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Investigation into the use of physical devices in teaching a unit of geometry. MacLean, Charles Fairbanks 1968

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AN INVESTIGATION INTO THE USE OF PHYSICAL DEVICES IN TEACHING A UNIT OF GEOMETRY by CHARLES FAIRBANKS  MacLEAN  B.P.E., UNIVERSITY OF BRITISH COLUMBIA, 1953  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF • MASTER OF ARTS, i n the Department of EDUCATION •  We accept t h i s t h e s i s as conforming  t o the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA September, 1968  In p r e s e n t i n g an the  thesis  advanced degree at Library  I further for  this  shall  the  in p a r t i a l  f u l f i l m e n t of  University  of  make i t f r e e l y  agree that  permission  s c h o l a r l y p u r p o s e s may  by  his  of  this  written  representatives. thesis  be  available  for  for extensive  g r a n t e d by  the  It is understood  for financial  gain  permission.  Department 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, Canada  British  Columbia  shall  requirements  Columbia,  Head o f my  be  I agree  r e f e r e n c e and c o p y i n g of  that  not  the  that  Study.  this  thesis  Department  c o p y i n g or  for  or  publication  allowed without  my  ABSTRACT T h i s study was  an attempt  to determine  the e f f e c t of the  use o f models i n l e a r n i n g the volume and t o t a l s u r f a c e a r e a of various polyhedra.  I t was  h y p o t h e s i z e d t h a t the c l a s s e s who  were allowed, to c o n s t r u c t models o f v a r i o u s space f i g u r e s would achieve s i g n i f i c a n t l y b e t t e r r e s u l t s on a t e s t o f the work covered than would those c l a s s e s who  were taught the same  t e r i a l by sketches and n o t e s on the chalkboard.  I t was  ma-  further  h y p o t h e s i z e d t h a t the c l a s s e s which had u t i l i z e d the models would s c o r e s i g n i f i c a n t l y h i g h e r on a r e t e s t which was approximately  two weeks a f t e r the i n i t i a l  held  test.  F i v e t e a c h e r s and e i g h t c l a s s e s were i n v o l v e d i n the study.  Two  t e a c h e r s taught a c o n t r o l group and an  experimental  group o f grade seven students w h i l e a t h i r d t e a c h e r had a cont r o l group and an experimental group of grade s i x s t u d e n t s . f o u r t h t e a c h e r taught one c o n t r o l group o f grade seven and the f i f t h  The  students  t e a c h e r taught one experimental group of grade  seven s t u d e n t s .  Mental age,  p r e v i o u s mathematical  achievement  ( t e a c h e r s ' E a s t e r grades) and s c o r e s on the geometry u n i t  test  and r e t e s t were r e c o r d e d f o r n i n e t y - f o u r c o n t r o l group students and n i n e t y - s e v e n experimental group s t u d e n t s .  Any  student  who  had not attended e i g h t y p e r c e n t of the t e a c h i n g p e r i o d s o r f o r whom no mental age o r p r e v i o u s mathematical a v a i l a b l e were not i n c l u d e d f o r purposes  achievement  of t h i s  was  study.  The geometry u n i t , which was;, o f two weeks d u r a t i o n , f o l l o w e d by a m u l t i p l e - c h o i c e t e s t o f t w e n t y - f i v e items and r e t e s t two weeks l a t e r .  The u s u a l c o r r e c t i o n f a c t o r was  p l i e d t o o f f s e t the e f f e c t o f g u e s s i n g by the examinees.  ap-  was a  ill A c o v a r i a t e a n a l y s i s was  then done i n which the two  independent  v a r i a b l e s , p r e v i o u s mathematical achievement and mental  age  were p a r t i a l l e d out and the a d j u s t e d means f o r the t e s t and r e t e s t were recorded. The  :  il  r e s u l t s I n d i c a t e d t h a t there was  no  significant  d i f f e r e n c e i n the mean s c o r e s o b t a i n e d by the two initial  groups on  the  t e s t - 14.647 f o r the c o n t r o l group and 14.6822 f o r the  experimental  - but t h a t there was  a significant difference i n  the mean scores o b t a i n e d by the two groups on the r e t e s t h e l d two weeks l a t e r .  The mean score o f the experimental group  was  14.4124 as compared to 1 3 * 9 2 5 5 f o r the c o n t r o l group. Although t h i s d i f f e r e n c e was  s i g n i f i c a n t a t the . 0 0 5 l e v e l i t was i r e -  cognized t h a t , f o r a l l p r a c t i c a l purposes,  a d i f f e r e n c e of h a l f  a p o i n t i n a t o t a l o f t w e n t y - f i v e items must be c o n s i d e r e d important  f o r the p r a c t i s i n g  un-  teacher.  A s u p e r f i c i a l examination  o f the d a t a i n d i c a t e d  two  q u e s t i o n s t h a t might be answered by f u r t h e r r e l a t e d study:  (1)  Would a l o n g e r p e r i o d between the t e s t and the r e t e s t show a g r e a t e r d i f f e r e n c e i n the mean scores, o f the two groups? and  (2)  Would the use o f models be of g r e a t e r b e n e f i t to a p a r t i c u l a r age group o r i to a homogeneous group o f h i g h o r low  achievers?  The r e s u l t s o b t a i n e d by t h i s and o t h e r r e l a t e d s t u d i e s seems to m i l i t a t e a g a i n s t any d e f i n i t i v e c o n c l u s i o n s as to the m e r i t o f u s i n g models i n the t e a c h i n g o f mathematics as a whole. j• appears t h a t i f any  It  s i g n i f i c a n t ,beneflt i s to be d e r i v e d from  t h e i r use i t w i l l be i n p a r t i c u l a r s u b j e c t areas t h a t w i l l determined  o n l y through  c a r e f u l l y designed  research.  be  iv  TABLE OF CONTENTS CHAPTER  PAGE i  i  I.  i  •  •  .  .  1  THE PROBLEM - Introduction - The Background - The Questions  II.  REVIEW OF THE LITERATURE . - Related  III.  8  Studies . . 16  DESIGN OF THE STUDY . . - Purpose and v a r i a b l e s , t o be c o n s i d e r e d - Content o f u n i t t o be taught - Design o f measuring  instrument  - Procedure f o r t e s t i n g and t a b u l a t i n g - Limitations IV. ' ANALYSIS AND DISCUSSION OF THE DATA  . .•>.  .22  - Procedure f o r A n a l y s i s - D i f f e r e n c e s i n s c o r e s o b t a i n e d on U n i t T e s t i  ' "  - D i f f e r e n c e s i n s c o r e s o b t a i n e d on Retest V.  CONCLUSIONS AND IMPLICATIONS FOR FURTHER STUDY . . .  .:2?  BIBLIOGRAPHY APPENDIX A.  24  S c a l e Drawings o f Models and I n f o r m a t i o n f o r 29  •' C o n s t r u c t i o n APPENDIX B. .< Geometry U n i t O u t l i n e and Guide f o r Teachers  33  APPENDIX C. !i Geometry U n i t T e s t land Answer Sheet . . . .  39  APPENDIX D.: Scores o b t a i n e d by:Experimental  46  Group . . .  APPENDIX E . , Scores o b t a i n e d by C o n t r o l Group . .  . . .  .49  LIST OF TABLES TABLE I.  PAGE D i f f e r e n c e s between Scores on T e s t 1 h e l d a t the c o n c l u s i o n o f the Geometry U n i t  II.  . . . . . . . . . .  D i f f e r e n c e s between Scores on Geometry Retest approximately Two Weeks a f t e r Test 1  III.  held . 2 3  A Comparison o f the High and Low A c h i e v e r s on the Geometry T e s t  IV.  23  . 2 5  A Comparison o f the High and Low A c h i e v e r s on the Geometry Retest  26  vi:. LIST OP FIGURES FIGURE 1.  PAGE  S c a l e Drawings o f a Right T r i a n g u l a r Prism and a Rectangular Prism  2.  . .  S c a l e Drawings o f a Cube and a Regular 31  Tetrahedron 3.  S c a l e Drawing o f a Right C i r c u l a r 32  Cylinder k.  Sketches o f a Right T r i a n g u l a r Prism and J6  a Rectangular Prism 5.  Sketch o f a Plane F i g u r e made up o f a Semi 37  c i r c l e , a Rectangle and a R i g h t T r i a n g l e 6.  30  Sketch o f a Space F i g u r e made up o f a Right T r i a n g u l a r Prism and one h a l f o f a Right Circular Cylinder  7.  38  .  Sketch o f a P e r s p e c t i v e o f a Rectangular Prism  , .  38  vii  Acknowledgements The w r i t e r would l i k e to render the customary g r a t i t u d e to Dr. E r i c D. MacPherson, Dr. B. C. Munro, Mr. T. Bates, Dr. J . D. Dennlson and Dr. D. McKle f o r t h e i r p a t i e n c e , encouragement, and guidance.  He i s a l s o most g r a t e f u l to the members  of the computer s c i e n c e s t a f f and to the t e a c h e r s who  collabo-  r a t e d w i t h him and without whose a s s i s t a n c e t h i s study c o u l d never have been i n i t i a t e d .  CHAPTER I  < '  THE PROBLEM INTRODUCTION  :  Of the many and d i v e r s e problems to be c o n s i d e r e d by the teachers of mathematics  i n our s c h o o l s today perhaps none  i s more apropos than the use of l e s s o n a i d s to f a c i l i t a t e l e a r n i n g of v a r i o u s mathematical  concepts.  From w i t h i n  the  this  g e n e r a l framework the use of p h y s i c a l models as an adjunct the t e a c h i n g of areas and volumes of v a r i o u s p o l y h e d r a i s subject  chosen f o r t h i s s t u d y .  The o b j e c t i v e w i l l be to  to the  eval-  uate the r e s u l t s o b t a i n e d from t e a c h i n g t h i s u n i t by two d i f f e r e n t methods and to measure t h e , d i f f e r e n c e s ,  i f any, i n the  amount l e a r n e d and r e t a i n e d by both groups.  BACKGROUND  S i n c e the i n c e p t i o n of the new mathematics  program i n  B r i t i s h Columbia s c h o o l s much work has been done to develop i n the students,,an a p p r e c i a t i o n of the s t r u c t u r e As the number of p o s t u l a t e s ,  of  mathematics.  d e f i n i t i o n s and theorems  the d i f f i c u l t y i n a s s i m i l a t i o n and r e t e n t i o n seems to  increases, rise  a c c o r d i n g l y . , The time a l l o t t e d f o r review and mastery of damentals has of n e c e s s i t y  fun-  been reduced and the hope i s t h a t  the  students w i l l l e a r n through understanding r a t h e r than r o t e memorization.  It  i s now commonly assumed t h a t a student  learns  best that which i s " m e a n i n g f u l " and f o r g e t s most q u i c k l y which i s n o t .  I f t h i s i s t r u e then the problem becomes how to  make the m a t e r i a l to be l e a r n e d m e a n i n g f u l . of the i n v e s t i g a t o r most students  In the  experience  seem a b l e to r e c a l l  satis-  f a c t o r i l y a r e l a t i v e l y short u n i t of work i n mathematics v i d e d they are  that  pro-  t e s t e d soon a f t e r the work has been completed.  The d i f f i c u l t i e s b e g i n when he i s r e q u i r e d to r e c a l l t h i s work i n order to master t h a t which f o l l o w s .  In time the a s t u t e t e a c h e r begins to r e a l i z e t h a t i t p o s s i b l e f o r most of h i s students  Is  to become g l i b w i t h symbols  and the new vocabulary without l e a r n i n g much of- the fundamental concepts  r e q u i r e d to understand and progress  a l g e b r a and geometry programs.  He r e a l i z e s  i n the  present  they can make the  same mistakes and misunderstand, the same b a s i c ideas as d i d t h e i r predecessors  on the o l d program.  It  p o i n t that one of two methods i s u t i l i z e d ; t r a c e h i s steps g i v i n g many more examples,  i s u s u a l l y at  this  the teacher may r e u n t i l most o f h i s  students are r e l a t i v e l y more s u c c e s s f u l on the q u i z z e s g i v e n , or he may press on, a s s u r i n g h i m s e l f and h i s students  that  understanding w i l l occur as they are exposed to more and more s o p h i s t i c a t e d theorems and problems r e l a t e d b a s i c concepts.  to the o r i g i n a l  The f i r s t method seems to s a t i s f y the  less  capable student and the second appears to be much more approp r i a t e f o r the more g i f t e d . .  r •  | .  .  '  I f the student i s unable' to f o r m u l a t e h i s own set concepts he may e a s i l y become confused and d i s c o u r a g e d .  of  Woodruff suggests t h a t ; '.i ii one of the t h r e e f u n c t i o n s of v e r b a l b e h a v i o r i s to p r o v i d e f o r concept f o r m a t i o n but o n l y i f the concepts to be communicated are a l r e a d y i n the p o s s e s s i o n of both the sender and the r e c e i v e r . Such communications can serve to h e l p each p a r t y review and r e o r g a n i z e h i s concepts but not to g i v e him concepts he does not a l r e a d y h a v e . i In the experience  of the i n v e s t i g a t o r  i n the ..current geometry program.  t h i s i s p a r t i c u l a r l y true  Most students have  little  d i f f i c u l t y i n p l o t t i n g ordered p a i r s on the C a r t e s i a n plane once they have l e a r n e d the agreements f o r p o s i t i v e and n e g a t i v e direction.  Very few f a i l to v i s u a l i z e the p o i n t (3»-2) as b e i n g  i n a p o s i t i o n three u n i t s to the r i g h t of the y - a x i s and two u n i t s below the x - a x i s .  The sketch of two p e r p e n d i c u l a r axes i s  simple f o r a l l to copy and the uniqueness of the p o s i t i o n of any point i s e a s i l y established.  The t h r e e dimensional problem i s  something q u i t e d i f f e r e n t and the attempt to i l l u s t r a t e conceptj o f t e n compounds the c o n f u s i o n . illustrate replicate  this  Even i f the t e a c h e r can  t h i s concept on the chalkboard the student can it.  S i m i l a r l y most students  rarely  can r e a d i l y understand  the development r e q u i r e d f o r d e t e r m i n i n g the a r e a of a t r i a n g l e , but most experience volume of a  great d i f f i c u l t y i n c o n c e p t u a l i z i n g the  tetrahedron.  The second f a c e t of t h i s problem i s d i r e c t l y r e l a t e d the f i r s t . cepts,  to  Even i f the students; can master the r e q u i s i t e c o n -  i s there any guarantee t h a t he, w i l l r e t a i n a  p o r t i o n of t h i s knowledge?  satisfactory  W i l l : he be a b l e to r e c a l l work t h a t  has been l e a r n e d e a r l i e r i n the year o r i n the p r e v i o u s year i n o r d e r to proceed to work which r e q u i r e s t h i s p r e v i o u s knowledge?  ^Because of the cumulative p a t t e r n of most of the work  i n h i g h , s c h o o l mathematics  t h i s problem i s of v i t a l concern  to  a l l who, teach t h i s s u b j e c t .  No teacher of grade twelve mathe-  matics would expect h i s students to have t o t a l r e c a l l of a l l that m a t e r i a l which was presented i n grade e l e v e n , but a b r i e f review as r e q u i r e d should be s u f f i c i e n t to f a c i l i t a t e of most of the b a s i c p r e r e q u i s i t e m a t e r i a l .  the  recall  Bigge suggests  that  m a t e r i a l which i s meaningful to students i s remembered much b e t t e r than that which i s n o t .  He goes on to say t h a t "meaning-  f u l n e s s c o n s i s t s of the r e l a t i o n between  facts-generalizations,  2 rules,  p r i n c i p l e s f o r which the students see some u s e . "  In the  case of areas and volumes of space f i g u r e s t h i s would appear be p a r t i c u l a r l y t r u e . meters,  areas,  to  P r e s e n t i n g a l i s t of formulae f o r p e r i -  s u r f a c e areas and volumes accompanied by a p o o r l y  c o n t r i v e d sketch may i n v i t e c o n f u s i o n f o l l o w e d by memorization by those who; are i n c l i n e d to do s o .  Pressey^ et  a l found t h a t  at the end of a course students remembered about three of the f a c t s c o v e r e d .  One year l a t e r  quarters  they c o u l d r e c a l l about  one h a l f the m a t e r i a l and two years l a t e r  about one q u a r t e r .  Although many p r a c t i s i n g t e a c h e r s may f e e l t h a t h i s f i n d i n g s too h i g h ,  Pressey goes on to suggest  t h a t t h e r e i s every reason  to suppose that the process of f o r g e t t i n g continues u n t i l , most types of " f a c t courses" clusion,  virtually a l l is lost.  for  In c o n -  he s t a t e s t h a t t h e r e i s no r e s e a r c h which f u r n i s h e s a  b a s i s f o r c o n t r a d i c t i n g the o p i n i o n t h a t many s c h o o l  courses  might as w e l l not be taken at a l l i f t h e i r v a l u e i s to be judged on the b a s i s of a n y t h i n g other than s h o r t - t e r m of f a c t s . ematics  are  j  retention  S t u d i e s such as t h i s must g i v e the t e a c h e r o f math-  caus^ f o r grave concern;  the awareness  t h a t what he i s  t e a c h i n g migljvt w e l l be f o r g o t t e n so soon must induce him to amine c a r e f u l l y h i s methods.  He must s t r i v e f o r some  alter-  ex-  5 n a t i v e which w i l l a l l o w the students to grasp the b a s i c p l e s of the work p r e s e n t e d .  princi-  Stephens sums up by s t a t i n g :  i f the m a t e r i a l i s s u f f i c i e n t l y meaningful there may be no f o r g e t t i n g whatever. An important governing p r i n c i p l e , l i k e the c o n s e r v a t i o n of energy, may so h e l p us o r g a n i z e the r e s t of our i d e a s t h a t i t stays w i t h us f o r l i f e . Content t h a t , i s not so b r i l l i a n t l y s t r u c t u r e d but which s t i l l has such meaning w i l l be remembered i n p r o p o r t i o n to i t s meaning. Nonsense m a t e r i a l i s headed f o r e x t i n c t i o n b e f o r e the l a s t s y l l a b l e i s uttered.^"  In the case of t e a c h i n g the areas and volumes of  space  f i g u r e s , much s e r i o u s thought must be g i v e n to the o r g a n i z a t i o n of the m a t e r i a l so t h a t a few b a s i c p r i n c i p l e s may be a p p r e c i - . ated.  The plane f i g u r e s , which are r e l a t i v e l y easy to l e a r n  and understand, should serve as a b a s i s f o r the i n t r o d u c t i o n . and l e a r n i n g of the space f i g u r e s . t a n g l e i s presented c l e a r l y ,  F o r example,  i f the  rec-  then the s u r f a c e a r e a of a  rec-  t a n g u l a r prism should f o l l o w n a t u r a l l y and serve to the same b a s i c concept.  reinforce  The volume of the p r i s m might then  f o l l o w as b e i n g the product of the r e c t a n g u l a r base and the height.  I f t h e student i s a l s o allowed to c o n s t r u c t ,  label  and manipulate a simple model of t h i s prism he might more r e a d i l y appreciate  the b a s i c r e l a t i o n s between the two and  three d i m e n s i o n a l f i g u r e s .  Similarly,  f i n d the a r e a w i t h i n a c i r c l e ,  a f t e r l e a r n i n g how to  he might then l o g i c a l l y proceed  to a model of the r i g h t c i r c u l a r c y l i n d e r to determine i t s f a c e a r e a (two c i r c l e s and a r e c t a n g l e ) and i t s volume (the duct of the ;area of i t s base and i t s  When Bruner, O l v e r ,  surpro-  height).  G r e e n f i e l d ^ et a l conducted t h e i r  s t u d i e s i n c o g n i t i v e growth, they found t h a t the number of  6 c h i l d r e n who l e a r n e d c o n s e r v a t i o n i n c o n d i t i o n s under which l a b e l l i n g and m a n i p u l a t i o n v a r i a b l e s were combined was markedly h i g h e r than when e i t h e r o r both of these v a r i a b l e s were m i s s i n g . They found t h a t 76% of the group ( 1 6 out of 21) were a b l e  to  l e a r n t h i s concept when both v a r i a b l e s were present but when no m a n i p u l a t i o n was allowed the number l e a r n i n g dropped to kQ% (8 out of 20) and when no l a b e l l i n g was done o n l y 30/6 (6 out of 20) were s u c c e s s f u l .  When n e i t h e r v a r i a b l e was present  the  number l e a r n i n g dropped to 2$% (5 out of 2 0 ) .  T h i s study f u r n i s h e s ample j u s t i f i c a t i o n f o r i n v e s t i g a t i n g the p o s s i b i l i t i e s f o r a b e t t e r method of t e a c h i n g u n i t o f geometry d i s c u s s e d h e r e i n .  the  The two main q u e s t i o n s to be  investigated' are: 1.  W i l l the i n t r o d u c t i o n of v a r i o u s geometric  space  f i g u r e s and the computation of t h e i r s u r f a c e  areas  and volumes be f a c i l i t a t e d by the c o n s t r u c t i o n , and l a b e l l i n g of simple p h y s i c a l models?  2.  W i l l those students :who have had the o p p o r t u n i t y to l e a r n by t h i s methodi r e t a i n more of what they have l e a r n e d than those students who have been taught by the use of sketches  on the  chalkboard?  7  FOOTNOTES CHAPTER I i  A. D. Woodruff, "Teacher E d u c a t i o n Programmes o f t h e F u t u r e " , (paper r e a d a t t h e Conference o f t h e P r o v i n c i a l Teac h e r s ' A s s o c i a t i o n s i n Western Canada, Vancouver, B r i t i s h Columbia, F e b r u a r y 6 t h , 1 9 6 6 ) , P. 3 - 4 (mimeographed). X  M. iL. B i g g e , L e a r n i n g Theory f o r Teachers, (New York: Harper and Row, 1 9 6 4 ) , p. 3 0 0 . 3 s . L. P r e s s e y , F. P. Robinson and J . E. H o r r o c k s , P s y c h o l o g y i n E d u c a t i o n , ( t h i r d e d i t i o n : New York: Harper and Row, 1 9 5 9 ) .  pp. 2 6 2 - 6 3 -  ^ J . M. Stephens, E d u c a t i o n a l P s y c h o l o g y , (Toronto: H o l t ,  R h i n e h a r t and W i n s t o n , 1 9 5 6 ) , •  5j.  s  .  pp. 4 2 7 - 2 8 .  B r u n e r , R. R. O l v e r , P. M. G r e e n f i e l d and o t h e r s , S t u d i e s i n C o g n i t i v e Growth, (New York: John' W i l e y and Sons,  1966), pp. 220-22.  CHAPTER I I A. REVIEW OF THE LITERATURE .  A review of the r e s e a r c h conducted i n the g e n e r a l of t e a c h i n g mathematics w i t h the a i d of p h y s i c a l d e v i c e s  area re-  v e a l e d a v a r i e t y of procedures and a number of c o n c l u s i o n s at some v a r i a n c e w i t h each o t h e r .  Most of the s t u d i e s were c o n -  cerned w i t h the d i f f e r e n c e i n achievement between two groups, one of which had used models as an a d j u n c t to the l e a r n i n g of some u n i t i n mathematics.  There appeared to be no g e n e r a l  agreement as to how the p h y s i c a l d e v i c e s were to be i n t r o d u c e d , how they were to be used and whether they were to be s u p p l i e d by the teacher o r c o n s t r u c t e d by the  Berger  1  and J o h n s o n  2  students.  suggest, t h a t i f t e a c h i n g a i d s  used p r o p e r l y the. r e s u l t should be more c o r r e c t  and more mean-  i n g f u l l e a r n i n g i n a s h o r t e r time wi;th b e t t e r r e t e n t i o n . f i r s t p a r t of t h i s statement  are  The  does not appear to be c o r r o b o r a t e d  by a review of the p e r t i n e n t s t u d i e s and the l a t t e r not seem to have enjoyed the same a t t e n t i o n of the  p a r t does investigators.  T h i s apparent l a c k of agreement as to the m e r i t of p h y s i c a l d e v i c e s might ; i n p a r t be e x p l a i n e d by B e r g e r ' s  2  suggestion, t h a t  p h y s i c a l d e v i c e s can be a waste of time i f not used e f f e c t i v e l y : i.e.  "what unique c o n t r i b u t i o n can t h i s m a t e r i a l make toward  b e t t e r l e a r n i n g t h a t cannot be made as w e l l o r b e t t e r without it?"  The q u e s t i o n then seems to. be hot whether models serve any  9 u s e f u l f u n c t i o n i n the l e a r n i n g of mathematics rather,  per se but  are there some models t h a t w i l l Improve the l e a r n i n g ,and  r e t e n t i o n of c e r t a i n t o p i c s i n  mathematics?  :i  •  •  , Osborne^, German**, Sanders^ and o t h e r s ^  •  seem to agree  t h a t the mere use of t e a c h i n g a i d s does not ensure t h a t objectives  f o r them w i l l be met.  They suggest  •  the  t h a t these a i d s  must be used at the r i g h t time and i n the r i g h t way i f they to be e f f e c t i v e i n promoting l e a r n i n g .  are  T h e i r excessive or i n -  d i s c r i m i n a t e use may l e a d to overdependence on c o n c r e t e ;or v i s u a l representation  on the one hand o r may o c c a s i o n a l l y l e a d  to mere, entertainment  on the o t h e r , r  Bruner? and h i s a s s o c i a t e s ? i n t h e i r study of c h i l d r e n ' s a b i l i t y to l e a r n the concept of c o n s e r v a t i o n demonstrated most c l e a r l y the n e c e s s i t y used.  of concern f o r how the models were to be  When the students were allowed to manipulate models  which were not l a b e l l e d o n l y 30% of the group l e a r n e d the c o n cept but when the models were l a b e l l e d the percentage of ners rose to a s t a r t l i n g 16%.  lear-  On the o t h e r hand o n l y k0% of  the  group who had access to l a b e l l e d models but were not allowed to manipulate them was s u c c e s s f u l i n l e a r n i n g t h i s concept.  This  study by a group l e d by one of the most r e s p e c t e d men i n the f i e l d of c o g n i t i v e growth must s u r e l y be g i v e n c a r e f u l c o n s i d e r a t i o n by any who would i n v e s t i g a t e  the m e r i t s of p h y s i c a l d e -  v i c e s as an a i d i n the teachingi;of mathematics. sequent s c r u t i n y of v a r i o u s r e s e a r c h " p r o j e c t s , t i o n was given by t h i s i n v e s t i g a t o r  In the submuch more  atten-  to the method by which the  models were ^introduced r a t h e r than r e s u l t s  o b t a i n e d by the coni-  ,10  t r o l or experimental groups.  ImmerzeelS i n a comparative i n -  v e s t i g a t i o n of the use and non-use of m a n i p u l a t i v e d e v i c e s i n the t e a c h i n g of seventh-grade  mathematics  found t h a t the  exper-  imental group made g r e a t e r progress as measured by s t a n d a r d i z e d tests,  teacher-made  t e s t s and d a i l y r e c o r d s .  Although i t was  stated; that the experimental group achieved s i g n i f i c a n t l y b e t t e r results  ($% l e v e l ) ,  there seemed t o ' b e  the study which might be mentioned.  some b a s i c weaknesses  in  Only two c l a s s e s were i n -  v o l v e d i n the study both of which were taught by the author and w h i l e i t was s t a t e d that they were equal i n a b i l i t y there was no mention of the method used to accomplish t h i s e q u a l i t y . l a c k i n g was any s p e c i f i c r e f e r e n c e  Also  as to how the models -  v a r i o u s r i g h t t r i a n g l e s which showed the squares on the l e g s and the hypotenuse - were used to a i d i n the l e a r n i n g of the P y t h a gorean  theorem.  ,. Anderson^ who d e a l t w i t h a much l a r g e r - 5^1 eigth-j-graders  sample p o p u l a t i o n  from three J u n i o r High Schools -  reported  much more s p e c i f i c a l l y on the method of p r e s e n t a t i o n of models " In e v a l u a t i n g t h e i r e f f i c i e n c y i n t e a c h i n g a r e a , Pythagorean r e l a t i o n s h i p .  volume and the  He r e p o r t e d that an eight-week i n -  s t r u c t i o n a l ' p e r i o d was g i v e n i n : w h i c h the experimental group had s i x t e e n p h y s i c a l d e v i c e s a v a i l a b l e to them at a l l times s e l f - h e l p and s t u d y .  I t was n o t e d , however,  for  t h a t no s p e c i f i c  i n s t r u c t i o n s , re t h e i r use was g i v e n and the students d i d not p a r t i c i p a t e fin t h e i r c o n s t r u c t i o n .  In h i s f i n d i n g s he r e p o r t e d  that w h i l e the experimental group achieved c o n s i s t e n t l y r e s u l t s , they> were not s i g n i f i c a n t l y s o .  He r e p o r t e d  higher  further  t h a t there was no r e l a t i o n between the amount the models were.  ;  - 11  '  used and the score a t t a i n e d . I In another s i m i l a r study, C o h e n  1 0  ' attempted to e v a l u a t e  a technique to Improve space p e r c e p t i o n a b i l i t i e s through the c o n s t r u c t i o n o f models by s e n i o r h i g h s c h o o l students i n a course ;in s q l i d geometry.  S i x t y - t h r e e matched p a i r s o f s t u d e n t s r  s p e c i a l i z i n g i n mathematics and s c i e n c e were s e l e c t e d and a f i v e month p e r i o d o f s o l i d and a n a l y t i c geometry was g i v e n .  The ex-  p e r i m e n t a l group c o n s t r u c t e d twenty-three a p p r o p r i a t e models w h i l e the c o n t r o l group was g i v e n the u s u a l treatment u s i n g chalkboard and diagrams.  A p r e t e s t on space p e r c e p t i o n and  post t e s t s on the Minnesota Paper Form Board and D i f f e r e n t i a l A p t i t u d e T e s t s on space p e r c e p t i o n were g i v e n . d i f f e r e n c e i n performance was noted.;  No  significant  He concluded t h a t , t h e r e  was no j u s t i f i c a t i o n f o r the c l a i m t h a t c o n s t r u c t i o n of.models by students d u r i n g t h e i r study o f geometry w i l l r  growth i n a b i l i t y to v i s u a l i z e . ,  ;, P i c t o n  further>their  a l s o attempted: to determine the e f f e c t on  f i n a l achievement o f an i n t r o d u c t o r y : u n i t based upon the development o f v i s u a l i z a t i o n and u n d e r s t a n d i n g through the cons t r u c t i o n and use o f models.  He developed a ten-day i n t r o d u c -  t o r y u n i t on the c o n s t r u c t i o n o f models and found no  significant  d i f f e r e n c e i n the f i n a l r e s u l t s n o f the experimental and,the , n  c o n t r o l groups.  r  .  .  12 Jamleson  attempted to, e v a l u a t e the e f f e c t i v e use o f a ;  v a r i a b l e - b a s e abacus i n t e a c h i n g c o u n t i n g i n numeration o t h e r than base t e n .  systems  He s e l e c t e d n i n e t y - f o u r seventh-grade n  12. students,  gave a p r e t e s t ,  post t e s t . abacus,  a f i v e - d a y i n s t r u c t i o n a l p e r i o d and a :  One group had the b e n e f i t o f a d e m o n s t r a t o r - s i z e  ;  a second group had the domonstrator and a small abacus  each w h i l e a t h i r d group had no a i d s but the b l a c k b o a r d and chalk.  He found no s i g n i f i c a n t d i f f e r e n c e i n the post  test  results. i  .  .•  In a s l i g h t l y d i f f e r e n t approach to a s i m i l a r problem, ' W i l l i a m s ^ i n v e s t i g a t e d the u t i l i z a t i o n of c o l o u r e d  transpa-  1  rencles  f o r the t e a c h i n g of d e s c r i p t i v e geometry to c l a s s e s of  engineering students.  Two i n s t r u c t o r s  each taught  one w i t h the a i d of the overhead p r o j e c t o r  two  and c o l o u r e d  parencies and the other w i t h the chalkboard o n l y .  classes, trans-  The r e s u l t s  i n d i c a t e d t h a t a marked advantage was gained by those i n the p e r i m e n t a l group;  the year-end average of the  ex-  experimental  group was 79*3% w h i l e the c o n t r o l group had an average of T h i s r e p r e s e n t e d a g a i n of s l i g h t l y more than h a l f a  7^.9$.  grade p o i n t . :  S i g n i f i c a n t a l s o was'the  result  that 70$ of  A ' s were found In the experimental group and 75% of the were i n the c o n t r o l group.  the  F's  While t h i s study does not appear  to  bear any d i r e c t r e l a t i o n s h i p to the use of models as an a i d to the learningc.of. mathematics  i t does l e n d credence  to  the.sug-"  g e s t l o n made rearlier t h a t many t e a c h e r s do have d i f f i c u l t y i n ;  r  s k e t c h i n g meaningful t h r e e - d i m e n s i o n a l diagrams on the board.  ,The c o l o u r e d t r a n s p a r e n c i e s  approval of the i n s t r u c t o r s  chalk-  met w i t h the "unanimous"  and students  a l l of whom agreed  that the t o p i c was more understandable and e n j o y a b l e when t h e r e was the a d d i t i o n a l time to d i s c u s s the t o p i c s as they were p r e sented.  . . / . . ' • ( . ' • . ' '  -i .. • .;  13 A f t e r r e v i e w i n g the l a s t  decade of s t u d i e s r e l a t e d  to  the use of p h y s i c a l models as an a i d i n the t e a c h i n g of mathematics,  f o u r c o n c l u s i o n s seem to be both important and v a l i d . 1.  Pew of the i n v e s t i g a t o r s  have evinced much concern  f o r the method of i n t r o d u c i n g the models nor have they e x h i b i t e d much c o n s i s t e n c y  i n d e c i d i n g what  a c t u a l purpose these a i d s should s e r v e .  2.  None of the s t u d i e s reviewed seemed to be concerned w i t h the v a l u e of the models i n h e l p i n g the r e t e n t i o n beyond the p o s t - u n i t  3.  students  test.  Most of those s t u d i e s which r e p o r t e d no s i g n i f i c a n t d i f f e r e n c e s i n achievement between the c o n t r o l and experimental groups had l i t t l e i f any emphasis upon the p r e p a r a t i o n of the i n s t r u c t o r s c o u l d be used to best i  ^.  so t h a t the model  advantage. i -  '  '  .  Considerable c o n f l i c t i n I f i n d i n g s m i l i t a t e d  against  any d e f i n i t i v e c o n c l u s i o n s as to the m e r i t of u s i n g models i n the t e a c h i n g of mathematics It  as a whole.  seems evident t h a t i f any s i g n i f i c a n t b e n e f i t  to be d e r i v e d from t h e i r use i t w i l l be i n p a r t i c u l a r s u b j e c t areas t h a t , w i l l be determined :only ; through c a r e f u l l y designed r e s e a r c h .  . •  is  14  FOOTNOTES CHAPTER II ^ J . E . Berger, " P r i n c i p l e s G u i d i n g the Use of Teacher Pupil-Made L e a r n i n g D e v i c e s " Twenty-second Yearbook of the N a t i o n a l C o u n c i l of Teachers of Mathematics. (Providence; American Mathematical S o c i e t y , 1 9 5 4 ) p p . 1 5 8 - 6 1 .  and  2  J . E . Berger and D . A . Johnson, A Guide to the Use and Procurement of Teaching A i d s f o r Mathematics. A b u l l e t i n p r e pared by the Secondary School C u r r i c u l u m Committe of the N a t i o n a l C o u n c i l of Teachers of Mathematics, ( A p r i l 1959) PP« 7 - 8 . . 3 . Osborne "The Use of Models i n Teaching The A r i t h m e t i c Teacher. (January 196l) p p . 22-24. R  Mathematics"  ' **F. W. German "What Laboratory Equipment f o r Elementary and High School Mathematics?" F o r t y - t h i r d Volume of School Science and Mathematics (1943) p p . 3 3 5 - 4 4 . ^W. J . Sanders, "The Use of Models i n Mathematics I n s t r u c t i o n " The A r i t h m e t i c Teacher (March 1964) p p . 1^6-6$ ^ N a t i o n a l C o u n c i l of Teachers of Mathematics, M u l t i Sensory A i d s i n the Teaching of" Mathematics. ( l 8 t h Yearbook, 1945) p . 18. ; ? J . S. Bruner, R. R. O l v e r , P . M. G r e e n f i e l d and o t h e r s , Studies" i n C o g n i t i v e Growth, (New York: John W i l e y and Sons, 1966) p p . 219-24. Q ° G . E . Immerzeel, "A Comparative I n v e s t i g a t i o n of the Use and Nonuse of M a n i p u l a t i v e Devices i n Teaching Seventh Grade Mathematics", (unpublished M a s t e r ' s t h e s i s , Iowa S t a t e Teachers C o l l e g e , 1956), p p . 35-41* 1 : G. JR. Anderson, " V i s u a l - T a c t u a l D e v i c e s : t h e i r E f f i c i ency i n Teaching A r e a , Volume and the Pythagorean R e l a t i o n " (unpublished D o c t o r a l t h e s i s , the P e n n s y l v a n i a S t a t e U n i v e r s i t y , 1957). p p . 187-94. g ,i y  L o u i s Cohen, "An E v a l u a t i o n of a Technique to Improve Space P e r c e p t i o n A b i l i t i e s Through the C o n s t r u c t i o n of Models by Students i n a Course i n S o l i d Geometry (unpublished D o c t o r a l t h e s i s , The New York Y a s h i v a U n i v e r s i t y , 1959). p p . 2 5 1 - 5 5 l 0  L J.-. 0 . P i c t o n , "The E f f e c t on F i n a l Achievement i n S o l i d Geometry of an I n t r o d u c t o r y U n i t based upon D e v e l o p i n g V i s u a l i z a t i o n and Understanding through the Use of Models" (unpublished D o c t o r a l t h e s i s , The Montana S t a t e C o l l e g e , Bozeman, 196.2), pp.' 1 2 4 - 2 6 . * ; 1 1  1  15 K . W. Jamieson, J r . , "The E f f e c t i v e n e s s of a V a r i a b l e Base Abacus i n Teaching Counting i n Numeration Systems other than Base T e n " , (unpublished D o c t o r a l T h e s i s , The George Peabody C o l l e g e , N a s h v i l l e , Tennessee 1 9 6 2 ) , p p . 1 9 8 - 2 0 5 . i 2  W. W i l l i a m s "An E v a l u a t i o n i n the U t i l i z a t i o n of Coloured T r a n s p a r e n c i e s f o r the Teaching of D e s c r i p t i v e Geom e t r y . " (umpublished D o c t o r a l T h e s i s , The U n i v e r s i t y of Texas, d i s s e r t a t i o n a b s t r a c t s , volume 24, 19&3) p'» 657*  CHAPTER  III  DESIGN OP THE STUDY  The purpose of the study was to compare two methods  of  t e a c h i n g a u n i t of Geometry i n v o l v i n g volume and s u r f a c e  area  of b a s i c  figures  space f i g u r e s .  In the f i r s t method the v a r i o u s  were i n t r o d u c e d by an i n f o r m a l d e s c r i p t i o n and an accompanying diagram on the c h a l k b o a r d , both of which were c o p i e d by the students.  T h i s was intended to r e p r e s e n t e n l i g h t e n e d t r a d i -  t i o n a l p r a c t i c e and was denoted as method A .  The formulae f o r  area and volume were then Introduced and r e l a t e d problems were presented f o r d i s c u s s i o n and s o l u t i o n .  The second method  (denoted as method B) was t y p i f i e d by the use of p h y s i c a l models which the students were used by the students  l e a r n e d to c o n s t r u c t .  group.  Appendix A i l l u s t r a t e s  used and how the models were  E i g h t c l a s s e s and f i v e t e a c h e r s , mentor,•were  i n v o l v e d i n the s t u d y .  each to have a ' p a i r  method A andione by method B .  '  the  i n c l u d i n g the  experi-  Two of the t e a c h e r s were one to be taught by  The experimentor  had two grade  one by each method, and the remaining two  t e a c h e r s taught s i n g l e c l a s s e s , method B .  to  constructed.  of grade seven c l a s s e s ,  s i x c l a s s e s to t e a c h ,  These  to h e l p l e a r n i n f o r m a t i o n i d e n t i c a l  that presented to the f i r s t materials  appropriate  one by method A and the o t h e r by w. "  1/7  All  c l a s s e s were grouped heterogeneously by the s c h o o l  a d m i n i s t r a t o r s at the s t a r t of the s c h o o l year and no f u r t h e r randomization was attempted. c a l achievement  Mental age and p r e v i o u s mathemati-  ( E a s t e r r e p o r t ) were recorded a l o n g w i t h scores  obtained i n the geometry u n i t t e s t and r e t e s t .  Because  this  u n i t was completely new to t h i s p a r t i c u l a r grade l e v e l i t was f e l t that there would be l i t t l e d i f f e r e n c e i n a b i l i t y or p r e ference of the t e a c h e r s to present felt  e i t h e r method.  I t was a l s o  that there would be l i t t l e chance of any p r e v i o u s m a t e r i a l  l e a r n e d by the students a f f e c t i n g t h e i r achievement i n t h i s area.  For t h i s second reason no p r e t e s t was g i v e n to  either  group,.  The vestigator  t e a c h e r s i n v o l v e d i n t h i s study met w i t h the i n p r i o r to the s t a r t of the experiment to d i s c u s s  u n i t to be taught and how best to present i t .  the  The teachers  were a p p r i s e d of the g e n e r a l l y accepted methods of d e r i v i n g maximum b e n e f i t of the models and of the most e f f e c t i v e way to sketch the t h r e e - d i m e n s i o n a l p o l y h e d r a to be s t u d i e d .  When the  teachers were c o n f i d e n t t h a t they had a t t a i n e d the r e q u i s i t e s k i l l i n both methods,  f u r t h e r time was spent i n a d i s c u s s i o n  of how best to teach the students to c o n s t r u c t r e q u i r e d space  o r sketch these  figures.  A f t e r agreement was reached as to content and method the experimentor then drew up a summary of the work to be covered (Appendix B) which a l l t e a c h e r s agreed to f o l l o w as c l o s e l y as p o s s i b l e . same time,  A l l classes  s t a r t e d a t approximately the  the t h i r d week i n May, and spent the same l e n g t h o f  18 time,  approximately two weeks,  on the u n i t .  Example questions  were i n c o r p o r a t e d i n the l e s s o n p l a n s and i t was agreed t h a t no formal.homework would be a s s i g n e d .  .The general format of  the  t e s t was a l s o i n c l u d e d i n t h i s o u t l i n e so that t h e r e would be no c o n f l i c t i n vocabulary, used.  The students  descriptions,  formulae and d e f i n i t i o n s  i n v o l v e d w i t h the models were not a l l o w e d  to remove them from the classroom nor were the students other group encouraged to p r a c t i s e  i n the  the s k e t c h i n g of any of  space f i g u r e s o u t s i d e of r e g u l a r c l a s s  the  time.  T h e : c o n t e n t of the u n i t began w i t h a d i s c u s s i o n of rectangle lateral  f o l l o w e d by the square,  t r i a n g l e and the c i r c l e .  the r i g h t t r i a n g l e ,  the  the equi-  In the f i r s t p a r t of the  periment n e i t h e r group used models;  a l l students  i n the u s u a l manner u s i n g r u l e r and compasses.  ex-  sketched them  A f t e r these  plane f i g u r e s were d e s c r i b e d and d i s c u s s e d the analogous  space  f i g u r e s were presented i n o r d e r - the r e c t a n g u l a r  the  cube,  prism,  the r i g h t t r i a n g u l a r p r i s m , the r e g u l a r t e t r a h e d r o n and  the r i g h t c i r c u l a r c y l i n d e r .  Time was a l l o t t e d i n the l e s s o n  p l a n n i n g f o r the teacher to e x p l a i n to the "method A" group how to make a t h r e e - d i m e n s i o n a l s k e t c h ,  d i s c u s s i n g such t h i n g s  as  o p t i c a l i l l u s i o n s and p e r s p e c t i v e . '  A s i m i l a r time was a l l o w e d  the "method :,B" group f o r c o n s t r u c t i o n and study o f the models.  Since ho s a t i s f a c t o r y able, names,  measuring instrument was a v a i l -  experimental q u e s t i o n s were developed to t e s t r e c a l l  of  d e s c r i p t i o n s and r e l a t e d formulae f o r the v a r i o u s f i g u r e s .  Other q u e s t i o n s were d e v i s e d to t e s t ^ t h e s t u d e n t s '  a b i l i t y to  s o l v e problems i n v o l v i n g areas and volumes, of the space i f i g u r e s .  A t h i r d s e t , o f q u e s t i o n s was designed t o t e s t the s t u d e n t s ' a b i l i t y to s o l v e more d i f f i c u l t problems i n v o l v i n g two or more of  the,figures discussed.  As the time a v a i l a b l e f o r t e s t i n g ,  purposes was r e s t r i c t e d to a f o r t y minute p e r i o d and t h e amount of work to be t e s t e d was c o n s i d e r a b l e , i t was d e c i d e d t h a t a m u l t i p l e - c h o i c e type o f t e s t would be most e f f i c a c i o u s . . F o r t h i s reason a f i v e - c h o i c e t e s t o f t h i r t y items was c o n s t r u c t e d . T h i s i s shown i n Appendix C.  B r i e f i n s t r u c t i o n s were g i v e n a t  the top o f the f i r s t page o f the t e s t , but because o f the r e l a t i v e n a i v e t y o f the examinees i t was agreed t h a t o r a l i n s t r u c t i o n s would precede t h e handing out o f t e s t papers. o r a l statement t o be made was,  The  "There w i l l be marks taken o f f  f o r wrong answers so do n o t guess u n l e s s you a r e r e a s o n a b l y sure t h a t the answer you s e l e c t i s the best one. Read each c h o i c e carefully.  No q u e s t i o n s w i l l be answered once the t e s t i s  handed t o you."  Each student was t o have two sharpened  pencils  and an e r a s e r and i t was agreed; t h a t t h e i n v e s t i g a t o r would have some e x t r a p e n c i l s i n case o f emergency.  No mention was made o f  the c o r r e c t i o n formula t o be a p p l i e d .  I P r i o r t o the a d m i n i s t r a t i o n o f the f i n a l examination a rough d r a f t iwas g i v e n t o a group o f grade t e n geometry students who had s t u d i e d and were f a m i l i a r w i t h t h i s p a r t i c u l a r u n i t . At  the I end o f t h i s t r i a l run, comments r e g a r d i n g a m b i g u i t i e s ,  time a l l o t m e n t and computational d i f f i c u l t i e s were s o l i c i t e d i from the group.  T h i s l a s t f a c t o r was o f c o n s i d e r a b l e concern,  i n t h a t the t e s t was supposed  t o be one o f understanding and r e -  c a l l r a t h e r i t h a n a t e s t o f computational s k i l l .  When t h i s ana-  l y s i s was completed and t h e l e s s d e s i r a b l e q u e s t i o n s e l i m i n a t e d .  t h i r t y questions remained,  the f i r s t f i v e of which would not be  counted.for the purposes of t h i s s t u d y .  These t e s t e d the  recall  of d e s c r i p t i o n s of the i n t r o d u c t o r y plane f i g u r e s which were taught by the s k e t c h i n g method to both groups.  I t was agreed,  however, that the scores on these items would be i n c l u d e d i n the r e s u l t s sent to the teachers of the c l a s s e s . t o use as saw f i t i n the e v a l u a t i o n of t h e i r s t u d e n t s  The  1  they  progress.  f i n a l d r a f t of the t e s t and the accompanying answer  sheet (Appendix C) was then d e l i v e r e d to the t e a c h e r s i n v o l v e d . I t was agreed that i t would be g i v e n on the day f o l l o w i n g  the  t e r m i n a t i o n of the u n i t and again approximately two weeks  later.  The  investigator  c o l l e c t e d and marked the examination and r e -  peated the procedure on the r e t e s t two weeks l a t e r . for  each group were recorded s e p a r a t e l y  the t e a c h e r s . relative The  The r e s u l t s  and c o p i e s were sent  to  No i n f o r m a t i o n was d i s t r i b u t e d concerning the  success of c o r r e s p o n d i n g c l a s s e s i n d i f f e r e n t  investigator  schools.  then d e l e t e d the score of any student who had  not attended a t l e a s t 8C$ of the l e s s o n s and any student  for  whom no mental age score o r E a s t e r r e p o r t mark was a v a i l a b l e . These f i n a l c o n s i d e r a t i o n s r e s u l t e d i n a p o p u l a t i o n drop of  six-  teen i n the method A group and n i n e t e e n i n the method B group.  21  LIMITATIONS  Probably one of the weakest aspects of t h i s study was the use of a teacher-made  t e s t the v a l i d i t y of which was depen-  dent upon the experience of the experlmentor. any weaknesses were at l e a s t with colleagues,  It  i s hoped t h a t  p a r t i a l l y overcome by c o l l a b o r a t i o n  the o t h e r t e a c h e r s i n v o l v e d i n the  experiment  and the grade ten c l a s s used i n the a n a l y s i s of the t r i a l run test.  Other weaknesses might be the d i s p a r i t y i n the a b i l i t y of  the teachers to present  the two methods of i n s t r u c t i o n , the i n -  t e r a c t i o n among members of the two groups o u t s i d e the and the r e l a t i v e l y short time between the t e s t and the  classroom retest.  As has been ,noted i n the d e s i g n of the experiment the i n v e s t i g a t o r attempted to keep the e f f e c t minimum.  of these weaknesses to a  CHAPTER IV, THE ANALYSIS AND DISCUSSION OP THE DATA ;  i  .  •<  There was a t o t a l o f 9 7 s u b j e c t s i n the method A group and 9 ^ s u b j e c t s i n the method B group f o r whom s c o r e s i n mental age, p r e v i o u s mathematical achievement, geometry u n i t t e s t and r e t e s t were r e c o r d e d .  A c o v a r i a t e a n a l y s i s was done by the  computer s c i e n c e department i n which the two independent v a r i a b l e s , p r e v i o u s mathematical achievement and mental age, were p a r t i a l l e d out and the adjusted'means f o r the f i r s t were o b t a i n e d .  These r e s u l t s a r e g i v e n i n T a b l e 1 .  unit  test  The i d e n -  t i c a l procedure was f o l l o w e d f o r the u n i t r e t e s t and these r e s u l t s are given i n Table 2 .  i.  Because the a d j u s t e d means oif the f i r s t  t e s t as shown  i n T a b l e 1 were so c l o s e - H f . 6 8 2 2 compared w i t h 1 ^ . 6 4 7 1 - i t was assumed:there was no s i g n i f i c a n t d i f f e r e n c e i n the s c o r e s o b t a i n e d b y the two groups on the f i r s t  test.  Because a d i f f -  erence was noted i n the means o f the two groups i n the r e t e s t , Table 2 a l s o shows the r e s u l t s i o b t a i n e d by comparing the s c o r e s of  the; two groups on t h e retest;.  I t was n o t e d t h a t a T - v a l u e o f  3 . 5 3 was o b t a i n e d which would i n d i c a t e t h a t t h e r e was a s i g n i f i c a n t d i f f e r e n c e i n the s c o r e s o f t h e two groups.  TABLE I DIFFERENCES BETWEEN SCORES ON TEST #1 HELD AT THE CONCLUSION OF THE GEOMETRY UNIT. ; A d j u s t e d Means  .N  Group A 14.6471  94  Group B 14.6822  97  TABLE I I DIFFERENCES BETWEEN SCORES ON GEOMETRY RETEST HELD APPROXIMATELY TWO WEEKS AFTER TEST #1. Adjusted Means Group A  13^9255  Degrees o f Freedom 190  0.49215  a ' Group B :  Mean Difference  * s i g n i f i c a n t at  S.D.  T-Value  0.13945 the  *3-53  .005 level  14.4124 Although the r e s u l t s o b t a i n e d i n T a b l e 2 i n d i c a t e  that  t h e r e was a s i g n i f i c a n t d i f f e r e n c e i n the s c o r e s o f the two groups on the r e t e s t i t must be r e c o g n i z e d t h a t the A  f o r a l l p r a c t i c a l purposes,  is  unimportant.  difference,  CHAPTER V CONCLUSIONS AND IMPLICATIONS FOR FURTHER STUDY  The purpose of the study was to determine  (l)  whether  the use of p h y s i c a l models would be b e t t e r than chalkboard sketches  f o r l e a r n i n g a u n i t i n t h r e e - d i m e n s i o n a l geometry and  (2) would they enhance the s t u d e n t ' s has learned?  a b i l i t y to r e t a i n what he  Judging from the r e s u l t s  obtained,  the answer  the f i r s t q u e s t i o n must be "no" and to the second, "yes".  to  a qualified  As was noted i n the A n a l y s i s of the Data, the  difference  i n r e t e n t i o n was s i g n i f i c a n t , but few p r a c t i s i n g t e a c h e r s would f e e l motivated to spend e x c e s s i v e time o r energy to r a i s e o v e r a l l mean of a mathematics  the  c l a s s by a mere h a l f - p o i n t o u t i o f  . t o t a l of 25 ( 2 # ) . :  ; Some i n t e r e s t i n g c o n j e c t u r e s might' l e a d to f u r t h e r r e l a t e d  a r i s i n g from t h i s  study.,  study  The f i r s t of these i s —  would the d i f f e r e n c e i n the scores of the two groups on the t e s t have been g r e a t e r had the ,time between the t e s t and the ;  r e t e s t been -longer?  A f o l l o w - u p on one of the  groups .^method B) taught by t h e • i n v e s t i g a t o r  almost  be noted that w h i l e t h i s appeared t o t h e i n v e s t i g a t o r remarkably high r e t e n t i o n there, w e r e , s e v e r a l tors;  (i)  :;  experimental  l a t e r produced a mean score of 1 2 . ^ on the same t e s t . (  re-  twojyears It  must  to be a  extenuating  fac-  the mean of t h i s group was c o n s i d e r a b l y h i g h e r than  the mean of the o v e r a l l experimental group ( 1 7 . 2 as opposed to 1^.6),  ( i i ) ,three students of the o r i g i n a l c l a s s had been r e 1  p l a c e d by f o u r newcomers whose background d i d not i n c l u d e t h i s work and were excused from the t e s t and  ( i i i ) t h e r e had been  some r e f e r e n c e to these geometric f i g u r e s i n the new seven mathematics programme.  grade  In s p i t e o f these c o n s i d e r a t i o n s  and the l a c k o f a c o n t r o l group i t might be i n f e r r e d t h a t the d i f f e r e n c e i n r e t e n t i o n would be g r e a t e r had the i n t e r v a l between the two  t e s t s been l o n g e r .  A s u p e r f i c i a l i n v e s t i g a t i o n o f the d a t a gave r i s e to the second c o n j e c t u r e :  "Would the use o f models o r p h y s i c a l  d e v i c e s be more b e n e f i c i a l t o a c e r t a i n type o f student o r class?*'  That i s , d i d the experimental procedure b e n e f i t  slower student r a t h e r than the b r i g h t e r one?  the  I t appeared  the more a b l e students ( j u d g i n g only; by the l e t t e r grade  that assig-  ned by the mathematics* teacher) b e n e f i t t e d more from the p e r i m e n t a l method than d i d the l e s s a b l e .  ,1'  r  'V'."  TABLE I I I :  : A COMPARISON OF THE HIGH AND LOW ACHIEVERS ON THE GEOMETRY TEST. ; l • High* Low* |i • •] - I Method A group 16.1 11.6 s :i * I • .  -  1  17.6  11.4  * those students w i t h l e t t e r grades o f A o r B 3  1  '>  ** those students w i t h l e t t e r grades o f D o r E  ex-  26  Despite the f a c t t h a t no c o n s i d e r a t i o n has been g i v e n to other p e r t i n e n t v a r i a b l e s such as mental age, i t might be i n f e r r e d t h a t c l a s s e s o f h i g h a c h i e v e r s i n mathematics c o u l d der i v e g r e a t e r b e n e f i t from the use o f p h y s i c a l models i n a geometry u n i t such as t h e one r e f e r r e d to i n t h i s  study.  D i r e c t l y r e l a t e d t o t h e second c o n j e c t u r e i s t h e t h i r d , "does the h i g h a c h i e v e r group r e t a i n i t s knowledge b e t t e r than does the low a c h i e v e r group?"  T h i s time t h e r e s u l t s appear t o  be q u i t e d i f f e r e n t from those noted i n the p r e c e d i n g  matrix.  TABLE IV A COMPARISON OP"THE HIGH AND LOW ACHIEVERS ON THE GEOMETRY RETEST. High  Low  Method A group  16.3  9»3  Method B group  17.1  11.6  Here i t w i l l be noted t h a t the h i g h a c h i e v e r s a r e v i r :  n  1'  t u a l l y i d e n t i c a l i n r e t e n t i o n whereas the low a c h i e v e r s i n the experimental  group gained markedly over those i n the c o n t r o l r• n.  group.  .  '• 1  26  D e s p i t e the f a c t t h a t no c o n s i d e r a t i o n has been g i v e n t o other p e r t i n e n t v a r i a b l e s such a? raental age, i t might be i n f e r r e d t h a t c l a s s e s o f h i g h achie%< r s i n mathematics  c o u l d de-  r i v e g r e a t e r b e n e f i t from the use >-r p h y s i c a l models i n a geometry u n i t such as the <~  D i r e c t l y re' "does the h i g h a c h i '  o f e r r e u t o i n t h i s study.  ^ the second c o n j e c t u r e i s the t h i r d , ^roup r e t a i n i t s knowledge b e t t e r than  does the low a c h i e v e ; group?"  T h i s time the r e s u l t s appear t o  be q u i t e d i f f e r e n t from those noted i n the p r e c e d i n g m a t r i x .  TABLE IV A COMPARISON OF THE HIGH AND LOW ACHIEVERS ON THE GEOMETRY RETEST. ;  High  Low  Method A group  16.3  9.3  Method B group  17.1  :  11.6 r  n  Here i t w i l l be noted t h a t the h i g h a c h i e v e r s a r e v l r t u a l l y i d e n t i c a l i n r e t e n t i o n whereas the low a c h i e v e r s i n the experimental group gained markedly over those i n the c o n t r o l group. '  27  BIBLIOGRAPHY Anderson, G. R. " V i s u a l - T a c t u a l D e v i c e s : t h e i r E f f i c i e n c y In Teaching A r e a , Volume and the Pythagorean R e l a t i o n " (unpubl i s h e d D o c t o r a l t h e s i s , the P e n n s y l v a n i a S t a t e . U n i v e r s i t y , 1957) Berger, J . E . " P r i n c i p l e s Guiding the Use of Teacher and P u p i l made L e a r n i n g D e v i c e s " . Twenty-second Yearbook of the N a t i o n a l C o u n c i l of Teachers of Mathematics. (Providence: American Mathematical S o c i e t y , 195^) Berger, J . E . curement pared by National  and Johnson, D . A . A Guide to the Use and P r o of Teaching A i d s f o r Mathematics. A b u l l e t i n p r e the Secondary School C u r r i c u l u m Committee of the C o u n c i l o f Teachers o f Mathematics, ( A p r i l 1 9 5 9 )  Bigge, M. L . L e a r n i n g Theory f o r Teachers (New York: Harper and Row, 19#n Bruner, J . S. O l v e r , R.R. G r e e n f i e l d , P . M . and o t h e r s , S t u d i e s i n C o g n i t i v e Growth (New York: John Wiley and Sons, 1966) Cohen, L o u i s , "An E v a l u a t i o n of a Technique to Improve Space P e r c e p t i o n A b i l i t i e s Through the C o n s t r u c t i o n of Models by Students i n a Course i n S o l i d Geometry (unpublished D o c t o r a l t h e s i s , the New York Y a s h i v a U n i v e r s i t y , 1 9 5 9 ) German, F . W. "What Laboratory Equipment f o r Elementary and High School Mathematics? " F o r t y - t h i r d Volume of School Science and Mathematics (19^3) I, . 'I ;" Immerzeel, G i E . "A Comparative j l n v e s t i g a t i o n o f the Use and Nonuse of M a n i p u l a t i v e Devices i n Teaching Seventh Grade Mathematics" (unpublished M a s t e r ' s t h e s i s , Iowa S t a t e . Teachers C o l l e g e , 1 9 5 6 ) : :  Jamieson, J . W. J r . , "The E f f e c t i v e n e s s of a V a r i a b l e Base Abacus i n Teaching Counting i n ' N u m e r a t i o n Systems o t h e r than Base T e n ' . (unpublished D o c t o r a l t h e s i s , the George Peabody C o l l e g e , N a s h v i l l e , Tennessee 1 9 6 2 ) N a t i o n a l C o u n c i l of Teachers of Mathematics, M u l t i - S e n s o r y A i d s . i n .the Teaching of Mathematics. i ( l 8 t h Yearbook, 19^+5; Osborne, R. "The Use of Models i n Teaching Mathematics" A r i t h m e t i c Teacher. (January 1 9 6 1 )  The  28  P i c t o n , J . 0. "The E f f e c t on F i n a l Achievement i n S o l i d Geometry of an I n t r o d u c t o r y U n i t based upon Deveoping V i s u a l i z a t i o n and Understanding through the Use of Models" (unpublished D o c t o r a l t h e s i s , the Montana S t a t e C o l l e g e , Bozeman, 1 9 6 2 ) Pressey, S. L . , Robinson, F . P . , and H o r r o c k s , J . E . , Psychology i n E d u c a t i o n , ( t h i r d e d i t i o n : New York: Harper and Row,  19591  Sanders,  W. J .  "The Use of Models i n Mathematics  The A r i t h m e t i c T e a c h e r  (March  Instruction"  1964)  Stephens, J . M. E d u c a t i o n a l Psychology, h a r t and Winston, 1 9 5 6 )  (Toronto:  Holt, Rhine-  W i l l i a m s , C. W. "An E v a l u a t i o n i n the U t i l i z a t i o n of Coloured Transparencies f o r the Teaching o f D e s c r i p t i v e Geometry" (unpublished D o c t o r a l t h e s i s , the U n i v e r s i t y o f Texas, d i s s e r t a t i o n a b s t r a c t s , volume 24, 1 9 6 3 ) Woodruff, A ; D . "Teacher E d u c a t i o n Programmes of the F u t u r e " (paper read at the Conference o f the P r o v i n c i a l T e a c h e r s ' A s s o c i a t i o n s i n Western Canada, Vancouver, B r i t i s h Columbia February 6 t h , 1 9 6 6 )  \  APPENDIX A  THE MODELS - SCALE £''=1"  30:  6"  RIGHT. TRIANGULAR PRISM  8' t ft  A"  RECTANGULAR PRISM  V  CUBE  REGULAR TETRAHEDRON  32  CXUNOER (Allow approximately | " f l a t spot on c i r c l e f o r hinge)  6  V  - Be sure a l l l e t t e r s and dimensions are marked c l e a r l y on the models so that they w i l l appear on the o u t e r s u r f a c e when model Is assembled. - L e t ' e a c h student put h i s name on the f l a p of each model so w i l l show when s t o r e d i n dissembled s t a t e . - Standard s i z e paper c l i p s w i l l h o l d the model n e a t l y securely. * - Paper and s u p p l i e s w i l l be s u p p l i e d i f  and  required.  - Have ;the name of each polyhedron p r i n t e d i n b l o c k l e t t e r s where i t can be read e a s i l y when model Is i n assembled s t a t e , (see CYLINDER above)' .  I  3  it  APPENDIX B  GEOMETRY UNIT-  3^  VOLUMES AREAS & PERIMETERS - F i g u r e s w i l l be d e s c r i b e d r a t h e r than d e f i n e d f o r m a l l y . - E x p o n e n t i a l n o t a t i o n w i l l n o t be used; r r , n o t r . - \ \ w i l l be represented as 22/7 r a t h e r than 3.14-.  FIGURES TO BE DISCUSSED A. Plane F i g u r e s (i)  (ii)  (suggested  descriptions)  rectangle  - a f o u r - s i d e d plane f i g u r e w i t h o p p o s i t e s i d e s equal and square corners.  square  - a special rectangle sides equal.  having a l l  ( i i i ) right triangle . - a t h r e e - s i d e d plane f i g u r e .; . >1 h a v i n g one square c o r n e r . • ' . 'i ( i v ) e q u i l a t e r a l t r i a n g l e . - a t r i a n g l e h a v i n g three equal sides. (v)  B.  circle  Space F i g u r e s (i)  (ii) (ill)  (iv) (v)  - a set of p o i n t s i n a plane i which are e q u i d i s t a n t from a given p o i n t . (suggested  rectangular  prism  cube  descriptions) - a space f i g u r e having three p a i r s of equal r e c t a n g u l a r faces. - a s p e c i a l r e c t a n g u l a r prism having s i x equal square f a c e s .  r i g h t t r i a n g u l a r prism - ' a i s p a c e f i g u r e having t h r e e r e c t a n g u l a r f a c e s and two l r i g h t t r i a n g u l a r bases. , r i g h t c i r c u l a r c y l i n d e r - two c i r c u l a r bases j o i n e d at . r i g h t angles by a curved w a l l . r e g u l a r tetrahedron  1  - a space f i g u r e having f o u r equilateral triangular faces.  35;  FORMULAE TO BE LEARNED Plane F i g u r e  Perimeter  Area  rectangle  P=2»(l+w)  A=l» w  square  P=4.s  A=s» s  right  P=a+b+c  triangle  equilateral  triangle  circle  Space F i g u r e  P=3vs  A=£*a.b  C=2.i?«r  A=ivr«r  T o t a l Surface Area ( T . S . A . )  Volume  r e c t a n g u l a r prism  A=2»(l»w+l»h+w«h)  V=l.w.h  cube  A=6.s.s  V=s - s .s  r i g h t t r i a n g u l a r prism  A=a •b+a.h+b-h+ch  V=i.a .b.h  r i g h t c i r c u l a r c y l i n d e r A=2tTr.r + 2 u r » h  V=\?r.r.h  r e g u l a r tetrahedron  V = l / 6 - a .b »h  A=2.a.b  . VOCABULARY ( i n f o r m a l ) INTRODUCE AS REQUIRED plane  - a flat  s u r f a c e h a v i n g no t h i c k n e s s  r i g h t angle i perpendicular  - square  corner  two l i n e s which meet a t r i g h t  r  angles  I*  i  equidistant  - an equal d i s t a n c e  p i o r TT  - comparison of the l e n g t h s of C and D ( 2 2 / 7 ) i,  circumference  - measure of the d i s t a n c e around a c i r c l e n :•  perimeter diameter radius volume  >i  - measure o f the d i s t a n c e around any plane figure - d i s t a n c e a c r o s s widest p a r t o f c i r c l e .  j,  - a d i s t a n c e from c e n t r e to c i r c l e ( h a l f of diam.) & - a space e n c l o s e d by any space f i g u r e h  n  '  3.6, - p a r t of plane enclosed by any plane  area  figure  t o t a l s u r f a c e a r e a - area of a l l f a c e s and bases of a space figure edge  - where any two  equilateral  - a l l s i d e s equal  vertex  - where two s i d e s of a plane f i g u r e o r edges of a space f i g u r e meet  f a c e s of a space f i g u r e meet  two  "altitude  - p e r p e n d i c u l a r d i s t a n c e from v e r t e x to o p p o s i t e s i d e of any a p p l i c a b l e plane figure  i  "height  - p e r p e n d i c u l a r d i s t a n c e between bases (ends) of any a p p l i c a b l e space f i g u r e  alt. base  base  face  - any w a l l o f • a space f i g u r e  "base  - top and bottom o f a space f i g u r e or the . s i d e p e r p e n d i c u l a r to the a l t i t u d e i n a plane f i g u r e (see sketch above)  regular•figure  - one having a l l s i d e s * angles and f a c e s  semi-circle  - half a c i r c l e .  equal ;  FORMAT OF TESfc Time:  40 minutes  multiple choice:  each item must be read c a r e f u l l y otherwise many o f the d i s t r a c t o r s w i l l seem a c c e p t a b l e . Students must f i n d best answer. 1  combination ,  ]  power and speed: w i l l extend the best student i n most c l a s s e s : ^ e x p l a i n t h a t there i s a p e n a l t y f o r wrong answers and encourage them not to guess i f they have no i d e a o f the c o r r e c t a' . answer, (do not giVe c o r r e c t i o n formula.) ;,  37 made up of three (i) (ii)  parts:  r e c a l l of d e s c r i p t i o n s g i v e n : q u e s t i o n s designed to encourage t h i n k i n g r a t h e r than r o t e memory. r e c a l l o f formulae f o r p e r i m e t e r , a r e a , volume and t o t a l s u r f a c e a r e a (remember: formulae w i l l be w r i t t e n u s i n g no exponents I)  SYMBOLS TO BE USED  ;  a = a l t i t u d e of plane f i g u r e b = base of plane f i g u r e c = hypotenuse of plane f i g u r e s = s i d e of a square o r edge of a cube l» = 2 2 / 7 h = height between bases o f space f i g u r e r = radius J  J *  .-;  :r'  d = diameter A = area P = perimeter T.S.A.  = total  surface  area  V = volume -  ;  (iii)  problems i n computation of a r e a s , perimeter and volume. They w i l l range i n d i f f i c u l t y , the most d i f f i c u l t being combinations of two or more of the figures. » eig.  r:  s  f i n d the area o f : f i n d the a r e a o f : -  (1) semi c i r c l e (2) rectangle (3) r i g h t t r i a n g l e  •tip f i n d  the. vplume o f :  (1) (2)  half a right circular cylinder r i g h t t r i a n g u l a r prism  Note that the problems w i l l not be d e s c r i b e d as shown i n the examples. There w i l l be a sketch o r a d e s c r i p t i o n but not b o t h .  PROCEDURE FOR TEACHING UNIT Both groups, must have the samefamount of i n s t r u c t i o n a l t i m e . The u n i t should take no more than 10 days (1 40 m i n . p e r i o d , per day) p l u s one p e r i o d f o r t e s t and one p e r i o d f o r the r e t e s t two weeks l a t e r . " Both groups w i l l have e x a c t l y the same i n s t r u c t i o n procedure f o r the plane f i g u r e s , u t i l i z i n g o n l y sketches on the b o a r d . Both groups should be g i v e n ample o p p o r t u n i t y to s o l v e r e l a t e d . problems. r When -introducing space f i g u r e s show group A how to sketch the t h r e e - d i m e n s i o n a l f i g u r e s and d i s c u s s problems of p e r s p e c t i v e and o p t i c a l i l l u s i o n e t c .  e.g.  .  r e c t a n g u l a r prism  Note that the p e r s p e c t i v e i s exaggerated here and you must be c e r t a i n that the c l a s s understands t h a t t h i s i s how i t l o o k s not how i t i s . To reduce the p e r s p e c t i v e e i t h e r move x n f a r t h e r away or move base B c l o s e r to base A . In the experimental group (B) use c o n s t r u c t i o n paper (not heavyweight grade) and l e t every student make h i s own models. Their c o n s t r u c t i o n should take no morie tim'e than group A uses f o r s k e t c h i n g . aThe f o l l o w i n g diagrams i n d i c a t e recommended dimens i o n s f o r the r e q u i r e d models. rNote t h a t they may be s t o r e d flat. ' ,  APPENDIX C  GEOMETRY UNIT TEST AND ANSWER SHEET  P l a c e an "X" o p p o s i t e the number ON THE ANSWER SHEET which i n d i c a t e s the best answer f o r each q u e s t i o n .  1.  2 . Read a l l c h o i c e s c a r e f u l l y before making your s e l e c t i o n . 3« F o o l s c a p w i l l be s u p p l i e d f o r rough work.  I . DESCRIPTION OF FIGURES 1.  Rectangle: a 4 - s i d e d plane f i g u r e h a v i n g : (a) k equal s i d e s and square c o m e r s (b) 4 equal s i d e s (c) o p p o s i t e s i d e s and o p p o s i t e angles equal. (d) 2 p a i r s of equal s i d e s *(e) o p p o s i t e s i d e s equal and a l l angles equal  2.  .Square: > (a) a plane f i g u r e w i t h 4 equal s i d e s *(b) a r e c t a n g l e w i t h a l l s i d e s equal ,(c) a plane f i g u r e w i t h 4 equal angles (d) a plane f i g u r e w i t h a l l s i d e s and a l l angles equal " (e) none of these J  3«  Right t r i a n g l e : a 3 - s i d e d plane f i g u r e h a v i n g : *(a) one square corner ;.(b) one square corner and three unequal s i d e s (c) three equal s i d e s and one square c o r n e r ' (d) three equal s i d e s (e) none of these  <  4.  E q u i l a t e r a l t r i a n g l e : a plane 1 3 - s i d e d f i g u r e h a v i n g : (a) a l l s i d e s equal and one square corner *(b) a l l angles equal (c) two equal angles and two equal s i d e s • (d) a l l s i d e s d i f f e r e n t (e) none of these  5.  C i r c l e : a plane f i g u r e : (a) shaped l i k e a hoop ~r~ I (b) shaped l i k e a p l a t e (c) having no corners ' (d) made up of a set of p o i n t s a l l of which are e q u i d i s tant from a g i v e n p o i n t *(e) none of these c • :  1  .'  6.  •  <•  Regular t e t r a h e d r o n : a space f i g u r e h a v i n g : (a) three e q u i l a t e r a l t r i a n g u l a r f a c e s (b) f o u r t r i a n g u l a r f a c e s (c) three e q u i l a t e r a l t r i a n g u l a r f a c e s and one square base c • c '(d) f o u r e q u i l a t e r a l f a c e s *(e) f o u r e q u i l a t e r a l t r i a n g u l a r bases  7. R e c t a n g u l a r P r i s m : a space f i g u r e h a v i n g : ( a ) equal o p p o s i t e f a c e s *(b) 3 p a i r s o f e q u a l r e c t a n g u l a r f a c e s (c) 6 equal f a c e s (d) a l l a n g l e s e q u a l (e) none o f these 8. Cube: - a space f i g u r e h a v i n g : ( a ) equal o p p o s i t e f a c e s (b) a l l edges e q u a l (c) a l l a n g l e s e q u a l (d) a l l square c o m e r s * ( e ) none o f t h e s e 9. Right C i r c u l a r C y l i n d e r : (a) a tube w i t h t h i n w a l l s *(b) two c i r c u l a r bases j o i n e d a t r i g h t a n g l e s by a c u r v e d wall (c) a t h r e e - d i m e n s i o n a l c i r c l e (d) a: s o l i d c i r c u l a r r o d (e) none o f t h e s e ' • ' .1. 10. R i g h t ] T r i a n g u l a r P r i s m : a space f i g u r e h a v i n g : * ( a ) y r e c t a n g u l a r f a c e s and 2 r i g h t t r i a n g u l a r bases (b) 5, r e c t a n g u l a r f a c e s i(c) 5 t r i a n g u l a r bases i (d) 3 t r i a n g u l a r and 2 r e c t a n g u l a r f a c e s (e) none o f these. :  1  ;  i i  FORMULAE:  ' .  •  (.  1. T o t a l S u r f a c e A r e a o f a R i g h t C i r c u l a r . C y l i n d e r : ( a ) "vT.r.r h+2."*o-r (b) XT-r»h+2.Tf.r * ( c ) 2vt\.r.r +2.(?.r.h •? \ (d) 2.>u.r.r+\i-r»h (e) (T*r.r+u-r«h 2.  • (  •  Total): S u r f ace A r e a o f a r e c t a n g u l a r p r i s m : (a) 1-w+l.h+w.h .(b) 1 •l+w»w+h»h i (c) l»w.h >• *(d) 2»(Lw+l.h+w.h) ' (e) 2i.w.h ' !  •  •  ' > • ' • ' " • ' • • • " . • • .  3 . T o t a l ; S u r f a c e A r e a o f a cube: *(a) 6?s «s (b) s»s.s (c) k'S »s (d) 6»s«s»s i (e) none o f t h e s e  ... : •  • -  \. • ' •  4. T o t a l Surface Area of a r e g u l a r tetrehedron: (a) 1/3. a -b -h (b) 4 . a «b (c) 3-a -b *(d) 2»a»b • (e) -|.a*b»h 1  5. Volume o f a r i g h t c i r c u l a r c y l i n d e r : , (a) 2«\vr»h , • • *(b) T\.r «r «h ' (c) J.n.r. r+a»7r»h (d) u.r.h (e)  2l?r» r «h  6. T o t a l S u r f a c e A r e a o f a r i g h t t r i a n g u l a r p r i s m : ' (a) -g »a»b«h (b) h(a+b+c) + ^a»b * ( c ) h.(a+b+c) + a»b (d) a .b»c «h (e) a«b+a»c+a»h i  ;  7. Volume o f a cube: (a) s»s • " (b) 6.s (c) 4.S» S (d) >6.s• s " ( e ) s»s*s  ' r,' : ' 1  -  1  : ,  8. Volume o f a R i g h t T r i a n g u l a r P r i s m : (a) a«b.h , "(b) Ifa.b.h (c) h(a+b+c) (d) l/6«a»b«h r, (e) none o f t h e s e 9 . 'Volume o f a R e g u l a r T e t r a h e d r o n : (a) area o f t h e base t i m e s t h e h e i g h t o f t h e t e t r a h e d r o n . (b) area o f t h e base t i m e s h a l f t h e h e i g h t o f t h e t e t r a hedron. " ( c ) one t h i r d t h e p r o d u c t o f t h e a r e a o f t h e base and t h e height (d) one h a l f t h e a r e a o f a t r i a n g u l a r p r i s m h a v i n g t h e same base and t h e same h e i g h t (e) none o f t h e s e 10. Volume o f a R e c t a n g u l a r P r i s m : (a) 6«1» w»h (b) £(l«w+l»h+w.h) . •'•(c) 6(l+w+h) (d) 4 » l . w h • "(e) l ' w h :  ' '  43  III  Problems ( l e t U =  22/7 wherever used)  1. The volume of a r e g u l a r t e t r a h e d r o n with a h e i g h t of 2 f e e t and the area of a f a c e 12 square i n c h e s i s : (a) 24 c u . i n . (b) 12 c u . i n . < (c) 48 c u . i n . *(d) 96 c u . i n . • (e) 8 c u . i n . 1  • 2. The volume of a r i g h t t r i a n g u l a r Prism w i t h an a l t i t u d e of 3 i ^ « a base of 4 i n . and a h e i g h t of 1 0 . i n . i s : (a) 120 c u . i n . (b) 70 c u . i n . *(c) 60 c u . i n . (d) 40 c u . i n . (e) 17 c u . i n . 3. -The t o t a l s u r f a c e area of a r e c t a n g u l a r prism w i t h l e n g t h 8 i n . , width 6 i n . and h e i g h t 2 f e e t i s : *(a) 5 1 / 3 s q . f t . (b) 768 s q . f t . (c) 21.6 s q . i n . ' ' • (d) 786 s q . f t . (e) 2 2/3 s q . f t . 1  4. The t o t a l s u r f a c e area of a r i g h t c i r c u l a r c y l i n d e r whose diameter i s 7 i n . ^and h e i g h t i s 3 i n . i s : (a) 77 s q . i n . • *(b) 143 sq.: i n . (c) 220 s q . i n . (d) $6 s q . i n . (e) 115 s q . i n . . 5. : I f the diameter of the base of a c y l i n d e r i s doubled and the h e i g h t i s h a l v e d , then the volume of the new c y l i n d e r . W i l l - be: y . • ; , [ . . (a) the same as the o r i g i n a l • . (b) h a l f as much as the o r i g i n a l *(c) twice as much as the o r i g i n a l (d) one q u a r t e r as much as the o r i g i n a l (e) f o u r times as much as the o r i g i n a l •  'r.  6.1 I f the l e n g t h , width and h e i g h t of a r e c t a n g u l a r p r i s m are a l l doubled then the t o t a l s u r f a c e area of the new prism w i l l be: (a) twice as l a r g e as the o r i g i n a l !i(b) e i g h t times as l a r g e as the o r i g i n a l (c) ^ i x times as l a r g e as the o r i g i n a l (d) n i n e times as l a r g e as the o r i g i n a l • *(e) f o u r times as l a r g e as the o r i g i n a l 'r' '  t »  'r  t  I  )i  »  '••  •  7- A r e c t a n g l e of l e n g t h 10 f t . and width 7 f t . has 2 semic i r c l e s f i t t e d e x a c t l y on to the two 7 f t . s i d e s . The perimeter of t h i s f i g u r e i s : (a) 2 7 f e e t . ' " (b) 108| f e e t . (c) 81 f e e t . *(d) 64 f e e t . * ' . . (e) 42 f e e t . 8.  The *(a) (b) (c) (d) (e)  area enclosed by the f i g u r e i n que. #7 above 108£ sq. f t . 27 sq. f t . 81 s q . f t . 64 s q . f t . 42 s q . f t .  is:  9« A l a r g e steamer trunk i s shaped l i k e a r e c t a n g u l a r prism w i t h the l i d i n the shape of h a l f a r i g h t c i r c u l a r cylinder. I f the dimensions of the r e c t a n g u l a r p a r t of the trunk are 2 f t . by 2 f t . by 7 f t . , what l e n g t h of s t r a p w i l l be needed to go around the trunk i n such a way as to ensure t h a t the l i d w i l l not come open even i f the l o c k f a i l s ? (Allow one e x t r a f o o t of s t r a p p i n g f o r the buckle and some o v e r l a p ) [ . (a) 18 3/7 f e e t . (b) 13 f e e t . , (c) 8 4/11 f e e t . *(d) .10 1/7 f e e t . • (e) 16 f e e t . 10.  A p a i n t can w i t h a diameter of 1 f o o t f i t s e x a c t l y i n t o a cube-shaped cardboard box so t h a t when the l i d of the box i s c l o s e d the can just, touches a l l s i x f a c e s of the box. The volume of the a i r space which i s i n s i d e the box but o u t s i d e the p a i n t can i s : ' (a). 1728 c u . i n . *(b) 3 1/14 c u . f t . . : r i; (c) 1420 c u . i n . • • • i (d) 11/14 c u . f t . (e) none of t h e s e . 3  .::  c' '  •  •  * r e p r e s e n t s the c o r r e c t c h o i c e f o r each q u e s t i o n  GEOMETRY UNIT TEST, Answer Sheet i  Name Please  print:  :  Grade School  A . D e s c r i p t i o n s of F i g u r e s (1) ;  (6)  B.  C.  (a) (b) (c) (d) Ce).  -  (a) (b) (°) (d) (e)  _ — _  ( 2 ) (a) (*) (c) (d) (e) (?)  __ -  (3)  _  (a) _(*> (c) ~ (d)-_ •(e) _  (8)  (a) (b) (c) (d) (e)  (4)  (a) (b) (c) (d) (e)  (a) (b) (°> U) (e)  (9)  (a) (b) .(c) •(d) (e)  Z Z  Formuale (1)  (a) • (b) •(c)  (2)  (6)  (a) (b) (c) (d) (e)  (7)  ,  (a) (3) (b) (c) (d) Z Z (e) (a) (8) (t>) Z Z (c) . (d) : (e)  (a) (b) (c) (d) (;e) (;a) (; ) ((c) (d) Ce) b  _ i _ W  _ _ _ _  (9)  (a) (b) (c) (d) (e) (a) (> (c) .(d) (e) b  Problems (1)  _ (2)  (a) (b) (c) (d)  —  (>  —  e  (6)  (a)  - L . (7)  ( > ZZ b  (c) (d) (le) I  _ _ -  ~_  (a) (b) (o)-~ (d) •(e) (a) (b) (c) (d) (e)  (3) :  I Z (8) _  _  (!a) •(b) (c) (d) Ce) Ga) ((c) (d) .. i . '  1  Z Z  z z —  _ (9) Z Z  z z  (a) (b) (c) (d) (e) (a) (b) (c) •(d) •(e)  )  APPENDIX D  SCORES OBTAINED BY THE EXPERIMENTAL GROUP  Subject Number 1 2 ' 3 4 5 6 7 8 9 10 11 12. 13 14 15 16 17 18 19 20  Letter Grade 6 4 •3 . 5 3. 6 1 7 • 5 3 5 2 5 4 '7 • 6 6 3 •1 4  Hen. - N e l . mental age  -  Geom. Test  18 ' '39 32 15 34 17 ' .14 43 ' 16 29 ••• . 4i 10 12 32 40 17 40 It 31 13 .45 . • 19 . 32 . 15 36 . 18 16 . 35 21 47 22 • 42'. 50 ,19 • 16 . 37 48 • •• • 12 4i :  •  21 . 22 23 • 24 25 26 27 28 29 30 31 32 33 . 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 51 52 53  4 • 3 6 7 - 3 • 4 6 5 5 6 3 6 4 7 2 4 • 5 5 4 4' 5  2 6 6'  1.  1  . 35 , 31 • 47 45 ' ' 36 . ' 39 ' 40 44 :. 34 37 . ' V . 32 • 43 40 37 . ' 32 . . 51 . 40 32 . *•' 39 43 ; ••' 37 .  :  ;  :  :  . s 4 6 i 4 5 .2 4  •'  •  •  ??  44 .29  40 39 39 43 • 35 39 '77 4 4 •-. 40  '•  Geom. Retest  Gain Score  17 14 18 15 \ 15 9 11 18 16 10 20 15 . 17 15 20 20 20 . 17 10 16 •  -1 . -1 • 1 1 :-l . 1 -l 1 2 -3 1 0 • -1 -1 -1 -2 1 1 -.2 1-  .  •••  5  15 13 12 18 14--. 10 17 16 14 9 : 13 18 15 . 19 14 17 . 29 7 10 V 15 17 • 18 J 3 .'. 5 "v ; :  1  14 13 12 ~ 15 7 17 19 18 25  •  16 11 12 : ' 19 ' ; 10 ' 1116 16 . 1 3 7  :  .;  io  • . 20 • 13 * ' 18.• .• ' 14 .. 20. 27 • 5 . .., 11 • 16 . 15 . 20 14 . 3 12' 13 10 • 16 . • 8 15 - •• 21 . 1626 •"'  1 ' -.2 ' 0 . 1 1 -1 ' 0 - 1 ". -2 -3 2. -2 -1 • 0 3 -2 -2 1 • 1 ' -2 2 -1 -2 -2 0 -2 • 1 1  t  -2 2 -2 . 1  Subject Number 5k 55  56 5? 58 59 60 6i 62 63 64' 65 66 67 68 69 70 71 72 73 • 74 75 76 77 78 79 • 80 81 82 • 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97  Letter Grade 4 5 3 5 5 2 1 3 5 7 4 4 5 5 6 3 4 2 2  Hen. - N e l . . mental age 3k  35 32  ?  •  40 30 39 . ' 35 .. k3  .  2 6 ' 5 • 3. 4 4 4 • 7 5 • 3 6 2 5 5 3 2 '7 4 1 5 ' 6 6 ' ' 4 5 7  '  6  .•  51 40 4l . 37 39 - 4l 31 34 32 36  ,  Geom. Test  Geom. Retest  ' 17 10 4 - 13 13 \ 12 9 ' 13  18 9 7 12 15 11 14 10 Ik 14 22 • 19 14 ••• 13 12 15 14 • 13 " 21. • 20 16 16 :8 9 12 ' 10 4 . . 7 6 9 , /  1 -1 3 •• -1 2 -1 5 -3 0 3 -l -3 . 1 ' -1 0 1 , -.2 3 -3  8 30 '• ' . 10 16 ,-. 38 , • 17 12 . 39 15 : ' 14 36 15 12 34 .'• 14 16 : ••••• 12 32 18 . 35 15 41 22 21 40 •" :.. 18 ; 16 34 17 18. ' 37 15 8 . 4 27 . , 34' . 17 13 . . 12 - k3 15 • . 10 . 11 33 . 11 19 .13 48 14 1416 12 37 14 .'• 32 13 . . • 20 43 40 20 " : 18 . 22 39 19 16 16 • 36 18 38 15 ' 4l 10 21 1  9  Gain Score  -2 •  - i ; -3 -1 2 -4 -3 1 • -2 -3 , . 3 -4 -4 3 : 1 2 0 -4 1 '. 1 • -2 , -3  0 3  1  APPENDIX E  SCORES OBTAINED BY CONTROL GROUP  Subject Number  4 • 4 2 6 5 6 3 ' 4 7 5 5 2 6 7 6 4 3 7 4 2  ' 1 2 3 4 5 . 6 7 8 9 10 11 12 13 14 15 16 17 • 18 19 20 . 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52  Letter Grade  ,  • .  6 5 5 4 6 4 3 4 4 3 4 7 5 : 4 4 5 ^ 6 4 3 7 5 2 2 3 1 7 7 4 6' 5 3 5  Geom. Test  Hen. - N e l . m e n t a l age  36 39 35 43 44 36 . 36 45 43 39 38 32 32 49 48 " 46 . 35 , 45 39 25  .  42 45 44 .37 44 36 46 39 44 40 *48 49 42 40 44 47 49 45 4l 43 46 35 38 40 31 50 47 45 4l 4l 36 35 -  .  .  12 18 11 13 , ; 16 14 . • 9 15 15 10 20 21 17 18 . 15 13 16 19 , 10 12  8 13 15 16 .14 8 10 17 . 19 19 • 20 21 16 18 16 22 13 15 • 16 10 17 14 6 - . 12 14 11  Geom. Retest  0 12 . -1 17 10 . -1 -2 15 14 ' -2 -2 12 10 1 12 -3 -2 13 11 1 17 -3 20 -1 -2 15 ' 18 0 16 1 14 1 14 . -2 -2 17 1.7 -3 10 -2 7  10 15 14 15 6 10 15 18 19 17 17 • 14 16 . • 19 20 15 16 • 15 .' 11 16 12 • 7 15 • 15 • 10  20 14 ' 16. • . 12 12 • • 15  Gain Score  17 16 16 15 ^ •' 10 16  -1-3 0 -2 1 -2 0 -2 -1 0 -3 -4 -2 -2 +3 -2 2 1 -1 1 1 -2 1 +3 1 . -1 - 3 '. 2 0 -3 ^2 .1  Subject Number  Letter Grade  53 55  56 57 58 59 . 6o 6l 62 63 64 65 66 67 68 69 . 70 71 72 73 7475 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94  •  .  5 ~ 2 3 7 .4 4 4 5 4 6 4 3 2 5 4  Hen. - N e l . mental age 38 36 32 39 .40 38 35 41 43 40 37 35 27 43 44 40 39 . 36.  .  ',5  3 5 2 4 4 3 7 4 2 5 4 6 6 7 1 '3 ' 5 5 .4 7 6 4 5 6 2  20 10 . 13 21 15 12 . 17 . 16 •17 21 14 ^ . 13 7 • 15 13 , ' :i4 1 22 18 , '  ••  .  -  3k  .  ,  30 36 38 40 , 37 43 ' 3k  39 36 33  •  ,  Geom. • Retest  Gain Score  19 6 13 22 \ 14 14 15 18 17 18 12 15 6 16 15 16 ' 23 17  -1 -4 0  10 : 6 7 14 . 15 18. 16 8 13 .10 18 21 21 9 ^ 14, . 11 .10 10 16 , 17 16. 13 • . . 18 •• 20 •' ; 17 ' 19 .• 16 13 14 15 17 15 ' 8 10 . 11 8 10 ' • . 13 19 16 17 . 10 14 15 17 . 20 19 .  3k  28 31 • 36 38 . . 47 41' 32 39 41 40 39 42  Geom. Test  •  l  -1 2 -2 2 0 -3 -2 2 -1 1 2 2 1 -1 -1 -2 -3 1  l  .2 -1 , -2 -1 -5 1 ' •1 '  2 -4 -3 0 •l  3 2 -2 -3 -1 -2 -2  .  

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