmon c h a r a c t e r i s t i c o f the  p r o f i c i e n t subjects i n the study was their tendency to complete  the c a l c u l a t i o n i n a l e f t - t o - r i g h t fashion. Most researchers Ball  (1956), Bidder  Mitchell  (1907),  (1856), Gardner  Scripture  (1977), Hunter  (1891), and Smith  including  (1962, 1979),  (1983) have  reported  s i m i l a r findings. The reasons for t h i s behaviour i s not c l e a r . Mitchell what  he c a l l e d  (1907) believed "convenience,"  that custom rather than e f f i c i e n c y , or sanctioned  the use o f l e f t - t o - r i g h t  procedures. He explained: ... i f , however, the calculator should a c c i d e n t a l l y form the habit o f beginning with the l a s t figure, i t i s hard to see where any r e a l inconvenience would r e s u l t . In mental as i n written arithmetic, much depends on custom and habit; i t i s hard t o see any great difference i n convenience beginning a t the r i g h t and beginning a t the l e f t , either i n mental or i n written m u l t i p l i c a t i o n , (p. 104) Thus, M i t c h e l l would accord no p a r t i c u l a r s i g n i f i c a n c e to t h i s feature of expert mental c a l c u l a t i o n . One i n t e r e s t i n g explanation proposed by Gardner (1977) explains why some expert c a l c u l a t o r s who earned a l i v i n g by performing mental calclulations  often preferred  a l e f t - t o - r i g h t method. The advantage  for a stage performer, according to Gardner, i s that: ...they can s t a r t c a l l i n g out a product while still c a l c u l a t i n g i t . This i s usually combined with other dodges to give the impression that computing time i s much l e s s than i t • r e a l l y i s . (p. 70) This clever deception may be true o f some professional  performers but  222  t h i s seems an u n l i k e l y explanation  f o r the l e f t - t o - r i g h t c a l c u l a t i v e  behaviour o f the subjects i n t h i s study. A c a r e f u l analysis o f l e f t - t o - r i g h t methods combined with the findings of several studies suggests that these methods are possibly l e s s demanding than r i g h t - t o - l e f t methods on short-term memory. There are several features about l e f t - t o - r i g h t c a l c u l a t i o n that may minimize the burden on short-term memory. First,  the l i k e l i h o o d o f forgetting the i n i t i a l  calculations  can be g r e a t l y lessened using l e f t - t o - r i g h t methods since portions o f the answer can be spoken before the e n t i r e c a l c u l a t i o n i s completed. The  skilled  answer  subjects  who used additive d i s t r i b u t i o n would state the  to 8 x 4211, f o r example,  "Thirty-two..., eighty-eight...  thirty-three  i n a series  thousand,  and...  o f stages  such as  s i x hundred  I t ' s thirty-three thousand s i x hundred  and...  eighty-eight."  This manner o f stating the answer may be e f f i c i e n t since Hitch (1978) demonstrated  that  computation w i l l utilized"  "interim  information  undergo rapid  produced  i n the course of  forgetting i f i t i s not immediately  (p. 306). On the other hand, i f r i g h t - t o - l e f t methods are  used, the answer cannot be stated  until  the e n t i r e c a l c u l a t i o n i s  completed. Furthermore, errors subjects  were made  Hitch's  that  i n mental  addition  fewer  i n the most s i g n i f i c a n t d i g i t s o f the sum when  proceeded  implications  finding  in  a  left-to-right  f o r studies o f estimation  manner  has  important  procedures. In the pursuit of  an estimate, e r r o r s i n c a l c u l a t i n g the value o f the most s i g n i f i c a n t d i g i t s are obviously  most undesirable.  Thus, i t can be hypothesised  that l e f t - t o - r i g h t methods w i l l improve the accuracy o f an estimate.  223 Further research i s needed to evaluate t h i s hypothesis. A second reason that l e f t - t o - r i g h t methods seem to be e f f i c i e n t i s that a transformation operation can be eliminated. Those subjects who favour  right-to-left  procedures  have an onerous task. They are  required t o r e t a i n not only a series of d i s c r e t e c a l c u l a t i o n s but, as an a d d i t i o n a l requirement,  the order o f the r e c a l l e d s e r i e s has to be  transformed  i n t o a l e f t - t o - r i g h t sequence more commensurate with the  Hindu-Arabic  numeration system. According to many u n s k i l l e d subjects,  transforming unskilled  the order  is a  subject expressed  somewhat  difficult  the d i f f i c u l t y  operation.  One  i n t h i s manner: "You're  going t h i s way and then you have to say i t that way." Since  left-to-right  calculation  seems  to correspond  c l o s e l y to place-value order, there i s no need f o r t h i s transformation.  more  "reversal"  The elimination o f any unnecessary mental operation  becomes e s p e c i a l l y important during p a r t i c u l a r l y lengthy c a l c u l a t i o n s . Further  research  i s needed  to determine  i f left-to-right  and  r i g h t - t o - l e f t methods o f mental c a l c u l a t i o n do impose d i f f e r e n t memory loads. 3. mental  Retaining  calculation  progressively  a single was  r e s u l t . Another  the tendency  feature o f p r o f i c i e n t  f o r subjects  to incorporate  the interim c a l c u l a t i o n s into a s i n g l e r e s u l t . In the  case o f d i s t r i b u t i o n , f o r example, the retention o f a single was accomplished  by c o n t i n u a l l y r e t r i e v i n g a sum, updating by adding a  newly c a l c u l a t e d p a r t i a l , factoring,  a  result  running  and storing  product  rather  t h i s new sum. In the case o f than  a  sum  was  modified  continually. What  seems  to be  the purpose  behind  this  technique?  The  224  proficient  subjects i n the study believed  that r e t a i n i n g a running  t o t a l was l e s s demanding on memory than the mathematically equivalent procedure o f computing a t o t a l i n the l a s t stages o f the c a l c u l a t i o n . "There i s too much to remember otherwise" said one s k i l l e d subject as he explained why he preferred t o keep a running sum. Those  who  have  studied expert mental  calculation  have made  s i m i l a r comments about the memory advantages o f t h i s technique. Bidder (1856)  explained  that  the  object  of  progressive  addition  in  d i s t r i b u t i o n was t o "have one fact and one f a c t only, stored away a t any  one time"  (p. 260). He believed  h i s rnemory demand could be  s i g n i f i c a n t l y reduced by focussing on only one r e s u l t . M i t c h e l l (1907) arguments were similar Bidder's: " . . . i t i s much easier to combine a t each separate stage, and r e l i e v e the memory o f the s t r a i n o f remembering the p a r t i a l r e s u l t s throughout the process" (p. 104). Since the unneeded calculations o f a p r o f i c i e n t calculator are  jettisoned  short-term  as  memory  so much  excess  i s supposedly  inemory  baggage,  lightened. Whether  technique does reduce the memory load as i t s proponents  the load or  on  not the  have claimed  w i l l have to be l e f t to the judgement o f an experimental study. Questions About R e t r i e v a l o f Numerical Equivalents Can i n d i v i d u a l s who d i f f e r i n mental c a l c u l a t i o n performance be characterized by the types of numerical equivalents they r e t r i e v e t o solve  a mental  skilled  mental  multiplication calculators  task? The evidence  suggests  had a more extensive l i b r a r y  numerical equivalents than the unskilled subjects.  that the o f useful  225 Basic ability  fact  recall.  to r e t r i e v e  multiplication? quickly  How  quickly  d i d the two groups and accurately  compare  the basic  on the  facts  of  The basic facts of m u l t i p l i c a t i o n were retrieved more  and mastered  u n s k i l l e d subjects.  to a higher degree by the s k i l l e d  than by the  The differences between the accuracy and speed of  r e c a l l o f each group were s t a t i s t i c a l l y s i g n i f i c a n t . However, neither difference  was great. Both s k i l l groups had attained  a considerably  high l e v e l o f mastery. A  weak  linear  relationship  between  mental  multiplication  performance and r e c a l l accuracy was found to e x i s t . A s i m i l a r l y weak relationship retrieve findings  between mental m u l t i p l i c a t i o n  the basic  facts  performance  of m u l t i p l i c a t i o n  also  and time to  existed.  indicate that the basic f a c t mastery o f young adults  an important factor contributing calculation  performance.  contributes  to  performance  to i n d i v i d u a l differences  Whether  individual  i n other samples  variation  differences  i n basic in  such as younger c h i l d r e n  i s not  i n mental  fact  mental  These  recall  calculation could be the  subject o f a future study. There were a few u n s k i l l e d subjects who were unable to retrieve a selected  s e t o f the basic  facts of m u l t i p l i c a t i o n .  Instead, these  p a r t i c u l a r f a c t s had t o be reconstructed either by a process o f skip counting or by applying which suggested that  reconstruction  c a l c u l a t i o n can contribute calculation. necessary  What  d i s t r i b u t i o n . There was some weak evidence  level  o f a basic  f a c t during a mental  to forgetting some important feature o f the of automaticity  o f basic  f o r completing mental c a l c u l a t i o n  another topic worthy o f future  investigations.  tasks  fact  recall i s  i s suggested as  226  R e c a l l o f large numerical equivalents. compare with respect in  a  mental  proposition  multiplication  d i d the s k i l l groups  to~ r e t r i e v i n g large numerical equivalents to a i d  calculation?  that  How  Was  there  any  evidence  p r o f i c i e n t calculators possess an  table?" The  to  support  the  "extended mental  s k i l l e d c a l c u l a t o r s were found to have a  more extensive "mental m u l t i p l i c a t i o n table" than the  10 x 10  basic  f a c t table possessed by the unskilled c a l c u l a t o r s . There were differences between the two groups i n the a b i l i t y to recall could  the  so-called  recall  "12's  table." Almost a l l the  these s p e c i f i c  types of  skilled  subjects  numerical information. On  the  other hand, almost none of these numerical facts could be r e c a l l e d by the u n s k i l l e d The  subjects.  extended  mental  multiplication  tables  c a l c u l a t o r s went only s l i g h t l y beyond the "12's  of  the  skilled  tables." The  majority  o f the s k i l l e d subjects could retrieve rather than c a l c u l a t e a number 2 2 2 2 of squares including 13 , 15 , 16 and 25 . This numerical information was  often  calculation  incorporated tasks  such  in as  subjects were able to solve  the  125  x  125.  In  of  contrast,  more  difficult  the  unskilled  a c a l c u l a t i v e task r a r e l y by  and accessing these large numerical HS,  calculation  "blocking"  equivalents.  the highly s k i l l e d subject,  had  by far the most extensive  mental m u l t i p l i c a t i o n table of those subjects who  p a r t i c i p a t e d i n the  study. She could r e t r i e v e rather than calculate the great majority of 2 - d i g i t squares and  a number of 3-digit and  4 - d i g i t squares as w e l l .  Many of her c a l c u l a t i o n s were solved either through a single r e t r i e v a l process or by arranging the c a l c u l a t i o n so that a r e t r i e v a l of a large numerical equivalent  would expedite the c a l c u l a t i o n .  227  Questions About Short-Term Memory Processes Can i n d i v i d u a l s who d i f f e r i n mental c a l c u l a t i o n performance be characterized  by the e f f i c i e n c y of their short-term memory systems?  The  suggests  evidence  that  the s k i l l e d  subjects  had greater  STM  c a p a c i t i e s , employed d i f f e r e n t memory devices, and were l e s s affected by a change from written  to mental methods of c a l c u l a t i o n than the  u n s k i l l e d subjects. The forgetting o f temporarily held information was found to be a major source o f error i n mental c a l c u l a t i o n . STM capacity.  Are there differences  among the s k i l l groups on  measures o f STM capacity? On each o f the four measures used to assess short-term memory processes, forward d i g i t span, backward d i g i t span, delayed  digit  span,  and l e t t e r  span,  there  was  a  statistically  s i g n i f i c a n t difference between the mean spans o f the two s k i l l groups. The STM capacity  o f the highly s k i l l e d mental c a l c u l a t o r HS was found  to be well within the normal range of memory span. A between measures capacity  statistically mental  significant  multiplication  o f STM capacity predominated  performance  was  as  a  but weak  found  linear  and each  to e x i s t .  predictor  of  relationship  of these  four  No one measure of  mental  multiplication  performance. Written  versus mental  calculation  e f f e c t on performance when mental rather used  to solve  statistically the  computational  non-significant  tasks?  performance.  What  i s the  than written methods must be There  difference  was  a  negligible  and  between the performances o f  s k i l l e d and u n s k i l l e d subjects on a written  multiplication test.  Thus, there seems t o be no relationship between the written and mental m u l t i p l i c a t i o n performance o f young adults.  228  A  differential  performance  was  dramatically  e f f e c t between mental and  noted.  while  the  The  unskilled  performance  group's  of  the  written  calculation  performance  skilled  r e l a t i v e l y undiminished by a change from a written  dropped  group  remained  to mental t e s t i n g  medium. Forgetting  of  information. What aspects of a c a l c u l a t i o n seem  l i k e l y to be forgotten during a mental calculation? An analysis of errors  made by  aspects  of  the  the  seemed to be  subjects  initially  in  stated  the  study  task  p a r t i c u l a r l y susceptible  c a l c u l a t i o n . The  following  types of  or  indicated the  that  interim  to forgetting forgetting  the  certain  calculations  during a mental  were i d e n t i f i e d :  (1)  forgetting the factors of the i n i t i a l task; (2) forgetting the numbers in  a  partially  completed  calculation;  d i r e c t i o n of the c a l c u l a t i o n ; (5)  forgetting  to  carry  for  carry e r r o r s ,  losing  track  of  the  (4) forgetting the order of the s e r i e s ;  or  "misalignment" during the  (3)  adding  additive  the  wrong  phase of  carry;  (6)  mental  the c a l c u l a t i o n . Except  these types of forgetting were not  evident during  written computation. Memory devices. Do memory devices forgetting evidence to  of  than the  s k i l l e d mental c a l c u l a t o r s employ d i f f e r e n t  unskilled  initial  suggest that  or the  mental interim  calculators  to  calculations?  minimize  There was  the some  s k i l l groups employed some cornmon, as  w e l l as d i f f e r e n t , memory devices to minimize the forgetting of some feature of s k i l l e d and  the  calculation. A  u n s k i l l e d subjects was  This memory device was stated  tactic  factors  and  used extensively  rehearsal,  by  both covert and  applied i n an attempt to remember the  the  numbers computed  at  both  various  the  overt.  initially  stages of  the  229 calculation. A v i s u o - s p a t i a l representation o f pencil-and-paper  computation  was another memory device used by most of the u n s k i l l e d c a l c u l a t o r s and  a few o f the s k i l l e d  c a l c u l a t o r s . Deciding  t o put the "larger  number on top and the smaller on the bottom" can be thought o f as a method o f helping a subject keep track o f a t l e a s t the i n i t i a l  stages  of a c a l c u l a t i o n . Another t a c t i c used by these subjects to help r e t a i n the order of a s e r i e s o f d i g i t s was imagining the s e r i e s of d i g i t s i n each p a r t i a l product as aligned i n "columns." The use o f fingers during a mental c a l c u l a t i o n was used by the majority o f u n s k i l l e d  mental c a l c u l a t o r s i n an apparent attempt to  enhance retention. This t a c t i l e a i d was used to write each factor and to record a l l stages of the c a l c u l a t i o n . One subject used her fingers to p o s i t i o n the d i g i t s i n each p a r t i a l product i n preparation for the final  additive phase. Whether  or not t h i s  tactile  behaviour  aids  retention can be determined only through future empirical studies. Because digit-by-digit, subjects, they  the  skilled  right-to-left  subjects  tended  not  strategy so favoured  to  use the  by the unskilled  employed a somewhat d i f f e r e n t set o f memory devices.  Chunking, or what one s k i l l e d subject described as "blocking," was a p a r t i c u l a r l y important device used by the s k i l l e d subjects. C e r t a i n l y , arranging a f a c t o r i n t o a s e r i e s of "blocks" as a preparatory step for the subsequent r e t r i e v a l o f a large numerical equivalent seemed t o be an e f f e c t i v e way t o reduce the amount of numerical d e t a i l . The  introspective  report  provided  by  one  skilled  subject  indicated that associations could be used t o foster retention o f the partial  products  during  a  difficult  calculation.  Usually  such  230  associations were based upon some number property apprehended by the subject. The data are too weak to conclude during  calculation  calculators. used,  and  i s a prevalent practice  However, Aitken claimed deeply  irrelevant  distrust.  associations a  (Smith, 1983,  that the use of mnemonics  They  mental  that, "iMnemonics I have never  merely  faculty  among p r o f i c i e n t  perturb  that should  be  with  alien  pure and  and  limpid"  p. 61). Further investigation of the memory devices of  mental c a l c u l a t o r s seems to be necessary before any firm conclusions about either t h e i r popularity or effectiveness can be Conclusions about STM the r o l e of the supposedly store  i n mental  a  short-term  process  about  performance?  The  types  of  forgetting  IV d i d seem to be consistent with the notion of  memory system  temporary  processes. What can be concluded  l i m i t e d capacity of the short-term memory  calculation  documented i n Chapter  reached.  with  a  information.  limited  Many  capacity  subjects  to  store  forgot  and  interim  c a l c u l a t i o n s , l o s t track of the c a l c u l a t i v e d i r e c t i o n , and could not remember the order of a series of stored d i g i t s . unskilled  subjects' computational  performance  The f i n d i n g that the dropped dramatically  when the external rremory-store served by the written page was  removed  seems to be a d d i t i o n a l support for the notion of a limited-capacity memory store. On  e  the other hand, the finding that only a weak r e l a t i o n s h i p  existed between mental c a l c u l a t i o n performance and STM capacity seems to  weaken  the  argument  for  the  short-term memory system supposedly  important  c o n t r i b u t i o n that  the  makes to t h i s type of reasoning  process. Several explanations for t h i s apparent discrepant finding can be presented.  231  On  the  t e c h n i c a l side, the low c o r r e l a t i o n s reported i n t h i s  study could have been attenuated by the somewhat r e s t r i c t e d range of the STM capacity scores determined for the subjects of the study. For example, i n the case of forward d i g i t span, the span estimate for a l l subjects including the highly s k i l l e d subject only varied from 5 to 9: well  within  the  expected  range  for normal subjects  (Miller,  1956;  Dempster, 1981). Perhaps, the r e l a t i o n s h i p between capacity and mental c a l c u l a t i o n performance becomes evident only when a greater range of span estimates  are considered:  those subjects with exceptionally low  span estimates  being more l i k e l y to be poor mental c a l c u l a t o r s than  those with p a r t i c u l a r l y high estimates. However, s t a t i s t i c a l one little  i n STM  memory  store  this  lack of  a  r e l a t i o n s h i p could  be  more than  produced s o l e l y by studying subjects who  differ  a  very  e f f i c i e n c y . Possibly the notion of a limited-capacity is  conceptually  inadequate  to  explain  individual  differences i n mental c a l c u l a t i o n performance. In strategy  the  present  were  left  study,  the  unconstrained.  subjects' choices A l l subjects  of  calculative  were l e f t  free  to  s e l e c t and use whatever mental methods that seemed suited to the task at  hand. Unconstrained  most experimental  choice of strategy has not been a hallmark  of  studies of mental c a l c u l a t i o n . -In some studies the  c o n t r o l over a subject's use of strategies has been accomplished  by  presenting tasks which can be attacked i n only very few ways. Control over  use  of  s t r a t e g i e s has  a l s o been exercised by  instructing  the  subjects to solve a task i n a p a r t i c u l a r manner. Jarman (1978, 1980)  has designated  those tasks where a v a r i e t y  of s o l u t i o n methods are possible as heterogenous. Those tasks which  232  are  structured  so  that very  little,  i f any,  s e l e c t i o n i s possible, he has designated  variation in  strategy  as homogeneous. Furthermore,  Hunt (1980) has argued that strong relationships between STM  capacity  and complex cognitive processes are evident only when homogenous tasks are used i n a research capacity  and  the  study. Thus, the strong  more homogeneous tasks  r e l a t i o n s h i p between  used by  most experimental  studies to measure mental c a l c u l a t i o n performance l i k e l y e x i s t s when there i s only a l i m i t e d opportunity  for a subject to employ a v a r i e t y  of strategies. The  tasks used i n t h i s study could be considered  heterogeneous  because they seemed to foster rather than i n h i b i t s t r a t e g i c v a r i a t i o n . Since,  as  Hitch  calculative  (1977,  1978)  s t r a t e g i e s can  vary  and  Hunter  i n terms of  (1978)  have  argued,  information-processing  requirements, the resourceful person with an average span can always f i n d a way of  to ease the burden of memory when presented with a series  heterogeneous tasks.  mental c a l c u l a t i o n . He mental  be  overcome. He  complicated,  (1856) made s i m i l a r arguments about  argued that only through the clever use of a  c a l c u l a t i o n strategy  calculations prolix,  Bidder  and  can  the  memory  explained,  inexpeditious,  "the  demands  of  difficult  process might  although  it  is  appear  actually  arranged with a view of affording r e l i e f to the memory" (p. 254). Subjects require  large  such as the highly s k i l l e d subject HS s t r u c t u r a l resources  c a l c u l a t i v e tasks. How in  recalling  a  calculation? The  large  to solve most types of  much short-term numerical  simply  memory processing  equivalent  memory demand must be  to  solve  do  not  everyday  i s involved a  mental  surely minimal. Through  the  judicious s e l e c t i o n of a c a l c u l a t i v e strategy, HS can get by with l e s s  233  s t r u c t u r a l resources than the s e l e c t i o n of more i n e f f i c i e n t strategies would necessitate. Like f r u g a l people who  learn not  to s t r a i n  their  l i m i t e d f i n a n c i a l resources, p r o f i c i e n t c a l c u l a t o r s l e a r n to r e l y on a clever choice o f strategy designed  to place no great s t r a i n on t h e i r  temporary memory resources. The  simultaneous-successive  information  processing  model  of  cognition described by Das, Kirby, and Jarman (1979) provides another perspective underlie  from  mental  which  to  view  calculation.  the  In  cognitive processes  brief,  information i s l i k e l y to be processed synthesis  r e f e r s to  (Jarman, 1980, retained  in  processing  the  model  which  proposes  may that  i n one of two ways. Successive  i n sequential  temporal-based  forms  p. 76). Any task where information must be arranged a  serial  order  requires  successive  and  processing.  Simultaneous integration r e f e r s to the synthesis of separate elements into  groups;  these  groups often  taking on  spatial  Kirby & Jarman, p. 49). Simultaneous processes  overtones  (Das,  are involved i n tasks  where the discernment of relationships and patterns are required. This  study  calculative  demonstrated  s t r a t e g i e s which  relationships  and  c a l c u l a t i o n . But  to  that  the  required  incorporate  the instruments  skilled  them this  to  subjects apprehend  information  used i n t h i s study  employed numerical into  to estimate  the STM  capacity involve mainly successive information processing. Thus, the lew  c o r r e l a t i o n s between mental c a l c u l a t i o n  capacity  reported  successive should  be  in  processing. included  this  study  may  apply  Perhaps measures of in  any  future  p r o f i c i e n c y and only  measures  simultaneous  investigations  differences i n mental c a l c u l a t i o n performance.  to  of  memory of  processing individual  234  Furthermore,  i t can be hypothesised that a r e l a t i o n s h i p between  simultaneous-successive processing and mental c a l c u l a t i o n performance may  vary  across  pencil-and-paper  calculative  mental  strategies.  analogue  and  Performance  additive  on  distribution  the would  l i k e l y require more successive than simultaneous processing since the user must be more concerned with remembering the order and l o c a t i o n of calculative  information  than  with  determining  useful  number  properties. Those i n d i v i d u a l s demonstrating high successive processing s k i l l s as estimated by measures such as forward d i g i t span and v i s u a l short-term memory would be expected to perform better with these types of  c a l c u l a t i v e s t r a t e g i e s than those i n d i v i d u a l s with low successive  processing s k i l l s . Conversely, processing  skills  those  i n d i v i d u a l s demonstrating  high  simultaneous  as estimated by measures such as Raven's Coloured  Progressive Matrices (Das, Kirby & Jarman, p. 52) would be expected to perform better with factoring strategies than those i n d i v i d u a l s with low  simultaneous  simultaneous  processing  processing  and  skills.  Weak  performance  relationships  with  the  between  pencil-and-paper  mental analogue and between successive processing and performance with factoring would provide further evidence that simultaneous-successive processing  requirements  vary  systematically  across  calculative  strategies. Further  research  into  the  relationship  between  cognitive  processes and performance on other types of mental c a l c u l a t i o n tasks is  also  needed.  An  operations suggests well  to  the  analysis  of  mental  addition  and  that neither operation lends i t s e l f  "shortcuts"  so  evident  in  this  study  subtraction particularly of  mental  235  multiplication.  Consequently,  these addition  and  subtraction  tasks  could be l e s s heterogenous than mental m u l t i p l i c a t i o n tasks. Attaining p r o f i c i e n c y i n mental addition, i t can be hypothesised, would require different  processing  skills  than a  similar  level  of attainment i n  mental m u l t i p l i c a t i o n performance. The recent studies of expert abacus operators (Hatano & Osawa, 1983; Hatano, Miyake & Binks, 1977) provide some i n d i r e c t support for t h i s hypothesis. These experts can c a l c u l a t e a mental sum of f i f t e e n 5-digit  to 9 - d i g i t addends presented o r a l l y . However, unlike expert  mental c a l c u l a t o r s , they d i d not appear to use any novel strategies or "short-cuts" Instead,  to  accomplish  according  to  their  the  calculative  experts'  verbal  feats  of  reports,  addition. sums  were  determined by c a l c u l a t i n g on a form of a v i s u a l i z e d "mental abacus." Not s u r p r i s i n g l y , the forward and backward d i g i t spans of these subjects were found contrast  to average  about  15  to the abnormally large d i g i t  and  14,  respectively.  spans, the l e t t e r  In  spans of  these i n d i v i d u a l s were about average. Hatano and Osawa concluded that the  enlarged d i g i t spans were developed i n response to the s p e c i f i c  information-processing requirements of the mental c a l c u l a t i o n that  these  experts  aspired  to  master.  A  similar  skills  reciprocal  r e l a t i o n s h i p between other non abacus-derived mental addition  tasks  and d i g i t span could be speculated to e x i s t . The f a c t that a subject had access to innumerable memory aids not  available  could  be  during  another  the assessments  reason why  STM  of STM  capacity was  processing found  efficiency  to be a poor  predictor of mental m u l t i p l i c a t i o n performance. There i s no possible way that a subject can re-create the e n t i r e series from a few r e c a l l e d  236  d i g i t s i n a conventional measure of d i g i t span because the s e r i e s have been designed the  case of  to discourage  these reconstructive process. However, i n  mental c a l c u l a t i o n ,  i f a subject  forgets a  s e r i e s of  stored c a l c u l a t i o n s , the luxury of r e - c a l c u l a t i o n e x i s t s for as long as the subject r e t a i n s the i n i t i a l factors. Thus, d i g i t span i s l i k e l y not an accurate  p a r a l l e l of the fundamental memory processes  needed  during a mental c a l c u l a t i o n . Mental  calculation  can  also  provide  the  subject  with  more  opportunities f o r chunking than provided by conventional rnemory tests. The  skilled  subject  S7,  whose  use  of  associative techniques  was  reported i n Chapter IV, could have g r e a t l y increased the retentiveness of  interim c a l c u l a t i o n s by  these  associations. The  information-processing  r e l y i n g on h i s long-term memory to make  use  of  long-term  memory  to  increase  the  e f f i c i e n c y of a subject's c a l c u l a t i o n s cannot  be predicted by the crude STM measures used i n t h i s study. In conclusion, mental c a l c u l a t i o n may  involve processes  similar  to those which underlie conventional memory t e s t s . But there i s a r e a l p o s s i b i l i t y that subtle, and Unfortunately,  in  l i k e l y important,  understanding  the  d i f f e r e n c e s do  everyday  uses  of  exist. mental  c a l c u l a t i o n , Cole, Hood and McDermott's conclusion that "the a n a l y t i c apparatus we bring to these psychology does not apply"  (everyday) environments from  experimental  (1982, p. 373), could be c o r r e c t .  237  Implications for  Instruction  Researchers such as Case (1975) and Shulman (1976) have argued that  novice-expert  t o new  insights  regarding mathematics teaching. This study which compared  unskilled  and  skilled  comparisons  mental  can often  calculation  lead  performance  has  implications  for  developing i n s t r u c t i o n a l programs designed t o improve the performance of lesser s k i l l e d i n d i v i d u a l s . Likely performance  the simplest would  be  route  f o r teachers  to improved  mental  to provide  opportunities  meaningful p r a c t i c e . A l l subjects who participated convinced that  calculation  i n the study were  the topic o f mental c a l c u l a t i o n was never taught i n  school. Even the most p r o f i c i e n t subjects i n the study believed for a l l p r a c t i c a l purposes, they were self-taught working were o f t h e i r own The  for  results  of  that,  and t h e i r rules of  invention. the  screening  phase  of  the  study  have  demonstrated that very few individuals reach even a moderate l e v e l of p r o f i c i e n c y i f mental c a l c u l a t i o n i s l e f t t o develop on i t s own. This seemingly obvious f a c t appears to have been overlooked by teachers i n their The  zealous pursuit  f a c t that  o f student competence i n written  computation.  so few teachers expect students to c a l c u l a t e  mentally  seems to explain why so many unskilled subjects could see no purpose to mental c a l c u l a t i o n .  "Why bother," said one subject,  "when I can  always use my c a l c u l a t o r . " I f any mathematics educators believe mental c a l c u l a t i o n program,  mental  should  form an i n t e g r a l part  calculation  activities  should  before the development of such negative attitudes.  that  o f the mathematics be  introduced  well  238  Since the skilled and unskilled mental calculators could be characterized by their different choices of calculative strategies, a clear implication for improving the performance of young adults would be to teach the strategies of the skilled to the unskilled. Further research  i s needed  to determine which calculative  strategies  are  "teachable." However, some educated guesses can be made. As has been discussed i n the study, left-to-right additive distribution  appears  digit-by-digit,  to be  a  more efficient  pencil-and-paper mental  strategy  analogue.  Even  than the the most  unskilled subjects should be able to master this strategy. There are several reasons to warrant this optimistic conclusion. First,  the number of  "mental  steps" needed  to complete a  calculation using distribution does seem to be less than the number needed  to  complete  the  digit-by-digit,  pencil-and-paper mental  analogue. Consequently, there seems to be no reason to suspect that additive  distribution involves more processing than the method so  favoured by the unskilled subjects. The  observation that some unskilled  pencil-and-paper  mental  analogue  and,  subjects abandoned the  i n an  apparent  burst  of  inspiration, used an incomplete form of distribution i s another reason for optimism. Perhaps with proper instruction and sufficient practice even the most unskilled mental calculators should be able to master the  complete  form of distribution to solve some types of mental  multiplications. How well the unskilled subjects can learn the other seemingly more complex strategies used by the skilled subjects remains to be seen. The fact that seme unskilled subjects began to apply factoring  239  by a l i q u o t parts during interviews  provides  the l a t t e r stages of these r e l a t i v e l y  suggestive  evidence that  brief  a reasonable l e v e l  of  p r o f i c i e n c y i s possible. The f i n d i n g that an exceptionally large STM capacity seemed not to be necessary f o r p r o f i c i e n t mental m u l t i p l i c a t i o n performance seems to be  more good news for those mathematics educators attempting  to  improve the mental c a l c u l a t i o n s k i l l s of young adults. The s e l e c t i o n of an e f f i c i e n t strategy seems to be a much more c r i t i c a l factor than memory  span  in  determining  the  level  of  mental  multiplication  performance. The  f i n d i n g that forgetting seemed to occur when basic f a c t s  were reconstructed rather than i n s t a n t l y r e c a l l e d has implications for i n s t r u c t i n g those subjects  such as younger c h i l d r e n who  do not have  complete mastery of the basic f a c t s . A great deal of reconstruction i s often necessary i n the e a r l y stages of basic f a c t mastery (Thornton, 1978). However, t h i s reconstruction could ability  to  calculation amongst  calculate  mentally.  performance  younger  and  children  The  basic  relationship  fact  should  be  factor  in  seriously l i m i t a child's  recall  the  between  speed  subject  of  mental  and  accuracy  an  empirical  investigation. Another  important  developing  mental  calculation  s k i l l s could be providing young children with ample opportunities explore  number  patterns.  Bidder,  for  example,  believed  that  to his  incredible powers had very humble o r i g i n s . As a young c h i l d , he would arrange concrete  materials such as peas and marbles i n t o rectangular  arrays.  the  Through  continued  exploration  of  the  number  patterns  suggested by these arrangements, he gradually acquired a repertoire of  240  strategies. Moreover, he reported he could r e c a l l many basic f a c t s of m u l t i p l i c a t i o n w e l l before he knew the word "multiply" p.  (Bidder,  1856,  258). He noticed that larger "sums" could be calculated by a clever  re-arrangement of a larger array into smaller s i z e arrays. Hence, the discovery  of  additive  explorations provided  d i s t r i b u t i o n followed  from  these  intuitive  by structured materials. His proposal that such  i n t u i t i v e notions be w e l l developed before "ciphering" i s taught seems very modern. The an  highly s k i l l e d subject's s k i l l s had developed l a r g e l y from  early  fascination  "practising  and  with  playing  with  talents were discovered child  was  about  discovery was  10  numbers  numbers."  quite by  years  manifested  old.  The  accident The  by  the  subject's  by  the  mother's  continual calculative  parents when  the  r e c o l l e c t i o n of  the  as follows:  We were i n the car one night asking ordinary times tables to some of the smaller c h i l d r e n . And she said she wanted to take part i n the game but wanted something harder. Only we kept making them harder and harder and she was coming back quicker and quicker than you can do on a c a l c u l a t o r . Although highly  this  study demonstrated that  skilled  daughter was A  now  mental  calculator,  the  the  young c h i l d  mother  "much slower and l e s s accurate  was  commented  still that  a her  too."  keen i n t e r e s t i n the properties of numbers seemed to be  common experience of many s k i l l e d subjects as w e l l . Those subjects  a who  could r e c a l l t h e i r e a r l y days i n elementary school commented that they l i k e d to "explore the m u l t i p l i c a t i o n tables for i n t e r e s t i n g patterns." These  intuitive  relationship  that  explorations the  child  occasionally would  later  would  develop  reveal into  a  some mental  241  calculation  technique.  Thus, teachers would do w e l l to use a c t i v i t i e s which encourage c h i l d r e n to explore number patterns and r e l a t i o n s h i p s . Developing understanding  of  number  relationships and  properties  e s t a b l i s h a stable network of associated and  could  an help  r e l a t e d concepts which  support each other. These inter-connected ideas and r i c h associations could  form  a  durable  base  to  which  new  but  related  numerical  information can be f i t t e d . Concluding Remarks Mathematics has hurt one it."  i f one  Perhaps,  c a l c u l a t i o n . The  been said to "contain much that w i l l neither  does not know i t nor help one this  epigram  is  an  apt  i f one  does not know  description  trend to create devices designed  of  mental  to replace most of  one's c a l c u l a t i v e e f f o r t s w i l l no doubt continue and l i k e l y there w i l l be  little  incentive  c a l c u l a t i o n . Why memorizing  to  acquire  considerable  expertise  in  mental  should one bother learning c a l c u l a t i v e strategies and  numerical  equivalents  when  this  information  can  be  purchased for a few d o l l a r s ? This study has demonstrated that p r o f i c i e n t mental c a l c u l a t i o n i s much more than a computational modern meaning  technology. and  tool which can be replaced e a s i l y by  P r o f i c i e n c y demands  understanding  and  any  that  process  the  user  search  for  which depends upon  the  integration of seemingly disconnected mathematical concepts and r u l e s is  well  worth  r e t a i n i n g as  a  goal  of  mathematics  education.  The  benefits of mental c a l c u l a t i o n , therefore, go beyond the a b i l i t y make quick c a l c u l a t i o n s . Max  Beberman made t h i s point over 25  to  years  242  ago: Mental arithmetic... i s one of the best ways of helping c h i l d r e n become independent o f techniques which are usually learned by s t r i c t memorization....Moreover, mental arithmetic encourages c h i l d r e n to discover computational short cuts and thus to gain deeper insight into the number system. (Sr. Josephina, 1960, p. 199) The u n s k i l l e d subjects' d i f f i c u l t i e s were not due to a lack o f a computational t o o l : rather, a lack o f awareness that d i f f e r e n t tasks called  for different  tools  appeared  to be at the root  of  their  problems. They attempted to solve c a l c u l a t i v e tasks much i n the same manner as an inexpert handyman might use a wrench instead of a hammer to drive a f i n i s h i n g n a i l . Though both tools can do the job, one seems g r e a t l y unsuited t o the task a t hand. HS's  thoughts  particularly  on  the purposes  of mental  calculation  seem  i n s i g h t f u l and w e l l worth reporting. When asked i f she  had ever been given a memory test, she remarked, "I'm not very good a t memorizing."  After  the testing  had been  completed,  this  researcher  commented that some people might regard her a b i l i t y t o r e c a l l squares and to c a l c u l a t e mentally as an exercise which required a great deal of  memory. She exclaimed, "But that's not memorizing. That's knowing  and thinking." To  close  demonstrated  on  that  a  somewhat  proficient  museful mental  note,  this  calculation  study involves  has a  sophisticated form of thinking that hopefully w i l l not be supplanted by  advances  calculation  i n calculator  technology. The destination of a mental  i s a correct answer but the path i s understanding. The  good mental c a l c u l a t o r i s free t o choose many paths; the poor mental c a l c u l a t o r can choose few paths. How to open more paths to more people  243 i s a question whose answer i s l e f t to future studies.  244 REFERENCES  Alf,  E.F., J r . and Abrahams, N.M. The use of extreme groups i n assessing r e l a t i o n s h i p s . Psychometrika, 1975, 40(4), 563-572.  Abrahams, N.M. and A l f , E.F. Relative costs and s t a t i s t i c a l power i n the extreme groups approach. Psychometrika, 1978, 43(1), 11-17. A l l e n , M.J., and Yen, W.M. Introduction to Measurement Monterey, C a l i f o r n i a : Brooks/Cole, 1 9 7 9 .  Theory.  Allport, D.A. Patterns and actions: Cognitive mechanisms are content-specific. In G. Claxton (ed.), Cognitive Psychology. London: Routledge & Kegan Paul, 1980a, 26-64. A l l p o r t , D.A. Attention Cognitive Psychology. 112-153.  and performance. In G. Claxton London: Routledge & Kegan Paul,  Anderson, J.R. Cognitive Psychology and Francisco: W.H. Freeman & Company, 1980.  (ed.), 1980b,  I t s Implications.  San  Atkinson, R.C. and S h i f f r i n , R.M. Human memory: A proposed system and i t s c o n t r o l processes. In K.W. Spence and J.T. Spence (eds.), The Psychology o f Learning and Motivation: Advances i n Research on Memory (Vol. 2). New York: Academic Press, 1968. Atkinson, R.C. and S h i f f r i n , R.M. The control of short-term memory. S c i e n t i f i c American, 1971, Aug., 225(2), 82-90. Atkinson, R.L., Atkinson, R.C, and Hilgard, E.R. Introduction to Psychology, (8th ed.). Toronto, Ont.: Harcourt Brace Jovanovich Ltd., 1983. Baddeley, A.D. and Hitch, G. Working memory. In G.H. Bower (ed.), The Psychology o f Learning and Motivation, ( v o l . 8). New York: Academic Press, 1974, 47-89. Baddeley, A.D. The concept o f working memory: A view of i t s current state and probable future development. Cognition, 1981, 10, 17-23. B a l l , W.W.R. C a l c u l a t i n g prodigies. In J.R. Newmann (ed.), The World o f Mathematics, (vol. 1) New York: Simon & Schuster, 1956, 467-487. Begle, E.G. and Gibb, E.G. Why do research? In R.J. Shumway (ed.) Research i n Mathematics Education. Reston, V i r g i n i a : NCTM, 1980, 3-19. Bidder, G.P. On Mental Calculation. Minutes o f Proceedings, I n s t i t u t i o n o f C i v i l Engineers (vol. 15). London, 1856, 251-280.  245  Bierce, A. The D e v i l ' s Dictionary. New 1911. Broadbent, Press,  D.E. 1958.  Perception  and  York: The World Publishing Co.,  Communication.  New  York:  Pergamon  Brockmann, E.M. Memory search processes used by second-grade c h i l d r e n i n the comprehension of place value (Doctoral d i s s e r t a t i o n , Fordham U n i v e r s i t y , 1978). D i s s e r t a t i o n Abstracts International, 1978, 39, 3, p. 1395A. (University Microfilms No. 781656) Brown, G. An area of neglect i n the study of arithmetic-mental arithmetic. Mathematics Teacher, 1957, 50, 166-169. Brown, H.L. and Kirsner, K.A. A within-subjects analysis of the r e l a t i o n s h i p between memory span and processing rate i n short-term memory. Cognitive Psychology, 1980, 12, 177-187. Carpenter, T., Coburn, T.G., Reys, R.E. and Wilson, J.W. Results from the F i r s t Mathematics Assessment of the National Assessment of Educational Progress. Reston, V i r g i n i a : NCTM, 1978. Carpenter, T.P., C o r b i t t , M.K., Kepner, H.S., Lindquist, M.M., and Reys, R.E. National Assessment. In E. Fennema (ed.) Mathematics Education Research: Implications for the 80's. Alexandria, V i r g i n i a : Association for Supervision and Curriculum Development, 1981, 22-40. Case, R. V a l i d a t i o n of a neo-Piagetian mental capacity construct. Journal o f Experimental C h i l d Psychology, 1972, 14, 287-302. Case, R. Mental s t r a t e g i e s , mental capacity, neo-Piagetian i n v e s t i g a t i o n . Journal of Psychology, 1974a, 18, 382-397.  and i n s t r u c t i o n : A Experimental Child  Case, R. Structures and s t r i c t u r e s : Some functional l i m i t a t i o n s on the course of cognitive growth. Cognitive Psychology, 1974b, 6, 544-573. Case, R. Gearing the demands of i n s t r u c t i o n to the developmental c a p a c i t i e s of the learner. Review of Educational Research, 1975, 45(1), 59-87. Case, R. A developmentally based theory and technology of i n s t r u c t i o n . Review of Educational Research, 1978, 48(3), 439-463. Case, R. and Globerson, T. F i e l d independence and space. C h i l d Development, 1974, 45, 772-778. Chase, W.G. and Simon, H.A. 1973, 4, 55-81.  c e n t r a l computing  Perception i n chess. Cognitive Psychology,  246  Claxton, G. Cognitive psychology: A suitable case f o r what sort o f treatment? In G. Claxton (ed.), Cognitive Psychology. London: Routledge & Kegan Paul, 1980, 1-25. Cockcroft, W.H. Mathematics Counts. London: HMSO, 1982. Cole, M., Hood, L. and McDermott, R. E c o l o g i c a l niche picking. In U. Neisser (ed.), Memory Observed. San Francisco, C a l i f o r n i a : W.H. Freeman and Osmpany, 1982, 366-373. Cole, M. and Means, B. Comparative Studies o f How People Cambridge, Massachusetts: Harvard U n i v e r s i t y Press, 1981.  Think.  Craik, F.I.M. and Lockhart, R.S. Levels o f processing: A framework f o r memory research. Journal o f Verbal Learning and Verbal Behaviour, 1972, 11, 671-684. Crowder, R.G. The demise of short-term 1982, 50, 291-323.  memory. Acta  Psychologica,  Dansereau, D.F. and Gregg, L.W. An information processing a n a l y s i s o f mental m u l t i p l i c a t i o n . Psychonomic Science, 1966, 6(2), 71-72. Das,  J.P., Kirby, J.R., and Jarman, R.F. Simultaneous and Successive Cognitive Processes. New York, N.Y.: Academic Press, 1979.  Dempster, F.N. Memory span and short-term memory capacity: A developmental study. Journal o f Experimental C h i l d Psychology, 1978, 26, 419-431. Dempster, F.N. Memory span: Sources of i n d i v i d u a l and developmental d i f f e r e n c e s . Psychological B u l l e t i n , 1981, 89(1), 63-100. Denmark, T. and Kepner, H.S. Basic s k i l l s i n mathematics: A survey. Journal f o r Research i n Mathematics Education, 1980, 11(2), 104-123. Ericsson, K.A., Chase, W.G. and Faloon, S. A c q u i s i t i o n o f a memory s k i l l . Science, 1980, 208, 1181-1182. Ericsson, K.A. and Simon, H.A. Verbal reports as data. Psychological Review, 1980, 87(3), 215-251. Feldt, L.S. The use o f extreme groups to test f o r the presence o f a r e l a t i o n s h i p . In P. Badia, A. Haber and R.P. Runyon (eds.), Research Problems i n Psychology. Reading, Massachusetts: AddisonWesley, 1970, 133-143. Flournoy, M.F. Developing a b i l i t y Teacher, 1957, 4, 147-150. Flournoy,  M.F.  Providing  mental  i n mental arithmetic. Arithmetic  arithmetic experiences.  Arithmetic  247  Teacher, 1959, 6, 133-139. Gagne, R.M. The Conditions o f Learning Rinehart & Winston, 1977.  (3rd ed.).  New York: Holt,  Gagne, R.M. Some issues i n the psychology o f mathematics i n s t r u c t i o n . Journal for Research i n Mathematics Education, 1983a, 14(1), 7-18. Gagne, R.M. A r e p l y t o c r i t i q u e s of some issues i n the psychology o f mathematics i n s t r u c t i o n . Journal f o r Research i n Mathematics Education, 1983b, 14(3), 214-216. Gagne, R.M. Learning a c t i v i t i e s and their e f f e c t s : Useful categories o f human performance. American Psychologist, 1984, 39(4), 377-385. Gardner, M. Mathematical C a r n i v a l , New York: Random House, 1977. Garg, R. An empirical comparison of three strategies used i n extreme group designs. Educational and Psychological Measurement, 1983, 43 (2), 359-371. Glennon, V. J . Variables i n a exceptional c h i l d r e n and Mathematical Education o f I n t e r d i s c i p l i n a r y Approach.  theory o f mathematics i n s t r u c t i o n f o r youth. In V.J. Glennon (ed.), The Exceptional Children and Youth: An Reston, V i r g i n i a : NCTM, 1981, 23-49.  Hatano, G., Miyake, Y. and Binks, M.G. Performance o f expert abacus operators. Cognition, 1977, 5, 47-55. Hatano, G. and abacus-derived  Osawa, K. D i g i t memory o f grand experts i n mental c a l c u l a t i o n . Cognition, 1983, 15, 95-110.  Hiebert, J . , Carpenter, T.P., and Moser, J.M. Cognitive development and c h i l d r e n ' s solutions to verbal arithmetic problems. Journal for Research i n Mathematics Education, 1982, 13(2), 83-98. Hitch, G. Mental arithmetic: Short-term storage and information processing i n a cognitive s k i l l . In A.M. Lesgold, J.W. P e l l e g r i n o , S. Fokkema, and R. Glaser (eds.), Cognitive Psychology and Instruction. New York: Plenum Press, 1977, 331-338. Hitch, G.J. The r o l e of short-term working rrtemory arithmetic. Cognitive Psychology, 1978, 10, 302-323.  i n mental  Hitch, G.J. Developing the concept of working rnemory. In G. Claxton (ed.), Cognitive Psychology. London: Routledge & Kegan Paul, 1980. 154-196. Howe, M.J. A. and Ceci, S.J. Educational implications o f rnemory research. In M.M. Gruneberg and P.E. Morris (eds.), Applied Problems i n Memory. London: Academic Press, 1979, 59-94. Hunt, E. I n t e l l i g e n c e as an information-processing concept.  248  B r i t i s h Journal o f Psychology, 1980, 71, 449-474. Hunter, I.M.L. An exceptional t a l e n t for c a l c u l a t i v e thinking. B r i t i s h Journal o f Psychology, 1962, 53(3), 243-258. Hunter, I.M.L. An exceptional memory. B r i t i s h Journal o f Psychology, 1977, 68, 155-164, Hunter, I.M.L. The r o l e of memory i n expert mental c a l c u l a t i o n s . In M.M. Gruneberg, P.E. Morris, and R.N. Sykes (eds.), P r a c t i c a l Aspects o f Memory. London: Academic Press, 1978, 339-345. Hunter, I.M.L. Memory i n everyday l i f e . In M.M. Gruneberg and P.E. Morris (eds.), Applied Problems i n Memory. London: Academic Press, 1979, 1-24. Jakobsson, S. Report on two prodigy mental arithmeticians. Acta Medica Scandinavica, 1944, 119(3), 180-191. James, W. The P r i n c i p l e s o f Psychology. New York: Dover Publications, 1950. Jarman, R.F. Comments on John B. C a r r o l l ' s "How s h a l l we study i n d i v i d u a l differences i n cognitive a b i l i t i e s ? - Methodological and t h e o r e t i c a l perspectives." Intelligence, 1980, 4, 73-82. Jarman, R.F. Level I and Level I I a b i l i t i e s : Some t h e o r e t i c a l r e i n t e r p r e t a t i o n s . B r i t i s h Journal o f Psychology, 1978, 69, 257-269. Josephina, Sr. Mental arithmetic Teacher, 1960, A p r i l , 199-207.  i n today's classroom.  Arithmetic  Keeves, J.P. and Bourke, S.F. Australian Studies i n School Performance (vol. I ) . L i t e r a c y and Numeracy i n A u s t r a l i a n Schools: A F i r s t Report. Canberra: A u s t r a l i a n Government Publishing Service, 1976. Krutetski, V. A. An i n v e s t i g a t i o n of mathematical a b i l i t i e s i n schoolchildren. In J . K i l p a t r i c k and I. Wirszup (eds.), Soviet Studies i n the Psychology o f Learning and Teaching Mathematics (vol. 2). The Structure o f Mathematical A b i l i t i e s . Chicago, 111.: University o f Chicago, 1969, 5-57. Lachman, R., Lachman, J.L., and B u t t e r f i e l d , E.C. Cognitive Psychology and Information Processing: An Introduction. H i l l s d a l e , N.J.: Lawrence Erlbaum Associates, 1979. Levin, J.A. Estimation techniques f o r arithmetic: Everyday math and mathematics i n s t r u c t i o n . Educational Studies i n Mathematics, 1981, 12, 421-434. Levine, D.R. Strategy use and estimation a b i l i t y o f c o l l e g e students. Journal f o r Research i n Mathematics Education, 1982, 13(5),  249 350-359. Lindsay, P.H. and Norman, D.A. Human Information Processing: An Introduction t o Psychology (2nd ed.). New York: Academic Press, 1977. Maier, E. Folk math. Instructor, Feb., 1977, 84-89, and 92. Martin, M. Assessment o f i n d i v i d u a l v a r i a t i o n i n memory a b i l i t y . In M.M. Gruneberg, P.E. Morris, and R.N. Sykes (eds.), P r a c t i c a l Aspects o f Memory. London: Academic Press, 1978, 334-362. McNemar, Q. A t randan: sense and nonsense. The American Psychologist, 1960, 15, 295-300. Merkel, S.P. and H a l l , V.C. The relationship between memory for order and other cognitive tasks. Intelligence, 1982, 6, 427-441. M i l l e r , G.A. The magical number seven, plus or minus two: Some l i m i t s on our capacity f o r processing information. The Psychological Review, 1956, 63(2), 81-97. M i t c h e l l , F.D. Mathematical prodigies. American Journal o f Psychology, 1907, 18, 61-143. Nason, R.A. and Redden, M.G. Mathematical learning d i s a b i l i t i e s : An information-processing view. Focus on Learning Problems i n Mathematics, 1983, 5(2), 57-77. National Advisory Committee on Mathematical Education (NACOME). Overview and Analysis o f School Mathematics. Grades K-12. Reston, V i r g i n i a : NCTM, 1975. National Assessment of Educational Progress. Math Technical Report: Exercise Volume. Denver, Colorado: NAEP, 1977. National Assessment of Educational Progress. and S k i l l s . Colorado: NAEP, 1979.  Mathematical Knowledge  National Assessment of Educational Progress. The Third National Mathematics Assessment: Results, Trends, and Issues. Denver, Colorado: NAEP, 1983a. National Assessment of Educational Progress. Spring, 1983b.  NAEP Newsletter,  16(2),  National Council of Teachers of Mathematics. An Agenda f o r Action. Reston, V i r g i n i a : NCTM, 1980. National Council o f Teachers of Mathematics. P r i o r i t i e s Mathematics (PRISM). Reston, V i r g i n i a : NCTM, 1981. Neisser,  U.  Memory:  What  are the important  questions?.  i n School In  M.M.  250  Gruneberg, P.E. Morris, and R.N. Sykes (eds.), P r a c t i c a l Aspects o f Memory. London: Academic Press, 1978, 5-24. Newell, A. and Simon, H.A. Human Problem Solving. Englewood N.J.: P r e n t i c e - H a l l Inc., 1972.  Cliff,  Nilsson, L.G. Functions o f rriemory. In L.G. N i l s s o n (ed.), Perspectives on Memory Research: Essays i n Honour o f Uppsala U n i v e r s i t y ' s 500th Anniversay. H i l l s d a l e , N.J.: Lawrence Erlbaum Associates, 1979, 3-15. Niss, M. Goals as a r e f l e c t i o n of the needs of society. In R. Morris (ed.), Studies i n Mathematics Education. P a r i s : UNESCO, 1981, 1-21. Norman, D.A. and Bobrow, D.G. On data-limited and resource-limited processes. Cognitive Psychology, 1975, 7, 44-64. Pascual-Leone, J . A mathematical model f o r the t r a n s i t i o n rule i n Piaget's developmental stages. Acta Psychologica, 1970, 32, 301-345. Pearson, K. Mathematical contributions to the theory o f evolution: XI. On the influence o f natural selection on the v a r i a b i l i t y and c o r r e l a t i o n o f organs. Transactions o f the Royal Society (London), 1903, Series A, 200, 1-66. Peters, C C . and Van Voorhis, W.R. Mathematical Bases. New  S t a t i s t i c a l Procedures and Their York: McGraw-Hill, 1940.  Reitman, J.S. S k i l l e d perception i n Go: Deducing memory strategies from inter-response times. Cognitive Psychology, 1976, 8(3), 336-356. Resnick, L.B. and Ford, W.W. The Psychology o f Mathematics f o r Instruction. H i l l s d a l e , N . J . : Lawrence Erlbaum Associates, 1981. Reys, R.E., Rybolt, J.F., Bestgen, B.J., and Wyatt, J.W. Processes used by good computational estimators. Journal f o r Research i n Mathematics Education, 1982, 13(3), 183-201. R o b i t a i l l e , D.F. and S h e r r i l l , J.M. Test Results: B r i t i s h Columbia Mathematics Assessment. V i c t o r i a , B.C.: M i n i s t r y o f Education, Learning Assessment Branch, 1977. Robitaille, D.F. Summary Report: B r i t i s h Columbia Mathematics Assessment. V i c t o r i a , B.C.: Ministry of Education, Learning Assessment Branch, 1981. Romberg, T.A. and C o l l i s , K.F. The assessment of children's M-space. Technical Report, Research and Development Center f o r Individualized Schooling, Wisconsin University, Madison. Washington, D.C.: HEW, NIE, 1981. (ERIC Document Reproduction  251  Service ED195331) Sauble, I. Development of a b i l i t y to estimate and to compute mentally. Arithmetic Teacher, 1955, 2, 33-39. Scardamalia, M. Information-processing capacity and the problem of horizontal decalage: A demonstration using combinatorial reasoning tasks. C h i l d Development, 1977, 48, 28-37. Scripture, E.W. Arithmetical prodigies. Psychology, 1891, 4(1), 1-59.  The  American  Journal o f  S h a l l i c e , T. Neuropsychological research and the f r a c t i o n a t i o n of memory systems. In L.G. Nilsson (ed.), Perspectives on Memory Research: Essays i n Honour of Uppsala University's 500th Anniversay. H i l l s d a l e , N.J.: Lawrence Erlbaum Associates, 1979, 257-277. Shulman, L.S. Psychology and mathematics education r e v i s i t e d : Address to a conference (origin unknown).  1976.  Skvarcius, R. The place of estimation i n the mathematics curriculum of the junior high school. Cape Ann Conference on Junior High School Mathematics. Boston: Physical Science Group. Washington, D.C.: HEW, NIE, 1973. (ERIC Document Reproduction Service No. Ed 085 257) Smith, S.B. The Great Mental Calculators: The Psychology, Methods, and L i v e s o f Calculating Prodigies, Past and Present. New York: Columbia University Press, 1983. Snedecor, G.W. and Cochran, W.G. S t a t i s t i c a l Methods (6th ed.). Ames, Iowa: Iowa University Press, 1967. Thornton, C.A. Emphasizing thinking strategies i n basic fact i n s t r u c t i o n . Journal for Research i n Mathematics Education, 1978, 9, 214-227. Trafton, P.R. Estimation and mental arithmetic: Important components o f computation. In M. Suydam and R.E. Reys (eds.) Developing Computational S k i l l s . Reston, V i r g i n i a : NCTM, 1978, 196-213. T y l e r , R.W. Basic P r i n c i p l e s o f Curriculum and Instruction. Chicago, I l l i n o i s : U n i v e r s i t y of Chicago Press, 1970. Wandt, E. and Brown, G.W. Non-occupational Arithmetic Teacher, 1957, 4, 151-154. Waugh, N.S. and Norman, D.A. 1965, 72, 89-104.  uses  of  mathematics.  Primary memory. Psychological Review,  Wechsler, D. Manual for the Wechsler Adult Intelligence Scale. York: The Psychological Corporation, 1955.  New  252  Wechsler, D. Manual f o r the Wechsler I n t e l l i g e n c e Scale f o r Children-Revised. New York: The Psychological Corporation, 1974. Whimbey, A., Fischhof, V. and S i l i k o w i t z , R. Memory span: A forgotten capacity. Journal o f Educational Psychology, 1969, 60(1), 56-58. Whimbey, A. and Lieblum, S.L. Individual differences i n memory span with and without a c t i v i t y intervening between presentation and r e c a l l . Journal o f Educational Psychology, 1967, 58(5), 311-314. White, A.L. Avoiding errors i n educational research. In R.J. Shumway (ed.), Research i n Mathematics Education. Reston, V i r g i n i a : NCTM, 1980, 47-65. Wingfield, A.. Human Learning and Memory: An Introduction. New York: Harper & Row, 1979. Woodward, D.M. Mainstreaming the Learning Rockville, Md.: Aspen Publications, 1981.  Disabled  Adolescent.  Wortman, C.B. and Loftus, E.G. Psychology. New York: A l f r e d A. Knopf, 1981. Zechmeister, E.B., and Nyberg, S.E. Human Memory: An Introduction to Research and Theory. Monterey, C a l i f . : Brooks/Cole Publishing Company, 1982.  253  REFERENCE NOTES  , J.A. A study of the mental c a l c u l a t i o n a b i l i t i e s of education students. Unpublished study, University of Saskatchewan, 1983.  254  APPENDIX A SUBJECT CONSENT FORM  256  APPENDIX B THE INSTRUMENTS  257  THE ITEMS OF THE SCREENING TEST CALl 1. 7 x 51  (B)  11. 12 x 500  (B)  2. 25 x 48  (A)  12. 16 x 45  (A)  3. 16 x 72  (A)  13. 30 x 200  (B)  4. 8 x 70  (B)  14. 15 x 48  (A)  5. 16 x 16  (A)  15. 7 x 511  (B)  6. 8 x 99  (B)  16. 12 x 12  (B)  7. 32 x 64  (A)  17. 25 x 65  (A)  8. 60 x 40  (B)  18. 70 x 90  (B)  9. 27 x 32  (A)  19. 15 x 64  (A)  10. 24 x 24  (A)  20. 2 x 592  (B)  * A and B refer t o the d i f f i c u l t and easy items, respectively.  258  THE ITEMS OF THE PROBING TEST CAL2 1. 9 x 742  (6678)  16. 12 x 16  (192)  2. 12 x 15  (180)  17. 23 x 27  (621)  3. 8 x 99  (792)  18. 13 x 13  (169)  4. 25 x 480  (12 000)  19. 32 x 32  (1024)  5. 9 x 74  (666)  20. 8 x 612  (4896)  6. 4 x 625  (2500)  21. 15 x 16  (240)  7. 12 x 81  (972)  22. 12 x 250  (3000)  8. 50 x 64  (3200)  23. 25 x 32  (800)  9. 8 x 625  (5000)  24. 15 x 48  (720)  10. 25 x 48  (1200)  25. 8 x 999  (7992)  11. 16 x 16  (256)  26. 25 x 65  (1625)  12. 25 x 25  (625)  27. 49 x 51  (2499)  13. 32 x 500  (16 000)  28. 24 x 24  (576)  14. 25 x 120  (3000)  29. 8 x 4211  (33 688)  15. 17 x 99  (1683)  30. 15 x 15  (225)  259 THE ITEMS OF THE CHALLENGE TEST CAL3 1. 75 x 75  (5625)  2. 32 x 64  (2048)  10. 36 x 72  (2592)  3. 18 x 72  (1296)  11. 64 x 250  (16 000)  4. 24 x 625  (15 000)  12. 87 x 23  (2001)  5. 48 x 64  (3072)  13. 89 x 91  (8099)  6. 27 x 81  (2187)  14. 73 x 83  (6059)  7. 48 x 48  (2304)  15. 64 x 64  (4096  8. 125 x 125  [15 625]  9. 75 x 240  (18 000)  260  THE ITEMS OF THE WRITTEN MULTIPLICATION TEST WPP 64  6.  X 15  48  X 65  7.  X 15  24  8.  X 16  27 X 32  9.  X 64  16  45 X 16  X 24  32  25  48 X 25  10.  72 X 16  261  THE ITEMS OF THE BASIC MULTIPLICATION FACT TEST BFR  9  X  3 =  1  X  2 =  7  X  0 —  7  X  2  2  X  7 =  0  X  9 =  4  X  8 =  6  X  9  7  X  8  8  X  7 =  5  X  3 =  5  X  0  6  X  0 =  3  X  4 =  5  X  5 =  1  X  5  4  X  5 =  0  X  5 =  9  X  1 =  7  X  3  4  X  3 =  1  X  1 =  4  X  2 =  2  X  5  6  X  3 =  2  X  1 =  6  X  2 =  1  X  6  0  X  8 =  5  X  4 =  7  X  9 =  6  X  5  4  X  7 =  0  X  4  -  6  X  4 =  1  X  7  2  X  8 =  1  X  3 =  0  X  3 =  0  X  1  4  X  1 =  3  X  8 =  5  X  7 =  6  X  1  3  X  9 =  1  X  4  8  X  8 =  4  X  4  9  X  7 =  7  X  5 =  9  X  0  =  2  X  3  1  X  9 =  7  X  7 =  2  X  4 =  8  X  3  7  X  4  6  X  8 =  3  X  5 =  6  X  7 =  5  X  8 =  9  X  9  =  0  X  6 =  0  X  7  2  X  0 =  9  X  4 =  8  X  2 =  6  X  6  8  X  0 =  2  X  2 =  3  X  7 =  7  X  1  8  X  6 =  4  X  9 =  0  X  2 =  8  X  4  9  X  2  =  5  X  6 =  1  X  8 =  9  X  5  7  X  6 =  8  X  1 =  9  X  8  =  5  X  1  5  X  2 =  4  X  6 =  5  X  9 =  9  X  6  3  X  0  -  2  X  6  -  3  X  6 =  3  X  1  3  X  3 =  8  X  5 =  2  X  9  4  X  0  0  X  0 =  3  X  2 =  1  X  0 =  8  X  9  262  THE ITEMS OF THE DELAYED DIGIT SPAN TEST DDS  LIST 1 Letter  LIST 4 Letter  D i g i t Series  1.  J  5,0,1,2,1, f l e a  2.  V  1,4,7,2,9,7,2,6, crab  3.  R  4. 5.  D i g i t Series  1.  H  3,0,4,7, f i s h  . 2.  L  2,9,8,2,4,0, b i r d  2,6,1,1,3,2, frog  3.  V  5,8,3,9,2, crab  H  1,8,6,9,8,5,9, f i s h  4.  Q  3,6,9,4,1,3,5, bear  Q  2,7,7,1, bear  5.  J  7,5,8,7,9,2,4,1, f l e a  LIST 2  Letter  LIST 5 Letter  D i g i t Series  D i g i t Series  1.  J  4,8,7,3,6,7, f l e a  1.  L  6,8,2,3,0,4,7, b i r d  2.  H  3,8,4,6,5,0,4,3, f i s h  2.  V  9,5,8,6,0,9, crab  3.  R  4,1,7,9,3,0,8, frog  3.  R  2,9,3,6, frog  4.  Q  6,1,5,7,2, bear  .4.  Q  6,8,7,5,3,1,8,4, bear  5.  L  4,5,3,4, b i r d  5.  H  8,9,8,1,2, f i s h  LIST 3 Letter  LIST 6  D i g i t Series  Letter  D i g i t Series  1.  V  4,6,1,9, crab  1.  L  3,7,0,8,5,3,4,2, b i r d  2.  J  2,5,6,3,6,7,3, f l e a  2.  H  9,4,3,4,2,0, f i s h  3.  R  1,9,2,5,6,3,2,7, frog  3.  R  5,1,9,2,7, frog  4.  Q  3,7,9,2,5,2, bear  4.  V  8,4,3,9,3,0,5, crab  5.  L  8,5,3,5,1, b i r d  5.  J  8,9,1,3, f l e a  263 THE ITEMS OF THE LETTER SPAN TEST LS  LIST 2  LIST 1 1. C,G,C  1.  2. J,M,K,F,G,M,M,F,J,F  2. FfCfC/J  3. J,H,G F  3. J,D,C,H,K,C,D,L,K  4. B,L,M,F,K,B,D,H  4. F,M,F,K,H,M,C,L,C,H,L  5. L,J,D,K,J  5. B,F,B,L,C,D,F  6. C,D,M,J,M,L  6. M,M,H,F,G,C,D,F  7. F,B C,L,F,L,F,G,H  7. F,J,C,K,C,M  8* L>>F,F,B C/JtH  8. H,K,H  f  f  f  9. H,J,L B,B M G G F,M J f  f  f  r  f  f  LIST 3  FiJrK/L,CfDfL/G,JiG  9. L,J,M,L,D  LIST 4  1. G,B,F,H,H B  1. M,L,F  2. B,K,F,M,J,C,C,B  2. C,J,D,F,H,J  3. C,F,H,J,F,B,J,B,M  3. C,L,J,C,J,C,D  4. K,L,B,B,F  4. K,J,B,L,B  5. K,D,L,C  5. H/FfKfKfD,J,K,J,L G  6. F,D,H,H,M,L,J  6. M,F,B,H,D,J,C,F,L,K,G  7. C,B,L  7. D,J H,L,K,C,D,G  f  f  f  8. F/B B,D,B KfJ/LfG/C/K  8. F,B,N,K D,G,F/H,D  9. D,M,B,H,M,K,C,G,K,C  9. K,G,D,G  f  r  f  264  APPENDIX C PORTION OF A CAL2 INTERVIEW WITH A SKILLED SUBJECT  265  PORTION OF A CAL2 INTERVIEW WITH A SKILLED SUBJECT  The following portion o f a CAL2 interview with a s k i l l e d subject has been included t o provide further c l a r i f i c a t i o n o f the procedures used t o gather information about the methods used by subjects to c a l c u l a t e mental products. This p a r t i c u l a r interview was selected because the subject employed a wide v a r i e t y o f c a l c u l a t i v e strategies. For the sake o f brevity, some items have been excluded from t h i s discussion. R and S refer to statements made by the researcher and subject, respectively. Comments about these statements follow each discussed item. The solution time for the subject's f i r s t attempt a t a c a l c u l a t i o n i s presented i n the parentheses. R: S: R: S: R: S: R: S:  R: S: R: S:  Try 9 times 742. 9 times 742 (repeats question to himself) 6 thousand, .six hundred and....78 (34 seconds). Good. How was that problem done? I j u s t m u l t i p l i e d i t out l i k e I would on p e n c i l and paper. 9 times 2 i s 18, and then carrying a 1? Yes. Why do you think i t took you so long to do that problem? Well, I f i n d i t much easier to do these things (mental calculation) i f I can see them on a piece of paper. I t ' s always there and I can't forget i t while I'm concentrating on multiplying a d i f f e r e n t p a i r o f numbers. Did you forget a c a l c u l a t i o n and have to do i t over again? I d i d i t several times to recheck. Any other better ways o f doing t h i s problem. No.  <X*fl_asiTS:  R: S: R: S: R: S: R:  This strategy was c l a s s i f i e d as the d i g i t - b y - d i g i t , r i g h t t o - l e f t , pencil-and-paper mental analogue P&PO.  T r y 12 times 15. 180 (2 seconds). How was that done? I j u s t happen t o know that one. Did you do any c a l c u l a t i o n s ? No. Why would you know t h i s fact?  266  S:  Well, b a s i c a l l y I know my times tables up to 15 times 15 a t l e a s t .  <Xk>lMENTS: This strategy was c l a s s i f i e d as a r e t r i e v a l of a numerical equivalent. He was the only s k i l l e d subject to claim r e c a l l for t h i s item. H i s claim that he "knew h i s times table t o 15 x 15" was not completely accurate." He needed t o c a l c u l a t e rather than r e c a l l the products 13 x 14, 15 x 13, and 15 x 14. In these cases, the c a l c u l a t i o n was very rapid. R: S: R: S: R: S:  How about 8 times 99? 792 (4 seconds). How was that one reasoned? I d i d 8 times 100 and then subtracted 8. When you subtracted 8 from 100, d i d you v i s u a l i z e the problem as you would i n c a l c u l a t i n g with a p e n c i l and paper? Well. I thought 792. I t was obvious.  CCMMEasiTS: This strategy was c l a s s i f i e d as subtractive d i s t r i b u t i o n . R: S: R: S: R: S: R: S:  25 times 480. 1200? No, 12 000. (18 seconds) What was your method? Well, I knew that 100 i s 25 times 4. So I divided the 480 by 4 t o get 120 and m u l t i p l i e d by 100. Why d i d you say 1200 a t f i r s t ? I l o s t track of the places..the number of zeroes when I was multiplying. Did you d i v i d e 48 by 4 instead of 480 by 4? Yes, I think so.  CCWMENTS: This strategy was c l a s s i f i e d as factoring by a l i q u o t parts. This apparent d i f f i c u l t y i n retaining zeroes was a problem common t o a l l subjects. The items 9 x 74, 4 x 625, 12 x 81, 50 x 64 were presented. They were a l l answered c o r r e c t l y and r a p i d l y . R: S: R: S: R: S:  How about 8 times 625? 5000 (4 seconds) What was your method? I remembered the 4 times 625 I calculated e a r l i e r ( r e f e r r i n g t o item 4 x 625 given previously) which i s 2500 so I doubled 2500 because 8 i s twice as much as 4. You j u s t doubled 2500? Yes.  CX1WMENTS: This strategy was c l a s s i f i e d as factoring by halfing-anddoubling. The s k i l l e d subjects often were able to r e c a l l some previous c a l c u l a t i o n and to incorporate t h i s value i n a l a t e r s o l u t i o n without having to resort to a re-calculation.  267 The item 25 x 48 was presented and answered c o r r e c t l y and r a p i d l y by using aliquot parts. R: S: R: S: R: S: R: S:  Try 16 times 16. 256 (1 second). How d i d you know that? I know powers of 2 pretty w e l l . So, i t ' s j u s t a f a c t and you don't have to multiply? Yes. Why would you remember powers of 2? Possibly with working with computers a l o t and the binary system.  CDMMEJNTS: This strategy was c l a s s i f i e d as a r e c a l l of a numerical equivalent. R: S: R: S: R: S: R: S: R:  Try 25 times 25. 625 (1 second). How d i d you determine that? Was i t just a fact? Yes. Why would you remember that one? I'm r e a l l y not sure. Maybe because i t seems to come i n t o common usage more than a l o t of other m u l t i p l i c a t i o n s , e s p e c i a l l y squares l i k e that. Are you f a m i l i a r with many squares? Some squares. The obvious ones up to about 20. I ' l l l i k e l y t e s t you on that l a t e r .  CCMffiNTS: This strategy was c l a s s i f i e d as r e c a l l of a numerical equivalent. The subject-demonstrated that he could r e c a l l these squares: 13 , 14 , 15 , 16 , 17 , 21 , z  25  2  and  36 . 2  The items 32 x 500 and 25 x 120 were presented and solved. R: S: R: S: R: S: R:  S: R: S: R: S:  Try 17 times 99. 1683 (9 seconds). How was that done? By taking 17 times 100 and then subtracting 17 from that. When you were subtracting 17 from 1700, what strategy d i d you use? I took 1700 and put i t on the top and then the 17 below. And then I subtracted i t out and then I made a quick check to make sure. 1683 and 17 added to make 1700. Now l e t me make sure I understand what you d i d to subtract. When you v i s u a l i z e d the 17 under the 1700, d i d you c a l c u l a t e the difference i n the same manner as i n pencil-and-paper calulation? No. Well 100 minus 17 I know i s 83. So 1700 minus 17 would be 1683. And then you checked by adding a 17. Yes. How d i d you add to check? Well, I can j u s t see numbers combining l i k e that when they  268  R: S:  add up to 100. 83 and 17, i t ' s quite obvious to me that the answer i s 100. Do you often check your answers i n mental arithmetic? Yes. I use another method i f i t ' s convenient to do i t that way.  CCMMENTS: This strategy was c l a s s i f i e d as subtractive d i s t r i b u t i o n . Interestingly, he completed the subtraction by a l i g n i n g the "17 under the 1700" but d i d not use a mental equivalent of the written algorithm. Perhaps t h i s v i s u a l process was used to help r e t a i n the minuend and subtrahend. The following discussion followed the presentation of 17 x 99. R: S: R: S:  R: S:  R:  When you were young do you remember looking at number patterns or doing mental c a l c u l a t i o n ? When I was quite young I used to p r a c t i s e writing out my own charts. I used to r e a l l y l i k e numbers. I'd write out my own m u l t i p l i c a t i o n tables and charts. Did you examine these charts for patterns and relationships? Yes. Thinking back..I know i t ' s a long time but can you remember any patterns you discovered? Well, one thing I remember discovering i s with squares. You can figure out the next square, i f you know another square. For example, i f you know 20 squared i s 400, you can f i g u r e out the next square 21 squared by adding the 20 + 21 which i s the next number to the square of 20. I think I know what you are trying to describe but, to make sure, write out the example for me. 2 (the subject took a p e n c i l and paper and wrote 21 = 400 + 20 + 21.)  R: S: R: S: R: S:  2 =20  + 20 + 21  Did you ever t r y to f i g u r e out why t h i s pattern works? No, I haven't r e a l l y . Can you remember when you discovered t h i s c a l c u l a t i n g pattern? I think i t was around grade 5 or so. And you discovered t h i s pattern simply from examining and w r i t i n g out number charts? Yes.  COMMENTS: His b e l i e f that an i n t e r e s t i n number patterns was the d r i v i n g force behind some of h i s mental c a l c u l a t i o n methods was common to the most p r o f i c i e n t c a l c u l a t o r s . His r u l e for c a l c u l a t i n g squares can be explained as: (x + 1) * = x^ + x + (x + 1) R: S: R: S:  Did you ever t r y to adapt your r u l e to other squares? Not r e a l l y . Can you think of a way of adapting the r u l e to solve 29 squared? Let's see 29 squared,..you could probably work that backwards. I  269  R: S:  guess I'd s t a r t with 900 and subtract the next number 30 and subtract 29. But you've never used t h i s rule i n the past? No, I j u s t thought of i t .  COMMENTS: The subject generalized t h i s rule: (x - I ) * = x -x - ( x - 1 ) z  R: S:  Any other patterns you discovered? Well, l e t ' s see. I remember one where i f you want to multiply, say 14 x 16, you'd think 15 squared i s 225... I noticed i f one number i s greater than the square and the other i s one l e s s than a square then the answer i s 1 l e s s than the square. So 14 x 16 i s 225 - 1 = 224. R: Do you ever remember getting any i n s t r u c t i o n i n mental arithmetic? S: No. I j u s t seemed t o learn i t on my own. COMMENTS: The r u l e he explained was the difference o f squares: (x + 1) (x - 1) = x - 1. As was the case with most s k i l l e d c a l c u l a t o r s , he could not r e c a l l any teacher i n s t r u c t i o n i n the topic of mental c a l c u l a t i o n . They were convinced that, f o r a l l p r a c t i c a l purposes, they were self-taught. R: S: R: S: R: S: R: S: R: S: R: S:  Try 12 times 16. Hmm. I ' l l have to calculate....192 (7 seconds). How was that done? Well, that one I j u s t decided to go ..well multiply by 2, four times. Just as a check, can you explain your reasoning again? Well 16 i s 2 to the power o f 4, so I d i d 12 times 2 i s 24, times 2 again i s 48, times 2 again i s 96, and times 2 again i s 192. Why d i d you choose t h i s method instead of the pencil-and-paper method you used on some of the e a r l i e r items? I t j u s t seemed easier than multipying i t out the long way. Do you change strategies depending on the problem? Yes. Do you ever f i n d that you t r y one strategy, go so f a r i n t o the c a l c u l a t i o n , and then abandon the strategy i n favour o f another? Yes..if I s t a r t getting l o s t or the numbers get confusing and I forget them. I j u s t kind of s t a r t another one.  COMMENTS: This strategy was c l a s s i f i e d as factoring by doubling-andh a l f i n g . Certain properties of the factors, i n t h i s case powers of 2, seemed to cue p a r t i c u l a r c a l c u l a t i v e strategies. R: S: R: S:  Try 23 times 27. 621 (7 seconds). How was that done? Well, using that t r i c k I had here (referring t o discussion of  270  R: S: R: S: R: S: R: S: R: S: R: S: R: S:  e a r l i e r "differences of squares" solution of 14 x 16). Do you mean 25 squared minus 4? Yes, but I'm not sure i f t h i s works or not? OK. Try another way to confirm the answer. 23 x 27? Yes. Yes, i t ' s 621 (3 seconds). What method d i d you use? I d i d 20 x 27 and 27 x 3. How d i d you c a l c u l a t e 27 x 3? Did you think 7 times 3 i s 21, carry 2. No for that one I thought immediately of 81 because i t ' s powers of 3. How about the 2 x 27? Thought immediately o f 54. Then 540. How d i d you add 540 and 81? Well, I thought o f dropping the 1 and then thought 62, so 621.  COMMENTS: This strategy was c l a s s i f i e d as quadratic d i s t r i b u t i o n (difference o f squares). His check involved a d d i t i v e d i s t r i b u t i o n and included the r e t r i e v a l o f the large numerical equivalents 3 x 27 and 2 x 27. This a b i l i t y to organize a c a l c u l a t i o n i n t o a series of e a s i l y determined "blocks" was a c h a r a c t e r i s t i c of the s k i l l e d subjects. He was presented the item 13 x 13 which he r e t r i e v e d as 169. R: Hew about 32 times 32. S: Hmm... 1024 (4 seconds). R: How was that done? S: Powers o f 2. R: Would you elaborate? What do you mean by powers o f 2? S: Well, 32 i s 2 to the 5th, times 2 to the 5th again, so 2 t o the 10th, which I j u s t know as a f a c t . I t ' s 1024. R: Any reason why you knew that 2 to the 10th i s 1024. S: I j u s t know some powers o f 2. R: I s t h i s probably related to your work with the computer? S: Possibly. COMMENTS: This strategy was c l a s s i f i e d as exponential f a c t o r i n g . He was the only s k i l l e d subject to use t h i s strategy to solve a CAL2 item. He was presented with the items 8 x 612 and 15 x 16 which were solved c o r r e c t l y and rapidly. R: S: R: S: R: S:  T r y 12 x 250. 750 (13 seconds). Whoops, that's not i t . I got l o s t . T r y again. O.K. 3000 (3 seconds). In your f i r s t attempt, what d i d you mean by "getting l o s t ? " Well, I thought 250 i s 1000 divided by 4. I'm not sure how I got l o s t but 12 divided by 4 i s 3. So probably I d i d 3 times 250 instead o f 3 times 1000.  271  R: S:  What d i d you do i n your second attempt? I d i d 10 times 250 i s 2500 and 2 times 250 i s 500, so 3000.  COMMENT:  His i n i t i a l strategy was c l a s s i f i e d as factoring by aliquot parts. This error o f using the incorrect factor 250 instead of 1000 was made by several subjects who used a l i q u o t parts. He was given the item 25 x 32 which he solved c o r r e c t l y and rapidly.  R: S: R: S:  How about 15 times 48? ..720 (7 seconds). What was your method? OK. I took the tens part f i r s t and thought 480. And since i t was 5 times 48 i t ' s j u s t h a l f o f 480 which i s 240. And then I added 480 and 240.  COMMENTS: This strategy was c l a s s i f i e d as f r a c t i o n a l d i s t r i b u t i o n . He was given the item 8 x 999 which he solved c o r r e c t l y and rapidly. R: S: R: S:  Try 25 times 65. 1625 (5 seconds). How was that done? Divided the 65 by 4, and got 16.25 and then moved the decimal place over two places.  COMMENTS: This strategy was c l a s s i f i e d as aliquot parts. He was the only subject who incorporated decimal arithmetic i n t o a calculation. R: Try 49 times 51. S: 2499 (4 seconds). R: How? S: 50 squared minus 1. COMMENTS: This strategy was c l a s s i f i e d as quadratic d i s t r i b u t i o n (difference o f squares). He was presented with 24 x 24 which he solved c o r r e c t l y i n 14 seconds using additive d i s t r i b u t i o n . He said he slowed down i n t h i s c a l c u l a t i o n because "I thought I knew that one but I had to c a l c u l a t e anyway." R: S: R: S: R:  How about 8 times 4211. 33 thousand, 6 hundred, 88 Straight m u l t i p l i c a t i o n . 8 times 11 i s 88 and 8 times 2 i s 16 carry the 1.. Which type o f problem do you prefer: doing a problem with a 1 - d i g i t factor l i k e t h i s or a problem l i k e 49 times 51? 49 times 51. I have a t r i c k for those but these other long ones l i k e 8 times 4211 I have to remember a s t r i n g of c a l c u l a t i o n s as I go along.  272  CXMffiNTS: This strategy was c l a s s i f i e d as pencil-and-paper mental analogue with one non basic f a c t r e t r i e v a l (8 x 11). His comment that he preferred problems with factors that have some discernable properties rather than those with "property-less" 1 - d i g i t by x - d i g i t factors was echoed by several s k i l l e d subjects. The l a s t presented item was 15 x 15 which he r e c a l l e d immediately as 225.  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share