Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

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

Item Metadata

Download

Media
831-UBC_1968_A8 M385.pdf [ 3.27MB ]
Metadata
JSON: 831-1.0104446.json
JSON-LD: 831-1.0104446-ld.json
RDF/XML (Pretty): 831-1.0104446-rdf.xml
RDF/JSON: 831-1.0104446-rdf.json
Turtle: 831-1.0104446-turtle.txt
N-Triples: 831-1.0104446-rdf-ntriples.txt
Original Record: 831-1.0104446-source.json
Full Text
831-1.0104446-fulltext.txt
Citation
831-1.0104446.ris

Full Text

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 thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1968 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I a g r e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and Study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada ABSTRACT This study was an attempt to determine the e f f e c t of the use of models i n learning the volume and t o t a l surface area of various polyhedra. I t was hypothesized that the classes who were allowed, to construct models of various space figures would achieve s i g n i f i c a n t l y better r e s u l t s on a test of the work covered than would those classes who were taught the same ma-t e r i a l by sketches and notes on the chalkboard. I t was further hypothesized that the classes which had u t i l i z e d the models would score s i g n i f i c a n t l y higher on a ret e s t which was held approximately two weeks a f t e r the i n i t i a l t e s t . Five teachers and eight classes were involved i n the study. Two teachers taught a control group and an experimental group of grade seven students while a t h i r d teacher had a con-t r o l group and an experimental group of grade s i x students. The fourth teacher taught one control group of grade seven students and the f i f t h teacher taught one experimental group of grade seven students. Mental age, previous mathematical achievement (teachers' Easter grades) and scores on the geometry unit test and r e t e s t were recorded f o r ninety-four control group students and ninety-seven experimental group students. Any student who had not attended eighty percent of the teaching periods or f o r whom no mental age or previous mathematical achievement was available were not included f o r purposes of t h i s study. The geometry unit, which was;, of two weeks duration, was followed by a multiple-choice test of twenty-five items and a retest two weeks l a t e r . The usual correction f a c t o r was ap-p l i e d to o f f s e t the e f f e c t of guessing by the examinees. i l l A covariate analysis was then done i n which the two independent variables, previous 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 test and re-test were recorded. : il The r e s u l t s Indicated that there was no s i g n i f i c a n t difference i n the mean scores obtained by the two groups on the i n i t i a l t est - 14.647 f o r the control group and 14.6822 f o r the experimental - but that there was a s i g n i f i c a n t difference i n the mean scores obtained by the two groups on the retest held two weeks l a t e r . The mean score of the experimental group was 14.4124 as compared to 1 3 * 9 2 5 5 f o r the control group. Although t h i s difference was s i g n i f i c a n t at the . 0 0 5 l e v e l i t was i r e -cognized that, f o r a l l p r a c t i c a l purposes, a difference of h a l f a point i n a t o t a l of twenty-five items must be considered un-important f o r the p r a c t i s i n g teacher. A s u p e r f i c i a l examination of the data indicated two questions that might be answered by further related study: (1) Would a longer period between the test and the retest show a greater difference i n the mean scores, of the two groups? and (2) Would the use of models be of greater benefit to a p a r t i c u l a r age group or i to a homogeneous group of high or low achievers? The r e s u l t s obtained by t h i s and other r e l a t e d studies seems to m i l i t a t e against any d e f i n i t i v e conclusions as to the merit of using models i n the teaching of mathematics as a whole. I t j • appears that i f any s i g n i f i c a n t ,beneflt i s to be derived 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 subject areas that w i l l be determined only through c a r e f u l l y designed research. i v TABLE OF CONTENTS CHAPTER PAGE i i i • • . . I. THE PROBLEM 1 - Introduction - The Background - The Questions I I . REVIEW OF THE LITERATURE . 8 - Related Studies I I I . DESIGN OF THE STUDY . . . . 1 6 - Purpose and variables,to be considered - Content of unit to be taught - Design of measuring instrument - Procedure f o r t e s t i n g and tabulating - Limitations IV. ' ANALYSIS AND DISCUSSION OF THE DATA . .•>. .22 - Procedure f o r Analysis - Differences i n scores obtained on Unit Test i ' " - Differences i n scores obtained on Retest V. CONCLUSIONS AND IMPLICATIONS FOR FURTHER STUDY . . . 24 BIBLIOGRAPHY . : 2 ? APPENDIX A. Scale Drawings of Models and Information f o r •' Construction 29 APPENDIX B. .< Geometry Unit Outline and Guide f o r Teachers 33 APPENDIX C. !i Geometry Unit Test land Answer Sheet . . . . 39 APPENDIX D.: Scores obtained by:Experimental Group . . . 46 APPENDIX E., Scores obtained by Control Group . . . . . . 4 9 LIST OF TABLES TABLE PAGE I. Differences between Scores on Test 1 held at the conclusion of the Geometry Unit . . . . . . . . . . 2 3 I I . Differences between Scores on Geometry Retest held approximately Two Weeks a f t e r Test 1 . 2 3 I I I . A Comparison of the High and Low Achievers on the Geometry Test . 2 5 IV. A Comparison of the High and Low Achievers on the Geometry Retest 2 6 vi:. LIST OP FIGURES FIGURE PAGE 1. Scale Drawings of a Right Triangular Prism and a Rectangular Prism . . 30 2. Scale Drawings of a Cube and a Regular Tetrahedron 31 3 . Scale Drawing of a Right C i r c u l a r Cylinder 32 k. Sketches of a Right Triangular Prism and a Rectangular Prism J6 5. Sketch of a Plane Figure made up of a Semi c i r c l e , a Rectangle and a Right Triangle 3 7 6. Sketch of a Space Figure made up of a Right Triangular Prism and one h a l f of a Right C i r c u l a r Cylinder . 38 7. Sketch of a Perspective of a Rectangular Prism , . 38 v i i Acknowledgements The writer would l i k e to render the customary gratitude 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 patience, encour-agement, and guidance. He i s also most gr a t e f u l to the members of the computer science s t a f f and to the teachers who collabo-rated with him and without whose assistance t h i s study could never have been i n i t i a t e d . CHAPTER I '< THE PROBLEM INTRODUCTION : Of the many and diverse problems to be considered by the teachers of mathematics i n our schools today perhaps none i s more apropos than the use of lesson a ids to f a c i l i t a t e the l e a r n i n g of various mathematical concepts. From within t h i s general framework the use of p h y s i c a l models as an adjunct to the teaching of areas and volumes of various polyhedra i s the subject chosen f o r t h i s study. The objec t ive w i l l be to e v a l -uate the r e s u l t s obtained from teaching t h i s u n i t by two d i f f e -rent 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 learned and retained by both groups. BACKGROUND Since the incept ion of the new mathematics program i n B r i t i s h Columbia schools much work has been done to develop i n the students,,an apprec ia t ion of the s t ructure of mathematics. As the number of postula tes , d e f i n i t i o n s and theorems increases , 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 r i s e 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 f u n -damentals has of necess i ty been reduced and the hope i s that the students w i l l l e a r n through understanding rather than rote memorization. I t i s now commonly assumed that a student learns best that which i s "meaningful" and forgets most q u i c k l y that which i s not . I f t h i s i s true then the problem becomes how to make the mater ia l to be learned meaningful . In the experience of the i n v e s t i g a t o r most students seem able to r e c a l l s a t i s -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 pro-vided they are tested soon a f t e r the work has been completed. The d i f f i c u l t i e s begin when he i s required to r e c a l l t h i s work i n order to master that which f o l l o w s . In time the astute teacher begins to r e a l i z e that i t Is poss ible f o r most of h i s students to become g l i b with symbols and the new vocabulary without l e a r n i n g much of- the fundamental concepts required to understand and progress i n the present algebra and geometry programs. He r e a l i z e s they can make the same mistakes and misunderstand, the same basic ideas as d i d t h e i r predecessors on the o l d program. I t i s u s u a l l y at t h i s point that one of two methods i s u t i l i z e d ; the teacher may r e -trace his steps g i v i n g many more examples, u n t i l most of h i s students are r e l a t i v e l y more successful on the quizzes given, or he may press on, assuring himself and h i s students that understanding w i l l occur as they are exposed to more and more sophis t i ca ted theorems and problems r e l a t e d to the o r i g i n a l basic concepts. The f i r s t method seems to s a t i s f y the l e s s capable student and the second appears to be much more appro-p 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 formulate h i s own set of concepts he may e a s i l y become confused and discouraged. Woodruff suggests that; '.i i i one of the three functions of verbal behavior i s to provide f o r concept formation but only i f the concepts to be com-municated are already i n the possession of both the sender and the r e c e i v e r . Such communications can serve to help each party review and reorganize h i s concepts but not to give him concepts he does not already h a v e . i In the experience of the i n v e s t i g a t o r t h i s i s p a r t i c u l a r l y true i n the ..current geometry program. Most students have l i t t l e 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 Cartesian plane once they have learned the agreements f o r p o s i t i v e and negative d i r e c t i o n . Very few f a i l to v i s u a l i z e the point (3»-2) as being i n a p o s i t i o n three uni ts 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 perpendicular 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 e s t a b l i s h e d . The three dimensional problem i s something quite d i f f e r e n t and the attempt to i l l u s t r a t e t h i s conceptj often compounds the confusion. Even i f the teacher can i l l u s t r a t e t h i s concept on the chalkboard the student can r a r e l y r e p l i c a t e i t . S i m i l a r l y most students can r e a d i l y understand the development required f o r determining the area of a t r i a n g l e , but most experience great d i f f i c u l t y i n conceptual iz ing the volume of a tetrahedron. The second facet of t h i s problem i s d i r e c t l y r e l a t e d to the f i r s t . Even i f the students; can master the r e q u i s i t e con-cepts, i s there any guarantee that he, w i l l r e t a i n a s a t i s f a c t o r y por t ion of t h i s knowledge? W i l l : he be able to r e c a l l work that has been learned e a r l i e r i n the year or i n the previous year i n order to proceed to work which requires t h i s previous know-ledge? ^Because of the cumulative pat tern of most of the work i n h igh ,school 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 subject . 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 mater ial which was presented i n grade eleven, but a b r i e f review as required should be s u f f i c i e n t to f a c i l i t a t e the r e c a l l of most of the basic p r e r e q u i s i t e m a t e r i a l . Bigge suggests that material which i s meaningful to students i s remembered much better than that which i s not . He goes on to say that "meaning-fulness consis ts of the r e l a t i o n between f a c t s - g e n e r a l i z a t i o n s , 2 r u l e s , 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 to be p a r t i c u l a r l y t r u e . Presenting a l i s t of formulae f o r p e r i -meters, areas, surface areas and volumes accompanied by a poorly contrived sketch may i n v i t e confusion followed by memorization by those who; are i n c l i n e d to do so. Pressey^ et a l found that at the end of a course students remembered about three quarters of the f a c t s covered. One year l a t e r they could r e c a l l about one h a l f the mater ia l and two years l a t e r about one quarter . Although many p r a c t i s i n g teachers may f e e l that h i s f i n d i n g s are too high , Pressey goes on to suggest that there 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 , f o r most types of " f a c t courses" v i r t u a l l y a l l i s l o s t . In con-c l u s i o n , he states that there i s no research which furnishes a basis f o r c o n t r a d i c t i n g the opinion that many school courses might as w e l l not be taken at a l l i f t h e i r value i s to be j judged on the basis of anything other than short-term r e t e n t i o n of f a c t s . Studies such as t h i s must give the teacher of math-ematics caus^ f o r grave concern; the awareness that what he i s teaching migljvt w e l l be forgot ten so soon must induce him to ex-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 a l t e r -5 native which w i l l allow the students to grasp the basic p r i n c i -ples of the work presented. Stephens sums up by s t a t i n g : i f the mater ia 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 conservation of energy, may so help us organize the res t of our ideas that i t stays with us f o r l i f e . Content that , i s not so b r i l l i a n t l y s tructured but which s t i l l has such meaning w i l l be remembered i n proport ion to i t s meaning. Nonsense mater ia l i s headed f o r e x t i n c t i o n before the l a s t s y l l a b l e i s uttered.^" In the case of teaching the areas and volumes of space f i g u r e s , much serious thought must be given to the organizat ion of the mater ia l so that a few basic 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 basis 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 . For example, i f the r e c -tangle i s presented c l e a r l y , then the surface area of a r e c -tangular prism should f o l l o w n a t u r a l l y and serve to r e i n f o r c e the same basic concept. The volume of the prism might then f o l l o w as being the product of the rectangular base and the height . I f the student i s a lso allowed to construct , l a b e l and manipulate a simple model of t h i s prism he might more r e a d i l y appreciate the basic r e l a t i o n s between the two and three dimensional f i g u r e s . S i m i l a r l y , a f t e r l e a r n i n g how to f i n d the area w i t h i n a c i r c l e , 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 sur -face area (two c i r c l e s and a rectangle) and i t s volume (the pro-duct of the ;area of i t s base and i t s h e i g h t ) . When Bruner, Olver , G r e e n f i e l d ^ et a l conducted t h e i r studies i n cogni t ive growth, they found that the number of 6 c h i l d r e n who learned conservation i n condit ions under which l a b e l l i n g and manipulation v a r i a b l e s were combined was markedly higher than when e i t h e r or both of these v a r i a b l e s were m i s s i n g . They found that 76% of the group (16 out of 21) were able to learn t h i s concept when both v a r i a b l e s were present but when no manipulation 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 only 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 ) . This study furnishes ample j u s t i f i c a t i o n f o r i n v e s t i -gat ing the p o s s i b i l i t i e s f o r a bet ter method of teaching the u n i t o f geometry discussed h e r e i n . The two main questions to be investigated' are: 1. W i l l the i n t r o d u c t i o n of various geometric space f i g u r e s and the computation of t h e i r surface areas and volumes be f a c i l i t a t e d by the construction, 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 opportunity to l e a r n by t h i s methodi r e t a i n more of what they have learned than those students who have been taught by the use of sketches on the chalkboard? 7 FOOTNOTES CHAPTER I i XA. D. Woodruff, "Teacher Education Programmes of the Future", (paper read at the Conference of the P r o v i n c i a l Tea-chers' 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 ) , P. 3 - 4 (mimeographed). M. iL. Bigge, Learning Theory f o r Teachers, (New York: Harper and Row, 1 9 6 4 ) , p. 3 0 0 . 3 s . L. Pressey, F. P. Robinson and J . E. Horrocks, Psychology i n Education, ( t h i r d e d i t i o n : New York: Harper and Row, 1 9 5 9 ) . pp. 262 - 6 3 -^ J . M. Stephens, Educational Psychology, (Toronto: H o l t , Rhinehart and Winston, 1 9 5 6 ) , pp. 427-28. • 5 j . s . Bruner, R. R. Olver, P. M. G r e e n f i e l d and others, Studies i n Cog n i t i v e Growth, (New York: John' Wiley and Sons, 1966), pp. 220-22. CHAPTER II A. REVIEW OF THE LITERATURE . A review of the research conducted i n the general area of teaching mathematics with the a i d of p h y s i c a l devices r e -vealed a v a r i e t y of procedures and a number of conclusions at some variance with each other . Most of the studies were con-cerned with 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 adjunct to the l e a r n i n g of some u n i t i n mathematics. There appeared to be no general agreement as to how the p h y s i c a l devices were to be introduced, how they were to be used and whether they were to be suppl ied by the teacher or constructed by the students. Berger 1 and Johnson 2 suggest, that i f teaching a ids are used properly the. r e s u l t should be more correct and more mean-i n g f u l l e a r n i n g i n a shorter time wi;th bet ter r e t e n t i o n . The f i r s t part of t h i s statement does not appear to be corroborated by a review of the pert inent studies and the l a t t e r part does not seem to have enjoyed the same a t t e n t i o n of the i n v e s t i g a t o r s . This apparent lack of agreement as to the merit of p h y s i c a l de-v i c e s might ; in part be explained by B e r g e r ' s 2 suggestion, that p h y s i c a l devices 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 mater ia l make toward better l e a r n i n g that cannot be made as w e l l or bet ter without i t ? " The question then seems to. be hot whether models serve any 9 u s e f u l funct ion i n the l e a r n i n g of mathematics per se but rather , are there some models that w i l l Improve the l e a r n i n g ,and retent ion of c e r t a i n topics i n mathematics? :i • • • • , Osborne^, German**, Sanders^ and others^ seem to agree that the mere use of teaching a ids does not ensure that the object ives f o r them w i l l be met. They suggest that these a ids must be used at the r i g h t time and i n the r i g h t way i f they are to be e f f e c t i v e i n promoting l e a r n i n g . T h e i r excessive or i n -discr iminate use may l e a d to overdependence on concrete ;or v i s u a l representation on the one hand or may o c c a s i o n a l l y l e a d to mere, entertainment on the other , r Bruner? and h i s associates? 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 conservation demonstrated most c l e a r l y the necessi ty of concern f o r how the models were to be used. When the students were allowed to manipulate models which were not l a b e l l e d only 30% of the group learned the con-cept but when the models were l a b e l l e d the percentage of l e a r -ners rose to a s t a r t l i n g 16%. On the other hand only 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 successful i n l e a r n i n g t h i s concept. T h i s study by a group l e d by one of the most respected men i n the f i e l d of cogni t ive growth must surely be given c a r e f u l conside-r a t i o n by any who would inves t iga te the merits of p h y s i c a l de-v i c e s as an a i d i n the teachingi;of mathematics. In the sub-sequent scrut iny of various research"projec ts , much more a t t e n -t i o n was given by t h i s i n v e s t i g a t o r to the method by which the models were ^introduced rather than r e s u l t s obtained by the coni-, 1 0 t r o l or experimental groups. ImmerzeelS i n a comparative i n -ves t iga t ion of the use and non-use of manipulative devices i n the teaching of seventh-grade mathematics found that the exper-imental group made greater progress as measured by standardized ; tes ts , teacher-made tests and d a i l y records . Although i t was stated; that the experimental group achieved s i g n i f i c a n t l y bet ter r e s u l t s ($% l e v e l ) , there seemed to 'be some basic weaknesses i n the study which might be mentioned. Only two classes were i n -volved i n the study both of which were taught by the author and while i t was stated 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 . Also l a c k i n g was any s p e c i f i c reference as to how the models -various r i g h t t r i a n g l e s which showed the squares on the legs and the hypotenuse - were used to a i d i n the l e a r n i n g of the Pytha-gorean theorem. ,. Anderson^ who deal t with a much l a r g e r sample populat ion - 5^1 eigth-j-graders from three Junior High Schools - reported much more s p e c i f i c a l l y on the method of presentat ion of models " In evaluating t h e i r e f f i c i e n c y i n teaching area, volume and the Pythagorean r e l a t i o n s h i p . He reported that an eight-week i n -s t r u c t i o n a l 'period was given in :which the experimental group had sixteen p h y s i c a l devices a v a i l a b l e to them at a l l times f o r s e l f - h e l p and study. I t was noted, however, that no s p e c i f i c ins t ruct ions , re t h e i r use was given 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 reported that while the experimental group achieved c o n s i s t e n t l y higher results, they> were not s i g n i f i c a n t l y so . He reported f u r t h e r that there was no r e l a t i o n between the amount the models were. ' - 11 used and the score attained. I In another s i m i l a r study, Cohen 1 0 ' attempted to evaluate a technique to Improve space perception a b i l i t i e s through the construction of models by senior high school students i n a course ;in s q l i d geometry. Sixty-three matched pairs of rstudents s p e c i a l i z i n g i n mathematics and science were selected and a f i v e month period of s o l i d and an a l y t i c geometry was given. The ex-perimental group constructed twenty-three appropriate models while the control group was given the usual treatment using chalkboard and diagrams. A pretest on space perception and post tests on the Minnesota Paper Form Board and D i f f e r e n t i a l Aptitude Tests on space perception were given. No s i g n i f i c a n t difference i n performance was noted.; He concluded that,there was no j u s t i f i c a t i o n f o r the claim that construction of.models by students during t h e i r study of geometry w i l l further>their growth i n a b i l i t y to v i s u a l i z e . , r ;, Picton also attempted: to determine the e f f e c t on f i n a l achievement of an introductory:unit based upon the deve-lopment of v i s u a l i z a t i o n and understanding through the con-struction and use of models. He developed a ten-day introduc-tory unit on the construction of models and found no s i g n i f i c a n t difference i n the f i n a l resultsnof the experimental and,the , control groups. n r . . 12 Jamleson attempted to,; evaluate the e f f e c t i v e use of a variable-base abacus i n teaching counting i n numeration systems other than base ten. He selected ninety-four seventh-grade n 12. students, gave a pretes t , a f i v e - d a y i n s t r u c t i o n a l period and : a post t e s t . One group had the benef i t of a demonstrator-size ; abacus, a second group had the domonstrator and a small abacus each while a t h i r d group had no aids but the blackboard 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 tes t r e s u l t s . 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 1 ^ inves t iga ted the u t i l i z a t i o n of coloured transpa-rencles f o r the teaching of d e s c r i p t i v e geometry to c lasses of engineering students. Two i n s t r u c t o r s each taught two c lasses , one with the a i d of the overhead projec tor and coloured t rans -parencies and the other with the chalkboard o n l y . The r e s u l t s indicated that a marked advantage was gained by those i n the ex-perimental group; the year-end average of the experimental group was 79*3% while the c o n t r o l group had an average of 7 ^ . 9 $ . This represented a gain of s l i g h t l y more than h a l f a grade p o i n t . : S i g n i f i c a n t also was'the r e s u l t that 70$ of the A ' s were found In the experimental group and 75% of the F ' s were i n the contro l group. 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 lend credence to t h e . s u g - " gest lon ; made r rearlier that many teachers do have d i f f i c u l t y i n sketching meaningful three-dimensional diagrams on the chalk-board. ,The coloured transparencies met with the "unanimous" approval of the i n s t r u c t o r s and students a l l of whom agreed that the topic was more understandable and enjoyable when there was the a d d i t i o n a l time to discuss the topics as they were p r e -sented. . . / . . ' • ( . ' • . ' ' -i .. • .; 13 A f t e r reviewing the l a s t decade of studies 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 teaching of mathe-matics, four conclusions 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 exhibi ted much consistency i n dec iding what actual purpose these aids should serve. 2. None of the studies reviewed seemed to be concerned with the value of the models i n h e l p i n g the students re tent ion beyond the p o s t - u n i t t e s t . 3. Most of those studies which reported 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 preparation of the i n s t r u c t o r s so that the model could be used to best advantage. i 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 conclusions as to the merit of using models i n the teaching of mathematics as a whole. I t seems evident that i f any s i g n i f i c a n t b e n e f i t i s to be derived from t h e i r use i t w i l l be i n p a r t i -cular subject areas t h a t , w i l l be determined :only ; through c a r e f u l l y designed research. . • 14 FOOTNOTES CHAPTER II ^ J . E . Berger, " P r i n c i p l e s Guiding the Use of Teacher and Pupil-Made Learning Devices" Twenty-second Yearbook of the  Nat ional Council of Teachers of Mathematics. (Providence; American Mathematical Society , 1954) pp. 1 5 8 - 6 1 . 2 J . E . Berger and D . A . Johnson, A Guide to the Use and  Procurement of Teaching Aids f o r Mathematics. A b u l l e t i n p r e -pared by the Secondary School Curriculum Committe of the N a t i o n -a l Counci l of Teachers of Mathematics, ( A p r i l 1959) PP« 7 - 8 . . 3 R . Osborne "The Use of Models i n Teaching Mathematics" The Ari thmetic Teacher. (January 196l) pp. 22-24. ' **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) pp. 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 Ari thmetic Teacher (March 1964) pp. 1^6-6$ ^National Counci l of Teachers of Mathematics, M u l t i - Sensory Aids i n the Teaching of" Mathematics. ( l 8 t h Yearbook, 1945) p . 18. ; ?J . S. Bruner, R. R. Olver , P. M. G r e e n f i e l d and others , Studies" i n Cognit ive Growth, (New York: John Wiley and Sons, 1966) pp. 219-24. Q °G. E . Immerzeel, "A Comparative Inves t iga t ion of the Use and Nonuse of Manipulative Devices i n Teaching Seventh Grade Mathematics", (unpublished Master 's t h e s i s , Iowa State Teachers Col lege , 1956), pp. 35-41* 1 : yG. JR. Anderson, " V i s u a l - T a c t u a l Devices : t h e i r E f f i c i -ency i n Teaching Area, Volume and the Pythagorean R e l a t i o n " (unpublished Doctoral t h e s i s , the Pennsylvania State U n i v e r s i t y , 1957). pp. 187-94. g ,i l 0 L o u i s Cohen, "An Evaluat ion of a Technique to Improve Space Perception A b i l i t i e s Through the Construct ion of Models by Students i n a Course i n S o l i d Geometry (unpublished Doctoral t h e s i s , The New York Yashiva U n i v e r s i t y , 1959). pp. 251-55 L 1 1 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 Introductory Unit based upon Developing V i s u a l i z a t i o n and Understanding through the Use of Models" (unpublished Doctoral t h e s i s , The Montana State Col lege , Bozeman, 196.2), pp.' 124-26. * ; 1 15 i 2 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 Doctoral Thes is , The George Peabody College , N a s h v i l l e , Tennessee 1 9 6 2 ) , pp. 1 9 8 - 2 0 5 . W. Will iams "An Evaluat ion i n the U t i l i z a t i o n of Coloured Transparencies f o r the Teaching of D e s c r i p t i v e Geo-metry." (umpublished Doctoral Thes is , 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 abstrac ts , volume 24, 19&3) p'» 657* CHAPTER III DESIGN OP THE STUDY The purpose of the study was to compare two methods of teaching a uni t of Geometry i n v o l v i n g volume and surface area of basic space f i g u r e s . In the f i r s t method the various f i g u r e s were introduced by an informal d e s c r i p t i o n and an accompanying diagram on the chalkboard, both of which were copied by the students. This was intended to represent enlightened 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 appropriate p h y s i c a l models which the students learned to construct . These were used by the students to help l e a r n information i d e n t i c a l to that presented to the f i r s t group. Appendix A i l l u s t r a t e s the materials used and how the models were constructed. Eight c lasses and f i v e teachers, i n c l u d i n g the e x p e r i -mentor,•were involved i n the study. Two of the teachers were each to have a ' p a i r of grade seven c lasses , one to be taught by method A andione by method B. The experimentor had two grade s i x c lasses to teach, one by each method, and the remaining two teachers taught s i n g l e c l a s s e s , one by method A and the other by method B. ' w. " 1/7 A l l c lasses were grouped heterogeneously by the school administrators at the s tar t of the school year and no fur ther randomization was attempted. Mental age and previous mathemati-c a l achievement ( E a s t e r report) were recorded along with scores obtained i n the geometry u n i t test and r e t e s t . Because t h i s uni 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 pre-ference of the teachers to present e i t h e r method. I t was also f e l t that there would be l i t t l e chance of any previous mater ia l learned 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 pretest was given to e i t h e r group,. The teachers involved i n t h i s study met with the i n -ves t igator p r i o r to the s ta r t of the experiment to discuss the uni t to be taught and how best to present i t . The teachers were apprised of the general ly accepted methods of d e r i v i n g max-imum benef i t of the models and of the most e f f e c t i v e way to sketch the three-dimensional polyhedra to be s tudied . When the teachers were confident that they had a t ta ined 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 construct or sketch these required space f i g u r e s . 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 teachers 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 . A l l c lasses s tar ted at approximately the same time, the t h i r d week i n May, and spent the same length of 18 time, approximately two weeks, on the u n i t . Example questions were incorporated i n the lesson plans and i t was agreed that no formal.homework would be assigned. .The general format of the test was also included i n t h i s o u t l i n e so that there would be no c o n f l i c t i n vocabulary, d e s c r i p t i o n s , formulae and d e f i n i t i o n s used. The students involved with the models were not allowed to remove them from the classroom nor were the students i n the other group encouraged to p r a c t i s e the sketching of any of the space f i g u r e s outside of regular c lass time. The:content of the u n i t began with a d i s c u s s i o n of the rectangle followed by the square, the r i g h t t r i a n g l e , the e q u i -l a t e r a l t r i a n g l e and the c i r c l e . In the f i r s t part of the ex-periment nei ther group used models; a l l students sketched them i n the usual manner using r u l e r and compasses. A f t e r these plane f i g u r e s were described and discussed the analogous space f i g u r e s were presented i n order - the rectangular prism, the cube, the r i g h t t r i a n g u l a r prism, the regular tetrahedron 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 lesson planning f o r the teacher to explain to the "method A" group how to make a three-dimensional sketch, d i s c u s s i n g such things 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 allowed the "method :,B" group f o r construct ion and study of the models. Since ho s a t i s f a c t o r y measuring instrument was a v a i l -able , experimental questions were developed to test r e c a l l of names, 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 various f i g u r e s . Other questions were devised to test^the students ' a b i l i t y to solve 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 set,of questions was designed to test the students' a b i l i t y to solve more d i f f i c u l t problems involving two or more of the,figures discussed. As the time avai l a b l e f o r testing, purposes was r e s t r i c t e d to a fo r t y minute period and the amount of work to be tested was considerable, i t was decided that a multiple-choice type of test would be most efficacious.. For th i s reason a five-choice test of t h i r t y items was constructed. This i s shown i n Appendix C. B r i e f instructions were given at the top of the f i r s t page of the test, but because of the re-l a t i v e naivety of the examinees i t was agreed that o r a l i n -structions would precede the handing out of test papers. The oral statement to be made was, "There w i l l be marks taken o f f for wrong answers so do not guess unless you are reasonably sure that the answer you select i s the best one. Read each choice c a r e f u l l y . No questions w i l l be answered once the test i s handed to you." Each student was to have two sharpened pencils and an eraser and i t was agreed; that the investigator would have some extra pencils i n case of emergency. No mention was made of the correction formula to be applied. I P r i o r to the administration of the f i n a l examination a rough draft iwas given to a group of grade ten geometry students who had studied and were f a m i l i a r with t h i s p a r t i c u l a r u n i t . At the I end of t h i s t r i a l run, comments regarding ambiguities, time allotment 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. This l a s t f a c t o r was of considerable concern, i n that the test was supposed to be one of understanding and re-c a l l ratherithan a tes t of computational s k i l l . When t h i s ana-l y s i s was completed and the l e s s desirable questions eliminated. 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 study. These tested the r e c a l l of descr ip t ions of the introductory plane f i g u r e s which were taught by the sketching method to both groups. I t was agreed, however, that the scores on these items would be included i n the results sent to the teachers of the c l a s s e s . t o use as they saw f i t i n the evaluation of t h e i r students 1 progress . The f i n a l d r a f t of the test and the accompanying answer sheet (Appendix C) was then d e l i v e r e d to the teachers i n v o l v e d . I t was agreed that i t would be given on the day f o l l o w i n g the termination of the u n i t and again approximately two weeks l a t e r . The inves t iga tor 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 . The r e s u l t s f o r each group were recorded separately and copies were sent to the teachers. No information was d i s t r i b u t e d concerning the r e l a t i v e success of corresponding c lasses i n d i f f e r e n t schools . The i n v e s t i g a t o r then deleted the score of any student who had not attended at l e a s t 8C$ of the lessons and any student f o r whom no mental age score or Easter report mark was a v a i l a b l e . These f i n a l considerat ions resul ted i n a population drop of s i x -teen i n the method A group and nineteen 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 test the v a l i d i t y of which was depen-dent upon the experience of the experlmentor. I t i s hoped that any weaknesses were at l e a s 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 with colleagues, the other teachers involved 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 t e s t . 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 -te rac t ion among members of the two groups outside the classroom and the r e l a t i v e l y short time between the test and the r e t e s t . As has been ,noted i n the design 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 of these weaknesses to a minimum. CHAPTER IV, THE ANALYSIS AND DISCUSSION OP THE DATA ; i . •< There was a t o t a l of 9 7 subjects i n the method A group and 9 ^ subjects i n the method B group f o r whom scores i n mental age, previous mathematical achievement, geometry unit test and retest were recorded. A covariate analysis was done by the computer science department i n which the two independent v a r i -ables, previous 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 unit test were obtained. These r e s u l t s are given i n Table 1 . The iden-t i c a l procedure was followed f o r the unit retest and these re-sul t s are given i n Table 2 . i. Because the adjusted means oif the f i r s t t est as shown in Table 1 were so close - Hf . 6 8 2 2 compared with 1 ^ . 6 4 7 1 - i t was assumed:there was no s i g n i f i c a n t difference i n the scores obtained by the two groups on the f i r s t t e s t . Because a d i f f -erence was noted i n the means of the two groups i n the retest, Table 2 also shows the r e s u l t s i o b t a i n e d by comparing the scores of the; two groups on the retest;. I t was noted that a T -value of 3 . 5 3 was obtained which would indicate that there was a s i g n i f -icant difference i n the scores of the two groups. TABLE I DIFFERENCES BETWEEN SCORES ON TEST #1 HELD AT THE CONCLUSION OF THE GEOMETRY UNIT. ; Adjusted Means . N Group A 14.6471 9 4 Group B 14.6822 97 TABLE II DIFFERENCES BETWEEN SCORES ON GEOMETRY RETEST HELD APPROXIMATELY TWO WEEKS AFTER TEST #1. Adjusted Degrees of Mean S . D . T-Value Means Freedom Difference Group A 1 3 ^ 9 2 5 5 190 0.49215 0.13945 *3-53 a ' * s i g n i f i c a n t at the . 0 0 5 l e v e l Group B 14.4124 : Although the r e s u l t s obtained i n Table 2 i n d i c a t e that there 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 scores of the two groups on the r e t e s t i t must beA recognized that the d i f f e r e n c e , f o r a l l p r a c t i c a l purposes, i s unimportant. 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 bet ter than chalkboard sketches f o r l e a r n i n g a u n i t i n three-dimensional geometry and (2) would they enhance the s tudent 's a b i l i t y to r e t a i n what he has learned? Judging from the r e s u l t s obtained, the answer to the f i r s t question must be "no" and to the second, a q u a l i f i e d " y e s " . As was noted i n the A n a l y s i s of the Data, the d i f f e r e n c e i n re tent ion was s i g n i f i c a n t , but few p r a c t i s i n g teachers would f e e l motivated to spend excessive time or energy to r a i s e the o v e r a l l mean of a mathematics 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 conjectures a r i s i n g from t h i s study might' lead to f u r t h e r r e l a t e d 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 r e -test have been greater had the ,time ;between the test and the :; re tes t been -longer? A fo l low-up on one of the experimental groups .^method B) taught by the•invest igator almost twojyears l a t e r produced a mean score of 1 2 . ^ on the same t e s t . I t must be noted that while t h i s appeared t o ( t h e i n v e s t i g a t o r to be a remarkably high re tent ion there, were ,several extenuating f a c -t o r s ; ( i ) the mean of t h i s group was considerably higher 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 placed by four newcomers whose background d i d not include t h i s work and were excused from the test and ( i i i ) there had been some reference to these geometric figures i n the new grade seven mathematics programme. In spite of these considerations and the lack of a control group i t might be i n f e r r e d that the difference i n retention would be greater had the i n t e r v a l between the two tests been longer. A s u p e r f i c i a l investigation of the data gave r i s e to the second conjecture: "Would the use of models or physical devices be more b e n e f i c i a l to a c e r t a i n type of student or class?*' That i s , d i d the experimental procedure benefit the slower student rather than the brighter one? I t appeared that the more able students (judging only; by the l e t t e r grade a s s i g -ned by the mathematics* teacher) benefitted more from the ex-perimental method than d i d the l e s s able. r ,1' TABLE III: 'V'." : A COMPARISON OF THE HIGH AND LOW -. ACHIEVERS ON THE GEOMETRY TEST. ; l • High* Low* 1 |i • •] - I Method A group 1 6 . 1 1 1 . 6 s :i * I • 1 7 . 6 1 1 . 4 * those students with l e t t e r grades of A or B 3 1 '> ** those students with l e t t e r grades of D or E 26 Despite the fact that no consideration has been given to other pertinent variables such as mental age, i t might be i n -ferred that classes of high achievers i n mathematics could de-ri v e greater benefit from the use of physical models i n a geo-metry unit such as the one referred to i n t h i s study. D i r e c t l y related to the second conjecture i s the t h i r d , "does the high achiever group r e t a i n i t s knowledge better than does the low achiever group?" This time the r e s u l t s appear to be quite d i f f e r e n t from those noted i n the preceding matrix. TABLE IV A COMPARISON OP"THE HIGH AND LOW ACHIEVERS ON THE GEOMETRY RETEST. High Low Method A group 1 6 . 3 9 » 3 Method B group 1 7 . 1 1 1 . 6 Here i t w i l l be noted that the high achievers are v i r -: n 1' t u a l l y i d e n t i c a l i n retention whereas the low achievers i n the experimental group gained markedly over those i n the control r • ' • n. . 1 group. 26 Despite the fact that no consideration has been given to other pertinent variables such a? raental age, i t might be i n -ferred that classes of high achie%< rs i n mathematics could de-ri v e greater benefit from the use >-r physical models i n a geo-metry unit such as the <~ oferreu to i n t h i s study. D i r e c t l y re' ^ the second conjecture i s the t h i r d , "does the high achi' ^roup r e t a i n i t s knowledge better than does the low achieve; group?" This time the r e s u l t s appear to be quite d i f f e r e n t from those noted i n the preceding matrix. TABLE IV A COMPARISON OF;THE HIGH AND LOW ACHIEVERS ON THE GEOMETRY RETEST. High Low Method A group 1 6 . 3 9 . 3 Method B group 1 7 . 1 1 1 . 6 : r n Here i t w i l l be noted that the high achievers are v l r -t u a l l y i d e n t i c a l i n retention whereas the low achievers i n the experimental group gained markedly over those i n the control group. ' 2 7 BIBLIOGRAPHY Anderson, G. R. " V i s u a l - T a c t u a l Devices : t h e i r E f f i c i e n c y In Teaching Area, Volume and the Pythagorean R e l a t i o n " (unpub-l i s h e d Doctoral t h e s i s , the Pennsylvania State .Univers i ty , 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 Learning Devices" . Twenty-second Yearbook of the  Nat ional Counci l of Teachers of Mathematics. (Providence: American Mathematical Society , 195^) Berger, J . E . and Johnson, D.A. A Guide to the Use and Pro- curement of Teaching Aids f o r Mathematics. A b u l l e t i n p r e -pared by the Secondary School Curriculum Committee of the Nat ional Counci l of Teachers of Mathematics, ( A p r i l 1959) Bigge, M. L . Learning Theory f o r Teachers (New York: Harper and Row, 19#n Bruner, J . S. Olver , R.R. G r e e n f i e l d , P .M. and others , Studies  i n Cognitive Growth (New York: John Wiley and Sons, 1966) Cohen, L o u i s , "An Evaluat ion of a Technique to Improve Space Perception A b i l i t i e s Through the Construction of Models by Students i n a Course i n S o l i d Geometry (unpublished Doctoral t h e s i s , the New York Yashiva U n i v e r s i t y , 1959) 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 lnves t igat ion of the Use and Nonuse of Manipulative Devices i n Teaching Seventh Grade Mathematics" (unpublished Master 's t h e s i s , Iowa State . Teachers Col lege , 1956) : 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 in'Numeration Systems other than Base T e n ' . (unpublished Doctoral t h e s i s , the George Peabody College , N a s h v i l l e , Tennessee 1962) National Council of Teachers of Mathematics, Mult i -Sensory Aids . i n .the Teaching of Mathematics. i(l8th Yearbook, 19^+5; Osborne, R. "The Use of Models i n Teaching Mathematics" The  Ari thmetic Teacher. (January 1961) 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 Introductory Unit based upon Deveoping V i s u a l i z a t i o n and Understanding through the Use of Models" (unpublished Doctoral t h e s i s , the Montana State Col lege , Bozeman, 1962) Pressey, S. L . , Robinson, F . P . , and Horrocks, J . E . , Psychology  i n Education, ( 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 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) Stephens, J . M. Educat ional Psychology, (Toronto: H o l t , Rhine-hart and Winston, 1956) Will iams, C. W. "An Evaluat ion i n the U t i l i z a t i o n of Coloured Transparencies f o r the Teaching of D e s c r i p t i v e Geometry" (unpublished Doctoral 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 abstrac ts , volume 24, 1963) Woodruff, A ; D. "Teacher Education Programmes of the Future" (paper read at the Conference of the P r o v i n c i a l Teachers' Associat ions i n Western Canada, Vancouver, B r i t i s h Columbia February 6 t h , 1966) \ APPENDIX A THE MODELS - SCALE £''=1" 3 0 : 6 " RIGHT. TRIANGULAR PRISM t ft A" RECTANGULAR PRISM 8' 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 outer surface when model Is assembled. - Let 'each student put h i s name on the f l a p of each model so i t w i l l show when stored i n dissembled s ta te . - Standard s ize paper c l i p s w i l l hold the model neat ly and securely . * - Paper and supplies w i l l be supplied i f r e q u i r e d . - Have ;the name of each polyhedron p r i n t e d i n block 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 ta te , (see CYLINDER above)' . I 3 APPENDIX B GEOMETRY UNIT-3^ VOLUMES AREAS & PERIMETERS - Figures w i l l be described rather than defined formally. -Exponential notation w i l l not be used; r r, not r . - \ \ w i l l be represented as 22/7 rather than 3.14-. FIGURES TO BE DISCUSSED A. Plane Figures (suggested descr ipt ions) ( i ) rectangle - a f o u r - s i d e d plane f i g u r e with opposite sides equal and square corners . ( i i ) square - a s p e c i a l rectangle having a l l sides equal . ( i i i ) r i g h t t r i a n g l e . - a three-s ided plane f i g u r e .; . >- 1 having one square corner. • ' . 'i (iv) 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 having three equal s i d e s . (v) c i r c l e - a set of points i n a plane i which are equidis tant from a given p o i n t . B. Space Figures (suggested descr ip t ions ) (i) rectangular prism - a space f i g u r e having three p a i r s of equal rectangular faces . ( i i ) cube - a s p e c i a l rectangular prism having s i x equal square faces . ( i l l ) 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 three rectangular faces and two l r i g h t t r i a n g u l a r bases. , ( iv) 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 joined at . r i g h t angles by a curved w a l l . (v) regular tetrahedron - a space f i g u r e having four 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 . 1 35; Plane Figure rectangle square r i g h t t r i a n g l e e q u i l a t e r a l t r i a n g l e c i r c l e FORMULAE TO BE LEARNED Perimeter P=2»(l+w) P=4 .s P=a+b+c P=3vs C=2 .i?«r Area A=l» w A=s» s A = £ * a . b A = i v r « r Space Figure T o t a l Surface Area ( T . S . A . ) rectangular prism A=2» ( l»w+l»h+w«h) cube A = 6 . 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 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 regular tetrahedron A = 2 . a . b Volume V=l.w.h V=s - s .s V=i .a . b . h V=\?r.r.h V=l/6 -a .b »h . VOCABULARY plane r i g h t angle i perpendicular i equidistant p i or TT i, circumference perimeter diameter radius volume (informal) INTRODUCE AS REQUIRED - a f l a t surface having no thickness - square corner r two l i n e s which meet at r i g h t angles I* - an equal distance - comparison of the lengths of C and D ( 2 2 / 7 ) - measure of the distance around a c i r c l e n :• >i - measure of the distance around any plane f i g u r e - distance across widest part of c i r c l e j, . - a distance from centre to c i r c l e (half of diam.) & - a space enclosed by any space f i g u r e h n ' 3.6, area - part of plane enclosed by any plane figure t o t a l surface area- area of a l l faces and bases of a space figure edge eq u i l a t e r a l vertex " a l t i t u d e "height - where any two faces of a space figure meet - a l l sides equal - where two sides of a plane figure or two edges of a space figure meet - perpendicular distance from vertex to i opposite side of any applicable plane figure - perpendicular distance between bases (ends) of any applicable space figure a l t . b a s e base face "base regular•figure semi-circle - any wall of • a space figure - top and bottom of a space figure or the . side perpendicular to the a l t i t u d e i n a plane figure (see sketch above) - one having a l l sides* angles and faces equal - hal f a c i r c l e . ; FORMAT OF TESfc Time: 40 minutes multiple choice: each item must be read c a r e f u l l y otherwise many of the d i s t r a c t o r s w i l l seem acceptable. Students must f i n d best answer. 1 ] combination power and speed: w i l l extend the best student i n most classes:^explain that there i s a penalty , f o r wrong answers and encourage them not to guess i f they have no idea of the correct a ' . answer, (do not giVe correction formula.) ;, 37 made up of three par ts : ( i ) r e c a l l of descr ip t ions given: questions designed to encourage t h i n k i n g rather than rote memory. ( i i ) r e c a l l of formulae f o r perimeter, area, volume and t o t a l surface area (remember: formulae w i l l be wri t ten using 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 = side of a square or edge of a cube l» = 22/7 h = height between bases of space f i g u r e .-; r = radius J J * :r' d = diameter A = area P = perimeter T . S . A . = t o t a l surface area V = volume -; ( i i i ) problems i n computation of areas, 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 f i g u r e s . » e i g . f i n d the area o f : r: s f i n d the area of : (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) h a l f a r i g h t c i r c u l a r c y l i n d e r (2) 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 described as shown i n the examples. There w i l l be a sketch or a d e s c r i p t i o n but not both. 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 ime. The uni t should take no more than 10 days (1 40 min. period, per day) plus one per iod f o r tes t and one per iod f o r the r e t e s t two weeks l a t e r . " Both groups w i l l have exactly 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 only sketches on the board. Both groups should be given ample opportunity to solve 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 . three-dimensional f i g u r e s and discuss problems of perspective and o p t i c a l i l l u s i o n e t c . e . g . rectangular prism Note that the perspective i s exaggerated here and you must be c e r t a i n that the c l a s s understands that t h i s i s how i t looks not how i t i s . To reduce the perspect ive e i t h e r move xnfar ther away or move base B c l o s e r to base A. In the experimental group (B) use construct ion paper (not heavy-weight grade) and l e t every student make h i s own models. T h e i r construct ion should take no morie tim'e than group A uses f o r sketching. aThe f o l l o w i n g diagrams i n d i c a t e recommended dimen-sions f o r the required models. rNote that they may be stored f l a t . ' , APPENDIX C GEOMETRY UNIT TEST AND ANSWER SHEET 1. Place an "X" opposite the number ON THE ANSWER SHEET which indica tes the best answer f o r each quest ion . 2 . Read a l l choices c a r e f u l l y before making your s e l e c t i o n . 3« Foolscap w i l l be supplied 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 having: (a) k equal sides and square comers (b) 4 equal sides (c) opposite sides and opposite angles equal. (d) 2 pa i rs of equal sides *(e) opposite sides equal and a l l angles equal 2 . .Square: > (a) a plane f i g u r e with 4 equal sides *(b) a rectangle with a l l s ides equal ,(c) a plane f i g u r e with 4 equal angles (d) a plane f i g u r e with a l l s ides 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 having: *(a) one square corner ;.(b) one square corner and three unequal s ides < (c) three equal s ides and one square corner ' (d) three equal sides (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 having: (a) a l l sides equal and one square corner *(b) a l l angles equal (c) two equal angles and two equal s ides • (d) a l l sides 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 la te (c) having no corners ' (d) made up of a set of points a l l of which are e q u i d i s -1 tant from a given point *(e) none of these c • .' • <• 6 . Regular tetrahedron: a space f i g u r e having: (a) three e q u i l a t e r a l t r i a n g u l a r faces (b) four t r i a n g u l a r faces (c) three e q u i l a t e r a l t r i a n g u l a r faces and one square base c • c '(d) four e q u i l a t e r a l faces *(e) four e q u i l a t e r a l t r i a n g u l a r bases 7. Rectangular Prism: a space f i g u r e having: ( a ) equal opposite faces *(b) 3 p a i r s of equal r e c t a n g u l a r faces (c) 6 equal faces (d) a l l angles equal (e) none of these 8. Cube: - a space f i g u r e having: ( a ) equal opposite faces (b) a l l edges equal (c) a l l angles equal (d) a l l square comers *(e) none of these 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 angles by a curved w a l l (c) a three-dimensional c i r c l e (d) a: s o l i d c i r c u l a r rod (e) none of these ' • ' .1. 10. R i g h t ] T r i a n g u l a r Prism: a space f i g u r e having: *(a) y: r e c t a n g u l a r faces 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 faces i(c) 51 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 faces (e) none of these. i i F O R M U L A E : ' . • (. 1. T o t a l Surface Area of a Right 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): Surf ace Area of a re c t a n g u l a r prism: (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 . Total;Surface Area of a cube: * ( a ) 6?s «s (b) s»s.s (c) k'S »s \. • ' • (d) 6»s«s»s i (e) none of these 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 • 1 (e) -|.a*b»h 5. Volume of 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 Surface Area of a r i g h t t r i a n g u l a r prism: ' (a) -gi»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 7. Volume of a cube: (a) s»s • " (b) 6.s (c) 4.S» S ' - 1 : , (d) >6.s• s r,' : ' "(e) s»s*s 1 8. Volume of a Right T r i a n g u l a r Prism: (a) a«b.h , "(b) Ifa.b.h (c) h(a+b+c) (d) l/6«a»b«h r, (e) none of these 9 . 'Volume of a Regular Tetrahedron: (a) area of the base times the height of the tetrahedron. (b) area of the base times h a l f the height of the t e t r a -hedron. "(c) one t h i r d the product of the area of the base and the height (d) one h a l f the area of a t r i a n g u l a r prism having the same base and the same height (e) none of these 10. Volume of a Rectangular Prism: ' ' (a) 6«1» w»h (b) £(l«w+l»h+w.h) . : •'•(c) 6(l+w+h) (d) 4»l.wh • "(e) l ' w h 4 3 III Problems ( l e t U = 22/7 wherever used) 1. The volume of a regular tetrahedron with a height of 2 feet and the area of a face 12 square inches i s : (a) 24 cu . i n . 1 (b) 12 cu . i n . < (c) 48 cu . i n . *(d) 96 cu . i n . • (e) 8 cu . i n . • 2. The volume of a r i g h t t r i a n g u l a r Prism with an a l t i t u d e of 3 i^« a base of 4 i n . and a height of 1 0 . i n . i s : (a) 120 cu . i n . -(b) 70 cu . i n . *(c) 60 cu . i n . (d) 40 cu . i n . (e) 17 cu . i n . 3. -The t o t a l surface area of a rectangular prism with length 8 i n . , width 6 i n . and height 2 fee t i s : *(a) 51 /3 sq . f t . (b) 768 sq. f t . (c) 21.6 sq . i n . ' ' • 1 (d) 786 sq . f t . (e) 2 2/3 sq . f t . 4. The t o t a l surface 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 height i s 3 i n . i s : (a) 77 sq . i n . -• *(b) 143 sq.: i n . (c) 220 s q . i n . (d) $6 sq . i n . (e) 115 sq . 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 height i s halved, 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 quarter as much as the o r i g i n a l (e) four times as much as the o r i g i n a l 'r. • 6.1 I f the length , width and height of a rectangular prism -are a l l doubled then the t o t a l surface area of the new prism w i l l be: (a) twice as large as the o r i g i n a l !i(b) eight times as large as the o r i g i n a l (c) ^ix times as large as the o r i g i n a l (d) nine times as large as the o r i g i n a l • *(e) four times as large as the o r i g i n a l 'r' ' )i » ' • • • t 'r » t I 7- A rectangle of length 10 f t . and width 7 f t . has 2 semi-c i r c l e s f i t t e d exactly 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) 27 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 area enclosed by the f i g u r e i n que. #7 above i s : *(a) 108£ sq . f t . (b) 27 sq . f t . (c) 81 sq . f t . (d) 64 s q . f t . (e) 42 s q . f t . 9« A large steamer trunk i s shaped l i k e a rectangular prism with 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 c y l i n d e r . I f the dimensions of the rectangular part of the trunk are 2 f t . by 2 f t . by 7 f t . , what length of strap w i l l be needed to go around the trunk i n such a way as to ensure that the l i d w i l l not come open even i f the lock f a i l s ? (Allow one extra foot of s trapping f o r  the buckle and some overlap) [ . (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 paint can with a diameter of 1 foot f i t s exactly in to a cube-shaped cardboard box so that when the l i d of the box i s closed the can just, touches a l l s i x faces of the box. The volume of the a i r space which i s i n s i d e the box but outside the paint can i s : ' (a). 1728 cu . i n . *(b) 3 1/14 cu . f t . . : r i; (c) 1420 cu . i n . • • • i (d) 11/14 cu . f t . (e) none of these. 3 .:: c ' ' • • * represents the correc t choice f o r each question GEOMETRY UNIT TEST, Answer Sheet i Name Please : p r i n t : Grade School A . Descr ipt ions of Figures (1) (a) - (2) (a) _ _ (3) (a) (4) (a) ;(b) (*) - (b) (b) (c) (c) (c) (c) (d) (d) (d) (d) Ce). (e) _ (e) (e) (6) (a) _ (?) (a) _ - ( 8 ) (a) (9) (a) (b) — (*> (b) (b) ( ° ) - (c) ~ (°> Z Z .(c) (d) _ ( d ) - _ U ) •(d) (e) •(e) _ (e) (e) B. Formuale (1) (a) • (b) •(c) (6) (a) (b) (c) (d) (e) C. Problems (1) (a) _ (b) (c) — (d) (e> — (2) (a) (b) ( o ) - ~ : (d) •(e) I Z (3) (!a) •(b) Z Z (c) z z (d) Ce) — (a) (b) (c) (d) (e) (6) (a) - L . (b> ZZ (c) (d) _ (le) _ I - ~_ (7) (a) (b) _ (c) (d) (e) _ (8) Ga) _ Z Z ((c) z z (d) .. i . ' 1 (9) (a) (b) (c) •(d) •(e) (2) (a) (3) (a) _ i _ W (a) (b) (b) (b) (c) (c) (c) (d) Z Z (d) (d) (e) (;e) (e) (7) (a) (8) (;a) _ (9) (a) (t>) Z Z (;b) _ (b> (c) . ((c) _ (c) (d) : (d) _ .(d) , (e) Ce) (e) ) APPENDIX D SCORES OBTAINED BY THE EXPERIMENTAL GROUP Subject L e t t e r Hen. - N e l . Geom. Geom. Gain Number Grade mental age Test . Retest Score 1 6 '39 18 ' • 17 -1 2 4 32 15 14 . -1 ' 3 •3 . - 34 17 18 • 1 4 5 43 ' .14 15 1 5 3. 29 ••• . ' 16 \ 15 : - l 6 6 4 i 10 9 . 1 7 1 32 12 11 - l 8 7 40 17 18 1 9 • 5 40 It 16 2 10 3 31 13 10 - 3 11 5 . 4 5 . • 19 . 20 1 12. 2 32 : . 15 15 . 0 13 5 36 . 18 17 • -1 14 4 35 16 . 15 -1 15 ' 7 • 47 21 20 -1 16 6 • 42'. 22 20 - 2 17 6 50 , 19 • 20 1 18 3 . 37 16 . 17 1 19 • 1 48 • •• • 12 10 -.2 20 4 4 i • • ' 1 5 ••• 16 • 1-21 . 4 • . 35 , : 15 16 : 1 22 3 31 13 11 ' -.2 ' 23 • 6 47 • 12 12 0 . 24 7 45 18 : ' 19 1 25 - 3 ' ' 36 . 14--. ' ; 10 .; 26 • 4 ' 39 10 ' 11- 1 27 6 ' 40 17 : 16 -1 ' 28 5 44 16 16 0 29 5 :. 34 14 . 1 3 -1 ". 30 6 37 . ' V . 9 : 7 - 2 31 3 32 • 13 i o - 3 32 6 43 18 • . 20 2 . 33 . 4 40 15 . • 13 * ' -2 34 7 37 -; . ' 19 18- .• .• ' -1 • 35 2 32 . : 14 14 .. 0 36 4 . 51 . : 17 . 2 0 . 3 37 • 5 40 29 27 - 2 38 5 32 . 7 • 5 - 2 39 4 *•' 39 10 V 11 . .., 1 • 40 4 ' 43 ; ••' 15 • 16 . 1 ' 4 i 5 37 . 17 • 15 . - 2 42 . s • ?? 18 20 - 2 43 4 44 J 3 .'. - 14 . -1 44 6 . 2 9 '• 5 " v 1 ; 3 - 2 45 i 4 40 14 12' - 2 46 5 • 39 13 13 0 47 . 2 39 12 ~ 10 • - 2 • 48 4 43 15 16 . 1 1 49 2 • 35 7 • 8 t 50 39 17 15 - •• - 2 51 6 '77 19 21 2 - 2 . 52 6' 44 •-. 18 . 16-53 1. 40 25 26 •"' 1 Subject L e t t e r Hen. - Nel. Geom. Geom. Gain Number Grade . mental age Test Retest Score 5k 4 3k 17 ' 18 1 55 5 35 10 9 -1 56 3 32 ' 4 - - 7 3 5? 5 ? 6 13 12 •• -1 58 5 40 13 \ 15 2 59 2 30 12 11 -1 60 1 39 . ' 9 14 5 6i 3 • 35 . . ' 13 10 - 3 62 5 k3 . • Ik 14 0 63 7 51 • 19 22 3 64' 4 40 14 ••• 13 - l 65 4 . 4l . 15 12 -3 66 5 37 • 13 " 14 . 1 67 5 39 2 1 . • 20 ' -1 68 6 - 4 l 16 16 0 69 3 31 : 8 9 , 1 70 4 34 , 12 ' 10 -.2 71 2 32 4 . . 7 3 72 2 36 9 , / 6 -3 73 • 2 30 '• ' . 10 8 -2 74 6 ' 38 , • 17 16 ,-. • - i ; 75 5 39 15 : 12 . -3 76 • 3 . 36 ' 14 15 -1 77 4 34 12 .'• 14 2 78 4 32 16 : ••••• 12 -4 79 • 4 . 35 18 15 -3 80 • 7 41 21 22 1 • 81 5 40 •" :.. 18 ; 16 -2 82 • • 3 34 17 -3 , 83 6 37 15 18. ' . 3 84 2 27 . 8 . 4 -4 85 5 , 3 4 ' . 17 13 -4 86 5 - k3 . . 12 15 • 3 : 87 3 33 . 10 . 11 1 88 2 19 . 11 .13 2 89 ' 7 48 14 14- 0 90 4 37 16 12 -4 91 1 32 13 14 .'• 1 92 5 ' 43 . 1 9 . • 20 '. 1 • 93 6 40 20 " : 18 . -2 94 6 ' ' 39 22 19 , -3 95 4 • 36 16 16 0 96 5 38 15 18 3 97 7 ' 4 l 10 21 1 APPENDIX E SCORES OBTAINED BY CONTROL GROUP Subject L e t t e r Hen. - N e l . Geom. Geom. Gain Number Grade mental age T e s t Retest Score ' 1 4 36 12 12 . 0 2 • 4 39 18 17 -1 3 2 35 11 10 . -1 4 6 43 13 , 15 - 2 5 5 44 16 ; 14 ' - 2 . 6 6 36 . 14 . • 12 - 2 7 3 36 9 10 1 8 ' 4 45 15 12 - 3 9 7 43 15 13 - 2 10 5 39 10 11 1 11 5 38 20 17 - 3 12 2 32 21 20 -1 13 6 32 17 15 ' - 2 14 7 49 18 18 0 15 6 48 15 . 16 1 16 4 " 46 . . 13 14 1 17 • 3 35 16 14 . - 2 18 7 , 45 19 , 17 - 2 19 4 39 10 1.7 - 3 20 . 2 25 12 10 - 2 21 6 42 8 7 -1-22 , 5 . 45 13 10 - 3 23 5 44 15 15 0 24 4 . 3 7 16 14 - 2 25 6 44 . 1 4 15 1 26 4 36 8 6 - 2 27 3 46 10 10 0 28 4 39 17 15 - 2 29 4 44 . 19 18 -1 30 3 40 19 19 0 31 4 *48 • 20 17 - 3 32 7 49 21 17 • - 4 33 5 : 42 . 16 14 - 2 34 4 40 18 16 . - 2 35 4 44 16 • 19 +3 36 5 ^ 47 22 20 - 2 37 6 49 13 15 2 38 4 45 15 • 16 1 39 3 4 l • 16 15 -1 40 7 43 10 . ' 11 1 41 5 46 17 16 1 42 2 35 14 12 - 2 43 2 38 6 • 7 1 44 3 40 - . 12 15 +3 45 • 1 31 14 • 15 • 1 46 7 50 11 10 . -1 47 . 7 47 20 17 - 3 '. 48 4 45 14 16 2 49 6' 4 l ' 16. • 16 0 50 5 4 l . 12 15 ^ •' - 3 51 3 36 12 • • 10 ^2 52 5 35 - 15 16 .1 Subject L e t t e r Hen. - N e l . Geom. Geom. Gain Number Grade mental age Test • Retest Score 53 5 ~ 38 20 19 -1 55 2 . 36 •• 10 6 - 4 3 32 . 13 13 0 56 7 39 21 22 l 57 .4 .40 15 \ 14 -1 58 4 38 12 14 2 59 . 4 35 . 17 15 - 2 6o 5 41 . 16 18 2 6l 4 43 •17 17 0 . 62 6 40 21 18 - 3 63 4 37 14 ^ 12 - 2 64 3 35 . 13 15 2 65 2 27 7 6 -1 66 . 5 43 • 15 16 1 67 4 44 13 15 2 68 ',5 40 , ' : i 4 16 ' 2 69 . 39 1 22 23 1 70 3 . 3 6 . 18 , ' 17 -1 71 5 3k 11 10 : -1 72 2 28 8 • 6 - 2 73 4 31 • 10 ' • 7 - 3 7 4 - 4 36 . 13 14 1 75 3 38 19 . 15 l 76 7 . . 47 16 18. . 2 77 4 41' 17 . 16 -1 78 2 32 10 8 , - 2 79 • 5 39 • 14 13 -1 80 4 41 15 . 1 0 - 5 81 6 40 17 18 1 ' 82 6 39 . 20 21 • 1 ' 83 7 42 19 . 21 2 84 1 . 3k ^ 9 - 4 85 '3 30 , 14, . 11 - 3 86 ' 5 - . 36 . 1 0 10 0 87 5 38 16 , 17 •l 88 . 4 40 , 13 • . . 16. 3 89 7 37 18 •• 20 •' ; 2 90 6 , 43 ' ' 19 17 - 2 91 4 3k . • 16 13 - 3 92 5 39 15 14 -1 93 6 36 17 15 ' - 2 94 2 33 10 . 8 - 2 

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.831.1-0104446/manifest

Comment

Related Items