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Two different instructional procedures for a multiplication algorithm and their transfer effects to a.. Hope, John Alfred 1972-12-31

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TOO DIFFERENT INSTRUCTIONAL PROCEDURES FOR A MULTIPLICATION ALGORITHM AND THEIR TRANSFER EFFECTS TO A HIGHER-ORDER ALGORITHM. by John A. Hope B.Sc,  U n i v e r s i t y o f B r i t i s h Columbia, 1965  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n t h e Department of Mathematics  Education  We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the required standard  THE UNIVERSITY OF BRITISH COLUMBIA May, 1972.  In  presenting  this  an advanced degree the I  Library  further  for  agree  in  at  University  the  make  it  partial  freely  that permission for  this  representatives. thesis  for  It  financial  gain  Department . o f  The U n i v e r s i t y  Date  of  B r i t i s h Columbia  Canada  ^Xt^vJLlX^  /  of  of  Columbia,  British for  extensive by  the  shall  not  the  requirements  reference copying of  Head o f  i s u n d e r s t o o d that  written permission.  Vancouver 8,  fulfilment  available  s c h o l a r l y purposes may be g r a n t e d  by h i s of  shall  thesis  I  agree  and this  be a l l o w e d  that  study. thesis  my Department  copying or  for  or  publication  w i t h o u t my  Abstract  T h i s was  a study t o determine the e f f e c t s o f two  instructional  p r o c e d u r e s f o r a m u l t i p l i c a t i o n p a l g o r i t h m on the a b i l i t y of e l e m e n t a r y s c h o o l c h i l d r e n to extend t h i s a l g o r i t h m t o the s o l v i n g of t a s k s i n v o l v i n g the use of a h i g h e r - o r d e r Each of two groups was  algorithm.  given preliminary i n s t r u c t i o n i n  s o l v i n g m u l t i p l i c a t i o n problems v i a the a p p l i c a t i o n of the law.  A f t e r t h i s r e a d i n e s s phase was  assigned  group.  The T l s u b j e c t s  a rote-type standard m u l t i p l i c a t i o n algorithm f o r  the s o l u t i o n of 2 x 1 and 3 x 1 was  distributive  completed, s t u d e n t s were randomly  t o e i t h e r a T l or T2 treatment  were taught  computational  products.  No e x p l i c i t  instruction  g i v e n t o i n d i c a t e the r e l a t i o n s h i p s between the two  learning  t a s k s , v i z . the a c q u i s i t i o n o f the d i s t r i b u t i v e law and the m u l t i p l i c a t i o n algorithm.  determining  standard  U n l i k e the T l i n s t r u c t i o n a l sequence, the  T2 i n s t r u c t i o n a l sequence was  designed  t o promote the l e a r n i n g of  the r e l a t i o n s h i p s between the s e r i e s of l e a r n i n g t a s k s . the T2 s u b j e c t s were taught  a standard m u l t i p l i c a t i o n  That i s ,  algorithm  t h a t r e q u i r e d the e x p l i c i t use of the d i s t r i b u t i v e law and acquired algebraic s k i l l s .  I t was  hypothesised  other  that t h i s c o n t i n u a l  i n t e g r a t i o n o f l e a r n i n g t a s k s would e n a b l e the T2 s u b j e c t s t o e x h i b i t s u p e r i o r i t y over the T l s u b j e c t s i n e x t e n d i n g m u l t i p l i c a t i o n a l g o r i t h m to computational an untaught h i g h e r - o r d e r  algorithm.  their  standard  t a s k s r e q u i r i n g the use  A t o t a l of 238 s u b j e c t s  8 t e a c h e r s were used i n a l l phases of the experiment.  and  of  A mixed model :-6f.analysis of variance was the performance hypothesis.  It was  used to validate  found that the T l subjects were  s i g n i f i c a n t l y better than the T2 subjects i n the performance of the standard m u l t i p l i c a t i o n algorithm.  An analysis of covariance was  performed to determine the v a l i d i t y of the transfer hypothesis. subject's score on the performance test was  A  used as a covariate i n  order to equate the disparate computational a b i l i t i e s of the T l and T2 subjects.  Although the mean score of the T2 subjects was  higher than that of the T l subjects on the transfer test, this difference was not s t a t i s t i c a l l y s i g n i f i c a n t .  i  TABLE OF CONTENTS Page LIST OF TABLES  i i  Chapter I OUTLINE OF,'-THE PROBLEM Introduction  . . . . . . . . . . .  1  '  1  G e n e r a l Statement of the P r o b l e m  3  D e f i n i t i o n of Terms  4  D i s c u s s i o n and S i g n i f i c a n c e of the Problem  6  Hypotheses  15  Chapter I I SURVEY OF THE LITERATURE  16  Chapter  III  DESIGN OF THE EXPERIMENT  20  The Sample  . 2 0  The I n s t r u c t i o n a l Sequences The M e a s u r i n g Instruments  20 . .  23  Chapter IV ANALYSIS OF THE DATA  27  The Performance H y p o t h e s i s The T r a n s f e r H y p o t h e s i s  27 . . . . . . . .  29  Chapter V CONCLUSIONS AND IMPLICATIONS FOR FURTHER RESEARCH  . . . .  32  Performance H y p o t h e s i s  32  Transfer Hypothesis  32  Problems f o r F u r t h e r Study  33  BIBLIOGRAPHY  34  APPENDICES  35  A.  The I d e n t i f i c a t i o n of A n o t h e r I n t e r n a l A l g o r i t h m . . .  35  B.  Readiness Phase Lesson P l a n s  C.  Treatment Phase Lesson P l a n s . . . . . . . . . . . . .  49  D.  The M e a s u r i n g Instruments  67  E.  The E x p e r i m e n t a l Data  . •. .  38  71  ii  LIST OF TABLES Page Table I A n a l y s i s of the Performance T e s t :  Point ,  • •  B i s e r i a l r f o r each i t e m  26  Table I I A n a l y s i s of the T r a n s f e r T e s t :  Point  . , ,,  B i s e r i a l r f o r each i t e m  26  Table I I I A n a l y s i s of V a r i a n c e ; Performance H y p o t h e s i s  29  Table IV A n a l y s i s of C o v a r i a n c e :  Transfer Hypothesis  . . . . . .  31  CHAPTER I OUTLINE OF THE PROBLEM  IN:T-R0D%:GTION>  Most modern a r i t h m e t i c programs a r e i n agreement t h a t t h e f i e l d p o s t u l a t e s f o r t h e system of a r i t h m e t i c s h o u l d part of a r i t h m e t i c content.  form an i n t e g r a l  Both m a t h e m a t i c i a n s and p s y c h o l o g i s t s  have a d v i s e d t h a t t h e u n d e r s t a n d i n g  o f many o f t h e s e p o s t u l a t e s be  i n c l u d e d as e l e m e n t a r y s c h o o l o b j e c t i v e s .  P a r t i c i p a t i n g mathematicians  a t t h e Cambridge C o n f e r e n c e on School Mathematics s t r e s s e d t h a t  students  be f a m i l i a r w i t h p a r t o f t h e " g l o b a l s t r u c t u r e " o f mathematics."''  They  f e l t t h a t a v e r y s o l i d m a t h e m a t i c a l s u p e r s t r u c t u r e can be e r e c t e d w h i c h w i l l h e l p p u p i l s i n more advanced m a t h e m a t i c a l f i e l d s .  Although  the i d e a o f . " g l o b a l s t r u c t u r e " was never c l e a r l y d e f i n e d t h e r e i s l i t t l e doubt, a f t e r examining t h e i r recommendations f o r c u r r i c u l u m  content,  3 t h a t t h e f i e l d p o s t u l a t e s formed p a r t o f i t . Jerome B r u n e r , a g a i n a v o i d i n g t h e k n o t t y problem o f d e f i n i t i o n , s t a t e d " t h e r e a r e a t l e a s t f o u r g e n e r a l c l a i m s t h a t can be made f o r t e a c h i n g t h e fundamental s t r u c t u r e o f a s u b j e c t , c l a i m s i n need o f G o a l s f o r S c h o o l M a t h e m a t i c s , (New York: Houghton M i f f l i n , 1963), p. 8. 2 I b i d . , p. 8. 3 I b i d . , p. 36.  2  d e t a i l e d study". 1.  He l i s t e d t h e f o l l o w i n g as s u p p o r t i v e c l a i m s ;  U n d e r s t a n d i n g fundamentals makes a s u b j e c t more com-  prehensible. 2. rapidly  Unless d e t a i l i s placed into a structured pattern i t i s  forgotten. 3.  U n d e r s t a n d i n g o f fundamental p r i n c i p l e s and i d e a s l e a d s  to t r a n s f e r of t r a i n i n g . 4.  By c o n s t a n t l y r e e x a m i n i n g m a t e r i a l taught i n elementary  and secondary s c h o o l s f o r i t s fundamental c h a r a c t e r , one i s a b l e t o 4 narrow t h e gap between "advanced"  knowledge and " e l e m e n t a r y " knowledge.  D a v i d A u s u b e l , c l a i m s t h a t " p r e c i s e and i n t e g r a t e d u n d e r s t a n d i n g s a r e , presumably, more l i k e l y t o develop i f t h e c e n t r a l , u n i f y i n g i d e a s o f d i s c i p l i n e a r e l e a r n e d b e f o r e more p e r i p h e r a l c o n c e p t s and i n f o r m a t i o n a r e i n t r o d u c e d " . ^ I n h i s o p i n i o n , " t h e most s i g n i f i c a n t advances t h a t have o c c u r r e d i n r e c e n t y e a r s i n t h e t e a c h i n g of such s u b j e c t s as mathematics,  c h e m i s t r y , p h y s i c s and b i o l o g y a r e p r e d i c a t e d on t h e  assumption t h a t e f f i c i e n t l e a r n i n g and f u n c t i o n a l r e t e n t i o n o f i d e a s and i n f o r m a t i o n a r e l a r g e l y dependent upon t h e adequacy o f c o g n i t i v e s t r u c t u r e , i . e . upon t h e adequacy o f an i n d i v i d u a l ' s e x i s t i n g o r g a n i z a t i o n , s t a b i l i t y and c l a r i t y o f knowledge i n a p a r t i c u l a r s u b j e c t - m a t t e r f i e l d " .  Jerome S. B r u n e r , The P r o c e s s o f E d u c a t i o n , (New York: V i n t a g e Books, 1963), p. 23-26. ^ D a v i d P. A u s u b e l , The P s y c h o l o g y o f M e a n i n g f u l V e r b a l L e a r n i n g , (New York: Grune and S t r a t t o n , 1963), p. 21. I b i d . , p. 26.  3  In this writer's opinion Ausubel supports the early understanding of the f i e l d postulates when he claims that:''  "the a c q u i s i t i o n of adequate  cognitive structure, i n turn, has been shown to depend upon both substantive and programmatic factors using for organizational and integrative purposes those substantive concepts and p r i n c i p l e s i n a given d i s c i p l i n e that have the widest explanatory power, inclusiveness, g e n e r a l i z a b i l i t y , and r e l a t a b i l i t y to the subject-matter content of that d i s c i p l i n e " . ^ Although much has been hypothesised about the pedagogical benefits of subject-matter structure, l i t t l e v a l i d a t i o n has been attempted.  Moreover, those studies that have been concerned with such  issues have rarely attempted to offer suitable psychological explanations of the role of subject-matter structure i n arithmetic understanding. Assuming that the f i e l d postulates form part of mathematical structure, the intent of this study i s to provide both plausible psychological explanations and empirical data related to the role of the understanding of these f i e l d postulates i n promoting arithmetic understandings.  GENERAL STATEMENT OF THE PROBLEM;  1  Since computational algorithms are commonly given l o g i c a l j u s t i f i c a t i o n by using the f i e l d postulates, i t i s hypothesised that the learning of the f i e l d postulates w i l l f a c i l i t a t e understanding, and through understanding, the learning of such algorithms.  More s p e c i f i c a l l y ,  this study w i l l attempt to determine under what i n s t r u c t i o n a l conditions  Ibid., p. 26.  4  the u n d e r s t a n d i n g  of the f i e l d p o s t u l a t e s promotes ease o f  to untaught c o m p u t a t i o n a l  algorithms.  extension  Moreover, an attempt w i l l  made t o p r o v i d e a p s y c h o l o g i c a l r a t i o n a l e f o r the i n c l u s i o n o f  be these  p o s t u l a t e s i n a contemporary a r i t h m e t i c program.  'DEFINITION OF TERMS  I n o r d e r t o a v o i d an ambiguous and h y p o t h e s e s i t was  f e l t necessary  Algebraic principles. axioms, f i e l d p r i n c i p l e s , and subset  l e n g t h y statement o f  t o d e f i n e the f o l l o w i n g terms: These a r e a l s o r e f e r r e d to as  f i e l d postulates.  In t h i s study  field the  of f i e l d p o s t u l a t e s w i t h w h i c h we a r e concerned i s the s e t o f  p o s t u l a t e s t h a t a p p l y to the whole numbers. Algorithm.  Any  r u l e or o r d e r e d  s e t of p r o c e d u r e s t h a t  be used t o produce a c o r r e c t s o l u t i o n t o a c o m p u t a t i o n a l  can  task  independent o f the i n d i v i d u a l u s i n g t h a t a l g o r i t h m ; f o r example, the u s u a l column a d d i t i o n a l g o r i t h m . Internal algorithm.  Any  a l g o r i t h m whose p r i m a r y  t h a t i t i s used i n the g e n e r a t i o n of o t h e r a l g o r i t h m s .  function i s  It i s internal  i n the sense t h a t i t i s c o n s i d e r e d a means t o an end r a t h e r than an end i n i t s e l f .  That i s , i t s prime i n s t r u c t i o n a l purpose i s t o  as an a l g e b r a i c p r e r e q u i s i t e f o r more complex c o m p u t a t i o n a l The writer«will use the term f o r m a i n l y not a t t r i b u t e any algorithms.  The  algorithms.  r e f e r e n t i a l purposes and  will  s p e c i a l p s y c h o l o g i c a l properties to i n t e r n a l i n t e r n a l a l g o r i t h m used i n t h i s study i s the  a l g o r i t h m ; the r e a d e r for  serve  annexation  s h o u l d examine the T2 I n s t r u c t i o n a l Sequence on page 9  an e x p l a n a t i o n o f t h i s a l g o r i t h m .  Appendix A d e s c r i b e s a n o t h e r  5  internal algorithm. Standard m u l t i p l i c a t i o n a l g o r i t h m .  F o r t h e purposes o f t h i s  study t h e s t a n d a r d m u l t i p l i c a t i o n a l g o r i t h m w i l l r e f e r t o those used to. compute p r o d u c t s  procedures  such as axb where e i t h e r a o r b has a one  d i g i t numeral and the o t h e r has a two o r t h r e e d i g i t numeral.  For  example: 12 x 9  132 x 9  7 x 18  6 x 132  H e r e a f t e r such p r o d u c t s w i l l be r e f e r r e d t o as 2x1 and 3x1 p r o d u c t s . Higher-order "higher-order products  algorithm.  F o r t h e purposes o f t h i s study a  a l g o r i t h m " w i l l r e f e r t o an a l g o r i t h m used t o compute  such as axb w h e r e . n e i t h e r  a n o r b hase> a one d i g i t numeral  and where e i t h e r a o r b may have more than two d i g i t s i n t h e numeral. For example: 1001 x 7 :  132 x 111  12 x 1002  These a l g o r i t h m s a r e " h i g h e r " i n the sense t h a t the s t a n d a r d  multiplication  a l g o r i t h m must be c o n c e p t u a l l y m o d i f i e d i n o r d e r t o compute n o v e l products.  Further e l a b o r a t i o n i s given l a t e r i n the chapter. Performance t a s k s .  T h i s r e f e r s t o those t a s k s r e q u i r i n g t h e  a p p l i c a t i o n of the standard m u l t i p l i c a t i o n algorithm.  Level of per-  formance was measured by a w r i t t e n t e s t d e s c r i b e d i n Chapter I I I . . Transfer tasks.  A s o l u t i o n of a t r a n s f e r task required the  s u c c e s s f u l e x t e n s i o n o f t h e p r e v i o u s l y taught algorithm.  standard  multiplication  L e v e l o f t r a n s f e r was measured by a w r i t t e n t e s t  i n Chapter I I I .  described  6  T l group. Sequence.  Those students who  completed  The reader i s referred to page 9  the T l Instructional  for d e t a i l s of this  sequence. T2 group. Sequence.  Those students who  completed  The reader i s referred to page 9  the T2 Instructional  for d e t a i l s of the  sequence.  The role of algorithms i n arithmetic programs has changed considerably over the past twenty years.  Previously, considerable  i n s t r u c t i o n a l time was devoted to increasing a student's proficiency with an algorithm rather than his understanding of that algorithm; Arithmetic content was treated as i f i t were a series of l o g i c a l l y unrelated algorithmic tasks rather than an integrated set of r e l a t i o n ships between r e l a t i v e l y simple concepts.  With advances i n technology  less stress has been placed on mere performance  of computational algorithms,  although computational algorithms, s t i l l form the main substance of most modern arithmetic programs.  Thus,the modern curriculum  developer  has been primarily concerned that children understand the rationale of an algorithm; i . e . concerned about the a b i l i t y of.children to explain the relationships between the algorithm and other previously acquired algebraic p r i n c i p l e s . t •  Since computational algorithms are l o g i c a l l y related to the properties of place value systems and the f i e l d p r i n c i p l e s , i t has frequently been claimed by some mathematics educators that these l o g i c a l  7 r e l a t i o n s h i p s enhance the u n d e r s t a n d i n g o f c o m p u t a t i o n a l a l g o r i t h m s . E r i c MacPherson e x p r e s s e s t h i s view when he s t a t e s , "the c h i l d understands a r i t h m e t i c i s t h e c h i l d who g f o l l o w s from t h e s e ^ p r i n c i p l e s " .  each a l g o r i t h m  I t would be erroneous to c o n c l u d e  from such statements t h a t c h i l d r e n who are  sees how  who  u n d e r s t a n d the f i e l d  principles  a b l e to d e r i v e s p o n t a n e o u s l y t h e u s u a l s t a n d a r d i z e d c o m p u t a t i o n a l  algorithms.  Rather such views i m p l y t h a t when a c h i l d  the r o l e o f t h e p r i n c i p l e i n an a l g o r i t h m , (e.g.  understands  r e c o g n i z e s an  i n s t a n c e of the p r i n c i p l e i n an a l g o r i t h m , ^ d e m o n s t r a t e s t h a t a ' s t e p ' i n an a l g o r i t h m i s a n o t h e r a p p l i c a t i o n of some p r e v i o u s l y l e a r n e d p r i n c i p l e , e t c . ) he i s more l i k e l y to u n d e r s t a n d the r a t i o n a l e of  other r e l a t e d algorithms.  However, what seems t o be l a c k i n g i n  the arguments o f " s t r u c t u r e a d v o c a t e s " i s a r e a s o n a b l e p s y c h o l o g i c a l i n t e r p r e t a t i o n of the r o l e of subject-matter s t r u c t u r e i n e f f e c t i n g understanding. of  More s p e c i f i c a l l y , i n what sense does u n d e r s t a n d i n g  the r o l e of the f i e l d p o s t u l a t e s i n s p e c i f i c a l g o r i t h m s promote  ease of e x t e n s i o n t o . u n t a u g h t r e l a t e d a l g o r i t h m s ? of  F o r the purposes  t h i s s t u d y , i t would seem t h a t o f the many l e a r n i n g p s y c h o l o g i s t s ,  D a v i d P. A u s u b e l and Robert M. Gagne a r e two whose views seem 1  particularly  relevant.  In o r d e r t o demonstrate  the r e l e v a n c e of t h e s e p s y c h o l o g i c a l  views t o t h i s s t u d y , i t i s n e c e s s a r y to r e f e r c o n s t a n t l y to s p e c i f i c i n s t r u c t i o n a l sequences  used i n t h i s s t u d y . 1  Hence i t seems a p p r o p r i a t e ,  f i r s t , to e x p l a i n the n a t u r e of t h e s e i n s t r u c t i o n a l sequences.  The  E r i c D. MacPherson, "The F o u n d a t i o n s of Elementary S c h o o l M a t h e m a t i c s " , Modern I n s t r u c t o r , Volume 33 (October 1964), p. 70.  8 reader i s referred  to Figure 1. on page 9  for a diagrammatic explanation  of these sequences.  A.  The T l Instructional  Sequence  This sequence i s t y p i c a l of many that occur i n modern textbooks.  The f i r s t s k i l l taught i n this sequence i s use of the  d i s t r i b u t i v e law.  A c h i l d i s assumed to understand the d i s t r i b u t i v e  law when he can: a)  use the d i s t r i b u t i v e law to solve such algebraic  expressions as 9x5=  (9 x 3) + (9 x 1) + (9 x Q )  8x6=  (2 x 6) + (2 x 6) .<+"' ( & x 6)  9x7=  (6 + 3) x (5 +2) = (6 x 5) + (6 x A)  + (3 x D ) + ( 3 x 2 )  (2 x 6) + (2 x 6) + ( Q x 6) = 8 x 6 (8 x 3) + (8 x 5) + (8 x 1) = 8 x C_ b)  compute products such as 9 x 8 by application  of the  d i s t r i b u t i v e law: 9 x 8 = 9 x (2 + 6) = 18 + 54 = 72 9 x 8 = 9 x |2 +: 5 ^l)= 18 + 45 + 9 = 72 r  :  ;  9x8=  (4 + 5) x (2 + 6) = 8 + 24 +10 + 30 = 72  The next objective i n the sequence i s the acquisition  of a  rote-type standard multiplication-,;algorithm, the algorithm i s being considered to be rote-type i n the sense that no attempt i s made e x p l i c i t l y to indicate  the relationships  mastered s k i l l and this algorithm.  between the previously  ,9  FIGURE 1.  THE INSTRUCTIONAL SEQUENCES  T l Instructional  Distributive  Sequence  Standard M u l t i p l i c a t i o n -  Law  Algorithm  L  i  *  ' »  i t  T2 Instructional  Sequence  Transfer Task: Higher-Order Algorithm  TFT Distributive Law  Annexation Algorithm  Associative Law  Standard M u l t i p l i c a t i o n Algorithm  ,10 B.  The T2 Instructional Sequence  As with the T l sequence, the T2 instructionalfeseqaience incorporates  the understanding of the d i s t r i b u t i v e law as an i n i t i a l  learning objective. considered necessary.  However, additional algebraic s k i l l s are also These s k i l l s involve the use of the associative  law and an i n t e r n a l algorithm, The  i n this case the annexation  algorithm.  c h i l d i s taught to compute products i n which 10 i s a factor  by "annexing the zeros".  For example the product of 7 x 200 i s  i n i t i a l l y computed by using the associative law i n the following manner: 7 x 200 = 7 x (2 x 100) = (7 x 2) x 100 = 14 x 100 = 1400 or  7 x 2 hundreds = ( 7 x 2 ) hundreds = 14 hundreds = 1400 Later computation simply involves d i r e c t annexation.  For  example, 7_ x 2 00 = 14 00  The standard m u l t i p l i c a t i o n algorithm  u t i l i z e d i n this sequence  validates procedural "steps" by e x p l i c i t l y pointing out instances of the p r i o r learned  skills.  This writer i s primarily interested i n the effects of each i n s t r u c t i o n a l sequences on the amount of transfer to computational tasks that involve an untaught higher-order algorithm.  As mentioned  e a r l i e r i n this chapter, the views of Gagne and Ausubel would seem to provide possible explanations of these transfer differences. Gagne has developed what he considers model.  9  a hierarchy-of-learning  • Before a s p e c i f i e d learning task can be mastered, Gagne would  Robert M. Gagne, The Conditions of Learning, (New York: Holt, Rinehart and Winston, 1970).  1.1 claim that a numbernof subordinate concepts must also be mastered. These concepts i n turn depend upon other subordinate concepts so that i t can be argued that Gagne*'s model ultimately resembles that of learning.  S—R  As Gagne explains, when such an4'.analysis (selecting appropriate prerequisite tasks) i s continued progressively to the point of delineating am .entire set of c a p a b i l i t i e s having an order r e l a t i o n to each other (in the sense that i n each case prerequisite c a p a b i l i t i e s are represented as subordinate i n position, i n d i c a t i n g they need to be previously learned), one has a learning hierarchy. The analytic process may be carried out i f desired, u n t i l the simplest kinds of learnings (Ss-^R's, chains, d i s criminations) are reached and i d e n t i f i e d . ^  Thus:, once the terminal task i s c l e a r l y s p e c i f i e d , the problem i s to select hypothesised prerequisites and arrange these i n a h i e r a r c h i c a l manner.  Although i n i t i a l l y these prerequisites are selected l o g i c a l l y  on an a p r i o r i basis, a hypothesised prerequisite i s concluded to be pedagogically explains:  necessary only a f t e r empirical investigation.  "a subordinate s k i l l i s determined to be  As Gagne  pedagogically  necessary i f i t f a c i l i t a t e s the learning of the higher-level s k i l l to which i t i s related.  In contrast, i f the subordinate s k i l l has  been previously mastered, there w i l l be no f a c i l i t a t i o n of the level s k i l l .  This l a t t e r condition does not mean that the  l e v e l skill^cannot be learned —  not  higher-  higher-  only that, on the average, i n the  group of students for whom a topic sequence has been designed, learning w i l l not be accomplished readily"."'"''" Thus i f transfer differences between the T l and T2 groups were observed, Gagne, rather than trying  10  Gagne, op. c i t . , p.  1:L  I b i d . , p. 239-240.  2^8'-  12 to explain the differences i n terms of any p a r t i c u l a r learning theory, would probably a t t r i b u t e these differences to the selections and arrangement of prerequisites, since he seems to be more concerned with the development of empirically validated,-hierarchies than the v a l i d a t i o n of contemporary psychological theories.  Hence this study could prove  to be valuable for the curriculum designer i f i t produced a more e f f e c t i v e i n s t r u c t i o n a l sequence for teaching i n i t i a l m u l t i p l i c a t i o n skills. Ausubel would view the p o t e n t i a l e f f i c i e n c y of each i n s t r u c t i o n a l sequence for promoting transfer i n quite a d i f f e r e n t sense than would Gagne.  For Ausubel, the amount of transfer brought to a learning  task depends on an individual's cognitive structure, where "cognitive structure" means an individual's organization, s t a b i l i t y , and  clarity  ' 12 of knowledge i n a p a r t i c u l a r subject-matter f i e l d at any given time. That i s e x i s t i n g cognitive structure i s regarded as the major factor influencing the learning and retention of p o t e n t i a l l y meaningful material in the same f i e l d .  new  According to Ausubel, a major c r i t e r i o n  determining whether learning material i s p o t e n t i a l l y meaningful i s i t s r e l a t a b i l i t y to the p a r t i c u l a r cognitive structure of a p a r t i c u l a r learner.  As Ausubel states: for meaningful learning to occur i n f a c t , i t i s not s u f f i c i e n t that the new material simply be relatable to relevant ideas . i n the abstract sense of the term. The cognitive structure of the p a r t i c u l a r learner must include a r e q u i s i t e i n t e l l e c t u a l . capacities, ideational content and experientfa^, background.  Ausubel, pp. c i t . , p. 'ibid. , p . 23.  26.  13 The key concern of this study i s the e f f e c t of these i n s t r u c t i o n a l sequences on cognitive structure.  That i s , which of the  T l and T2 sequences might be best integrated by the learner and i n what sense t h i s act of integration promotes greater transfer to tasks requiring the use of an untaught higher-order According*to  algorithm.  Ausubel, new learning i s sometimes incorporated 14  into cognitive structure by c o r r e l a t i v e subsumption.  This psychological  phenomenon occurs when a learner somehow determines that new learning material i s related to relevant cognitive subsumers v i a some general principle.  Thus new learning material may be best incorporated into  an individual's cognitive structure i f those p r i n c i p l e s which require the least extension act as subsumers.  In Ausubel's terms one might  suppose that the learning of the algebraic p r i n c i p l e s of arithmetic may a f f e c t the learning of l o g i c a l l y related computational in the same sense as 'advance organizers'.  algorithms  Thus i t i s hypothesised that  the T2 i n s t r u c t i o n a l tasks might form r e l a t i v e l y stronger subsumers than the T l tasks, f o r future transfer tasks requiring the use of a higher-order  algorithm.  For example, consider the possible d i f f e r i n g complexity of extension  from.the standard m u l t i p l i c a t i o n algorithm to the higher-  order algorithm that each treatment group must make for successful solution of such a transfer task as 107 x 11.  Ausubel, op. c i t . , p. 77.  14  A t y p i c a l solution that might be exhibited by the T2 group could be as follows: 11 x 107  ^  10+1 100 + .7  ( p a r t i t i o n i n g both factors into (binary sums involving powers of ten  10 + 1 100 + 7 7 70 100 1000  ( a p p l i c a t i o n of d i s t r i b u t i v e /law and annexation algorithm  It i s assumed that no 'new' concept or s k i l l i s required for successful 1  extension from the standard m u l t i p l i c a t i o n algorithm to the higherorder algorithm. (The s k i l l of p a r t i t i o n i n g both factors, rather than just one factor, before application of the d i s t r i b u t i v e p r i n c i p l e was  included i n both i n s t r u c t i o n a l sequences.) The extension of the rote-type standard m u l t i p l i c a t i o n  algorithm to the standardized higher order algorithm by the T l procedure seems a very remote p o s s i b i l i t y : 11 x 107 "  11 ^ x 107 107 107  f"move over one space to the l e f t ^when multiplying by a factor of ten"  Suppose a T l group member attempts to compute such products as 107 x 11 by considering the 11 as 'one d i g i t ' and proceeds^as with the standard m u l t i p l i c a t i o n algorithm: 107 x 1 = 107 x d-0 7J 7  ll x (L0_7J 1177 1  Q  as with standard m u l t i p l i c a t i o n algorithm, place 'seven'.and 'carry ten'  (107 x 1) + 10 = 117  place 'seven' and  'carry 11'  I  15 '  Although such p a r t i a l products as 7 x 11 could be computed by using the T l standard m u l t i p l i c a t i o n algorithm,this extended procedure.' seems much more d i f f i c u l t than the hypothesised T2 procedure. Transfer to tasks involving 3 x 3  and 4 x 3  products would seem  even more unlikely considering the complexity of extending the T l standard algorithm.  HYPOTHESES  Most textbooks andypractitioners are being urged by curriculum s p e c i a l i s t s to promote the understanding of algebraic principles.  The arguments for the inclusion of such p r i n c i p l e s are  based on the b e l i e f that much of arithmetic, and especially computational algorithms, may be better understood through the learning of algebraic p r i n c i p l e s .  Hence from both a p r a c t i c a l and'ahtheoretical  point of view, i t seems worthwhile to investigate the v a l i d i t y of the following hypotheses: •  Hypothesis One  —The  T l group w i l l score  higher than the T2 group i n the performance  significantly  of the standard  m u l t i p l i c a t i o n algorithm, as measured by the performance Hypothesis Two —  test.  The T2.group w i l l score s i g n i f i c a n t l y  higher than the T l group on the test of transfer from the standard m u l t i p l i c a t i o n algorithm to a higher-order algorithm.  CHAPTER II SURVEY OF THE LITERATURE  In reviewing the l i t e r a t u r e , one soon r e a l i z e s that very few studies have been concerned with children's acquisition or use of the f i e l d of postulates to generate algorithms. Children's understanding of the f i e l d postulates without formal i n s t r u c t i o n was studied by Crawford i n 1964.^  Using a  multiple choice test of 45 items, he tested each of the eleven f i e l d axioms once at each l e v e l of Bloom's taxonomy. the  He found that  mean'scores increased s i g n i f i c a n t l y , from one even numbered grade  to the next, ..in. a l i n e a r manner.  Students exposed to 'modern  mathematics' content i n grades 9 and 10 had scores s i g n i f i c a n t l y superior to those of students i n a l l other programs at the same level.  This study seems important i n that i t provides data on  developmental processes which were occuring without e x p l i c i t teaching. A study by H a l l attempted to determine whether the rote learning of c e r t a i n m u l t i p l i c a t i o n combinations could be accomplished more e f f e c t i v e l y through teaching procedures emphasising the commutative and ordered pair :approach i n conjunction with practice on related 16' combinations.  This procedure was compared to teaching procedures  ."^Douglas Crawford, "An Investigation of Age-Grade Trends i n Understanding.the F i e l d Axioms," Dissertation Abstracts, Syracuse University, 1964. "^Kenneth D. H a l l , "An Experimental Study of Two Methods of Instruction.for Mastering M u l t i p l i c a t i o n Facts at the Third-Grade Level," Doctoral Dissertation, Duke University, 1967.  17  employing the t r a d i t i o n a l approach with practice on commuted combinations. He found no s i g n i f i c a n t difference between the groups on both arithmetic computation and achievement  in multiplication.  This  result lends support to the notion that there i s no advantage i n the mere acquisition of a f i e l d postulate. Gray.,, i n 1964, t r i e d to determine how a method of teaching introductory m u l t i p l i c a t i o n which stressed development of an understanding of the d i s t r i b u t i v e law would r e l a t e to pupil development as measured i n terms of achievement,  transfer, retention and progress  toward maturity of understanding of multiplication."*"^ treatment groups. cOne group, T l was  He used two  taught according to what was  judged to be. the ..best of current .methods.  The other group, T2,.was  provided with introductory m u l t i p l i c a t i o n using an understanding of the d i s t r i b u t i v e p r i n c i p l e . were covaried.  Pre-experimental achievement  and  I.Q.  He constructed written pre-test, post-test, retention,  and transfer.tests.  Individual interviews of 110 random subjects  measured maturity of understanding.  His results warranted the follow-  ing conclusions: 1.  A program of arithmetic i n s t r u c t i o n which introduces  m u l t i p l i c a t i o n by a method which stressed understanding of the d i s t r i b u t i v e property produced r e s u l t s superior to those of current methods. 2.  Understanding of. the d i s t r i b u t i v e property enables  children to proceed independently to the finding of products of  Roland.F. Gray, "An Experimental Study of Introductory M u l t i p l i c a t i o n , " D o c t o r a l Dissertation, University of C a l i f o r n i a , Berkeleyi 1964. •  18  novel m u l t i p l i c a t i o n combinations,  t o a g r e a t e r e x t e n t than  c h i l d r e n not i n t r o d u c e d t o the d i s t r i b u t i v e 3.  principle.  These c h i l d r e n appeared not t o d e v e l o p an  o f the d i s t r i b u t i v e p r o p e r t y u n l e s s i t was  There have been r e l a t i v e l y  Jf ew ;  those  understanding  specifically  taught.  s t u d i e s w h i c h have been  concerned w i t h the r e l a t i o n s h i p between u n d e r s t a n d i n g  o f the  p o s t u l a t e s and l e a r n i n g of c o m p u t a t i o n a l  In most s t u d i e s  algorithms.  t h e a l g o r i t h m s were i l l u s t r a t e d u s i n g p h y s i c a l d e v i c e s . S c h r a n k l e r t r i e d t o e v a l u a t e the e f f e c t i v e n e s s o f two treatments.in. combination  field  However,  pre-algorithm  w i t h two a l g o r i t h m s f o r t e a c h i n g  the 18  m u l t i p l i c a t i o n o f whole numbers at t h r e e i n t e l l i g e n c e  levels.  E f f e c t i v e n e s s was  skills,  e v a l u a t e d i n terms of c o m p u t a t i o n a l  i n computation, understanding  speed  of the m u l t i p l i c a t i o n p r o c e s s , problem  s o l v i n g and r e t e n t i o n of t h e f o u r p r e v i o u s c r i t e r i a .  The  readiness  phase p l a c e d emphasis on the 100 m u l t i p l i c a t i o n f a c t s f o r group Emphasis was  p l a c e d on t h e commutative, a s s o c i a t i v e and  p r o p e r t i e s f o r group H^.  F o l l o w i n g t h i s p e r i o d , these  were s u b d i v i d e d i n t o a l g o r i t h m i c groups. taught  Group  B^.  distributive groups  s u b j e c t s were  the i n d e n t u n i t - s k i l l s  algorithm: 57 x 28 . 456 114 1596  • W i l l i a m . S c h r a n k l e r , "A Study o f the E f f e c t i v e n e s s o f Four Methods o f Teaching M u l t i p l i c a t i o n of-.Whole Numbers i n Grade F o u r , " D i s s e r t a t i o n A b s t r a c t s , U n i v e r s i t y o f M i n n e s o t a , 1966.  19  Group  subjects were taught the p a r t i a l products algorithm: 57 x 28 56 400 140 1000 1596  No mention was made of the use of the annexation algorithm i n the p a r t i a l products algorithm. A  1 1' 2 1' 1 2' 2 2' B  A  B  A  B  intelligence.  A  B  w  e  r  e  Students i n each of the treatment groups,  i d e n t i f i e d at one of three levels of  Schrankler found that the A^B^  group tested higher on  the test of understanding than the other groups.  This same group  also tested higher on the retention test of understanding.  The  fact that theiA^B^ group was found to be superior to the A^B^  group  on the test of understanding of the m u l t i p l i c a t i o n algorithm i s of particular.interest.  This r e s u l t suggests that the understanding  of computational algorithms i s best promoted by the e x p l i c i t application of.previously acquired algebraic p r i n c i p l e s .  Studies  such as Schrankler's have b een r e s t r i c t e d to examining the use of algebraic p r i n c i p l e s i n promoting understanding of already acquired computational algorithms.  No:studies were found which examined  the use of algebraic p r i n c i p l e s i n promoting transfer to untaught higher-order algorithms.  CHAPTER I I I DESIGN OF THE EXPERIMENT  THE SAMPLE The e x p e r i m e n t e r d e c i d e d t o use grade t h r e e s t u d e n t s as s u b j e c t s i n t h e study s i n c e they had.had some e x p e r i e n c e w i t h m u l t i p l i c a t i o n b u t had n o t as y e t been taught t h e s t a n d a r d m u l t i p l i c a t i o n algorithm.  E i g h t grade t h r e e c l a s s e s were s e l e c t e d from s i x B r i t i s h  Columbia s c h o o l s .  A l l e i g h t o f the t e a c h e r s i n v o l v e d i n t h e s t u d y  were v o l u n t e e r s . A f t e r t h e r e a d i n e s s phase, w h i c h w i l l be d e s c r i b e d i n t h e n e x t s e c t i o n , s t u d e n t s i n each c l a s s r o o m were randomly a s s i g n e d t o e i t h e r t h e Tl.'or T2 group.  A s t u d e n t ' s t e s t s c o r e s were o m i t t e d  from t h e study i f more than one treatment l e s s o n was m i s s e d .  A  t o t a l of 238 s u b j e c t s were used, t o o b t a i n t h e f i n a l s e t o f d a t a ; 44 s u b j e c t s were used f o r t e s t a n a l y s i s , and t h e r e m a i n i n g 194 s u b j e c t s for  t e s t i n g t h e hypotheses.  THE INSTRUCTIONAL SEQUENCES A.  The Readiness  Phase.•  D u r i n g t h i s phase, a l l t h e s u b j e c t s were taught the s k i l l s w h i c h were c o n s i d e r e d t o be p r e r e q u i s i t e s f o r the t r e a t m e n t phase.  21  A set of lesson plans was provided f o r each teacher involved i n the study.  B r i e f l y , these lessons stressed: -  the relationship  between m u l t i p l i c a t i o n  and arrays.  For example, 3x4 -  means a "3 by 4" array  the d i s t r i b u t i v e law; both the right hand and the l e f t hand.  This was to be accomplished by breaking arrays into  the "sums" of smaller arrays. 4x5  4 x  (2 + 3)  For example: (4 x 2)  +  (4 x 3)  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  the application  of the d i s t r i b u t i v e law to m u l t i p l i c a t i o n  problems. Only the techniques of breaking a product into the sums of smaller products was stressed and no attempt was made to have children provide a f i n a l numerical answer. 28 x 19  For a f u l l description  20+8 x 19  For example, 20+8 x 19 19 x 8 + 19 x 20  of these lessons, the reader i s referred to  Appendix B. In order to p a r a l l e l t y p i c a l teaching practices  and thus  increase the g e n e r a l i z a b i l i t y of this study, the writer did not demand a fixed c r i t e r i o n of mastery of the d i s t r i b u t i v e law. Rather, a l l teachers were instructed  to terminate this phase when, i n t h e i r  judgement, the students indicated  a mastery of the d i s t r i b u t i v e law.  22  The t e a c h e r s r e p o r t e d t h a t t h i s phase g e n e r a l l y took about f i v e hours o f c l a s s r o o m  B.  instruction.  The Treatment Phase. Every t e a c h e r was p r o v i d e d w i t h a s e t o f w r i t t e n l e s s o n p l a n s  s u i t a b l e f o r each treatment  lesson.  The l e s s o n s c o n t a i n e d the g e n e r a l ,  d i a l o g u e , examples and seatwork t o be used.  The t e a c h e r s met w i t h  the w r i t e r t w i c e d u r i n g t h i s phase t o ensure t h a t they the l e s s o n m a t e r i a l s .  To m i n i m i z e  the e f f e c t of teacher d i f f e r e n c e s  each t e a c h e r t a u g h t b o t h groups w i t h i n h e r c l a s s . i n t e r a c t i o n , i t was a r r a n g e d treatment  lesson.  understood  To m i n i m i z e  pupil  t o have t h e groups s e p a r a t e d d u r i n g a  A l l p u p i l s were t o l d by t h e i r t e a c h e r they  were i n v o l v e d i n an experiment.  To m i n i m i z e  they  outside influences,,  t e a c h e r s were i n s t r u c t e d t o g i v e no;;homework during t h i s phase. the T l and T2 groups had a p p r o x i m a t e l y f o u r hours o f treatment A b r i e f d e s c r i p t i o n o f b o t h treatments  Both time.  i s provided i n the following  s e c t i o n b u t t h e r e a d e r i s r e f e r r e d t o Appendix C f o r t h e l e s s o n p l a n s used.  The T l I n s t r u c t i o n a l Sequence  The T l group was taught t h e r o t e - t y p e a l g o r i t h m d e s c r i b e d i n Chapter  I.  The a l g o r i t h m was r e s t r i c t e d t o 2 x 1 and 3 x 1  products.  To c o n v i n c e t h e s t u d e n t s o f t h e - l e g i t i m a c y o f t h i s . - ,  a l g o r i t h m , a l l answers were i n i t i a l l y checked .using t h e d i s t r i b u t i v e law.  F o r example^ the check might be made as f o l l o w s :  23  x  12 9  *  C 2 + 2 + 2 + 3 + 3 x 9  18 + 18 + 18 + 27 + 27 = 108 *  In contrast to the T2 Instructional Sequence, no e x p l i c i t application of the d i s t r i b u t i v e p r i n c i p l e was stressed.  Once the students were  convinced that this rote-type algorithm yielded correct products, the objective of the succeeding lessons was merely to provide further practice.  The T2 Instruction Sequence The T2 subjects were f i r s t taught the annexation algorithm. A l l computation  of 2 x 1 and 3 x 1 products were accomplished by  using the d i s t r i b u t i v e p r i n c i p l e . i n conjunction with the annexation algorithm.  The teachers were instructed to use the same examples and  seatwork with both  groups.  THE MEASURING INSTRUMENTS  Both the performance and transfer tests were written tests constructed by the experimenter.  The teachers knew the general  nature of each test p r i o r to the treatment phase but did not see the actual test items u n t i l the test administration date.  Teachers were  instructed to give students ample time to complete both tests. solution by repeated addition was disregarded f o r both tests. reader i s referred to Appendix D f o r the actual tests used.  Any The  24  A. .Performance Test. This test consisted of twenty items that required the use of the standard m u l t i p l i c a t i o n algorithm. responses was  The t o t a l number of correct  considered a measure of an individual's performance.  In  order to delete items that were either excessively d i f f i c u l t or easy, a point b i s e r i a l correlation was  calculated for every item.  decided to reject an item i f the point b i s e r i a l r was  It was  less than  .20  19 in magnitude.  As a result of this analysis, a l l items of the o r i g i n a l  test were retained.  Since this test was designed to measure a very  s p e c i f i c t r a i t , ( v i z . the a b i l i t y to use the standard m u l t i p l i c a t i o n algorithm), i t was determined. was  f e l t that a measure of item homogeneity should be  Thus a KR20 was  found to be .93.  calculated for the twenty item test and  This value indicated that the performance test  was high i n item homogeneity.  The results of the items analysis can  be found i n Table I. B> Transfer Test. This test consisted of fourteen items which were intended to measure the a b i l i t y to compute novel products requiring the use of a higher-order algorithm.  Neither the T l group nor the T2 group had  been previously exposed to any of these items. correct responses was  The t o t a l number of  considered a measure of the a b i l i t y to extend  the standard m u l t i p l i c a t i o n algorithm.  As with the performance test,  Nunnally, J.C., Psychometric Theory, (New York: H i l l Book Company, 1967), p. 242.  McGraw-  25  a KR20 was c a l c u l a t e d ' t o e v a l u a t e i t e m homogeneity. f i n a l f o u r t e e n i t e m t e s t was found t o be .78.  The KR20 o f t h e  I t i s p o s s i b l e that the  KR20 might have been i n c r e a s e d i n magnitude by i n c l u d i n g a d d i t i o n a l t e s t items.  However, t h i s l e n g t h e n i n g p r o c e d u r e was f e l t t o be  i n a p p r o p r i a t e s i n c e a v e r y l e n g t h y t e s t might have had t h e u n d e s i r a b l e e f f e c t o f i n c r e a s i n g t e s t a n x i e t y o f such young and ' t e s t immature' students.  The r e s u l t s o f t h e i t e m a n a l y s i s can be found i n T a b l e I I .  26  TABLE I ANALYSIS OF THE PERFORMANCE TEST  Item Number 1 2 3 4 5 6 7 8 9 10  Point Biserial  Item Number.  Point Biserial  .45 .61 .50 .77 .64 .75 .74 .70 .54 .64  11 12 13 14 15 16 17 18 19 20  .53 .73 .74 .72 .49 .67 .72 .64 .74 .67  TABLE II  ANALYSIS OF THE TRANSFER TEST  Item Number  1 2 3 4 5 6 7 8 9 10 11 12  -  Deleted items.  Point Biserial  Item Number  .50 .59 .43 .59 .65 .65 0.0 * .56 .46 .43 .65 .51  13 14 15 16 17 18 19 20 21 22 23  '  Point Biserial  0.0 * 0.0 * 0.0 * 0.0 * 0.0 * 0.0 * 0.0 * .24^ .47 0.0 * .69  CHAPTER IV ANALYSIS OF THE DATA  EXPERIMENTAL RUN A. ;  The Performance H y p o t h e s i s . The s t a t i s t i c a l , h y p o t h e s e s t o be t e s t e d were: H :  There w i l l be no 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 means  c  of t h e T l and T2 groups as measured by t h e performance test. That i s : ^ H^:  T  =/(T  2  The mean o f t h e T l group w i l l be s i g n i f i c a n t l y g r e a t e r than t h e mean o f t h e 1^ group as measured by t h e performance t e s t . That i s :  T  ±  ^> J4^ T  2  Each c l a s s r o o m t e a c h e r taught b o t h t h e T l and T2 groups i n her classroom.  Thus a s u b j e c t i n a c l a s s r o o m was g i v e n e i t h e r t h e  T l o r T2 i n s t r u c t i o n by h i s o r h e r u s u a l c l a s s r o o m t e a c h e r . experimenter  The  c o n s i d e r e d t h e d i f f e r e n c e s i n t e a c h e r performance t o be  a random e f f e c t , w h i l e d i f f e r e n c e i n treatment were c o n s i d e r e d to be a f i x e d e f f e c t .  I n o t h e r words, a mixed a n a l y s i s o f v a r i a n c e  model was f e l t t o be t h e most a p p r o p r i a t e s t a t i s t i c a l model t o t e s t the h y p o t h e s i s .  The l i n e a r model chosen was:  i n d i c a t e s t h e i t h t e a c h e r (random e f f e c t ) and r e p r e s e n t s t h e j t h treatment l e v e l ( f i x e d  effect).  28  The experimenter made the usual assumptions underlying an ANOVA but did not test for these as the F test i s reasonably robust to v i o l a t i o n s 20 of these assumptions. a)  The assumptions made were:  the teachers, used i n the experiment, were randomly selected from a normal population, i . e . ±  are NID (0, ^  W i j b)  I  D  C  ) and  ° ' ^ >  the£"^_.^ are normally distributed, i . e .  £ c)  a r e N  2  Co, 6£),  a r e N I D i j k  the treatment variances are homogeneous, i . e .  Cf  T 1  =  G  ,  n  The reader i s reminded that the denominator i n the test f o r treatment  (fixed) effects i n a mixed model i s the interaction term 21  and not the usual error term.  The n u l l hypothesis was considered  to be rejected i f the probability of obtaining an F value, under the n u l l hypothesis, was less than or equal to ^  = .05.  All  calculations were done at the University of B r i t i s h Columbia Computer Centre using the BMD-X64 program.  This program allows for d i f f e r i n g  numbers of subjects i n a c e l l by using the least squares estimate technique.  The results of this analysis are summarized i n Table I I I . 20  Lindquist, E.F., Design and Analysis of Experiments in Psychology and Education (Boston: Houghton, M i f f l i n Company, 1953), pp. 78-90. and W i n s t o n ^ o i f ? S ^ 3 - 4 4 t  B t  *  B t  *  n p  (  N  e  W  Y  °  r  k  :  H  o  l  t  '  R  i  n  e  h  a  r  t  29 TABLE I I I ANALYSIS OF VARIANCE:  Source o f Variation Teacher Treatment Interaction Error  PERFORMANCE HYPOTHESIS  df  Sum o f Squares  7 1 7 178  812.197 1775.630 57.679 3927.752  Mean Squares 116.028 1775.630 8.240 22.066  5.258 215.491 0.373  .0000083  Mean f o r T l group was 14.418 Mean f o r T2 group was 8.330 S i n c e t h e p r o b a b i l i t y o f o b t a i n i n g an F - v a l u e o f 215.491 was c a l c u l a t e d t o be f a r l e s s * t h a n  .05, t h e n u l l h y p o t h e s i s H  c  was r e j e c t e d  and the a l t e r n a t e h y p o t h e s i s H^ was a c c e p t e d .  Bu,  The T r a n s f e r H y p o t h e s i s . The s t a t i s t i c a l hypotheses t o be t e s t e d were: H : Q  There w i l l be no 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 means o f t h e T l and T2 group as measured by t h e transfer test. That i s :  H^:  J^  =  T±  J  L  (  T  2  The mean o f t h e T2 group w i l l be g r e a t e r than t h e mean of t h e T l group as measured by t h e t r a n s f e r t e s t . That i s :  y6^  T 2  >  /<  T 1  O r i g i n a l l y t h e e x p e r i m e n t e r had hoped t o t e r m i n a t e t h e treatment phase o n l y when b o t h groups had reached a s p e c i f i e d p e r formance c r i t e r i o n .  That i s , u n t i l t h e r e were no s i g n i f i c a n t d i f f e r e n c e s  between the two groups on t h e performance o f t h e s t a n d a r d m u l t i p l i c a t i o n  algorithm.  Thus, i f any degree o f . c o r r e l a t i o n e x i s t e d between  performance and t r a n s f e r t a s k s , t h i s p r e l i m i n a r y e q u a t i n g  the -  would  m i n i m i z e any d i f f e r e n c e s between t h e groups on t h e t r a n s f e r t e s t t h a t might be a r e s u l t o f d i f f e r e n c e s between t h e means on t h e performance t a s k s .  However, t o b r i n g about t h e e q u a l i t y o f t h e  groups on t h e performance t e s t , performance s c o r e s were with transfer scores.  covaried  Thus t h e l i n e a r model used t o t e s t t h e  t r a n s f e r h y p o t h e s i s was:  Y  +T3  i j k  <xv  ±i  +  a...  - x.. ) k  +e.. k  where T'j 1  € w'  a  i s an e s t i m a t e  n  d £ „  were p r e v i o u s l y  k  o f t h e common p o p u l a t i o n  defined; regression  coefficient; X...  i s a s u b j e c t ' s performance s c o r e and  X...  i s t h e grand mean o f t h e t o t a l sample on t h e performance test.  In a d d i t i o n t o t h e n e c e s s a r y assumptions u n d e r l y i n g an ANOVA t h a t were discussed  i n the previous  s e c t i o n , t h e use o f t h i s model n e c e s s i t a t e s  the f o l l o w i n g a d d i t i o n a l a s s u m p t i o n s : a)  the population w i t h i n - c e l l r e g r e s s i o n c o e f f i c i e n t s are homogeneous, i . e .  8  w  =  £w..  f  0  r  3  1  1  Because l i t t l e i s known about t h e F t e s t w i t h r e s p e c t t o v i o l a t i o n of the foregoing  a s s u m p t i o n , i t was d e c i d e d t o t e s t t h i s  assumption a t a l e v e l o f s i g n i f i c a n c e e q u a l t o .10. U s i n g t h e BMD-X82 computer program, w h i c h a d j u s t s f o r d i f f e r i n g numbers o f s u b j e c t s  31 i n a c e l l , an F o f 1.06, w i t h a numerator and denominator o f 15 and 162 degrees o f freedom r e s p e c t i v e l y , was o b t a i n e d .  Since the  p r o b a b i l i t y o f o b t a i n i n g such an F, under t h e n u l l h y p o t h e s i s i s .398, homogeneity o f t h e r e g r e s s i o n c o e f f i c i e n t s was assumed.  b)  the pooled e s t i m a t e ^ ' i s not zero. w  In t e s t i n g t h i s assumption a t t h e . 0 5 l e v e l o f s i g n i f i c a n c e , an F o f 99.14, w i t h a numerator and denominator o f 1 and 177 degrees of freedom r e s p e c t i v e l y , was o b t a i n e d .  Since the p r o b a b i l i t y of o b t a i n i n g —8  such an F, under t h e n u l l h y p o t h e s i s , i s l e s s than 10 of z e r o s l o p e was e a s i l y  the hypothesis  rejected.  The r e s u l t s o f t h e s t a t i s t i c a l a n a l y s i s  o f the t r a n s f e r  h y p o t h e s i s a r e summarized i n T a b l e I V . TABLE IV ANALYSIS OF COVARIANCE:  Source o f Variation  ^  Teacher Treatment Interaction Error  7 1 7 177  TRANSFER HYPOTHESIS  Adjusted Sum Square  179.984 24.363 62.570 1149.077  Mean Square  25.712 24.363 8.939 6.492  3.961 2.725 1.377  .141  A d j u s t e d mean f o r T l group was 3.776 A d j u s t e d mean f o r T2 group was 4.544 S i n c e t h e p r o b a b i l i t y o f o b t a i n i n g an F o f 2.725 i s .141, t h e n u l l h y p o t h e s i s was a c c e p t e d .  That i s , the: mean o f t h e T2 group was h i g h e r ,  but n o t s i g n i f i c a n t l y h i g h e r , than t h e mean o f t h e T l group.  1  CHAPTER V CONCLUSIONS AND IMPLICATIONS FOR FURTHER STUDY  DISCUSSION OF CONCLUSIONS A./  Performance Hypothesis. With respect to the performance hypothesis, i t was found  that subjects taught a rote-type algorithm did s i g n i f i c a n t l y better on tasks requiring the use of a standard m u l t i p l i c a t i o n algorithm than did the subjects taught a standard m u l t i p l i c a t i o n algorithm using previously learned algebraic p r i n c i p l e s .  In fact, the performance  l e v e l of the T2 group was so i n f e r i o r to that of the T l group that this researcher suspected thafc.t^ei'dfaihep'teacliers hadlnot-followed the " recommended treatment procedures.  It was quite possible that, since  most teachers had never used an i n s t r u c t i o n a l sequence l i k e the T2 sequence, they may have had an experimental bias towards the rote T l sequence.  Perhaps more frequent observations of teacher  performance would have eliminated such a bias towards treatment.  B,  Transfer Hypothesis. With respect to the transfer hypothesis, i t was found that  subjects taught a standard m u l t i p l i c a t i o n algorithm using algebraic principles appeared to exhibit superior p o s i t i v e transfer to tasks requiring the use of a higher-order algorithm.  However, this difference  33  in the amount of transfer was not statistically significant at the oC  =  level of significance. Because of the nature of the treatments, a T2 subject needed  more time fo format correctly a computational problem than did a Tl subject.  Thus, teachers were instructed to give students  at least one hour to attempt a l l fourteen items of the transfer test. However, after a brief discussion with the teachers, i t was noted that some had allowed students about thirty minutes to complete this test.  In fact, one teacher who obviously misunderstood the intent of  the transfer test, stated that she gave children about fifteen minutes on this test because "the students weren't taught to compute such large products".  This^situation could not be remedied by another  test administration because school holidays immediately followed the test administration date.  PROBLEMS FOR FURTHER STUDY Since the results of this study must remain inconclusive because of important uncontrolled factors, a replicate study employing controls to minimize teacher misunderstandings should be conducted. This writer also suggests that a study be conducted to examine the effect of instructional sequences that use algebraic principles to teach computational algorithms on a student's attitude toward arithmetic.  It is postulated that instructional sequences that  maximize the use of previously learned algebraic principles may enable a student to view arithmetic as a series of integrated tasks.  This inte-  grated view of arithmetic, might, in turn, have a positive effect on a student's attitude towards arithmetic.  BIBLIOGRAPHY  A u s u b e l , D a v i d P. The P s y c h o l o g y o f M e a n i n g f u l V e r b a l L e a r n i n g . New »>York: Grune and St r a t t o n , 1963. Bruner, Jerome S. Books, 1963.  The P r o c e s s o f E d u c a t i o n .  New Y o r k :  Vintage  The Cambridge Conference on S c h o o l Mathematics. Goals f o r S c h o o l Mathematics. New York: Houghton M i f f l i n Company, 1963. Crawford, Douglas Houston. "An I n v e s t i g a t i o n of Age-Grade Trends i n U n d e r s t a n d i n g the F i e l d Axioms." D i s s e r t a t i o n A b s t r a c t s , Syracuse U n i v e r s i t y , 1964. E i c h o l z , Robert E., e t a l . Elementary S c h o o l Mathematics, Book 3. Don M i l l s , O n t a r i o : Addison-Wesley (Canada) L t d . , 1966. Gagne, Robert M. The C o n d i t i o n s o f L e a r n i n g . R i n e h a r t and W i n s t o n , 1970.  New  York:  Holt,  Gray, Roland F r a n c i s . "An E x p e r i m e n t a l Study of I n t r o d u c t o r y M u l t i p l i c a t i o n . " D o c t o r a l D i s s e r t a t i o n , U n i v e r s i t y of C a l i f o r n i a , B e r k e l e y , 1964. H a l l , Kenneth Dwight. "An E x p e r i m e n t a l Study of Two Methods of I n s t r u c t i o n f o r M a s t e r i n g M u l t i p l i c a t i o n F a c t s at t h e T h i r d Grade L e v e l . " D i s s e r t a t i o n A b s t r a c t s , Duke U n i v e r s i t y , 1967. Hays, W i l l i a m L. 1963.  Statistics.  New  York:  H o l t , R i n e h a r t and  Winston,  L i n d q u i s t , E.F. Design and A n a l y s i s of Experiments i n P s y c h o l o g y and E d u c a t i o n . B o s t o n : Houghton M i f f l i n Company, 1953. MacPherson, E r i c . D . "The Foundations o f • E l e m e n t a r y S c h o o l Mathematics." The Modern I n s t r u c t o r , V o l . 33, September 1 9 6 4 - A p r i l 1965. M a c S h e l l , Leo. "Two A s p e c t s of I n t r o d u c t o r y M u l t i p l i c a t i o n : The A r r a y and the D i s t r i b u t i v e P r o p e r t y . " D i s s e r t a t i o n A b s t r a c t s , S t a t e U n i v e r s i t y of Iowa, 1964. N u n n a l l y , J.C. P s y c h o m e t r i c Theory. Company, 1967.  New Y o r k :  M c G r a w - H i l l Book  S c h r a n k l e r , W i l l i a m Jean. "A Study of the E f f e c t i v e n e s s o f Four Methods f o r Teaching M u l t i p l i c a t i o n of Whole Numbers i n Grade F o u r . " D i s s e r t a t i o n A b s t r a c t s , U n i v e r s i t y of M i n n e s o t a , 1966.  APPENDIX A THE IDENTIFICATION OF ANOTHER INTERNAL ALGORITHM  THE DIVISION OF FRACTIONS ALGORITHM AND THE EQUAL FACTORS ALGORITHM  In addition to the mere rote, performance of an algorithm, most modern programs attempt to provide some rationale of that algorithm.  Perhaps the most d i f f i c u l t algorithm to explain  reasonably to the average elementary school c h i l d i s the d i v i s i o n of fractions algbarithm.  In an attempt to provide this rationale, 22  a t y p i c a l approach i s as follows: 1 - 3 Step 1 5 y T--^ = W  The work below shows how to 1 3 11 divide 5TT by — . Use — as another 2 4 2  a  name for oy. Step 2  ^  Express the "divisiori-vr. i n this way. 3.  .'4 Step 3  4 4  3  '  11 — x—  3  x — 11  4  d i v i s o r , you multiply — x — to get 1.  3  X  Step 4  F i r s t you need to get 1 for the  11 9 _J 4_  4  J  4  has been multiplied by —. So you  must also multiply —  x —•  4  —— x —  You do not need to write the d i v i s o r when i t i s 1.  11 Step 5  4  —11 x -4 '' 2 S  5  I  2  X  3  i 3 * 4  =  So now you can write the computation  44 6  7^  •3 7  in t h i s way 7j  You found 7^- by multiplying -j by  M a u r i c e L. Hartung, et al.„ Seeing Through Arithmetic 6, Scott, Foresman and Co., Chxcago-7-p-r-198^ 22  37 One apparent assumption t h a t has been made i s t h a t t h e p r o c e d u r e s t a k e n i n Steps 3 and 4 can be f o l l o w e d by the e l e m e n t a r y school c h i l d .  However, t h e v a l i d i t y o f t h e s e two s t e p s must be  b l i n d l y a c c e p t e d by the c h i l d s i n c e no p r e l i m i n a r y work has been done t h a t c o u l d be used t o j u s t i f y these s t e p s .  One wonders what  advantages t h i s modern treatment has over the r o t e " i n v e r t and m u l t i p l y " a l g o r i t h m because a p p a r e n t l y we have m e r e l y s u b s t i t u t e d a long r o t e a l g o r i t h m f o r a short rote  algorithm.  What i s needed t o v a l i d a t e s t e p s 3 and 4 i s an i n t e r n a l a l g o r i t h m ; the e q u a l f a c t o r s a l g o r i t h m .  This a l g o r i t h m s t a t e s that  i f the d i v i s o r and d i v i d e n d a r e m u l t i p l i e d o r d i v i d e d by any nonzero r a t i o n a l number, the q u o t i e n t remains unchanged.  F o r example:  (8 4- 4) = (2 x 8) 4- (2 x 4) I f t h i s i n t e r n a l a l g o r i t h m i s mastered, the d i v i s i o n of f r a c t i o n s becomes much more r e a s o n a b l e Step 1  11 . 3  ~2  T  to the elementary school ,11  ~k ' ~2~ =  X  4 "J  N  . ,3 4"  X  4  child.  N  3"  equal f a c t o r s algorithm  Step 2  (—  x —; T (•£• x —) = ( —  x —) T 1  multiplication of r e c i p r o c a l s  Step 3 Step 4  (—  x —; T 1 = —  11 3 2 11 2 ' 3 ~~ 2  X  x —  _ 3  -  property  o f one  APPENDIX B  READINESS PHASE LESSON PLANS  THE READINESS PHASE  These three lessons should; enable most students to acquire the necessary prerequisite s k i l l s before the actual experimental treatment begins.  The teacher w i l l find that a l l lesson plans are  quite detailed including examples to, use, questions to ask, answers one can expect, and seatwork problems to be used after each lesson. In order to minimize any misunderstanding  that may  r e s u l t , w i l l the  teachers please observe closely the following instructions: 1.  Carefully read the lesson plans at least a day before the presentation.  If you have any questions  or suggestions, please don't hesitate to contact me. 2.  The phone number i s 736-0595.  Try to give the answers to a l l seatwork questions before the students leave school for that day.  Give  NO HOMEWORK as outside influences must be discouraged. 3.  Record any absenteeisms on the l i s t  provided.  4.  If more examples are needed to i l l u s t r a t e any  concept  before the seatwork i s attempted, please f e e l free to do more. 5.  If you f e e l that another period may be necessary, extend this phase for another period.  then  40  LESSON 1:  MULTIPLICATION AND ARRAYS  The basic objectives of the lesson are:. A.  To introduce the concepts of an array and i t s relationship to m u l t i p l i c a t i o n .  B.  To i l l u s t r a t e the commutative p r i n c i p l e for m u l t i p l i c a t i o n ; a x b = b x a ( i n this case, an a x b array, though drawn d i f f e r e n t l y , has the same number of elements as a b x a array).  1.  Introduction of an Array "Today we w i l l see how we can multiply using an array." (Write the word array on the board). "Here i s an example of an array." xxx xxx "This array i s c a l l e d a 2 x 3 array since i t has 2 rows of 3 crosses." „ 2x3 0  xxx  row 1  xxx  row 2  "We usually write the words '2 by 3' as '2 x 3'." Draw a 4 x 3 array on the board; ask children to give reasons for their responses. X  X  X  xxx xxx X  X  X  "This i s a 4 x 3 array because i t has 4 rows of 3."  41  xxx  row 1  4x3  xxx  row 2  array  xx x  row 3  xxx  row 4  Draw the following examples on the board (one at a time) and ask the children to name each.  Ask children to give reasons f o r t h e i r  responses. xxx xxx xxx xxx  x x x x x x  xxx  x x x x x x.  x x x x x x  2x6  1x6  5x3  array  (5 rows of 3)  array  (1 row of 6)  (2 rows of 6)  "Here i s the name of an array."  array  (Put 3 x 6 on the board).  "This time, t r y to draw what this array would look l i k e . " (Give children a few moments and then check i n d i v i d u a l pupil's work) Answer: 3 x6 array  x x x x x x  row 1  x x x x x x  row 2  x x x x x x  row 3  Ask the children to draw the following arrays: 1x4 2x5 3x7 Check p u p i l s ' work and ask reasons f o r t h e i r responses. 2.  The Commutative P r i n c i p l e for M u l t i p l i c a t i o n ; a x b = b x a "Can^anyone come up to the'board and draw a 2 x 4 array?" (Have a p u p i l come to the board and draw the array; ask the  c h i l d how many crosses are i n this array).  42  "Can^y anyone come up to the board and draw us a 4 x 2 o n  array? Draw attention to the fact that a 2 x 4 array and a 4 x 2'array have the same number of elements but are drawn d i f f e r e n t l y . array.  Repeat the same procedure using the 5 x 4  array and a 4 x 5  Draw an 8 x 4 array on the board. x x x x x x x x x x x x x x x x x x x x x x x x x x x x X  X  X.  x  Ask the children i f they can find another array which would have the same number of crosses, but would be drawn d i f f e r e n t l y . Note:  several answers are possible, but draw attention to  the fact that i f we rotate the array we end up with a 4 x 8 array. x x x x x x x x x x x x x x x x x x x x  x x x x x x x x  x x x x  x x x x x x x x  x x x x  x x x x x x x x  x x x x  x x x x x x x x  8x4  array  4x8  array  "Here i s a very large array (13 x 8).  Can*'anyone t e l l me  another array that would be drawn d i f f e r e n t l y but would have the same number of crosses?"  (Answer:  8 x 13).  43  If needed, do other examples to emphasize the point that an axb  3.  array has the same number of elements as a b x a array.  Seatwork These series of questions are to provide additional practice  with the concepts covered i n Lesson 1.  Please allow enough time f o r  marking the seatwork as this w i l l enable you to determine i f most of your class w i l l be ready for Lesson 2. A.  Name the following arrays. 1.  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  2.  :x x x x  3.  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  (Answer:  5x3)  (Answer:  1x7)  (Answer:  3x5)  (Answer:  7x1)  x X X X X X X  B.  Draw the following a-rrays. 1.  2x6  2.  8x2  3.  10 x 4  4. 5  1 x 11 11 x 1  44 C.  Name or draw another array which would have the same number of x's but would look d i f f e r e n t . 1.  2x6  2.  xxx xxx xxx xxx xxx  45 LESSON 2:  THE DISTRIBUTIVE  LAW  The basic objective of this lesson i s : To introduce both the l e f t hand and the right hand d i s t r i b u t i v e law.  The l e f t hand law states that a x (b + c) = (a x b) + (a x c ) . For example:  4 x 7-= 4 x (4 + 3) = (4 x 4) + (4 x 3).  The right hand law states that (b + c) x a = (b x a) + (c x a). For example:  8x6=  ( 3 + 5 ) x 6 = (3 x 6) + (5 x 6).  The teaching of both p r i n c i p l e s w i l l be accomplished bydividing an array into smaller arrays. Please do not use the terms right hand and l e f t hand d i s t r i butive laws with the children as this only leads to confusion. 1.  Review a)  Draw a 6 x 7 array on the board and ask the children the name of t h i s array..  Children should give reasons  for their answers. Example: b) 2.  there are s i x rows of seven x's.  Have a c h i l d come to the board and draw a 4 x-2 array.  "Let us look at the following array." x x x x x x x K't-X X X X  X X X  X X  X-  X  X  X  X  X  X X  X X  X X  X X  X  X  X X  "What i s the name of this array?"  (5x7)  "How could we f i n d out how many crosses there are i n that array?"  46  (Children w i l l probably o f f e r suggestions such as counting the  individual elements, adding 5 seven's e t c . ) . " A l l of these methods are fine, but here i s another interesting way.  Let's break up the 5 x 7  array into  smaller arrays l i k e t h i s . " Step 1 5  X  7  Step 2  Step 3  5 x (4 + 3)  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  (5  X  4) + (5 x 3  X  X  X  X  X  X  X  X  X  xxx  X  X  X  X  X  X  X  X  xxx  X  xxx  X  X  X  X  X  X  X  X  xxx  X  X  xxx  X  v.X  X  X  X  X  X  X  xxx  X  X  X  X  X  X  X  X  X  X  X  xxx  X  7 array equals a 5  X  X  "Notice that the 5 5x3  array.  X  X  X  4 array  Can any of you think of other ways of breaking  up this array?" (Let For  children suggest other p o s s i b i l i t i e s ) . example:  Step 1  Step 2  5x7  Step 3  5 x. (2 + 2 + 3)  (5 x 2) + (5 x 2) + (5 x 3)  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  X  Allow children to break up a 7 x 8 array.  Try to emphasize  Step 1, 2 and 3. For example: 7 x 8 = 7 x (3 + 5) = (7 x 3) + (7 x 5).  47 3.  The Right Hand D i s t r i b u t i v e Law "Here i s another array." x x x x x x x x x x x x  5x4  x x x x x x x x "We have been breaking these arrays up by renaming the second number." For example:  5 x 4 = 5 x (5. x 2) + (5 x 2)  "We can also/break an array into smaller arrays by renaming t  ^  ie  ^ i r s t number."  For example: Step 1  Step 2  x x x x  Step 3 3x4  3 x x x x x x x x x x x x  2  ^^ ^  2x4  x x x x 5x4 Ask children for further ways of breaking up t h i s array by renaming the f i r s t number. For example: 5x4=  (1 + 1 + 2 + 1) x 4 = (1 x 4) + (1 x 4) + (2 x 4) + (1 x 4) . "Now we should be able to break up any>array into smaller arrays be renaming the second number or renaming the f i r s t number." Note:  stage.  Several more examples w i l l probably be needed at t h i s  The teacher .should emphasize the techh^iques or renaming both  the f i r s t number and the second number.  48  4.  Seatwork A.  Break up each of these arrays by renaming the' second number. 1) . 6 x 7 = 2)  B.  3 x 8=  Break mpseach of the following arrays by renaming the f i r s t number. 1) :6 x 7 = 2)  C.  8 x.4 =  Provide the numeral which makes the sentence true. 1)  6  X  7 = 6 x (4 + 3) == (6 x ?) + (6 x 3)  2)  4  X  8 = :• (4 x 2) + 0? x 6).  3)  3  X.  8 = ( 3 x 2 ) + (3 x 2) + (3 x ?:)  4)  7  X  5 = (4 + ?) x 5 = (4 x 5) + (? x 5)  5)  7  X  9 = (7 x 8) + (7  x  ?)  APPENDIX C  TREATMENT PHASE LESSON PLANS  TREATMENT ONE  LESSON 1: MULTIPLICATION ,;OF TWO BY ONE PRODUCTS The objective of t h i s less.on i s to teach the rote m u l t i p l i c a t i o n algorithm for 2 by 1 products; both with and without carrying. A.  Without Carrying "Let us look at the following m u l t i p l i c a t i o n problem" 11 x  6 ,  "Can anyone suggest a way of solving t h i s problem by renaming the top number?" (One possible answer might be): 11 5 + 5 + 1 x 6x 6 :  5 + 5 + 1 x 6 6 x 1 =6 + 6 x 5 =30 + 6 x 5 =30 66  The teacher should leave the work for 11 x 6 on the board and write down 11 x 6 somewhere else. "Today we w i l l learn another way that i s probably faster than breaking up a m u l t i p l i c a t i o n problem.  We merely have  to work i n the following way." 11 x  6 ;  "We f i r s t ask ourselves what i s 6 x 1? 6 ones i n t h e ones p o s i t i o n . " 11 x 6 (  Then we place the  51 "Then we ask ourselves again what i s 6 x 1?  This time  we have 6 tens and must place the 6 tens i n the tens position."  x  11 6 66  To confirm the answer, the teacher should refer to the problem 11 x. 6. done by the d i s t r i b u t i v e p r i n c i p l e  ( f i r s t example) .  At this point the teacher should ask one part of the T l group to t r y the problem, 11 x by renaming the top number.  n  The other half should t r y the new algorithm.  When both groups have finished, the answers should be  compared.  If needed, t r y the problem of 11 x 9 i n the same suggested manner. B. With Carrying 12 , on the board. x 6_ Ask,for suggestions as to how to solve this, problem by  Write the problem.  renaming the top number.  x  12 6_  One suggestion might be:  6 + 6 x 6  6 + 6 x 6 6x6= 36 + 6 x 6 =+36 72  "We can solve this problem using our new  x "What i s 6 x 2?  way."  126  This time we have 12 ones.  this up into 1 ten and 2 ones. the one's place as before."  Let's break  Now we can place the 2 i n  52  x  12 6  'We should place the 1 ten i n the ten's place." i: 12 x 6  _2  "Now  we ask ourselves—what  6 tens.  i s 6 x 1?  This time we get  But since we have another group of ten under-  neath, we must add i t to the 6.  Then we place the 7 tens  in the ten's place."  x  12 6  I  2  The teacher should then try a problem l i k e 6 x 22 which invoices thegplacement of a 1 i n the hundreds place.  Use the same  steps as before. Using the new  algorithm, the pupils should attempt the  following: x  21 6  x  11 9  x  14 5  During this time help can be given to individuals as needed.  C.  One by Two  Products  This involves the handling of a problem such as: 6 x 23 Since i n the readiness phase the commutative law for m u l t i p l i c a t i o n was  taught, i t should be easy to convince the c h i l d that  53  w i t h t h i s type of problem we merely 6 x 23 Now  D.  " t u r n i t u p s i d e down."  x  23 6  the c h i l d s h o u l d be a b l e t o s o l v e t h i s type o f problem.  Seatwork I t must be emphasized a g a i n t h a t o n l y these l i s t e d problems  s h o u l d be attempted.  I t i s a l s o i m p o r t a n t t h a t the answers be  g i v e n t o the c h i l d r e n b e f o r e they l e a v e s c h o o l f o r t h a t  day.  Multiply: 1.  65 x 3  7. x  6 41  2.  7 x 13  3.  89 x5  4.  15 x8  5.  49 x4  6.  99 x2  54  LESSON 2:  MULTIPLICATION OF THREE BY ONE PRODUCTS  The objective of this lesson i s to teach the rote m u l t i p l i c a t i o n algorithm for 3 x 1 carrying.  products; both with and without  Since the procedures for 3 x 1  products are very simple  extensions of those for 2 by 1 products, a detailed lesson would be redundant. computation A.  However, the teacher i s urged to r e s t r i c t a l l  to only the examples given.  Review Examples to use: 3 x 37  x  11 8  x  78 3  Emphasize the steps taken to get the f i n a l answer. B.  Without Carrying Ask for suggestions to solve the problem 132 x 3 Most children w i l l probably suggest extending the pro-  cedures used to solve 2 x 1  products.  A t y p i c a l explanation of the procedures to use might go as follows: 132 x 3 "Multiply the 3 x 2; we get 6 ones and have to place this 6 in the one's p o s i t i o n . " 132 x 3  55 "Multiply the 3 x 3; we get 9 tens and place this 9 in the ten's p o s i t i o n . " 132 x 3 96 " F i n a l l y , multiply the 3 x 1; we get 3 hundreds and place the 3 i n the hundred's p o s i t i o n . " 132 x 3 396 Children should attempt:  1Q2 x 3  412 x 2  210 x 4.  After s u f f i c i e n t time, ask the children to explain the procedure i n addition to the f i n a l answer. C.  With Carrying Example to use:  213 x 7  Again children w i l l probably extend procedures for products.  Go through steps as i n problem.without  2x1  carrying, but  stress breaking up 7 x 3 = 21 ones = 2 tens + 1 one. Children should attempt:  D.  Solution of  120 x 7  108 x 5  223 x 6  6 x 142  Again, as i n 2 x 1 products, children should be urged to turn problem "upside down" and then solve. 6 x 142  142 x 6  56 E.  Seatwork  5.  222 x 7  2.  107 x 7  3.  24 x 101  636 x 5  6.  101 x 4  7.  2 x 191  27 x 550  57  LESSON 3:. A REVIEW OF THE ROTE TYPE ALGORITHM FOR  3x1  AND  2 x 1  PRODUCTS  T h i s l e s s o n i s needed t o a l l o w t h e T2 group t o f i n i s h t h e i r treatment.  S i n c e most o f t h e s t u d e n t s i n t h e T l group w i l l have  mastered the r o t e - t y p e a l g o r i t h m , t h i s l e s s o n i s p r o b a b l y f o r t h i s group.  unnecessary  However, i t i s e s s e n t i a l f o r t h e purposes of  study and can be used as merely a p r a c t i c e l e s s o n . use o n l y the problems g i v e n i n t h i s l e s s o n . problems to those who  f i n i s h early.  this  The t e a c h e r  should  P l e a s e do not g i v e e x t r a  The t e a c h e r s h o u l d have ample  time t o g i v e i n d i v i d u a l h e l p d u r i n g t h i s p e r i o d .  I n a d d i t i o n to  g i v i n g answers t o the problems, the t e a c h e r s h o u l d e x p l a i n t h e p r o cedures used t o get the f i n a l answer i n 3 or 4 problems.  Multiply: 62 6  x 11.  16.  1 x 49  x  3  8 280  x  57 x 7  12.  99 x 4  17.  23 6  x  108 x 2  949 523 x 4  13.  45 x 9  18. x  730' x 5  14.  9 30  19.  x  40 2  425 x 5  39 x 3  10.  4 x 49  9 424  15.  61 x 5  253 x 4  20.  208 x 8  58  TREATMENT TWO LESSON 1:  THE BEGINNINGS OF THE ANNEXATION ALGORITHM  The o b j e c t i v e o f t h i s l e s s o n i s t o t e a c h c h i l d r e n a t e c h n i q u e f o r m u l t i p l y i n g any number by 10, 100, o r 1000.  The t e a c h e r s h o u l d f i r s t q u i c k l y r e v i e w m u l t i p l i c a t i o n as repeated  addition. e.g.  3 x 8 = 8 + 8 + 8  = 24  L i s t t h e f o l l o w i n g s e r i e s o f q u e s t i o n s somewhere on t h e board. 2 x 10 = ?  6 x.10 = ?  3 x 10 = ?  7 x 10 = ?  4 x 10 = ?  8 x 10 = ?  5 x 10 = ?  9 x 10 = ?  S t a r t i n g w i t h 2 x 10 ='? a s k c h i l d r e n how t o s o l v e by a d d i n g (10 + 1 0 ) . S o l v e each p r o b l e m by a d d i n g . I t s h o u l d befpfflinted out t o t h e s t u d e n t s t h a t i n each problem t h e one d i g i t number has changed p l a c e s . F o r example 2_ x 10 = 20 "The 2 was o r i g i n a l l y i n t h e one's p l a c e b u t a f t e r m u l t i p l i c a t i o n by t e n i t s h i f t e d t o t h e t e n ' s p l a c e and a z e r o was p l a c e d t o the r i g h t . "  59  The children should quickly r e a l i z e that to multiply by 10 we merely place a zero to the right of the other m u l t i p l i e r . The following series of questions should then be placed on the board. 10 x 11 = ? 12 x 10 = ?  10 x 21 = ? , 18 x 10 = ?  13 x 10 = ? Solve at least 2 or 3 problems by adding.  Again have the  students note that when multiplying by 10 the d i g i t s of the other m u l t i p l i e r a l l s h i f t to the l e f t and a zero i s placed to the r i g h t .  2. ClQtoeuMh'dVe^dVCas a Factor Again the teacher should l i s t a series of questions such as: 2 x 100 = ?  4 x 100 = ?  3 x 100 = ?  5 x 100 = ?  Solve each by adding.  This time i t should be noted that  students should recognize that the d i g i t s have shifted two places (from the one's place to the hundred's place) and two zero's are then placed to the right. The following series of questions should then be placed on the board. 11 x 100 = ?  13 x 100 = ?  12 x 100 = ?  26 x 100 = ?  After ..solving the f i r s t problem or so by adding, the children should be able to quickly generalize that 26 x 100 = ? can be solved by ''placing two zeros to the right of the 26."  60  (26 x 100 = 2600) I t i s p r o b a b l y a d v i s a b l e t o show the p u p i l s how the 2 d i g i t and 6 d i g i t o f 26 have s h i f t e d  3-  two p l a c e s t o t h e l e f t .  ^ o W I r i S l a s ^ F a c t or By now the s t u d e n t s s h o u l d be a b l e t o g e n e r a l i z e t o  problems such a s : 6 x 1000 = ?  - - -  12 x 1000 = ?  6000 12000  To c o n v i n c e some p u p i l s of t h e l e g i t i m a c y o f t h i s t e c h n i q u e i t may be n e c e s s a r y t o s o l v e a problem o r two by a d d i n g . A g a i n t h e p u p i l s . s h o u l d r e a l i z e t h a t t h e d i g i t s have  shifted  t h r e e p l a c e s t o t h e l e f t and t h r e e z e r o s have been p l a c e d to t h e r i g h t .  Seatwork 1.  8 x  10 = ?  2.  10 x  3.  15 x  100 = ?  4.  12 x  100 = ?  5.  10 x 1000 = ?  6.  100 x  9 = ?  7.  1000 x 8  8.  12 x 100  9.  10 x 12  10,  100 x 9  11.  72 x  12.  98 x  13.  100 x 13  14.  1000 x 3  15.  28 x 10  10 = ?  ' 12  = ?  .100 = ?  When marking the t e a c h e r s h o u l d have s t u d e n t s e x p l a i n how they determined  t h e i r f i n a l answers.  61 LESSON 2:  THE ANNEXATION ALGORITHM  The objectives of this lesson are: 1)  to complete the annexation algorithm 5.x 80 = 240  2)  to begin applying the annexation algorithm and d i s t r i b u t i v e p r i n c i p l e to solve 2 x 1  1.  products.  Review a)  ask children to multiply the following: 3 x  10 = ?  100 x 11 = ?  18 x 100 = ?  1000 x 19 = ?  Explain the procedures used to get f i n a l answer. Example:  (multiply by 100; we place two zeros to the right  of the other >fae.t.6.r.';ier';' e t c . ) . b)  review breaking up a productxinto  the sum of smaller  products by renaming the top or bottom number.  Final  answer not important. 12 x 7  2.  10 + 2 x 7 7x2 + 7 xlO  and  Put the following series of questions 3 x 20 = • ? 70 x  5 = ?  110 x ' 3 = ? 20 x 3  3 = ?  7 x 13  7 x 10 + 3 3x7 + 10 x 7  on the board. 300 x 4 = ? 110 x 2 = ?  x 200 = ?  Show the children how to solve any of the above i n the following manner. For example:  3 x 20 = ?  62  a)  "How many tens are there i n 20? (Ans. 2) "we can rewrite 20 as 2 x 10 3 x 20 = 3 x 2 x 10  b)  Now the order i n which we multiply i n a question does not matter, so; 3 x 20 = 3 x 2 x 10 = 6_ x 10.  c)  We have already learned how to multiply a problem such as t h i s "  (place a zero to the r i g h t ) .  3 x 20 = 3 x 2 x 10 = 6 x 10 = 60 d)  To have the children see the emerging pattern for the series of questions, the teacher should underline the following: 3 x 20 = 60  If the teacher does a few more examples i n the above manner i t i s hoped that the.child w i l l see.how to multiply 3 x 200 = ? (Simply multiply 3 x 2 and place 2 zeros to the right 3_ x 200 = 600). Note:  In solving 70 x 5 = ? the teacher should r e -  write 70 as 10 x 7 rather than 7 x 10 since: 70 x 5 = 7 x 10 x 5 (have to commute 7 and 10 to solve) 70 x 5 = 10 x 7 x 5 = 10 x 35 = 350 The next series of questions should be assigned to the pupils.  This w i l l enable the teacher to quickly determine whether  or not the class i s ready to continue.  I f not, more example should  be used to increase the competency with the annexation 30 x  6 = ?  9 x 20 = ? 110 x  3 = ?  200 x  8 = ?  100 x 10 = ?  algorithm.  63  M u l t i p l i c a t i o n <bf>, 2 by 1 products "Now we are ready to do some d i f f i c u l t m u l t i p l i c a t i o n problems like a)  x  rename "top" number as x  b)  12 6  10+2 x 6  we know how to multiply this type; x  c)  , 6  12 6  10+2 x6 6x2 + 6 xlO  Now i t becomes easy since 6 x 2 = 12 and we know that 6. x 10 = 60 x  12 6_ 72  10+2 x6 6x2= 12 + 6 xlO = +60 72  The teacher should demonstrate:  x  21 6  20+1 x 6 6 x 1 = 6 + 6 x 20 = 120_ 126  Allow children to t r y  13 x „A :  28 x  3  If class appears to be acquiring some mastery and i f time s t i l l permits, continue to the next section. 4.  Renaming bottom numbers Problems such as  manner.  ^ ^  should be attacked i n the following  64 6 x 23  6 20+3 3x6= 18 + 20 x 6 = 120 138  Other examples to use might be: 4 x 31 5.  8 x 51  Seatwork Multiply the following: 1.  2 x 120 = ?  2.  3.  2 x 600 = ?  4.  5.  11 x 300 = ?  6.  7. 9.  7 x 13  8.  15  .10.  x 8  3 x  70 = ?  100 x 100 = ?  x  65 3  _x  89 5 6  x 41  The next series of questions should be assigned i f some students appear to have mastered the 2 x 1 products. 11. 6 .12. 100 + 1 0 + J 13. x 102 x 7  140 x 2  65  LESSON 3:  SOLUTION OF 3 BY 1 PRODUCTS  Hopefully, this should be the l a s t period of treatment for the T2 group.  The objective of this lesson i s to teach the techniques  for solving several types of 3 by 1 products. 1.  Review Pupils should be assigned the following: 8 x 20 = ? x  34 6_  9 x 23  In addition to f i n a l answers,ftthe procedures used to solve each should be re reviewed. For example: 34 6  30 + 4 x6 6x4= 24 + 6 x30 = +180  9 x 23  9 20+3 3x9= 27 +20 x 9 = +180  x  2.  8 x 20 = 160  Renaming the Top Number  Put the problem 102 on the board. Ask for suggestions f o r x 3 possible solution. Rename the top number i n the followingsway. The rationale for each step should be explained i n d e t a i l . 102 x 3  100 + 2 x 3  100+2 ' • • x 3 3x2 = 6 + 3 x.100 = +300 306  m u l t i p l i c a t i o n fact m u l t i p l i c a t i o n by 100  Other examples that should be demonstrated by the • teacher are:  66  3.  a)  134 x 3  100 + 20 + 4 x 3  b)  240 x 4  240 + 40 x 4 4. x 40 -= 160 + 4 x 200 = +800 960  100 + 30 + '-4 x 3 3 x 4 = 12 + 3 x 30 = +190 + 3 x 100 = +300 402  Renaming the Bottom Number The pupils should r e a l i z e that sometimes i t i s advantageous  to rename the bottom number.  These examples should i l l u s t r a t e the  techniques to be used. a)  8 x 101  8 x 100 + 1 1x8= 8 + 100 x 8 = 800 808  b)  3 x 246  3 x 200 + 40 x 6  7 x 120  7 100 + 20  c)  3 40+6 6x3= 18 + 40 x 3 = +120 + 200 x 3 = +600 738  x 200 +  7 100 + 20 20 x 7 = 140 +&100 x 7 = +.700 840  The preceding examples should be enough to enable most students to acquire some proficiency for solving 3 x 1 products. The seatwork to be assigned w i l l be good practice for a l l students and i s lengthy enough to allow the teacher to give i n d i v i d u a l help.  67  Seatwork Multiply :  6.  3 x 280  2.  730 x 5  3.  201 x 6  4.  485 x 5  5.  9 x.424  61 x 5  7.  208 x 5  8.  9 x 14  9.  45 x 3  10.  8 x 101  APPENDIX D  THE MEASURING INSTRUMENTS  Performance Test  Name  School First  Last  Part A - Multiply the following Please show a l l work  1.  87 x5  2.  53 x6  3.  6.  3 x;:41  7.  .4 x 15  8.  11.  732 x6  12.  623 x2  16.  4 x 606  17.  4 x 433  60 x 2,  4.  93 x2  7 x 23  9.  4 x 68  13.  201 x4  14.  840 x2  18.  6 x 218  19.  5 x*330  70  T r a n s f e r Test  Name First  School  Last  P a r t B - M u l t i p l y the  following  P l e a s e show a l l work  1001 x 6  x  3 1234  6 :x. 1100  3461 x 3  x  12 11  x  13 64  x  25 12  1001 x 11  x  10.  x  12 2010  11.  1111 x 15  12. x  11 26  16 1100  13.  Ill x 101  14.  203 x 122  15.  120 x 102  16.  113 x 201  17.  1001 x 101  18.  1200 x 110  19.  101 x 1111  20.  122 x 12  21.  101 x 18  22.  16 x 211  23.  11 xyl03  APPENDIX E  THE EXPERIMENTAL DATA  73  TABLE V  EXPERIMENTAL DATA  TREATMENT  T2  Tl Subject  Performance  Transfer  1  15  4  2  16  3  Subject  Performance  Transfer  1  8  2  3  2  8  5  18  2.  3  9  0  4  11  1  4  14  8  5  11  8  5  8  0  6  16  4  6  14  0  TEACHER  7  16  4  7  6  0  1  8  16  4  7  1  0  9  15  3  9  4  0  10  11  2  10  9  2  11  5  2  11  1  0  12  6  1  12  0  0  13  12  3  13  7  3  14  19  4  14  0  0  74  TABLE V EXPERIMENTAL DATA  TREATMENT  Tl Subject  Performance  T2 Transfer  Subject  Performance  Transfer  1  20  11  1  5  0  2  10  1  2  14  0  3  18  8  3  14  0  4  10  AH  4  10  3  TEACHER  5  3  2  5  14  9  2  6  13  3  6  0  0  7  3  1  7  0  0  8  13  3  8  7  0  9  9  2  9  0  1  10  5  1  10  0  0  75  TABLE V  EXPERIMENTAL DATA  TREATMENT  T2  Tl Subject  Performance  Transfer  Performance  Transfer  1  18  4  1  17  4  2  15  9  2  15  7  3  14  12  3  7  5  4  11  Of  4  8  5  5  18  4  5  2  2  6  15  6  6  0  1  7  12  5  7  0  0  8  13  2  8  7  0  Subject  TEACHER 3  76  TABLE V  EXPERIMENTAL DATA  TREATMENT  Tl  T2  Subject  Performance  Transfer  Subject  1  19  5  1  17  8  2  17  7  2  17  8  3  19  10  3  15  9  4  20  4  4  16  3  5  12  5  5  14  6.  TEACHER  6  16  3  6  11  12  4  7  17  4  7  6  3  8  19  12  8  9  1  9  19  4  9  6  3  10  15  4  10  10  3  11  14  3  11  2  3  12  14  5 '  12  8  4  Performance  Transfer  77  TABLE V EXPERIMENTAL DATA  TREATMENT  Subject  ;i,  Subject  Performance  Performance  Transfer .  Transfer  l  20  12  1  19  12  2  19  14  2  12  11  3  18  8  3  19  10  4  18  7  4  17  12  5  17  12  5  14  8  6  16  9  6  6  5  7  19  12  7  13  7  8  19  10  8  15  6  9  19  9  9  14  7  10  18  12  10  13  5  11  19  8  11  20  6  12  16  11  12  10  4  13  12  6  13  8  2  14  17  11  14  6  1  15  16  4  15  4  1  16  12  1  16  4  0  17  16  17  1  0  TEACHER 5  1  78  TABLE V  EXPERIMENTAL DATA  TREATMENT T2  Tl Subject  TEACHER 6  Performance  Transfer ;  1  15  6  2  19  3  Subject  Performance  Transfer  1  8  4  14  2  12  7  19  9  3  15  9  4  11  4  4  7  2  5  18  17  5  9  3  6  14  4  6  9  1  7  4  3  7  11  5  8  16  7  8  6  0  9  16  3  9  13  4  10  9  4  10  3  0  11  16  3  11  0  0  12  17  3  12  3  0  13  9  2  13  0  0  14  0  0  14  0  0  79  TABLE V  EXPERIMENTAL DATA  TREATMENT  Tl  T2  Performance  Transfer  1  18  5  2  19  3  Subject  Performance  Transfer  1  5  2  8  2  5  6  12  3  3  11  5  4  17  2  4  14  7  5  18  4  5  0  0  6  12  4  6  0  0  7  8  2  7  9  2  8  6  0  8  0  1  Subject  TEACHER 7  80  TABLE V EXPERIMENTAL DATA  TREATMENT Tl Subject  TEACHER 8  Performance  T2 Transfer  Subject  Performance  Transfer  1  18  13  1  15  10  2  15  4  -2  12  7  3  19  4  3  16  8  4  13  4  4  3  0  5  17  4  5  7  7  6  12  3  6  7  2  7  15  1  7  11  0  8  17  3  8  15  1  9  19  4  9  9  3  10  18  2  10  9  7  11  12  3  11  11 .  0  12  12  3  12  7  0  13  14  4  13  4  0  14  11  2  14  7  0  

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