Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Inquiry training in elementary mathematics as related to sixth grade pupils' ability to analyze and solve… Weinstein, Marian S. 1970

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

Item Metadata

Download

Media
831-UBC_1971_A8 W45.pdf [ 3.2MB ]
Metadata
JSON: 831-1.0101859.json
JSON-LD: 831-1.0101859-ld.json
RDF/XML (Pretty): 831-1.0101859-rdf.xml
RDF/JSON: 831-1.0101859-rdf.json
Turtle: 831-1.0101859-turtle.txt
N-Triples: 831-1.0101859-rdf-ntriples.txt
Original Record: 831-1.0101859-source.json
Full Text
831-1.0101859-fulltext.txt
Citation
831-1.0101859.ris

Full Text

INQUIRY TRAINING IN ELEMENTARY MATHEMATICS AS RELATED TO SIXTH GRADE PUPILS' ABILITY TO ANALYZE AND SOLVE PROBLEMS. by Marian S. W e i n s t e i n B.A., A d e l p h i U n i v e r s i t y , 1968  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE  REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS  i n the Department of Mathematics E d u c a t i o n  We accept t h i s t h e s i s as conforming t o the required  standard.  THE UNIVERSITY OF BRITISH COLUMBIA October, 1970  In p r e s e n t i n g an  this  thesis  advanced degree at  the  Library  shall  the  in p a r t i a l  fulfilment of  University  of  make i t f r e e l y  I f u r t h e r agree t h a t permission for  s c h o l a r l y p u r p o s e s may  by  his  of  this  written  representatives.  be  available  granted  gain  permission.  Depa r t m e n t  Date  for  for extensive by  the  It i s understood  thesis for financial  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  British  Columbia  shall  requirements  Columbia,  Head o f my  be  I agree  r e f e r e n c e and copying of  that  not  the  that  study.  this  thesis  Department  copying or  for  or  publication  allowed without  my  Abstract T h i s was training alized  a study  to determine the e f f e c t s  of inquiry  i n e l e m e n t a r y m a t h e m a t i c s , i n b o t h g r o u p and  situations.  Three experimental  One g r o u p o f g r a d e s i x s t u d e n t s  individu-  c o n d i t i o n s were  r e c e i v e d p r o b l e m s on  used.  area  c a l c u l a t i o n and a r i t h m e t i c r a t e p r o b l e m s i n w h i c h n o t enough i n f o r m a t i o n was asking  given,  questions  that purpose. reply.  r e q u i r i n g them t o a c q u i r e more d a t a  on b l a n k c a r d s  t o them i n d i v i d u a l l y f o r  T h e y were t h e n g i v e n p r e p a r e d  response cards i n  A s e c o n d g r o u p r e c e i v e d t h e same p r o b l e m s w i t h  information, but a l l questions orally  i n a classroom  taneously A third  s o t h a t a l l c l a s s members c o u l d  g r o u p r e c e i v e d t h e same p r o b l e m data  present,  short training  and no new  sheets,  subjects with  but with a l l  i n f o r m a t i o n was g i v e n  s e s s i o n s were h e l d d e a l i n g w i t h  total  the nature  simulrequested.  area  c u l a t i o n and s o l u t i o n o f a r i t h m e t i c r a t e p r o b l e m s t o the  n o t enough  a n d a l l r e s p o n s e s w e r e made  r e c e i v e t h e i n f o r m a t i o n w h i c h one s t u d e n t  necessary Two  given  by  them. cal-  familiarize  o f the l e a r n i n g m a t e r i a l s .  A  o f 64 s u b j e c t s p a r t i c i p a t e d i n a l l p h a s e s o f t h e  experiment. All parallel  s u b j e c t s were g i v e n  to the f i n a l  cirterion  an IQ t e s t and a p r e t e s t , test,  c o n s i s t i n g o f problems  o f b o t h a n a r i t h m e t i c and g e o m e t r i c n a t u r e , calculation.  The c r i t e r i o n  test,  as w e l l as  area  a t the c o n c l u s i o n o f the  i n s t r u c t i o n a l p e r i o d , c o n s i s t e d o f t h e same t h r e e  subdivisions,  including  items b o t h  materials. t e s t was  similar  and d i s s i m i l a r  E a c h i t e m on b o t h  t o the t r a i n i n g  the p r e t e s t and t h e c r i t e r i o n  answerable by "not enough i n f o r m a t i o n " as w e l l as by  three numerical  possibilities.  Both p r e t e s t and  criterion  t e s t s were m a r k e d t w i c e , o n c e t o g i v e an a c t u a l s c o r e and o n c e to  d e t e r m i n e t h e c o r r e c t use o f t h e "not enough i n f o r m a t i o n "  response. and  area c a l c u l a t i o n  test, as  Each o f the three s u b t e s t s - a r i t h m e t i c ,  as w e l l ,  - was m a r k e d s e p a r a t e l y f r o m t h e t o t a l  and t h e y were t h e n  scored again  to consider  them  t e s t s m e a s u r i n g t h e c o r r e c t u s e o f " n o t enough i n f o r m a t i o n " .  Thus a t o t a l ing  geometric,  o f e i g h t separate  c r i t e r i o n scores with  correspond-  p r e t e s t s c o r e s were i n v o l v e d . An a n a l y s i s o f c o v a r i a n c e was p e r f o r m e d u s i n g t h e IQ  and  p r e t e s t s c o r e s as c o v a r i a t e s .  the group i n q u i r y t r a i n i n g of  The r e s u l t s  a p p r o a c h was  the t h r e e approaches i n t h a t these  indicated  the most s u c c e s s f u l  students  scored  c a n t l y h i g h e r marks on.--fiver: o f t h e e i g h t s c o r e s . c a n t d i f f e r e n c e s were f o u n d  that  No  signifisignifi-  among t h e o t h e r t h r e e s c o r e s .  i  TABLE OF CONTENTS Page i i i  LIST OF TABLES Chapter 1.  2.  THE PROBLEM  1  Background  1  The Problem  3  D e f i n i t i o n o f Terms  4  Justification  4  Hypotheses To Be Tested  6  A REVIEW OF RELATED LITERATURE .  8  Introduction General Opinions On D i s c o v e r y and I n q u i r y  8 . . .  Research o f General Relevance To The Problem  . .  Research o f S p e c i f i c Relevance To The Problem. Summary o f Research F i n d i n g s 3.  PROCEDURE AND DESIGN  .  8 11 15 18 20  Design  20  Subjects  21  Controls  22  Instructional Materials  23  Measuring Instruments  24  I.Q. Scores  24  Pretest  24  C r i t e r i o n Test  25  ii Chapter  4.  5.  Page Procedures  26  S t a t i s t i c a l Procedures  27  RESULTS OF THE STUDY  28  Results  28  Discussion o f Results  35  SUMMARY AND CONCLUSIONS  38  The Problem  38  The Findings  38  Implications  39  Limitations  40  Suggestions For F u r t h e r Research  41  BIBLIOGRAPHY  43  APPENDICES  45  iii  LIST OP TABLES Table 1.  2.  3. 4.  Page Means and Standard D e v i a t i o n s f o r the Three Experimental C o n d i t i o n s on T o t a l P r e t e s t , C r i t e r i o n T e s t , and IQ  28  Means and Standard D e v i a t i o n s on A r i t h m e t i c , Geometry, and Area P r e t e s t and C r i t e r i o n Tests  29  Means and Standard D e v i a t i o n s on T o t a l Information P r e t e s t and C r i t e r i o n T e s t s  29  . .  Means and Standard D e v i a t i o n s on A r i t h m e t i c , Geometry, and Area Information P r e t e s t and C r i t e r i o n T e s t s  30  Regression C o e f f i c i e n t s , Standard E r r o r s , and t-Values f o r the Two C o v a r i a t e s , P r e t e s t and IQ, on each T e s t  30  Group Means, Adjusted Group Means, and Standard E r r o r s f o r E i g h t Tests  31  7.  P a i r - w i s e t - T e s t s f o r Adjusted Group Means . .  33  8.  T-Values f o r C o n t r a s t s i n Adjusted Group Means t o T e s t I n d i v i d u a l versus Group and I n q u i r y Training Effects  34  5.  6.  Chapter THE  1  PROBLEM  Background Few  educators would argue w i t h Bruner"s c o n v i c t i o n  t h a t the g r e a t e r the p a r t i c i p a t i o n i n the l e a r n i n g p r o c e s s on the p a r t o f the student, the g r e a t e r w i l l be the  transfer  o f t r a i n i n g and the more l i k e l y w i l l be development o f intuitive thinking.^  Under the p r e s e n t e d u c a t i o n a l system  i n North America, students u s u a l l y l e a r n g e n e r a l i s a t i o n s which they cannot use e f f e c t i v e l y because these ideas are 2 not t h e i r own  and connected  t o t h e i r own  realities.  have been made t o overcome t h i s impediment through i n c o r p o r a t i o n o f v a r i o u s d i s c o v e r y techniques, o  over the l a s t f i f t e e n y e a r s . began u s i n g such techniques  3  However, M a r i a  Attempts the  particularly Montessori  i n the f i r s t decade o f t h i s  c e n t u r y when she attempted t o d e s i g n an e d u c a t i o n a l system to g i v e each c h i l d freedom t o develop h i m s e l f i n a prepared Jerome S. Bruner, On Knowing (Cambridge: Harvard U n i v e r s i t y P r e s s , 1962), p . 85. 2  i n g Co.,  J o h n H o l t , How C h i l d r e n F a i l 1964), p . 128.  (New  York: D e l l P u b l i s h -  M.C. W i t t r o c k , "The L e a r n i n g by D i s c o v e r y Hypothesis," L e a r n i n g by D i s c o v e r y : A C r i t i c a l A p p r a i s a l , eds. Lee S. Shulman and Evan R. K e i s l a r (Chicago: Rand McNally and Co., 1966), pp. 38-39. 3  environment.  4  In 1910 Dewey stressed the importance o f turn-  ing from l e c t u r i n g to passive students to active confrontations. Wertheimer, i n 1945, emphasized the importance o f a l t e r i n g the teacher's main r e s p o n s i b i l i t y from i n s t r u c t i o n to cooperation. In accord with Bruner's b e l i e f that discovery learning i s b e n e f i c i a l i n increasing i n t e l l e c t u a l powers, Suchman developed  the concept of inquiry t r a i n i n g to attempt to increase  the quantity o f student inquiry by posing problems which provide 7  the student wxth the opportunity to ask for more information. The technique i s expected to i n s t i l l autonomy and d i s c i p l i n e d thinking, both of which are valuable assets i n the problem g  s o l v i n g process.  At the same time the technique i s meant to  afford the teacher a better opportunity to be aware of some o f the cognitive processes a t work i n h i s students.  The  9  student receives practice i n hypothesis construction and testing and learns t o ask questions more e f f e c t i v e l y . ^ 1  Suchman  R.C. Orem (ed.), A Montessori Handbook (New York: Capricorn Books, 1966), p. 13. 4  John D. Dewey, How We Think (New York: D.C. Heath and Co., 1910), p. 198. 5  ^Max Wertheimer, Productive Thinking (New York: Harper and Row, 1945), p. 276. •7  J . Richard Suchman, "A Model f o r the Analysis of Inquiry," Analyses of Concept Learning, eds. Herbert J . Klausmeier and Chester W. Harris (New York: Academic Press), 1966, p. 177. J . Richard Suchman, "Inquiry T r a i n i n g : Building S k i l l s for Autonomous Discovery," Merrill-Palmer Quarterly, VTI (July, 1961), 158. 8  9  1 0  I b i d . , p. 152.  Ibid.  5  3 hoped to provide new goals for the students i n t h e i r thinking of the environment as a p o t e n t i a l source of information and i n t h e i r learning to seek out and organize information i n productive ways.  He also believed these techniques would  reduce student passivity.*"'" White pointed out that part of the aim of better i n s t r u c t i o n i s to reduce excessive l e v e l s of tension without 12 promoting boredom.  Therefore he claimed that i t i s necessary  to promote those types of a c t i v i t i e s , which "although p l a y f u l and exploratory i n character, at the same time show d i r e c t i o n , s e l e c t i v i t y , and persistence i n interacting with the environment.  Discovery learning has t h i s q u a l i t y of exploration  as i t s aim, and inquiry t r a i n i n g s p e c i f i c a l l y s a t i s f i e s t h i s requirement for better teaching. I f the value of learning by discovery i s to be thoroughly tested, i t would seem that inquiry t r a i n i n g should be better researched, as learning to ask questions i s the crux of discovery work. The Problem The purpose of t h i s study was  to explore the e f f e c t i v e -  ness of inquiry t r a i n i n g i n both group and  individualized  settings i n producing students better able to analyze and  i:L  I b i d . , p. 151.  R o b e r t W. White, "Motivation Reconsidered: The Concept of Competence," Psychological Review, LXVT, 5(1959), 313. 12  1 3  I b i d . , p. 329.  4 solve problems i n elementary mathematics.  Examined were both  the a b i l i t y to solve problems accurately and the a b i l i t y to determine whether a problem provided s u f f i c i e n t information to enable a s o l u t i o n .  Problems used were based i n several  areas of mathematics, including arithmetic and elementary geometry. D e f i n i t i o n o f Terms Inquiry t r a i n i n g s h a l l mean a process of investigating i n which students are given problems with i n s u f f i c i e n t informat i o n to solve, and during which the students are allowed to ask questions to gather the data to arrive a t solutions. Group inquiry w i l l involve inquiry t r a i n i n g i n which any student i n the class can ask as many questions as desired, while a l l other students hear the questions and the responses to them. Mass training s h a l l be the designation for those periods, before problem sets have been d i s t r i b u t e d , during which the experimenter explains to a l l groups the basic instructions of operation for the duration of the study as well as reviews mathematical  concepts needed i n the course of the  study. J u s t i f i c a t i o n f o r the Study Aside from i t s use as a further t e s t of discovery learning and the inquiry t r a i n i n g techniques, t h i s study serves two other functions. They are the examination o f the role of mathematics as the vehicle f o r inquiry t r a i n i n g and the  5  comparison  on i n d i v i d u a l and group approaches  i n inquiry  training. Blank and Covington, i n a study o f i n q u i r y ing, u t i l i z e d a training  train-  program designed t o encourage  q u e s t i o n a s k i n g through the use o f programmed  instruction.  As the p r e s e n t study does, they looked a t the r e s u l t o f the t r a i n i n g i n terms o f performance  on a s c h o o l - s u b j e c t  14 oriented test.  However, they d i d n o t attempt t o t r a i n  i n a s p e c i f i c content area.  C o n s i d e r i n g the p r a c t i c e o f  s c h o o l s , i t i s much more f e a s i b l e be attempted  f o r i n q u i r y t r a i n i n g to  on a s u b j e c t b a s i s than on the more g e n e r a l  type problem b a s i s used i n the Blank and Covington study. Secondly, t h e r e i s no reason to b e l i e v e t h a t p o s i t i v e r e s u l t s due to t r a i n i n g on a g e n e r a l i z e d program would a l s o r e s u l t from t r a i n i n g i n a s u b j e c t area, and the p r e s e n t study may g i v e some i n d i c a t i o n s as t o how s u b j e c t m a t e r i a l affects inquiry training  effects.  I t i s e q u a l l y important t h a t a study i n c o r p o r a t e a comparison between i n d i v i d u a l and group approaches  to inquiry  S t a n l e y S. Blank and M a r t i n Covington, "Inducing C h i l d r e n t o Ask Questions i n S o l v i n g Problems," J o u r n a l o f E d u c a t i o n a l Research, LIX (September, 1965), 21.  6 training.  Intrinsically  t h a t t h e group  approach  t h e two a p p r o a c h e s  are different i n  may i m p l y t h a t many s t u d e n t s d o n o t  g e t a chance t o i n q u i r e , w h i l e on t h e o t h e r hand, i t s e f f i c i e n c y may be s o much g r e a t e r t h a n t h e i n d i v i d u a l a p p r o a c h might  be compensation  f o r the lack o f t o t a l  that there  participation.  Furthermore,  many c l a s s r o o m s d o n o t o p e r a t e on a n i n d i v i d u -  alized basis  and i t i s i m p o r t a n t t o s e e w h e t h e r t h e s e  rooms, t o o , c a n b e n e f i t  from an i n q u i r y t r a i n i n g  class-  approach.  Hypotheses t o be T e s t e d Four major hypotheses H^: On a t e s t m e a s u r i n g  a r e t o be t e s t e d  the a b i l i t y  i n this  study:  to solve s p e c i f i e d  mathe-  m a t i c a l problems w i t h s u f f i c i e n t subjects ficient  trained  through  i n f o r m a t i o n , those  the use o f problems w i t h  information w i l l perform s i g n i f i c a n t l y  insuf-  better  than those n o t so t r a i n e d . H2:  On a t e s t m e a s u r i n g g i v e n problem tion, with  the a b i l i t y  c o n t a i n s enough i n f o r m a t i o n t o p e r m i t  those s u b j e c t s t r a i n e d insufficient  better  through  information w i l l  solu-  the use o f problems  perform  significantly  than those n o t so t r a i n e d .  H 3 : On a t e s t m e a s u r i n g m a t i c a l problems, perform  t o determine whether a  the a b i l i t y  group  inquiry trained  significantly better  subjects.  to solve specified  mathe-  subjects w i l l  than i n d i v i d u a l l y  trained  On a t e s t measuring the a b i l i t y t o determine whether  a  g i v e n problem c o n t a i n s enough i n f o r m a t i o n to p e r m i t s o l u t i o n , group t r a i n e d s u b j e c t s w i l l perform s i g n i f i c a n t l y b e t t e r than i n d i v i d u a l l y t r a i n e d s u b j e c t s .  8  Chapter 2 A REVIEW OP RELATED LITERATURE  Introduction The use o f t r a i n i n g which demands l e a r n e r p a r t i c i p a t i o n i n the form o f q u e s t i o n a s k i n g has been examined previously i n educational research l i t e r a t u r e , f o r use  i n examination  particularly  o f problem s o l v i n g s t r a t e g i e s .  Some  o f t h i s r e s e a r c h i s subsumed under the heading o f r e s e a r c h on the m e r i t s o f d i s c o v e r y l e a r n i n g and  inquiry training,  whereas g e n e r a l l y the p e r t i n e n t s t u d i e s d e a l w i t h the q u e s t i o n o f techniques  employed by u n t r a i n e d s u b j e c t s to s o l v e problems  whose content  i s not t h a t o f t r a d i t i o n a l s c h o o l s u b j e c t s .  General Opinions The  on D i s c o v e r y and I n q u i r y T r a i n i n g  term d i s c o v e r y l e a r n i n g has been so w i d e l y used  t h a t a t t h i s p o i n t i t has assumed many d i f f e r e n t meanings. Most d e f i n i t i o n s agree t h a t d i s c o v e r y l e a r n i n g i n v o l v e s techniques  o f r e a r r a n g i n g data i n a manner which enables  l e a r n e r to go beyond the e x i s t e n t data to g a i n new  the  insights.^  In so d o i n g , the ideas b e i n g l e a r n e d are made more comprehensible.  2  I t i s a type o f approach which r e q u i r e s  Jerome S. Bruner, On Knowing U n i v e r s i t y Press, 1962), p. TTT. 2  I b i d . , p.  101.  "flex-  (Cambridge:Harvard  9 ibility  i n t h i n k i n g and d i v e r g e n t , . . . ,  thought  processes."  3  I n t e r e s t i n d i s c o v e r y l e a r n i n g i s based on numerous d i f f e r e n t m o t i v e s , but Bruner has p o i n t e d out four of the most probable viz.,  b a s i c b e n e f i t s t o be expected from d i s c o v e r y l e a r n i n g ,  "an i n c r e a s e i n i n t e l l e c t u a l p o t e n c y t h e  l e a r n i n g of  the h e u r i s t i c s o f l e a r n i n g . . . a s h i f t from e x t r i n s i c to i n t r i n s i c rewards,... and as an a i d i n c o n s e r v i n g memory." Bruner s t a t e s t h a t the emphasis on d i s c o v e r y l e a d s t o t h i n k i n g designed  t o d i s c o v e r i r e g u l a r i t y and a v o i d  organized  "information  4  drift,"  thus t r a i n i n g good guessers,  and so s t i m u l a t i n g the  " a b i l i t y t o go beyond the i n f o r m a t i o n g i v e n to r e c o n s t r u c t i o n s o f other e v e n t s . "  5  probable  Suchman notes t h a t data  from  a c t i v e i n f o r m a t i o n g a t h e r i n g as i s i n v o l v e d i n i n q u i r y t r a i n i n g should be more u s e f u l t o and b e t t e r r e t a i n e d by the l e a r n e r than the p a s s i v e r e c e p t i o n o f i n f o r m a t i o n because of the rewarding experience  o f i n f o r m a t i o n g a t h e r i n g i n i t s e l f , because o f  the  s e l f - c o n f i d e n c e engendered i n c r e a t i o n o f hypotheses, and because o f the p r a c t i c e i n the use o f l o g i c a l  i n d u c t i v e processes.**  I t should a l s o be noted t h a t not a l l educators w i t h those who  advocate d i s c o v e r y l e a r n i n g techniques  agree as  the  John D. Cunningham, " R i g i d i t y i n C h i l d r e n ' s Problem S o l v i n g , " School, Science, and Mathematics, LXVT ( A p r i l , 1966), 377. ^Bruner, op. c i t . , pp.  85-95.  ^Bruner, "Going Beyond the Information Given," Contemporary Approaches t o C o g n i t i o n , eds. Jerome S. Bruner, and others (Cambridge: Harvard U n i v e r s i t y Press, 1957), p. 67. 6 J . R i c h a r d Suchman, "Inquiry T r a i n i n g : B u i l d i n g S k i l l s f o r Autonomous D i s c o v e r y , " M e r r i l l - P a l m e r Q u a r t e r l y , VTI ( J u l y , 1961), 148.  10 primary  t e a c h i n g approach.  Newton p o i n t s out t h a t i n q u i r y  t r a i n i n g can be d i s h o n e s t i n t h a t i t i s "not consonant w i t h the demonstrated needs o f a d o l e s c e n t s , . . . not honest t i o n f o r the college-bound,...  prepara-  does not h o n e s t l y r e f l e c t  nature o f s c i e n c e , . . . has not been analyzed  the  adequately,...  7  and i s o f t e n i n e f f e c t i v e and  inefficient."  Wittrock asserts  t h a t d i s c o v e r y l e a r n i n g i n i t s assumption t h a t each c h i l d  can  b e s t be taught by one method i s as much a t f a u l t as other monistic theories of teaching.  In a d d i t i o n , because d i s c o v e r y  l e a r n i n g i s f r e q u e n t l y time consuming and because the b a s i s o f c u l t u r e i s the t r a n s m i s s i o n o f knowledge o f o t h e r ' s d i s c o v e r ies,  t h e r e i s reason to c o n s i d e r the value o f demanding every  i n d i v i d u a l t o l e a r n by d i s c o v e r y an h y p o t h e s i s s t i l l  i n need  9  of  testing.  Some o f t h i s t e s t i n g has been going on i n the work  o f study groups such as the U n i v e r s i t y o f I l l i n o i s Committee on School Mathematics and the School Mathematics Study Group i n mathematics and has gone on i n the p a s t i n the work o f 10 11 Montesson and Dewey. David E . Newton, "The Dishonesty o f I n q u i r y Teaching," School, S c i e n c e , and Mathematics, LXVIII (December, 1968) p . 807. M.C. W i t t r o c k , "The L e a r n i n g by D i s c o v e r y Hypothesis," L e a r n i n g by_ D i s c o v e r y : A C r i t i c a l A p p r i a s a l , eds. Lee S. Shulman and Evan R. K e i s l a r (Chicago: Rand McNally and Co., 1966), p. 36. 8  9  Ibid.  R . C . Orem (ed.), A M o n t e s s o r i Handbook (New C a p r i c o r n Books, 1966), p. 13. 10  J o h n D. Dewey, How 1910), p . 193.  1 1  Co.,  We  Think  (New  York: D.C.  York: Heath and  11 Research of General Relevance to the Problem Studies dealing with the i d e n t i f i c a t i o n and evaluation of problem-solving strategies have been conducted by several researchers. John u t i l i z e d a machine c a l l e d the PSI (problem solving 12 instrument). The subject's task was to learn the method to achieve a stipulated output from a specified input on the 13 electromechanical c i r c u i t .  John designed the machine to  t r y to approximate best the ideal format for studying problem solving strategies using subjects i n advanced study i n various d i s c i p l i n e s ; he was giving the subjects a minimum of informat i o n to s t a r t and allowing them to structure t h e i r behavior 14 without s p e c i f i c content to consider.  In comparing technique  differences used i n solving two problems i d e n t i c a l with the exception that the second depended on use of the same pattern i n two d i f f e r e n t ways, variables measured included time to completion, number of questions asked, complexity of questions asked, rate, e f f o r t , and redundancy.  John found that s c i e n t i s t s  performed s i g n i f i c a n t l y better than students i n other d i s c i p l i n e s on most measures and that as the problem became more d i f f i c u l t , time needed and number of questions asked increased, as did 15 r i g i d i t y and stereotypy of questions. 12  Erwin Roy John, "Contributions to the Study of the Problem Solving Process," Psychological Review, LXXI, 447 (1957), p. 5. I b i d . , p. 6. 1 3  *^Ibid., p. 5. 1 5  I b i d . , pp. 35-37.  R i m o l d i , i n a d e s c r i p t i o n o f h i s techniques i n g problem s o l v i n g , d i s c u s s e d i n f o r m a t i o n s e e k i n g where the s u b j e c t was  f o r studyquestions  given a choice of p o s s i b l e questions  to  16 ask.  Measured were three p r o p e r t i e s o f the items: u t i l i t y  index, median v a l u e , and d i s p e r s i o n o f items, and s c o r i n g was  done i n s e v e r a l ways i n c l u d i n g s c o r e s based on  number o f items asked, the c o r r e c t n e s s o f the f i n a l 17 and a type o f q u a l i t a t i v e a n a l y s i s o f s t r a t e g y . t h i s study was  solution, Although  d e s c r i p t i v e r a t h e r than experimental  ques are o f r e l e v a n c e i n examination  utility,  i t s techni-  o f problem s o l v i n g .  U t i l i z i n g a twenty-questions-game approach and  two  types o f s i t u a t i o n s , ore i n which the c h i l d r e n were shown f o r t y - t w o p i c t u r e s o f common o b j e c t s and asked one  the experimenter  had  i n mind and one  were g i v e n a s i t u a t i o n and asked  to f i n d  i n which the  the children  to f i n d a cuase, Mosher and  Hornsby i n v e s t i g a t e d problem s o l v i n g s t r a t e g i e s and d i s c o v e r e d two b a s i c s t r a t e g i e s - c o n s t r a i n t seeking and  hypothesis  18 scanning.  In c o n s t r a i n t s e e k i n g , the s u b j e c t assumes  e q u a l l y l i k e l y a l t e r n a t i v e s ' and e l i m i n a t e s by h a l v e s , whereas i n h y p o t h e s i s scanning each guess t e s t s a s e l f s u f f i c i e h t H . J . A . R i m o l d i , "A Technique f o r the Study o f Problem S o l v i n g , " E d u c a t i o n a l and P s y c h o l o g i c a l Measurement, XV, (1955), 451. 16  1 7  I b i d . , p.  454.  F r e d e r i c k A. Mosher and Joan Rigney Hornsby, "On Asking Q u e s t i o n s , " S t u d i e s i n C o g n i t i v e Growth, eds. Jerome S. Bruner, and others (New York: John W i l e y and Sons, Inc., 1966), p . 88. 1 8  13 hypothesis.  Generally constraint seeking, which involves more  e f f i c i e n c y than hypothesis scanning and which insures better coverage of a l l p o s s i b i l i t i e s , involves more s t r a i n . * ^  However,  the constraint seeking approach i s generally preferred by older 20 children. In h i s investigation of two types of c r e a t i v i t y v a r i ables, i n complex problem solving, one defined i n terms of cognitive structure integrative complexity, as measured by scores on the Paragraph Completion Inventory, and one i n terms of emphasis on breadth of association, as determined by the Remote Associates Test and l a b e l l e d associative c r e a t i v i t y , Karlins u t i l i z e d an information-seeking  approach i n which each  subject was required to learn about an unfamiliar South Seas 21 island culture i n order to solve a  situation-problem.  Information was made available i n a 57-category program deck. Here the subject was active i n every phase of the learning u t i l i z a t i ofrom n ofactive data received. were ask process, informationSubjects gathering topermitted processingtoand 22 for information which would be provided  i f avilable.  He  found that the two groups of subjects termed a s s o c i a t i v e l y 19 Ibid., p. 89. I b i d . , p. 90. *Marvin K a r l i n s , "Conceptual Complexity and RemoteAssociative Proficiency as C r e a t i v i t y Variables i n a Complex Problem Solving Task," Journal of Personality and Social Psychology, VI, 3 (1967), 268. 2 0 2  2 2  I b i d . , p. 274.  14 creative, but d i f f e r i n g i n integrative complexity,  showed no  s i g n i f i c a n t differences i n terms of breadth of information search, evenness of information search, or willingness to d i r e c t l y explore, but the i n t e g r a t i v e l y complex subjects, regardless of associative c r e a t i v i t y scores, did show s i g n i f i 23 cantly better r e s u l t s than noncomplex subjects i n a l l areas. In a study to analyze the influence of l o g i c a l structure and problem language i n thinking processes and t h e i r r e l a t i o n to age, Rimoldi and his associates presented a problem and requested questions  the subjects to choose from a l i s t of possible to ask those they considered c r u c i a l to the s o l u t i o n  of the problem, continuing u n t i l they either solved the problem or decided to s t o p . ^ 2  A l l problems were i n d i v i d u a l l y admin-  istered without a time l i m i t and content was standard school material.  not based on  Scoring was based on the number  of questions asked, scheme, and correctness of the answer. They found that a s i g n i f i c a n t improvement comes through aging and that there i s some s i g n i f i c a n t difference due to i n t e r action of problem language and l o g i c a l structure.  Also they  found that there i s increased agreement among subjects' t a c t i c s with age and that l o g i c a l structure and language are not experimentally independent v a r i a b l e s . 2 3  I b i d . , p.  ?4  2 5  277.  H.J.A. Rimoldi, M. Aghi, and G. Burger, "Some E f f e c t s of Logical Structure, Language, and Age i n Problem Solving i n Children," Journal of Genetic Psychology, CXII (1968), 127. I b i d . , p. 142. 2 5  15 In the area of the use of problems which d i f f e r  from  standard textbook problems i n terms of either overabundance  or  i n s u f f i c i e n c y of information for solution, two studies are most relevant.  James found that on a t e s t measuring correctness of  responses to arithmetic problems, poorer performance  resulted  from problems with too much information than with standard 26  textbook problems.  At the other end of the spectrum, O'Brien  and Shapiro found that introducing the alternative of "not enough information" i n a t e s t designed to examine l o g i c a l thinking i n children s i g n i f i c a n t l y lowered marks from the same 27  t e s t without these p o s s i b i l i t i e s .  This tends to indicate  that a deficiency now exissfcs i n recognizing the i n s u f f i c i e n c y of information for problem s o l u t i o n . Research of Specific Relevance to the Problem One of the most popular devices i n the research involving the asking of questions by subjects i s the tab item o r i g i n a l l y developed by Glaser for use i n the A i r Force to replace more expensive procedures for problem solving observa28  t i o n without s a c r i f i c i n g v a l i d i t y i n examining the behavior. Jim Butler James, "A Comparison of Performance of Sixth Grade Children i n Three Arithmetic Tasks: Typical Textbook Verbal Problems, Revised Verbal Problems Including Irrelevant Data, and Computational Exercises, "Dissertation Abstracts, 28:2030 B, November, 1964. ^Thomas C. O'Brien and Bernard J . Shapiro, "The Development of Logical Thinking i n Children, " American Educational Research Journal, V (November, 1968),.537. 2  R o b e r t Glaser, Dora E. Damrin, and Floyd M. Gardner, "The Tab Item: A Technique for the Measurement of Proficiency i n Diagnostic Problems Solving Tasks," Educational,and Psychologic a l Measurement, XXV, 2(1954), 283. 28  16 The  t a b item,  as a t e c h n i q u e  information gathering described by Glaser the  l e a r n e r was  i n measuring v a r i a b l e s i n  i n the course  regarding  o f p r o b l e m s o l v i n g , was  i t s u s e i n d i a g n o s t i c work where  f o r c e d t o "perform a s e r i e s o f procedures i n  w h i c h t h e r e s u l t s , o f one p r o c e d u r e y i e l d  information  to supply  29 a cue f o r . the s e l e c t i o n o f the n e x t i t e m . " chooses a tabbed information and  item,  of information.  the weighted s e l e c t i o n e f f i c i e n c y . g r o u p s w e r e made b y G l a s e r Suchman,  as  3  this  showed  and p r e s e n t e d  physics  them w i t h t h e  illustrated resulted  e n d , he i n s t r u c t e d them t o  o b t a i n more i n f o r m a t i o n b y a s k i n g q u e s t i o n s answered  i s b a s e d on  No c o m p a r i s o n s b e t w e e n  the demonstrations  To a c h i e v e  inadequate,  i n this description.  t o students  o f e x p l a i n i n g why  they d i d . ^  Scoring  i n his inquiry training,  e x p e r i m e n t s on f i l m  subject  each c o n s i s t i n g o f s e v e r a l pages o f  i n c l u d i n g both r e l e v a n t , redundant, 30  i r r e l e v a n t types  task  The  " y e s " o r "no" i n a t h r e e - s t a g e  which could  process  be  of analysis  o f the episode, determination o f relevance o f v a r i o u s c o n d i t i o n s n e c e s s a r y f o r t h e outcome, and i d e n t i f i c a t i o n o f c o n s t r u c t s t o 32 explain  t h e phenomena.  u n l i k e t h e above s t u d i e s w h i c h were  c a r r i e d * o u t o n a n i n d i v i d u a l i z e d b a s i s , Suchman h a d a l l 2 9  Ibid.,  p . 284.  3 0  Ibid.,  p . 289.  Suchman, "Inquiry T r a i n i n g : B u i l d i n g S k i l l s f o r Autonomous D i s c o v e r y , " p . 1 5 1 . J X  3 2 i b i d . , p . 159.  q u e s t i o n i n g done o r a l l y i n a group and each p a r t i c i p a n t was 33 allowed  t o h o l d the f l o o r as l o n g as d e s i r e d .  During a  twenty-four week p e r i o d Suahman found t h a t questions  became  b o t h more abundant and more p r e c i s e as time passed and p r i n 34 c i p l e s were b e t t e r l e a r n e d than i n a c o n t r o l group.  However,  Suchman d i d n o t examine the d i r e c t e f f e c t s o f t h i s procedure on other types o f problem s o l v i n g . Utilizing and  the v e h i c l e o f programmed i n s t r u c t i o n , Blank  Covington attempted to induce q u e s t i o n a s k i n g i n s o l v i n g  problems through the use o f problems i n which i n s u f f i c i e n t i n f o r m a t i o n was p r o v i d e d .  The aim o f the i n v e s t i g a t o r s was to  i n c r e a s e q u e s t i o n asking among a group o f grade s i x students and  to determine the e f f e c t s o f t h i s i n f o r m a t i o n - s e e k i n g  i n g technique  on achievement t e s t scores  i n an area  train-  unrelated  t o the t r a i n i n g , as w e l l as on both an o r a l and w r i t t e n t e s t 35 s i m i l a r t o the t r a i n i n g m a t e r i a l .  U n l i k e the other s t u d i e s  which were more i n t e r e s t e d i n o b s e r v i n g q u e s t i o n  asking  s t r a t e g i e s . Blank and Covington examined the f i n a l o f the q u e s t i o n i n g .  effects  The t r a i n i n g problems were those o f a  type l i k e l y t o occur i n the d a i l y r o u t i n e , as, f o r example, a c h a l l e n g e t o determine how a t r u c k might g e t through a t u n n e l too low f o r i t . The students I b i d . , p . 163. 3 3  3 4  f i r s t examined sample problems  I b i d . , p . 167.  S t a n l e y S. Blank and M a r t i n Covington, "Inducing C h i l d r e n t o Ask Questions i n S o l v i n g Problems," J o u r n a l o f E d u c a t i o n a l Research, L I X (September, 1965), 21. 3 5  w i t h p e r t i n e n t questions which they might have asked t o s o l v e them.  Then they performed  on t h e i r own.  compared a group t r a i n e d f o r ten days,  The r e s e a r c h e r s  f o r t y - f i v e minutes per  day, on the above method mentioned, w i t h a group exposed to the same problems w i t h a l l i n f o r m a t i o n p r o v i d e d and no o p p o r t u n i t y f o r q u e s t i o n a s k i n g and w i t h a c o n t r o l group g i v e n no exposure to these problems.  They found t h a t i n a l l cases the group  t r a i n e d on the data-seeking items s i g n i f i c a n t l y  outperformed  36 the other  groups.  Summary o f Research  Findings  In much r e s e a r c h , d i s c o v e r y methods have proved t o be e f f e c t i v e t o o l s i n the b e t t e r i n s t r u c t i o n o f students from a m o t i v a t i o n a l viewpoint and  both  from the p o i n t o f view o f b e t t e r  37 retention.  Suchman and Blank and Covington  s p e c i f i c a l l y the i n q u i r y technique proved  found t h a t  to be v a l u a b l e i n  terms o f p r o d u c i n g students more i n c l i n e d to ask q u e s t i o n s and  t o perform b e t t e r on achievement t e s t s i n other a r e a s .  The work o f James and O'Brien and Shapiro i n d i c a t e s t h a t problems d i f f e r i n g  from standard t e x t s p r e s e n t new  t o students u n t r a i n e d i n an i n q u i r y Although  challenges  setting.  the concept o f examination  of question-  a s k i n g s t r a t e g i e s has been researched by many experimenters, l i t t l e a t t e n t i o n has been p a i d to the outcome o f t h i s except by Suchman and Blank and Covington. I b i d . , p.  However, Suchman  25.  M.C. W i t t r o c k , "The L e a r n i n g by D i s c o v e r y pp. 50-53. 37  training  Hypothesis,"  d i d n o t seek t o measure t r a n s f e r to other areas o f study and Blank and Covington d i d n o t examine i n q u i r y t r a i n i n g  applied  to s p e c i f i c content areas nor i n group s i t u a t i o n s . T h e r e f o r e , i t seems important t h a t r e s e a r c h be done i n b o t h group and i n d i v i d u a l i n q u i r y t r a i n i n g i n c o n t e n t a r e a s .  20  Chapter 3 PROCEDURE AND DESIGN Pes ign The present study employed three experimental  conditions  with the three treatment groups l a b e l l e d as T-^ T-,and T-.. The subjects were trained through individual question asking to seek the a d d i t i o n a l information required to find solutions to problems stated with i n s u f f i c i e n t information.  subjects  were trained as a group on the same problems with i n s u f f i c i e n t information as i n T,, and T 1  subjects were trained on the same 3  problems as the other two groups with the exception that the problems were stated with a l l information necessary to s o l u t i o n provided. Scores on a group IQ t e s t and on a c r i t e r i o n pretest were f i r s t gathered  for a l l pupils to be used as covariates to  insure equality of groups with respect to i n t e l l i g e n c e ,  initial  a b i l i t y i n recognizing problems with i n s u f f i c i e n t information, and i n i t i a l problem solving a b i l i t y i n three areas to be tested, v i z . , finding areas of polygonal figures, solving geometric reasoning problems, and solving arithmetic reasoning problems. A l l three groups then p a r t i c i p a t e d i n a mass t r a i n i n g of two sessions of twenty and forty minutes respectively on the topic of finding areas of polygons using the technique of decomposition.  Discussion of what information would be  21 needed t o f i n d areas o f v a r i o u s g e n e r a l  f i g u r e s was  not brought  up by the experimenter, b u t r e f e r e n c e s t o t h i s , i f made by s u b j e c t s , were allowed  and b r i e f l y d i s c u s s e d .  T-, were i n s t r u c t e d together w h i l e group T-, was m a t e r i a l as a group, but Each group met  Groups taught  and the same  separately.  twice, f o r f o r t y minutes each s e s s i o n ,  a t which time the s u b j e c t s worked on problem s e t s i n v o l v i n g f i n d i n g the area o f polygons by F o l l o w i n g the two  decomposition.  s e s s i o n s on area, the groups a g a i n  p a r t i c i p a t e d i n a mass t r a i n i n g s e s s i o n , i n which examples o f v e r b a l r a t e problems were examined.  This session l a s t e d only  f i f t e e n minutes because the concepts were f a m i l i a r to a l l s u b j e c t s i n i t i a l l y and r e q u i r e d l i t t l e e x p l a n a t i o n . s e s s i o n was  followed by two  The  separate meetings where the  s u b j e c t s worked on problem s e t s i n v o l v i n g the s o l u t i o n o f r a t e problems. A c r i t e r i o n t e s t to measure a b i l i t y to problems w i t h i n s u f f i c i e n t i n f o r m a t i o n and  recognize  to a c c u r a t e l y f i n d  s o l u t i o n s to problems i n a r i t h m e t i c , geometry, and l a t i o n was  administered  area c a l c u -  to a l l groups on the f i n a l  day.  Subjects I n i t i a l l y one  c l a s s o f 35 grade s i x c h i l d r e n was  a r b i t r a r i l y s e l e c t e d to be p l a c e d under experimental  condition  T  assigned  2  and  35 students  t o T j and T . 3  from another c l a s s t o be randomly  A f t e r students  absent f o r the p r e t e s t and/or the  c r i t e r i o n t e s t were e l i m i n a t e d , and  f o l l o w i n g the random a s s i g n -  22 ment o f s u b j e c t s to groups were  f i x e d a t 33 i n T , 2  and T , 3  the numbers i n each group  14 i n T^, and 17 i n T .  A l l subjects  3  were i n two a r i t h m e t i c c l a s s e s i n a Vancouver  elementary s c h o o l .  Controls P r e c a u t i o n s were taken i n attempting t o c o n t r o l f o r s e v e r a l types o f v a r i a b l e s : i n t e l l i g e n c e and  achievement,  experimenter b i a s , Hawthorne type e f f e c t s , and feedback  differ-  ences between the groups. IQ s c o r e s and c r i t e r i o n p r e t e s t s c o r e s were used i n an a n a l y s i s o f c o v a r i a n c e on the c r i t e r i o n p o s t t e s t s c o r e s to c o n t r o l f o r i n i t i a l d i f f e r e n c e s between the groups i n the areas o f i n t e l l i g e n c e and problem s o l v i n g a b i l i t y .  No  attempt  was made t o c o n t r o l f o r reading a b i l i t y due to Balow's and o t h e r s ' f i n d i n g s t h a t when IQ i s c o n t r o l l e d , computational a b i l i t y appears more s i g n i f i c a n t than r e a d i n g a b i l i t y f o r problem solving.^To minimize the e f f e c t s o f experimenter b i a s and teacher d i f f e r e n c e s , the same i n s t r u c t o r , the experimenter, was used i n a l l groups, and minimal c o n t a c t between the experimenter and the s u b j e c t s d u r i n g the problem  solving  a c t i v i t i e s was maintained a t a l l times. Although the n o v e l t y o f a strange i n s t r u c t o r e x c e p t i o n a l circumstances was  under  a f a c t o r i n the experiment, a l l  I r v i n g H. Balow, "Reading and Computational A b i l i t y as Determinants of Problem S o l v i n g , " The A r i t h m e t i c Teacher, XI (January, 1964), 22.  23 groups experienced  t h i s same e f f e c t and t h e r e f o r e i t can be  assumed t h a t a l l groups were a f f e c t e d e q u a l l y .  Care was  taken  t o see t h a t s u b j e c t s d i d n o t know who was i n which group or who was expected to perform b e s t , and no p a r t i a l r e s u l t s were d i s c u s s e d d u r i n g the p e r i o d o f the study. F i n a l l y , s u b j e c t s i n the group T-, were g i v e n the opport u n i t y t o ask questions  t o seek i n f o r m a t i o n , although  no data  were g i v e n them, merely cards which i n d i c a t e d t h a t they possessed  already  s u f f i c i e n t data t o s o l v e the problem, i n order t o  c o n t r o l f o r any advantage t o the respect to personal  feedback.  o r the T  2  groups w i t h  A l l cards looked a l i k e so t h a t  T3 s u b j e c t s were n o t aware t h a t t h e i r answers were any d i f f e r ent than those  f o r T^.  The m o t i v a t i o n  d e r i v e d from Smith and Right's  f o r t h i s approach was  finding that personalized  feed-  back can markedly improve problem s o l v i n g e f f i c i e n c y and insight.  2  Instructional Materials The  f i r s t mass t r a i n i n g s e s s i o n s , f o r a l l groups, was used  t o review the a l r e a d y e x i s t i n g knowledge o f how t o f i n d area o f rectangles. and  The concepts o f f i n d i n g areas o f p a r a l l e l o g r a m s  t r i a n g l e s were  then i n t r o d u c e d and e x p l a i n e d .  t h i s , s e v e r a l more complicated  polygonal  Following  f i g u r e s were d i s c u s s e d  i n terms o f decomposition i n t o more p r i m i t i v e f i g u r e s to f i n d  ^Ewart E . Smith and S t a n f o r d S. K i g h t , " E f f e c t s o f Feedback on I n s i g h t and Problem S o l v i n g E f f i c i e n c y i n T r a i n i n g Groups," J o u r n a l o f A p p l i e d Psychology, X L I I I , 3 (1959), 211.  24 their areas.  A l l subjects were given opportunities to ask  questions at these sessions. The two problem sheets consisted of f i v e problems each. Subjects were presented figures with some but not a s u f f i c i e n t number o f dimensions to solve the problems and asked to f i n d the areas by requesting other dimensions i f needed.  Only  units of measurement familiar to the students and only scale drawings were used.  A l l of the dimensions were given as whole  numbers, to avoid unnecessary d i f f i c u l t i e s with f r a c t i o n s . After a b r i e f discussion of the solution of an example rate problem, the subjects began work on two sets of arithmetic rate problems sheets, each consisting of s i x problems.  The  problems were c l a s s i f i a b l e i n terms of both r a t e - d i v i s i v e and r a t e - m u l t i p l i c a t i v e types, that i s , some requiring d i v i s i o n work and others m u l t i p l i c a t i o n . Measuring  Instruments  I.Q. Scores.  IQ scores f o r a l l subjects were available  from school records based on the Henmon-Nelson Test of Mental A b i l i t y , Revised 1958 e d i t i o n , published by Houghton-Mifflin Company, and administered throughout  the Vancouver School  district. C r i t e r i o n Pretest.  The c r i t e r i o n pretest consisted of  twelve multiple-choice problems.  The pretest was f i r s t  administered to a p i l o t group of 28 students for determination of r e l i a b i l i t y c o e f f i c i e n t s . .81 was obtained.  A t e s t - r e t e s t r e l i a b i l i t y of  Four items dealt with finding areas o f  polygons, three were of a nonmetric geometric type and f i v e  25 were a r i t h m e t i c p r o b l e m s . (d) was  the  "not enough i n f o r m a t i o n "  were i n s t r u c t e d t o use insufficient Wording o f basis  Of  of p i l o t  100  i t e m s were c h o s e n f r o m t h e the c r i t e r i o n The  test  i f they  original  item p o o l used  i n an e f f o r t  felt  that  provided.  v e r s i o n on The  the  pretest  in construction  to maintain restricted  o f s e v e r a l o f them and  subjects  been  grade s i x students.  number o f i t e m s was  involved nature  only  t o s o l v e the p r o b l e m had  a l t e r e d f r o m an  runs w i t h  choices  f o r a l l p r o b l e m s , and  this possibility  information  i t e m s was  four response  equivalence. t o t w e l v e due  the  of  to  time r e q u i r e d  the  for  solution. No this  standardized  l e v e l was  experimenter  test  deemed s u i t a b l e .  p e r i o d , a seventeen item  metric  r a t e problems.  o f the  criterion  t e s t was  a pilot  eight with  the  test  was  determine c l a r i t y  experimental  sample o f  28  four with  of non-  a r i t h m e t i c problems, i n c l u d i n g  one  " n o t enough  students.  t e s t e d on  of wording.  administered,  calculation,  A test-retest reliability  items were f i r s t  constructed.  area  Each i t e m had  response choice.  test,  A t t h e end  items d e a l t w i t h  g e o m e t r y , and  b a s e d on  Therefore,  at  constructed.  Criterion Test.  which f i v e  i n v o l v i n g geometric problems  100  This  of  information" .92  As w i t h  was  the  criterion  grade s i x students test,  t o o , was  computed  to  experimenter  pre-  Procedure B e f o r e t h e e x p e r i m e n t a l p r o c e d u r e s were under random a s s i g n m e n t  t o groups  class.  IQ t e s t h a d b e e n p r e v i o u s l y a d m i n i s t e r e d t o  all  The g r o u p  subjects.  a n d T^ was made from a  way, single  The p r e t e s t was a d m i n i s t e r e d o n t h e f i r s t  o f the experiment,  f o l l o w e d by a twenty  review o f polygonal area concepts. o f p o l y g o n s was e x t e n d e d  minute  Next,  mass  day  training  the concept o f area  through a f o r t y minute  training  session i n a l l groups. Two d a y s were d e v o t e d t h e s u b j e c t s were a s k e d figures All  after  t o two p r o b l e m  to find  sets  i n which  the areas o f v a r i o u s polygonal  t h e q u e s t i o n a s k i n g p r o c e d u r e was e x p l a i n e d .  t h r e e g r o u p s were g i v e n t h e same f i g u r e s ; h o w e v e r , o n l y T 3  s u b j e c t s were g i v e n a l l o f t h e n e c e s s a r y i n f o r m a t i o n a t t h e start.  Each  student i n  and T  3  was  supplied with a s e t of  c a r d s , b l a n k e x c e p t f o r t h e o u t l i n e s o f t h e f i g u r e s on t h e 1  worksheets.  He was  i n d i c a t i n g which then handing  t h e n p e r m i t t e d t o f i n d needed d a t a b y  d i m e n s i o n he r e q u i r e d  on t h e b l a n k c a r d and  t h e c a r d t o t h e e x p e r i m e n t e r , who  immediately  r e t u r n e d t o him a p r e v i o u s l y prepared a p p r o p r i a t e response card. card  Students indicating  available. The All  i n T^ who  asked q u e s t i o n s always  that s u f f i c i e n t  received  i n f o r m a t i o n was a l r e a d y  No h i n t s w e r e p r o v i d e d t o a n y s u b j e c t s ,  s u b j e c t s met a s a c l a s s  a  only data.  t o work on t h e i r p r o b l e m  sheets.  q u e s t i o n s a n d a n s w e r s were o r a l , w i t h a n s w e r s marked i n  large  f i g u r e s a t t h e f r o n t o f t h e room; i t may b e assumed  t h a t a l l g r o u p members t h u s h e a r d a n d b e n e f i t t e d  from  each  27 other's q u e s t i o n s . groups were found  S o l u t i o n s t o a l l problems i n a l l three individually.  F o l l o w i n g these two pated  s e s s i o n s , the groups again p a r t i c i -  i n a f i f t e e n minute mass t r a i n i n g s e s s i o n reviewing v e r b a l  r a t e problems and s o l u t i o n s . Again  the c a r d system was  Two  problem s e t s were d i s t r i b u t e d .  employed.  As b e f o r e , o n l y T3 s u b j e c t s  i n i t i a l l y r e c e i v e d s u f f i c i e n t data to s o l v e the problems. The  c r i t e r i o n t e s t was  the f i n a l day  o f the  administered  t o a l l groups on  study.  In s c o r i n g both the c r i t e r i o n p r e t e s t and p o s t t e s t s , t o t a l s c o r e s were c a l c u l a t e d as w e l l as s c o r e s i n each o f the three topic s u b d i v i s i o n s .  In a d d i t i o n , t o t a l s c o r e s and  sub-  d i v i s o n scores were obtained on the b a s i s o f the c o r r e c t or i n c o r r e c t use o f answer (d) - "not enough i n f o r m a t i o n " . Statistical The  Procedures a n a l y s i s to f o l l o w i s c o n s i d e r e d i n terms o f  e i g h t separate marks: t o t a l s c o r e , a r i t h m e t i c s c o r e , geometry s c o r e , area s c o r e , t o t a l i n f o r m a t i o n s c o r e , a r i t h m e t i c i n f o r m a t i o n s c o r e , geometry i n f o r m a t i o n s c o r e , and  area  i n f o r m a t i o n s c o r e , each w i t h i t s corresponding p r e t e s t s c o r e . A l l data were analyzed a t the U n i v e r s i t y o f B r i t i s h Columbia Computing Centre u s i n g an a n a l y s i s o f program.  covariance  3  H e a l t h Sciences Computing F a c i l i t y , Department of P r e v e n t i v e Medicine, U n i v e r s i t y o f C a l i f o r n i a , "BMDX82A n a l y s i s of Covariance," Los Angeles, C a l i f o r n i a .  28  Chapter 4 RESULTS OF THE STUDY Results The means and standard deviations f o r the scores on the eight separate pretest and c r i t e r i o n t e s t marks are given i n Tables 1, 2, 3, and 4. Table 1 Means and S.D.s f o r the Three Experimental Conditions on Total Pretest, C r i t e r i o n Test and IQ  Number o f Questions Correct Pretest  Criterion  IQ  Mean  S.D.  Mean  S.D.  Mean  S.D.  T  l  3.57  1.60  6.36  2.50  112.14  11.47  T  2  5.39  1.71  7.67  2.44  11-5.21  10.79  T  3  3.82  1.88  5.06  2.30  111.24  12.83  N.B.  T^ i s the individualized-inquiry trained group; T  2  i s the group-inquiry  trained group;  T^ i s the individualized-non-inquiry trained group.  29 Table 2 Means and Standard Deviations on Arithmetic, Geometry, and Area Pretest and C r i t e r i o n Tests.  Arithmetic - Pretest  Arithmetic  Geometry  Criterion  Pretest  Criterion  Pretest  Criterion  Mean S.D.  Mean S.D.  Mean S.D.  Mean S.D. 2.43 1.40  Mean S.D.  Mean S^D.  Geometry  Area  Area  T  x  1.50 1.29  2.93 1.64  .64  .84  1.00  .55  1.43 1.16  T  2  1.91 1.04  2.52 1.33  .61  .60  1.24  .94  1.21  .96 2.27 1.04  T  3  1.71 1.16  2.24 1.44  .88  .86  .65 .79  1.59  1.23 1.94 1.14  Table 3 Means and Standard Deviations on Total Information and C r i t e r i o n Test.  Pretest  Criterion  Mean  S.D.  Mean  S.D.  T  x  7.79  1.52  8.71  2.33  T  2  8.97  1.19  9.97  2.19  T  3  8.24  1.44  7.47  2.12  Pretest  30 Table 4 Means and Standard Deviations on Arithmetic, Geometry, and Area Information Pretest and C r i t e r i o n Tests Arithmetic  Geometry  Area  Pretest  Criterion  Pretest  Criterion  Pretest  Criterion  Mean S.D.  Mean S.D.  Mean S.D.  Mean S.D.  Mean S.D. Mean S.D.  Tj  3.14 1.17  4.07 1.21  1.14  .95  1.64  .74  3.50  .65 3.07 1.38  T  2  4.09  .95  4.90 1.33  .27  .45  2.03  .92  3.73  .57 3.12  T  3  3.59 1.12  3.71 1.65  .82  .80  1.29  .85  3.82  .39 2.53 1.28  .82  Table 5 Regression Coefficients, Standard Errors, and t-values for the Two Covariates, Pretest and IQ, on each Test.  Type of Test  Covariate  Regr. Coeff.  S.E.  t-value  t o t a l test  pretest IQ  .90 .05  .04 .02  23.80*( .01) 2.17*( .02)  a r i t h test  pretest IQ  .43 .05  .13 .01  3.25 4.19*( .01)  geom. test  pretest IQ  .09 -.01  .15 .01  .60 -.98  area t e s t  pretest IQ  .24 .03  .14 .01  1.72 2.06*( .05)  total info. test  pretest IQ  .10 .03  .21 .02  .46 1.34  arith. info. test  pretest IQ  .21 ,00  .17 .02  1.24 .21  geom. i n f o . test  pretest IQ  -.17 .00  .17 .01  -1.02 - .20  area i n f o . test  pretest IQ  .17 .02  .25 .01  .69 2.28*( .05)  Examination of the regression c o e f f i c i e n t s and t - t e s t values for the two covariates, pretest and IQ, for each of the eight scores presented i n Table 5 indicates that IQ was s i g n i f i cantly related to score on both the t o t a l and arithmetic tests beyond the .02 l e v e l , and pretest score was related to these two marks beyond the .01 l e v e l .  IQ was also s i g n i f i c a n t on  both the area and area information scores at the .05 l e v e l . Because the covariates were s i g n i f i c a n t i n several instances, the analysis of covariance, rather than analysis o f variance, was required.  Therefore, adjusted group means were  f i r s t calculated f o r a l l groups for scores on the c r i t e r i o n test.  These adjusted scores are indicated i n Table 6. Table 6  Group Means, Adjusted Group Means, and Standard Errors for Eight Tests Gp. Mean  Adj. Gp. S.E. Mean  Total Test T  l  T  2  T  3  Gp. Mean  Adj. Gp. Mean  S.E.  Total :Info. Test 6.36  7.34  .59  8.71  8.83  .61  7.67  6.84  .38  9.97  9.87  .40  5.06  5.86  .53  7.47  7.57  .54  A r i t h :Info. Test  A r i t h Test T  l  2.93  3.11  .32  4.07  4.20  .39  T  2  2.51  2.36  .21  4.91  4.83  .25  T  3  2.24  2.38  .29  3.71  3.75  .34  Geom Info. Test  Geom Test T  l  T 2  1.00  .99  .22  1.64  1.73  .25  1.24  1.26  .15  2.03  1.98  .16  .65  .61  .20  1.29  1.33  .21  32 Table 6 (Continued)  Group  Mean  A d j . Mean  S.E.  Area T e s t Tl T  2  Mean  Ad j . Mean  S.E.  Area I n f o . T e s t 2 A3  2.44  .29  3 .07  3.14  .29  2.27  2.26  .19  3.12  3.07  .19  1.94  1.94  .26  2.53  2.57  .26  33 Table 7 P a i r w i s e t - T e s t s f o r Adjusted Group Means  Pair  Regular T e s t s Total  T  1" 2  T  1" 3  T  T  T,-T2  3  .70  Arith  I n f o . Tests  Geom  Area  Total  Arith  Geom  Area  1.97 -1.02  .52  -1.39  -1.31  -.79  .19  1.30  1.57  .89  1.27  1.47  1.87  1.73  1.26  1.48  -.04  2.58* (.02)  .97  3.36* (.01)  2.51* 2.36* 1.59 (.02) (.05)  N.B. T^ i s the i n d i v i d u a l i z e d - i n q u i r y t r a i n e d group; T i s the g r o u p - i n q u i r y t r a i n e d group; T3 i s the i n d i v i d u a l i z e d - n o n - i n q u i r y t r a i n e d group. 2  The p a i r w i s e t - t e s t s f o r a d j u s t e d group means shown i n Table 7 i n d i c a t e s t h a t p a i r w i s e d i f f e r e n c e s o c c u r r e d i n the geometry as w e l l as the i n f o r m a t i o n t e s t s , a t the .02 l e v e l f o r the geometry t e s t , a t the .01 l e v e l f o r the t o t a l  informa-  t i o n t e s t , a t the .02 l e v e l f o r a r i t h m e t i c i n f o r m a t i o n t e s t , and a t the .05 l e v e l  f o r the geometry i n f o r m a t i o n t e s t .  No  s i g n i f i c a n t d i f f e r e n c e s occurred i n the area i n f o r m a t i o n t e s t . In a l l cases o n l y the T - ^ c o m p a r i s o n i s s i g n i f i c a n t . 2  It  t h e r e f o r e appears t h a t an i n t e r a c t i o n e f f e c t o f the two f a c t o r s , amount o f i n f o r m a t i o n g i v e n t r a i n i n g ) and the group versus  (the use o f i n q u i r y  individual factor i s significant  i n f a v o r o f the group approach w i t h n o t enough i n f o r m a t i o n  34  in  terms o f a b i l i t y t o determine whether problems p r o v i d e  s u f f i c i e n t i n f o r m a t i o n t o enable  solution.  To t e s t the main  hypotheses themselves, two t - v a l u e s were computed f o r each o f the e i g h t t e s t s - one t o examine group versus i n d i v i d u a l and one t o examine the n o t enough i n i t i a l sufficient initial  information  effect  versus  information e f f e c t . Table 8  t-Values f o r C o n t r a s t s I n Adjusted Group Means to T e s t I n d i v i d u a l versus Group and I n q u i r y T r a i n i n g E f f e c t s Contrast  Test  Tl  Total  & T3VS . To  Ti & T  T T  T  l  T  3  Total Info. Total Info.  -2.80* (.01) 2.62* (.05)  Arith. Info. Arith. Info.  -2.26* (.05) 1.75  Geom. Info. Geom. Info.  -1.81  Area Info. Area Info.  -.82  vs. A r i t h .  T  3  T  T  2 3  2  T3  v s . Geom. v s . Geom. v s . Area  T  1.03  2  v s . Area  vs.  T  3  vs.  2  T  3  Ti &  X.14*  T, &  T  T  T  3  T  2  T  3  2  3  Ti &  -.26  2  3  -2.18* (.05) (.05)  T  & 2 vs.  x  T  T  5  T, &  1.27  T  3  T, &  & T3 v s .  T  2  Ti & T  Ti  Ti &  1.89  3  & T  t-Value  T  2  Ti & T  2 vs. Total  T  T  Test  T  & 2 vs. A r i t h .  X  -.43  Contrast  3  & T  T  t-Value  2  T, & T  1.30  x  T3  2  vs. vs. vs. vs.  1.82  1.60  Table 8 i n d i c a t e s o n l y flare c s i g n i f i c a n t d i f f e r e n c e s among the s i x t e e n c o n t r a s t s t e s t e d , three o f which confirm the hypothesis  t h a t students  t r a i n e d as a group outperform  students  not so t r a i n e d and two o f which favor i n q u i r y t r a i n e d students.  Scores  i n group T  0  were s i g n i f i c a n t l y b e t t e r than  scores i n groups l e v e l , the  and  T  3  on the geometry t e s t a t the  t o t a l i n f o r m a t i o n t e s t a t the  a r i t h m e t i c i n f o r m a t i o n t e s t a t the T^  and  T  2  l e v e l , and  level.  the  Scores i n groups  were s i g n i f i c a n t l y g r e a t e r than s c o r e s i n group T3  on the geometry t e s t and .05  .05  .01  .05  l e v e l of  on  the  t o t a l information t e s t at  the  significance.  Discussion of Results The  data presented i n the p r e s e n t r e s e a r c h i n d i c a t e  group t r a i n i n g i s p r e f e r a b l e to i n d i v i d u a l t r a i n i n g , i n combination w i t h the  inquiry  problems and  especially  approach, i n terms o f  ment i n a b i l i t y to a r r i v e a t a c c u r a t e s o l u t i o n s  to  that  improve-  specified  to determine s u f f i c i e n c y o f i n f o r m a t i o n i n these  problems. Hypothesis H^ which p r e d i c t e d s i g n i f i c a n t i n a b i l i t y to a r r i v e at accurate solutions  differences  to elementary mathe-  m a t i c s problems as a r e s u l t o f i n q u i r y t r a i n i n g was  confirmed  o n l y i n the case of the geometry s u b t e s t .  be  a b l e to the no was  attribut-  i n t e r a c t i o n w i t h the group f a c t o r , however, i n  s i g n i f i c a n t differences removed.  T h i s may  were found when the  T h e r e f o r e , no d i f f e r e n c e s  t r a i n i n g were found. compared t o the  The  group  factor  based s o l e l y on  s h o r t d u r a t i o n of the  study  established  may  be  inquiry  as  l o n g e r p e r i o d o f time i n which p a t t e r n s  problem s o l v i n g behavior are  that  of  a factor i n  the  l a c k o f s i g n i f i c a n t changes. Hypothesis H  2  which p r e d i c t e d s i g n i f i c a n t  i n a b i l i t y t o c o r r e c t l y determine the i n a g i v e n problem due  to i n q u i r y  differences  s u f f i c i e n c y of  t r a i n i n g was  information  confirmed only  on the t o t a l i n f o r m a t i o n t e s t , although not on any o f the subtests.  T h i s seems to i n d i c a t e merely an accumulation  n o n s i g n i f i c a n t scores on the s u b t e s t s and significant in itself.  of  t h e r e f o r e i s not  Perhaps, here too, long term p a t t e r n s  r e s i s t a n t t o a l t e r a t i o n or too l i m i t e d exposure was  responsible  f o r l a c k o f change. Hypothesis H^ which p r e d i c t e d h i g h e r marks f o r groupt r a i n e d s u b j e c t s than i n d i v i d u a l l y - t r a i n e d  ones on  tests  measuring a b i l i t y t o a r r i v e a t problem s o l u t i o n s was in part.  confirmed  S i g n i f i c a n t l y h i g h e r marks d i d r e s u l t on the geometry  t e s t , although not on the a r i t h m e t i c or area c a l c u l a t i o n p o r t i o n s . Perhaps t h i s may was  the one  be accounted f o r i n t h a t the geometry  l e a s t f a m i l i a r to the s u b j e c t s and  content  so most amenable  t o change, p a r t i c u l a r l y when i n f o r m a t i o n g a t h e r i n g became more e f f i c i e n t , as can be presumed to be the case i n the group s e t t i n g and when i n q u i r y t r a i n i n g was  involved.  Hypothesis H^ which p r e d i c t e d h i g h e r marks f o r groupt r a i n e d s u b j e c t s than i n d i v i d u a l l y - t r a i n e d  ones on a t e s t  measuring a b i l i t y t o determine the s u f f i c i e n c y o f i n f o r m a t i o n i n a g i v e n problem was  a l s o confirmed  i n part.  Significantly  h i g h e r scores d i d r e s u l t on the a r i t h m e t i c i n f o r m a t i o n and  the t o t a l i n f o r m a t i o n t e s t .  Perhaps the k i n d o f  subtest  interaction  i n v o l v e d i n group i n f o r m a t i o n g a t h e r i n g proved to be more u s e f u l to the i n d i v i d u a l group members' a b i l i t y t o judge s u f f i c i e n c y of i n f o r m a t i o n i n a r i t h m e t i c problems as compared to geometry or area c a l c u l a t i o n ones because o f the b e t t e r working vocabulary  o f the students w i t h t h i s type o f problem  so a l l o w i n g f o r f a s t e r b e n e f i t s from t h i s t r a i n i n g .  The h i g h e r  marks on the t o t a l i n f o r m a t i o n t e s t can be accounted f o r subs t a n t i a l l y by the h i g h e r marks on the a r i t h m e t i c p o r t i o n ; geometry and area s c o r e d i f f e r e n c e s  also  favoured the group  approach, although n o t s i g n i f i c a n t l y . While no s i g n i f i c a n t d i f f e r e n c e s from i n q u i r y  training,  t h a t group i n q u i r y inquiry  directly  the p a i r w i s e comparisons d i d i n d i c a t e  training  i s s u p e r i o r to an i n d i v i d u a l non-  approach, b u t n o t s i g n i f i c a n t l y s u p e r i o r t o an  individual-inquiry  approach.  P a r t o f the cause f o r t h i s  r e s t w i t h the f a c t t h a t i n c o n t r o l l i n g to feedback i n the i n d i v i d u a l control  resulted  may  f o r the advantage due  situation,  the s u b j e c t s i n the  group experienced some o f the same c o n d i t i o n s connected  w i t h i n f o r m a t i o n s e e k i n g , although n o t a l l o f them, as the e x p e r i m e n t a l group.  T h e r e f o r e , i t appears t h a t  t r a i n i n g may have a u s e f u l  inquiry  r o l e t o p l a y i f combined w i t h the  more e f f i c i e n t group approach.  These r e s u l t s do not c o n f i r m  those o f Blank and Covington i n t h e i r study where trained  individually-  s u b j e c t s were more s u c c e s s f u l than i n d i v i d u a l s  trained,  a f a c t which may be a t t r i b u t a b l e  learning  a s p e c t o f the t r a i n i n g o r t o the nonsubject-  oriented  content.  not so  t o the programmed  1  S t a n l e y S. Blank and M a r t i n Covington, "Inducing C h i l d r e n t o Ask Questions i n S o l v i n g Problems," J o u r n a l o f E d u c a t i o n a l Research, LIX (September, 1965), 27.  38  Chapter 5 SUMMARY AND CONCLUSIONS  The  Problem The purpose o f t h i s study was t o determine whether an  i n q u i r y t r a i n i n g approach would be b e n e f i c i a l t o students i n e i t h e r a group or i n d i v i d u a l i z e d s e t t i n g i n terms o f i n c r e a s i n g a b i l i t y t o a n a l y z e s p e c i f i e d mathematical problems w i t h r e s p e c t t o competence i n d e t e r m i n i n g whether a problem p r o v i d e s s u f f i c i e n t i n f o r m a t i o n f o r s o l u t i o n , and i n p r o c e e d i n g t o s o l u t i o n where p o s s i b l e . The  Findings The r e s u l t s o f the d a t a a n a l y s i s i n d i c a t e t h a t t h e  group i n q u i r y approach i s more s u c c e s s f u l than t h e i n d i v i d u a l approach i n p r o d u c i n g students b e t t e r a b l e t o a r r i v e a t s o l u t i o n s t o problems o f a n o v e l nature and b e t t e r a b l e t o examine the adequacy o f i n f o r m a t i o n p r o v i d e d i n problems n o t p r e v i o u s l y examined  i n this light.  Furthermore, p a i r w i s e  comparisons  i n d i c a t e t h a t the group i n q u i r y approach i s s i g n i f i c a n t l y s u p e r i o r i n four out o f e i g h t comparisons t o the i n d i v i d u a l n o n - i n q u i r y approach. I t a l s o may be t h a t a s h o r t i n q u i r y t r a i n i n g p e r i o d had no s i g n i f i c a n t e f f e c t s on the a b i l i t y t o improve  performance  on types o f m a t e r i a l a l r e a d y v e r y f a m i l i a r t o the s t u d e n t s .  Implications As a consequence o f the f i n d i n g s o f t h i s study,  several  i m p l i c a t i o n s can be drawn. First,  i n q u i r y t r a i n i n g appears t o be e f f e c t i v e i n  p r o d u c i n g b e t t e r problem s o l v e r s i f employed w i t h a group approach and n o t used on an i n d i v i d u a l b a s i s .  This r e s u l t bears  out Suchman's f i n d i n g s ; * i t does n o t completely agree w i t h those 2 o f Blank and Covington.  Since no s u b s t a n t i a l d i f f e r e n c e s i n  d u r a t i o n e x i s t between the p r e s e n t study and Blank and Covington's study, i t would appear t h a t the d i f f e r e n c e s between the two  f i n d i n g s must r e s u l t from one o f two causes: the i n t e r a c t i o n  e f f e c t s o f i n q u i r y t r a i n i n g and programmed l e a r n i n g o r the use of generalized problems.  as opposed t o s p e c i f i c c o n t e n t - o r i e n t e d  I f the cause i s the former, then i t i s apparent  t h a t through the use o f programmed, m a t e r i a l ,  inquiry training  can be made e f f e c t i v e i n i n d i v i d u a l l y s t r u c t u r e d and  training  c o u l d be implemented i n many s e t t i n g s .  classrooms,  I f , on the other  hand, the cause i s the content i n the t r a i n i n g , perhaps what i s i n d i c a t e d i s the need t o t r a i n students t o recognize and s o l v e problems i n t h e i r everyday e x p e r i e n c e s f i r s t ,  before  i n t r o d u c i n g problems w i t h i n s u f f i c i e n t d a t a i n s u b j e c t  area  which might merely confound and f r u s t r a t e the s u b j e c t s . The  f a c t t h a t the group i n q u i r y approach was favoured  over the i n d i v i d u a l i n q u i r y approach seems t o i n d i c a t e  that  J . R i c h a r d Suchman, "Inquiry T r a i n i n g : B u i l d i n g S k i l l s f o r Autonomous D i s c o v e r y , " M e r r i l l - P a l m e r Q u a r t e r l y , VTI ( J u l y , 1961), 152. 2  Blank and Covington, l o c . c i t .  40 the e f f i c i e n c y o f i n f o r m a t i o n g a t h e r i n g outweighs any disadvantage fully participate  i n the group  situation  t h a t c e r t a i n g r o u p members d o n o t  i n the i n q u i r i n g .  N o t i c i n g t h a t mean s c o r e s w e r e , i n . g e n e r a l , h i g h e r on the  information p r e t e s t s than  indicates t h a t students  already possess  whether a problem c o n t a i n s but  apparently  this  on t h e c o r r e s p o n d i n g  ability  sufficient  pretests  some a b i l i t y  information  t o analyze  to solve i t ,  c a n be f u r t h e r improved  through  training. This  study  tends t o support  training  c a n be a v a l u a b l e  distinct  situations.  tool  the contention that i n q u i r y  i n mathematics e d u c a t i o n i n  Limitations The  basic limitations  categories:  of this  study  fall  t h e d u r a t i o n o f the r e s e a r c h and t h e v a l i d i t y o f  the measuring  instruments.  I n terms o f d u r a t i o n , s u c h a s h o r t - t e r m best give only  Secondly,  s m a l l s a m p l e , some d o u b t  c o n c e i v a b l y b e c a s t on t h e a c c u r a c y Unfortunately,  l e v e l deals with this  instruments  and because r e l i a b i l i t y c o -  e f f i c i e n t s were b a s e d on a r e l a t i v e l y  Certainly,  can a t  examination.  b e c a u s e two o f t h e m e a s u r i n g  were e x p e r i m e n t e r - c o n s t r u c t e d ,  significance.  study  i n d i c a t i o n s o f the e f f e c t s o f r e l e v a n t v a r i a b l e s ,  which r e q u i r e longer s t u d i e s f o r f u r t h e r  can  i n t o two  o f the scores  no s t a n d a r d i z e d  and t h e i r  test a t this  a l l o f the m a t e r i a l involved i n the study. should  be r e c t i f i e d  once t h e s t u d y  o f geometry  a t t h e e l e m e n t a r y l e v e l becomes more w i d e s p r e a d and p e r h a p s  41  then  the experiment  can be  r e r u n u s i n g t h e new  measuring  instruments. One  last alteration  u s e f u l w o u l d be later  t o e x a m i n e t h e same s t u d e n t s  t o determine  inquiry  o f t h i s design which might  the r e t e n t i o n e f f e c t s ,  prove  s e v e r a l months  i f any,  of  the  training.  Suggestions  f o r Future  Of o b v i o u s  Research  m e r i t w o u l d be  a l o n g term  study d e a l i n g  with  t h e same i s s u e s a s a r e  Only  t h e n c o u l d one  this  t r a i n i n g s y s t e m on a n y t h i n g o t h e r t h a n a s h o r t t e r m  This experiment  involved i n this research project.  make a t r u e e v a l u a t i o n o f t h e m e r i t s  could s t i l l  a p p r o a c h e s as w e l l as  encompass g r o u p and  i n q u i r y t r a i n e d and  of basis.  individual  noninquiry  trained  methods. I n a d d i t i o n , a s t u d y u s i n g programmed l e a r n i n g training inquiry  a s compared t o some o t h e r training  individualized  s i t u a t i o n s h o u l d be  conducted  w h e t h e r t h e programmed l e a r n i n g a s p e c t i s a n individual  inquiry  To d e t e r m i n e influences training  determine  important  one  in  the e x t e n t to which s u b j e c t c o n t e n t  the experimental  o u t c o m e s , a s t u d y t o compare  inquiry  i n v o l v i n g g e n e r a l i z e d p r o b l e m s as compared t o a p r o g r a m  treatments  utilize  learning  training.  w i t h s p e c i f i c mathematical two  to  inquiry  c o n t e n t m i g h t be  conducted,  employ programmed m a t e r i a l and  a different  form  of individualized  two  where  treatments  training.  42  S t u d i e s t o determine whether v a r i o u s g r o u p - t r a i n i n g approaches d i f f e r from a g r o u p - i n q u i r y approach i n both i z e d s e t t i n g s and s p e c i f i c content s e t t i n g s might be  general-  devised  t o decide whether the i n t e r a c t i o n e f f e c t s o f i n q u i r y t r a i n i n g and group l e a r n i n g are more powerful other  techniques.  than group l e a r n i n g v i a  43 BIBLIOGRAPHY Balow, Irving H. "Reading and Computation A b i l i t y as Determinants of Problem Solving," The Arithmetic Teacher, XI (January, 1964), 18-22. Blank, Stanley S., and Martin Covington. "Inducing Children to Ask Questions i n Solving Problems," Journal of Educational Research, LIX (September, 1965), 21-27. Bruner, Jerome S. Press, 1962.  On Knowing. Cambridge: Harvard University  , and others. Contemporary Approaches to Cognition. Cambridge: Harvard University Press, 1957. , and others. Studies i n Cognitive Growth. John Wiley and Sons, Inc., 1966.  New York:  Cunningham, John D. " R i g i d i t y i n Children's Problem Solving," School, Science, and Mathematics, LXVT ( A p r i l , 1966), 377-387. Dewey, John D. 1910.  How We Think.  New York: D.C. Heath and Co.,  Glaser, Robert, Dora E. Damrin, and Floyd M. Gardner. "The Tab Item: a Technique f o r the Measurement of Proficiency i n Diagnostic Problem Solving Tasks," Educational and Psychological Measurement, XIV, (1954), 283-293. Holt, John. How Children F a i l . Co., 1964.  New York: D e l l Publishing  James, Jim Butler. "A Comparison of Performance of Sixth Grade Children i n Three Arithmetic Tasks: Typical Textbook Verbal Problems, Revised Verbal Problems, Including Irrelevant Data, and Computational Exercises," Dissertation Abstracts, 28:2030 B, November, 1967. John, Erwin Roy. "Contributions to the Study of the Problem Solving Process," Psychological Monographs, LXXJ, 447 (1957). K a r l i n s , Marvin. "Conceptual Complexity and Remote-Associative Proficiency as C r e a t i v i t y Variables i n a Complex ProblemSolving Task," Journal of Personality and Social Psychology, VT, 3 (1967), 264-278. Kirk, Roger E. Experimental Design: Procedures for the Behavioral Sciences. Belmont, C a l i f . , : Brooks/Cole Publishing Co., 1968.  43  Klausmeieer, Herbert J . , and Chester w. H a r r i s . Analyses Concept Learning, New York: Academic Press, 1966. Newton, David E. "The Dishonesty of Inquiry Teaching," Science, and Mathematics, LXVTII (December, 1968), 807-810.  of  School,  O'Brien, Thomas C , and Bernard J . Shapiro. "The Development of L o g i c a l Thinking i n Children," American Educational Research Journal, V (November, 1968), 531-541. Orem, R.C. (ed.) Books, 1966.  A Montesorri Handbook.  New York: Capricorn  Rimoldi, H.J.A. "A techniques for the study of Problem Solving, Educational and Psychological Measurement, XV, (1955), 450-461. , M. Aghi, and G. Burger. "Some E f f e c t s of L o g i c a l Structure, Language, and Age i n Problem Solving i n Children, The Journal of Genetic Psychology, CXII (1968), 127-143. Shulman, Lee S. and Evan R. Keisler (eds.). Learning by Discovery: A C r i t i c a l Appriasal. Chicago: Rand McNally and Co., 1966. Smith, Ewart E., and Stanford S. Kight. "Effects of Feedback on Insight and Problem Solving E f f i c i e n c y i n Training Groups," Journal of Applied Psychology, XLIII, 3 (1959), 209-2111. Suchman, J . Richard. "Inquiry Training; Building S k i l l s for autonomous Discovery," Merrill-Palmer Quarterly, VII (July, 1961), 147-169. Wertheimer, Max. Row, 1945.  Productive Thinking.  New York: Harper and  White, Robert W. "Motivation Reconsidered: The Concept of Competence," Psychological Review LXVI, 5 (1959), 297-331. f  APPENDIX A  CRITERION PRETEST  46 Pretest INSTRUCTIONS: FOR EACH PROBLEM, CIRCLE THE ANSWER THAT YOU THINK IS MOST CORRECT. CHOOSE ONLY ONE ANSWER FOR EACH PROBLEM. USE ANSWER (d) ONLY IF YOU THINK THE PROBLEM DOES NOT GIVE YOU ENOUGH INFORMATION TO FIND AN ANSWER. THERE IS NO TIME LIMIT, SO TRY TO ANSWER AS MANY OF THE QUESTIONS AS YOU CAN.  1. Find the area o f the figure below. 3 ft. 4 ft. (a)7 sq. f t .  (b)12 sq. f t .  (c)6 sq. f t .  (d) not enough information  2. These two c i r c l e s touch a t two points. I f there are two c i r c l e s , one with a radius o f 2 yards and i  one with a radius o f 1 yard, a t how many points do they touch? (a) 0 3.  (b) 1  (c) 2  (d) not enough information  Using four points, a t most how many l i n e s can be drawn connecting them two a t a time?  (a) 4  (b) 8  (c) 6  (d) not enough information  4. I n a g r o u p o f c h i l d r e n , 3 o u t o f 5 a r e g i r l s . 50 b o y s , how many g i r l s (a)  25  (b) 75  5. F i n d t h e a r e a  are there?  ( c ) 30  (b) n o t enough  (b)25 s q . f t .  and a s o d a c o s t 30C. (a) 3 0 *  (b) 3 5 *  ••  ?  (d) n o t enough information  c o s t 85C.  One s a n d w i c h  How much d o e s a s a n d w i c h  ( c ) 25$  S it  ft/  (c)20 s q . f t .  6. Two s a n d w i c h e s a n d a s o d a t o g e t h e r  information  / " ' 'i]  o f the f i g u r e below.  5  (a) 15 s q . f t .  I f there are  (d) n o t enough  cost?  information  7. I n a c o l l e c t i o n o f c o i n s w o r t h 93$ made u p o f n i c k e l s , d i m e s , and p e n n i e s , t h e r e as  dimes.  (a) 4 8.  (a) 3  a s many n i c k e l s  How many n i c k e l s a r e t h e r e ?  (b) 6  (c) 8  (d) n o t enough  H e r e i s t h e way t o draw a t r i a n g l e  Using  9.  a r e 13 p e n n i e s and t w i c e  on t h r e e  information points.  f i v e p o i n t s , how many t r i a n g l e s c a n b e drawn? (b) 5  ( c ) 10  Find the area  (d) n o t enough  information  o f the f i g u r e below: — P lift.  (a)  10 s q . f t .  (b)8 s q . f t .  ( c ) l 2 s q . f t . (d) n o t enough information  43 10. There were 100 t r e e s i n an o r c h a r d . pear t r e e s .  H a l f o f the t r e e s were  o f t h e t r e e s i n the o r c h a r d d i e d .  A t most,  how many pear t r e e s d i e d ? (a) 50  (b) 40  (c) 35  (d) n o t enough i n f o r m a t i o n  11. Which o f the f i g u r e s below has the g r e a t e s t  area?  c r  Aft. J3  (a) i  (b) i i  (c) the same  (d) n o t enough i n f o r m a t i o n  12. A j a r c o n t a i n s 2 q u a r t s o f m i l k and 6 q u a r t e r s o f water. The mixture  i s completely blended.  % o f the mixture i s  poured out and the j a r i s r e f i l l e d w i t h m i l k .  How much  m i l k i s i n the j a r now? (a) 4 q t s .  (b) 6 q t s .  (c) 5 q t s .  (d) n o t enough i n f o r m a t i o n  APPENDIX B  INQUIRY TRAINING WORKSHEETSARITHMETIC AND GEOMETRIC  54 T  3  Problem II - Sheet I .  1.  John had 15 boxes of 30 p e n c i l s each and 23 boxes of 60 pencils each.  2.  How many p e n c i l s d i d John have?  On a t r i p of 180 miles. B i l l s * father drove 60 miles at a speed of 30 miles per hour, and the r e s t of the distance at a speed of 60 miles per hour.  3.  How long d i d the t r i p take?  For each 5 acres of beans and 8 acres of corn Mr. Jones planted, Mr. Smith grew only 3 acres of beans and 6 acres of corn.  I f Mr. Jones planted 90 acres of beans and 240  acres of corn, how much of each was Mr. Smith growing? 4.  Mr. Adams paid Bob $6.40 for washing. i28 windows and r e painting a window s i l l .  Bob's rate for window washing  was $1 for each five windows. How much d i d Bob charge f o r repainting the window s i l l ? 5.  Out of each $8 that he earned, Jim put away $2 for saving. He u s u a l l y earned $10 a week.  How many weeks would have  he to work to pay for a $75 t e l e v i s i o n set, i f he didn't use h i s savings? 6.  Mr. Smith drove 208 miles on 16 gallons of gasoline.  How  many gallons would he use at t h i s rate t o go 338 miles?  5§ Ti & T  2  Problem I I - Sheet I  1.  John had some boxes o f 30 p e n c i l s each and some boxes o f 60 p e n c i l s each.  2.  How many p e n c i l s d i d John have?  On a c e r t a i n t r i p . B i l l ' s f a t h e r drove 60 m i l e s a t a speed o f 30 m i l e s p e r hour, and the r e s t o f the d i s t a n c e a t a f a s t e r speed.  3.  How l o n g d i d the t r i p  take?  For each 5 acres o f beans and 8 acres o f c o r n Mr. Jones p l a n t e d , Mr. Smith grew o n l y 3 acres o f beans and a few more acres o f c o r n .  I f Mr. Jones p l a n t e d some beans and  240 acres o f c o r n , how much o f each was Mr. Smith growing? 4.  Mr. Adams p a i d Bob $6.40 f o r washing windows and r e p a i n t ing  a window s i l l .  Bob's r a t e f o r window washing was $1  f o r each f i v e windows.  How much d i d Bob charge f o r  r e p a i n t i n g the window s i l l ? 5.  Out o f each $8 t h a t he earned, Jim p u t away some o f i t f o r saving.  He u s u a l l y earned $10 a week.  How many weeks  would he have t o work t o pay f o r a t e l e v i s i o n s e t i f he d i d n ' t use h i s s a v i n g s ? 6.  Mr. Smith drove 208 m i l e s .  How many gaBons o f g a s o l i n e  would he use a t the r a t e o f h i s o r i g i n a l t r i p t o go 338 miles?  56 T  3  Problem  I I - Sheet I I  1. A r e c t a n g l e 162  i s t w i c e as l o n g as i t i s w i d e .  sq. units.  How l o n g  The a r e a i s  i s the rectangle?  2. The number o f s q u a r e i n c h e s i n a c e r t a i n s q u a r e ' s a r e a i s t h r e e t i m e s t h e number o f i n c h e s a r o u n d t h e s q u a r e . long  i s a side o f the square?  3. One p i p e c a n f i l l fill the  a tank i n 3 h o u r s and a n o t h e r p i p e can  the tank i n 6 h o u r s .  How l o n g w i l l  i t take t o f i l l  t a n k i f b o t h p i p e s a r e u s e d a t t h e same t i m e ?  4. I f J i m c a n mow lawn  How  a lawn  i n 20 m i n u t e s  i n 30 m i n u t e s , how l o n g w i l l  a n d Bob c a n mow t h e  i t t a k e t h e two b o y s  working together? 5. I f a b a n k c h a r g e s 16C a week o n e a c h $100 b o r r o w e d , man b o r r w s will 6.  $175 a n d p a y s  are to  two weeks l a t e r ,  how much  he b e c h a r g e d ?  T h e r e a r e 40 p u p i l s If  i t back  and a  % o f t h e boys i n the music these  clubs?  i n Jane's c l a s s  - 16 b o y s  and 24 g i r l s .  a r e i n t h e math c l u b a n d 2/3 o f t h e g i r l s c l u b , how many c l a s s members do n o t b e l o n g  57 T  l  &  T  2  Problem  1.  I I - Sheet I I  A rectangle  i s longer than i t i s wide.  square u n i t s . 2.  i s the r e c t a n g l e ?  The number o f s q u a r e i n c h e s i n a c e r t a i n a multiple long  3.  How l o n g  The a r e a i s 162  s q u a r e *s a r e a i s  o f t h e number o f i n c h e s a r o u n d  the square.  How  i s a side o f the square?  One p i p e c a n f i l l can f i l l  i t even  a t a n k i n a few h o u r s and a n o t h e r p i p e faster.  How l o n g w i l l  i t take t o f i l l the  t a n k i f b o t h p i p e s a r e u s e d a t t h e same t i m e ? 4.  I f J i m c a n mow the  lawn  will 5.  a lawn  in a little  b i t l o n g e r amount o f t i m e , how  long  i t t a k e t h e two b o y s w o r k i n g t o g e t h e r ?  I f a b a n k c h a r g e s i n t e r e s t e a c h week o n e a c h $100 b o r r o w e d , and a man b o r r o w s how much w i l l  6.  i n p a r t o f a n h o u r and Bob c a n mow  o v e r $100 a n d p a y s  he be  i t back  i n two weeks,  charged?  There a r e 40 p u p i l s  i n Jane's c l a s s , b o t h boys  If  a r e i n t h e math c l u b and 2/3 o f t h e  some o f t h e b o y s  girls  and g i r l s .  a r e i n t h e n r a s i c c l u b , how many c l a s s members d o n o t  belong t o these  clubs?  APPENDIX C  CRITERION TEST  5.9 Criterion  Test  INSTRUCTIONS; FOR EACH PROBLEM, CIRCLE THE ANSWER THAT YOU THINK IS MOST CORRECT. PROBLEM.  CHOOSE ONLY ONE ANSWER FOR EACH  USE ANSWER (d) ONLY I F YOU THINK THE  PROBLEM DOES NOT GIVE YOU ENOUGH INFORMATION TO FIND AN ANSWER.  THERE IS NO TIME LIMIT, SO TRY  TO ANSWER AS MANY OF THE QUESTIONS AS YOU CAN.  1. What i s the diameter o f the l a r g e s t c i r c l e t h a t c a n be f i t i n t o a square w i t h an area o f 4 square f e e t ? (a) 4 f t .  (b) 1 f t .  2. A bathtub can f i l l The  (c) 2 f t .  (d) n o t enough i n f o r m a t i o n  i n 15 minutes and empty i n 5 minutes.  tub holds 30 g a l l o n s o f water.  How much water w i l l be  l e f t i n the t u b a f t e r t h r e e minutes i f t h e t u b s t a r t s out f u l l and the water d r a i n s a t the same time t h a t more i s coming i n ? (a) 24 g a l .  (b) 12 g a l .  (c) 18 g a l .  3. Sue had two quarts o f honey.  A l i c e took % o f t h i s , and then  B i l l took h a l f o f what Sue had l e f t . (a) 3/4 q t .  (b) 1 q t .  (d) n o t enough i n f o r m a t i o n  (c) ±h q t .  How much d i d B i l l  (d) n o t enough i n f o r m a t i o n  4. F i n d the area o f the f i g u r e below.  (a)10 s q . f t .  (b)14 s q . f t .  get?  (c)8 s q . f t . (d) n o t enough information  60 5. Find the area o f the figure below.  (a)33 sq. f t . (b)42 sq. f t .  /  ,  (c)30sq. f t . (d) not enough information  6. These two c i r c l e s have three separate regions, or pieces  How many separate regions are there contained i n three c i r c l e s of the same size which each touch the other two a t exactly two p o i n t s ? (a) 6  (b) 7  (c) 5  7. Joe's age i s three times Susan's.  (d) not enough information Four years from now Joe  w i l l be only twice as old as Susan. (a) 12  (b) 6  (c) 15  How old i s Joe now?  (d) not enough information  8. I f a man can drive 84 miles on 6 gallons of gasoline and the t r i p costs S3.06, how many miles can he drive for $16.83? (a) 462 mi.  (b) 235 mi.  (c) 182 mi.  (d) not enough information  9. Find the area of the figure below.  (a) 80 sq. f t . (b) 128 sq. f t . (c) 120 sq. f t . (d) not enough information  10. F i n d the area o f the f i g u r e below.  (a) 108 s q . f t . (b) 100 s q . f t . (c) 96 s q . f t . (d) n o t enough information 11. A t a p e t shop there were 10 more dogs than c a t s . s o l d 2 dogs and 2 c a t s . as c a t s . (a) 12  The shop  Now there were twice as many dogs  How many dogs does the shop have now?  (b) 10  (c) 20  (d) n o t enough  information  12. One s i d e o f a t r i a n g l e i s twice as long as another, and the t h i r d s i d e i s three inches l e s s than the sum o f the o t h e r two. is  I f the perimeter  o f the t r i a n g l e i s 39 inches what  the l e n g t h o f the t h i r d  (a) 21  (b) 18  (c) 12  side?  (d) n o t enough i n f o r m a t i o n  13. F i n d the area o f the f i g u r e below. 4 ft.  (a) 36 s q . f t .  (b) 38 s q . f t . (c) 34 s q . f t . (d) n o t enough information  14. John, who had $1 and some n i c k e l s had twice as much money as B i l l who had three times as many n i c k e l s as John.  How  much money d i d B i l l have? (a) 50$  (b) 45<r  (c) 60*  (d) n o t enough  information  62 15. Four p o i n t s A,B,C, and D a r e on a l i n e and B i s between A and C and B i s between D and C, then (a) A-D-C  (b) D-B-C  (c) B-D-A  (d) d o t enough i n f o r m a t i o n  16. I f f o u r p o i n t s A,B,C, and D a r e drawn on a p i e c e o f paper w i t h p o i n t s A and B, B and C, C and D, and D and A a l l four inches a p a r t , the r e s u l t i n g (a) a square  (b) a l i n e  f i g u r e ABCD i s  (c) a p a r a l l e l o g r a m  (d) n o t enough information  17. A t 50* an hour how many hours w i l l B i l l have t o work t o earn enough bo buy a s e t o f t r a i n s , where each  train  c o s t s $3.? (a) 6  (b) 12  (c) 9  (d) n o t enough  information  APPENDIX D  SAMPLE RESPONSE CARDS  Problem I I - Sheet I question  2-  Smith's a c r e s o f beans Jones' a c r e s o f beans 6 acres of corn  54 90  f o r each  ratio  f o r corn  total  acres  o f Smith -  234  total  acres  o f Jones -  330  Smith's acres  -  8  4/3  of corn  -  180  Problem I I - Sheet I I question  3-  first  pipe  second p i p e  - 3 hours to  fill  - 6 hours t o  average f i l l i n g  t i m e - 4%  rate  of f i r s t  pipe  rate  o f second pipe  fill hours  - 200 g a l l o n s p e r h o u r - 100 g a l l o n s p e r h o u r  c a p a c i t y o f t a n k - 600 g a l l o n s  APPENDIX E ANALYSES OF VARIANCE FOR ADJUSTED GROUP MEAN SCORES  A n a l y s i s o f V a r i a n c e f o r T o t a l T e s t Scores  Source o f V a r i a n c e  D.F.  Sum o f Sq.  Mean Sq.  E q u a l i t y o f Adjusted Group Means  2  18.23  9.12  Zero Slope  2  2819.34  1409.67  59  282.15  4.78  4  33.59  8.40  55  248.56  4.52  Error E q u a l i t y o f Slopes Error  F-Value 1.9062 294.78*(.01  1.86  A n a l y s i s o f V a r i a n c e f o r A r i t h m e t i c T e s t Scores  Source o f Variance  D.F.  Sum o f Sq.  Mean Sq.  E q u a l i t y o f Adjusted Group Means  2  5.92  2.96  Zero Slope  2  42.97  21.48  59  81.26  1.38  4  1.91  .48  55  79.35  1.44  Error E q u a l i t y o f Slopes Error  F-Value 2.15 15.60*(.01  .3305  68  A n a l y s i s o f V a r i a n c e f o r Geometry T e s t Scores  Source o f V a r i a n c e  D.F.  Sum o f Sq.  Mean Sq.  E q u a l i t y o f Adjusted Group Means  2  4.63  2.32  Zero Slope  2  1.03  .52  59  40.91  .69  4  3.31  .83  55  37.60  .68  Error E q u a l i t y o f Slopes Error  F-Value 3.43*(.05) .74  1.21  A n a l y s i s o f V a r i a n c e f o r Area T e s t Scores  Source o f V a r i a n c e  D.F.  Sum o f Sq.  Mean Sq.  F-Value .89  E q u a l i t y o f Adjusted Group Means  2  2.03  1.02  Zero Slope  2  13.70  6.85  59  67.21  1.14  4  3.86  .96  55  63.36  1.15  Error E q u a l i t y o f Slope Error  6.01*(.01)  .84  69  A n a l y s i s o f V a r i a n c e f o r T o t a l I n f o . T e s t Scores  Source o f V a r i a n c e  D.F.  Sum o f Sq.  Mean Sq.  E q u a l i t y o f Adjusted Group Means  2  55.04  27.52  Zero Slope  2  9.19  4.59  59  286.87  4.86  4  17.14  4.28  55  269.74  4.90  Error E q u a l i t y o f Slopes Error  F-Value 5.66*( .0 .95  .87  A n a l y s i s o f Variance f o r A r i t h m e t i c I n f o . T e s t Scores  Source o f V a r i a n c e  D.F.  Sum o f Sq.  Mean Sq.  E q u a l i t y o f Adjusted Group Means  2  12.69  6.35  Zero Slope  2  3.06  1.53  59  116.12  1.97  4  5.19  1.30  55  110.93  2.02  Error E q u a l i t y o f Slopes Error  F-Value 3.22*( .0 .78  .64  70  A n a l y s i s o f V a r i a n c e f o r Geometry I n f o . T e s t Scores  Source o f V a r i a n c e  D.F.  Sum o f Sq.  Mean Sq.  F-Value  E q u a l i t y o f Adjusted Group Means  2  4.26  2.13  2.80  Zero Slope  2  .79  .39  .52  59  44.92  .76  4  6.07  1.52  55  38.85  .71  Error E q u a l i t y o f Slopes Error  2.15  A n a l y s i s o f V a r i a n c e f o r Area I n f o . T e s t Scores  Source o f V a r i a n c e  D.F.  Sum o f Sq.  Mean Sq.  F-Value  E q u a l i t y o f Adjusted Group Means  2  3.40  1.70  1.51  Zero Slope  2  6.37  3.19  2.83  59  66.31  1.12  4  1.58  .40  55  64.72  1.18  Error E q u a l i t y o f Slopes Error  .34  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0101859/manifest

Comment

Related Items