AN INVESTIGATION OF THE EFFECTS OF THE GEOMETRIC SUPPOSER SOFTWARE ON GEOMETRIC PROOF WRITING AT THE GRADE 10 LEVEL By JOSEPHINE REGINA WORSTER B.A., The University of B r i t i s h Columbia, 1960 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in THE FACULTY OF GRADUATE STUDIES (Department of Mathematics and Science Education) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July, 1989 ©Josephine Regina Worster, 1989 In presenting this degree at the thesis in partial fulfilment of the requirements University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that copying of department this thesis for scholarly or by his or her purposes may be representatives. It is for an permission for extensive granted by the understood head of my that publication of this thesis for financial gain shall not be allowed without permission. Department of Mat.hpmatirs anrf The University of British Columbia Vancouver, Canada DE-6 (2/88) SPIPTIPP advanced Education copying or my written ABSTRACT The purpose of t h i s study was to determine i f the use of a computer program c a l l e d the Geometric Supposer would result i n improved proof writing by grade 10 geometry students. The researcher studied 44 students enrolled i n a grade 10 geometry course. The students were divided into two classes; one class used the Geometric class d i d not. Supposer computer program while the other Both classes were taught at the same time every day and both classes covered the same content. The researcher kept i n close contact with the teacher of the noncomputer group regarding the content, the assignments, and the o v e r a l l progress of the students. Both classes were given two tests (an introductory geometry test and the van Hiele geometry test) at the beginning of the course. the end of the course At (one semester i n length) three tests were given to both c l a s s e s — t h e same van Hiele geometry test (measures geometric thought l e v e l s ) , a proof test, and an attitude t e s t . Weekly interviews were conducted with each of five students from the computer group. Two students from the noncomputer group were each interviewed twice near the end of the course. chosen based on t h e i r van Hiele l e v e l s . These students were The interviews provided the researcher with a better understanding of how some students approach and write geometric proofs. The data gathered from the introductory geometry test, the proof t e s t , and the attitude test were each analyzed using the independent t-test. The median test was applied to the pre van Hiele geometry - ii - t e s t r e s u l t s and t o t h e p o s t van H i e l e t e s t r e s u l t s . The s i g n t e s t was u s e d t o a n a l y z e t h e p r e and p o s t v a n H i e l e d a t a . A c h i square t e s t o f a s s o c i a t i o n was a l s o a p p l i e d t o t h e van H i e l e l e v e l s and tests. A .05 l e v e l o f s i g n i f i c a n c e was u s e d i n each o f t h e s e t e s t s . The r e s u l t s i n d i c a t e t h a t t h e group o f s t u d e n t s u s i n g t h e computer program, Geometric Supposer, p e r f o r m e d significantly better on t h e p r o o f t e s t t h a n t h e group o f s t u d e n t s who d i d n o t use t h e computers. The p r e v a n H i e l e geometry t e s t r e s u l t s i n d i c a t e t h a t more t h a n 50% o f s t u d e n t s e n t e r i n g t h e grade 10 geometry c o u r s e a r e a t a 0 or 1 l e v e l . T h i s l e v e l i s t o o low t o b e g i n t h e s t u d y o f g e o m e t r i c proof writing. The p o s t v a n H i e l e geometry t e s t r e s u l t s t h a t , a f t e r a semester indicate o f geometry, s t u d e n t s do move up i n t h e v a n H i e l e l e v e l s , w i t h o r w i t h o u t t h e use o f computer programs l i k e t h e Geometric Supposer. The r e s u l t s from t h e a t t i t u d e t e s t i n d i c a t e t h a t t h e r e was no d i f f e r e n c e between t h e two groups o f s t u d e n t s . Both c l a s s e s v a l u e t h e s t u d y o f mathematics i n g e n e r a l , and geometry i n particular. I n summary, t h e computer, w i t h a p p r o p r i a t e s o f t w a r e and t e a c h e r commitment, c a n c o n t r i b u t e t o r e d u c i n g t h e d i f f i c u l t y g e n e r a l l y e x p e r i e n c e d by s t u d e n t s i n m a s t e r i n g t h e w r i t i n g o f g e o m e t r i c p r o o f s . - iii - TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS viii Chapter I. INTRODUCTION TO THE PROBLEM 1 Statement o f t h e Problem 5 S i g n i f i c a n c e o f t h e Study II. . 7 REVIEW OF THE LITERATURE L i t e r a t u r e R e l a t i n g t o H i g h S c h o o l Geometry 9 . L i t e r a t u r e R e l a t i n g t o t h e van H i e l e T h e o r i e s . 22 L i t e r a t u r e R e l a t i n g t o t h e Geometric Supposer. 36 I I I . PROCEDURES IV. 9 41 The S u b j e c t s 41 D e s i g n o f t h e Study 43 Data C o l l e c t i o n 52 DATA ANALYSIS 61 Assessment o f t h e Groups 61 Geometric Thought L e v e l s P r i o r t o Treatment. . 62 Changes i n Geometric Thought L e v e l s 63 Written Proofs 65 Attitudes 67 I n t e r v i e w Data 69 A d d i t i o n a l Data 76 Data Summary 78 - i v- Chapter V. Page, SUMMARY AND DISCUSSION 79 Summary o f t h e Problem, Methodology, and R e s u l t s . 79 Interpretation of the Findings 80 L i m i t a t i o n s o f t h e Study 86 Suggestions f o r F u r t h e r Research 87 Implications 88 Conclusions . . . . References 90 91 Appendices A P e r m i s s i o n L e t t e r sent t o Parents/Guardians 97 B I n t r o d u c t o r y Geometry Test 100 Item A n a l y s i s 109 C D E F P e r m i s s i o n l e t t e r from Z. U s i s k i n I l l van H i e l e Geometry Test 113 Proof Test 124 Item A n a l y s i s 130 A t t i t u d e Test 131 Item A n a l y s i s 136 Summary o f Item S t a t i s t i c s 138 Permission Letter f o r Interviews - v - 139 LIST OF TABLES labia Page 1.1. Comparison between LOGO and t h e Geometric Supposer. 4 2.1. The Complementary A n g l e Theorem 11 2.2. The Supplementary A n g l e Theorem (a) 13 2.3. The Supplementary A n g l e Theorem (b) 14 2.4. The van H i e l e Model 24 4.1. Means, S t a n d a r d D e v i a t i o n s , and Statistical Comparison o f Groups: I n t r o d u c t o r y Geometry P r e t e s t . 62 4.2. Median T e s t : van H i e l e P r e t e s t 63 4.3. S i g n T e s t : P r e and P o s t van H i e l e Test Data . . . . 64 4.4. Means, S t a n d a r d D e v i a t i o n s , and Statistical Comparison o f Groups: P r o o f T e s t 4.5. Means, S t a n d a r d D e v i a t i o n s , and 65 Statistical Comparison o f Groups: A t t i t u d e T e s t 67 4.6. W r i t t e n Comments 68 4.7. S t u d e n t s ' van H i e l e L e v e l s 77 4.8. I n t e r v i e w e e s ' P r e and P o s t van H i e l e L e v e l s and t h e i r P r o o f T e s t S c o r e s - vi - 78 LIST OF FIGURES 'Figure. Ease. 1. The Opening screen of the Geometric Supposer . . . 45 2. The Right Triangle 46 3. An Acute Triangle with Altitudes 46 4. An Obtuse Triangle with Altitudes 46 5. Information Recorded on Angle Measurement 6. Supplementary Angles 7. P a r a l l e l Lines with Alternate Interior Angles 8. Relationship Between the Exterior Angle and .... 47 70 . . 70 Remote Interior Angles 70 9. Proving Two Segments Congruent 71 10. Proof #1 72 11. Proof #2 12. Proof #3 74 13. Proof #4 75 14. Overlapping Triangle Proof 76 .... ' 73 - vii- ACKNOWLEDGEMENTS I would l i k e to thank the members of my thesis committee, Dr. Marv Westrom, Dr. Doug Owens, and Dr. Harold R a t z l a f f f o r t h e i r guidance and assistance. Secondly, I would l i k e to express my appreciation to the grade ten students and the mathematics teacher who p a r t i c i p a t e d i n t h i s study. Lastly, I thank my husband, B i l l , f o r h i s encouragement, patience, and constant support. - viii j CHAPTER 1 INTRODUCTION TO THE PROBLEM I t is the glory of geometry that from so few principles, from without, it is able to accomplish fetched so much. Newton D e s p i t e Newton's a d m i r a t i o n f o r t h i s b r a n c h o f mathematics, the r o l e o f geometry i n the mathematics c u r r i c u l u m has been the o f many c o n t r o v e r s i a l d i s c u s s i o n s and d i v e r g i n g o p i n i o n s . subject "It is easy t o f i n d f a u l t w i t h the t r a d i t i o n a l course i n geometry, but a d v i c e on how sound t o remedy t h e s e d i f f i c u l t i e s i s h a r d t o come by." ( A l l e n d o e r f e r , 1969, p. 165) A l l e n d o e r f e r a l s o s t a t e d , " I n geometry . . . t h e r e i s not even agreement as t o what the s u b j e c t i s about." (p. 165) Project I n 1970 the U n i t e d S t a t e s Comprehensive S c h o o l Mathematics (CSMP) s p o n s o r e d a c o n f e r e n c e on the t e a c h i n g o f geometry and r e p o r t e d , "Of a l l t h e d e c i s i o n s one must make i n a c u r r i c u l u m development p r o j e c t w i t h r e s p e c t t o c h o i c e o f c o n t e n t , u s u a l l y t h e most c o n t r o v e r s i a l and l e a s t d e f e n s i b l e i s the d e c i s i o n about geometry." ( M o r r i s , 1986, Geometry and the van The p. 9) Hieles S o v i e t U n i o n had not o n l y i d e n t i f i e d the "geometry problem" much e a r l i e r but i n the 1960's, r e v i s e d the geometry s e c t i o n o f t h e i r mathematics c u r r i c u l u m . T h i s r e v i s i o n was b a s e d on a t h e o r y d e v e l o p e d by a Dutch husband and w i f e team who teachers—the van H i e l e s . Although were mathematics not w i d e l y known by t e a c h e r s i n N o r t h A m e r i c a today, the work o f the van H i e l e s p l a y s a major p a r t i n this investigation. - 1 - The I n t e r n a t i o n a l Commission on M a t h e m a t i c a l I n s t r u c t i o n h e l d an i n t e r n a t i o n a l seminar i n Kuwait on t h e m a t h e m a t i c a l i n 1986. Their discussions centered c u r r i c u l u m f o r t h e 1990's and a g a i n , geometry- appeared as t h e c o n t r o v e r s i a l t o p i c . "No p a r t i c u l a r mathematical a r e a w i t h i n t h e s c h o o l c u r r i c u l u m arouses so much c o n c e r n amongst mathematicians as does geometry, . . . " ( H o w s o n & W i l s o n , 1986, p. 58) . The l i t e r a t u r e c o n t a i n s c r e d i b l e reasons f o r geometry c o n t i n u i n g t o be a major t o p i c i n t h e mathematics c u r r i c u l u m . There i s , however, some g e n e r a l debate as t o whether geometry s h o u l d be i n t e g r a t e d throughout t h e mathematics c u r r i c u l u m o r s h o u l d be t a u g h t as a one y e a r c o u r s e , u s u a l l y a t t h e grade t e n l e v e l . A more • s p e c i f i c i s s u e a r i s e s from t h e r o l e o f p r o o f s and d e d u c t i o n s . Many s t u d e n t s i n h i g h s c h o o l geometry have d i f f i c u l t y w i t h d e d u c t i o n and p r o o f . "They don't u n d e r s t a n d t h e r o l e o r meaning o f an a x i o m a t i c system. D e s p i t e o u r b e s t e f f o r t s t o t e a c h them, even t h e most c a p a b l e a l g e b r a s t u d e n t s may s t r u g g l e and g e t t h r o u g h geometry by sheer w i l l p o w e r and m e m o r i z a t i o n b u t w i t h little u n d e r s t a n d i n g o f t h e l o g i c a l system we have been d e v e l o p i n g a l l year." (Shaughnessy & Burger, 1985, p. 419) We s h o u l d t h e r e f o r e not be s u r p r i s e d by t h e f a c t t h a t many s t u d e n t s t e n d t o d i s l i k e geometry—in p a r t i c u l a r , w r i t i n g p r o o f s ( F a r r e l l , 1986; H o f f e r , 1981; Senk, 1985; U s i s k i n , 1980). The v a n H i e l e s a l s o e x p e r i e n c e d f r u s t r a t i o n s w h i l e t e a c h i n g geometry. They were f a m i l i a r w i t h t h e work o f P i a g e t and from t h i s , P i e r r e van H i e l e developed h i s system o f thought To h e l p s t u d e n t s r a i s e t h e i r thought - 2 - l e v e l s i n geometry. l e v e l s , t h e van H i e l e system s p e c i f i e d a sequence of phases that moved from very direct i n s t r u c t i o n to the students becoming independent from t h e i r teachers (Hoffer, 1983). Mayberry (1981) summarized two consequences of these l e v e l s : - a student cannot function adequately on a given l e v e l unless he has passed through and learned to think i n t u i t i v e l y on each preceding l e v e l . - I f i n s t r u c t i o n , that i s , the language of the i n s t r u c t o r , problems i n the textbook, or pedagogical techniques assume the student to be on one l e v e l while i n fact the student i s on a lower l e v e l , there w i l l be serious communication problems between the instructor and the student because t h e i r geometric knowledge i s organized d i f f e r e n t l y (p. 6). Further, van Hiele i n his 1959 a r t i c l e , stated, "The bad results of the teaching of geometry must almost e n t i r e l y be a t t r i b u t e d to the disregard of the l e v e l s . The learning process i n geometry, as we have seen, covers many l e v e l s , but appreciation of t h i s has s t i l l so l i t t l e penetrated into the teaching world that one even encounters teaching methods i n which beginners are confronted with modes of reasoning based on symbols of the t h i r d (formal deduction) l e v e l . " (p. 21) Wirszup (1976) also stated, "The majority of our high school students are at the f i r s t l e v e l of development i n geometry, while the course they take demands the fourth l e v e l of thought. It i s no wonder that high school graduates have hardly any knowledge of geometry, and that t h i s irreparable deficiency haunts them continually l a t e r on." (p. 96) Geometry and the Microcomputer Geometry i s c l e a r l y a v i s u a l subject, yet much of the student's time i s spent writing. If students are to have an opportunity to - 3 - think i n t u i t i v e l y t h e y need a f a s t e r , l e s s cumbersome medium i n which to experience i t . The microcomputer has t h e s e c h a r a c t e r i s t i c s b u t has o n l y been u s e d i n a l i m i t e d way i n N o r t h American geometry classes. Can t h e microcomputer be u s e d t o reduce t h e d i f f i c u l t i e s s t u d e n t s have w i t h p r o o f s ? s o f t w a r e i s used. that T h i s w i l l be p o s s i b l e o n l y i f a p p r o p r i a t e The r e s e a r c h e r i n i t i a l l y c o n s i d e r e d u s i n g t h e programming language LOGO t o t e a c h g e o m e t r i c c o n c e p t s . This notion was d i s c a r d e d when t h e Geometric Supposer s o f t w a r e was brought t o t h e researcher's attention. The advantages o f t h e Geometric Supposer o v e r LOGO a r e l i s t e d i n T a b l e '1.1. Table 1.1 Comparison between LOGO and t h e Geometric Supposer LOGO GEOMETRIC SUPPOSER 1. Need t o l e a r n language before studying geometric concepts. E a s y - t o - u s e menu d r i v e n programs. 2. Need t o d e v e l o p e x p e r i m e n t s / e x e r c i s e s t h a t demonstrate geometric concepts. B u i l t on g e o m e t r i c shapes and r e l a t i o n s h i p s . 3. Need t o d e v e l o p a l l s u p p o r t materials. Some t e a c h i n g and s t u d e n t learning material available. T h i s s t u d y i n v e s t i g a t e d t h e e f f e c t s on a grade t e n geometry class of i n c l u d i n g t h e use o f microcomputers and t h e Geometric Supposer software i n the course. F u r t h e r , t h e c l a s s was compared t o a second c l a s s which was t a u g h t t h e same c o u r s e c o n t e n t i n t h e t r a d i t i o n a l , p r o o f w r i t i n g manner. - 4- Statement o f t h e Problem Because o f c r i t i c i s m o f h i g h s c h o o l geometry, G e a r h a r t c o n d u c t e d a n a t i o n w i d e s u r v e y o f secondary (1975) s c h o o l mathematics t e a c h e r s . " P r o o f was r e g a r d e d as an i m p o r t a n t t o p i c b y n e a r l y a l l teachers." (p. 490) However, t h e t e a c h e r s " a l s o i n d i c a t e d t h a t many s t u d e n t s do i n f a c t have t r o u b l e w i t h t h e m a t e r i a l and do n o t l i k e it" (p. 490). More r e c e n t l y , Suydam's (1985) r e p o r t on t h e NCTM (1981) s u r v e y i n d i c a t e d t h a t mathematics t e a c h e r s p r e f e r r e d t h e geometry c u r r i c u l u m be kept i n t a c t w i t h t h e f o c u s on E u c l i d e a n geometry. When s t u d e n t s were asked what t h e y d i s l i k e d most about t h e i r geometry c o u r s e , "There i s o n l y one s t r o n g answer: proof." (Usiskin, 1980, p. 419) D e s p i t e t h e emphasis on p r o o f s i n t e n t h grade geometry, s t u d e n t s seem t o emerge from t h i s c o u r s e w i t h o n l y a l i m i t e d a b i l i t y t o generate p r o o f s and n o t much u n d e r s t a n d i n g about the nature o f proof. Senk (1985) r e p o r t e d t h e r e s u l t s o f t h e C o g n i t i v e Development and Achievement i n Secondary S c h o o l Geometry P r o j e c t — a p p r o x i m a t e l y 30% o f t h e s t u d e n t s i n geometry c o u r s e s i n which p r o o f i s t a u g h t r e a c h a 75% mastery l e v e l i n p r o o f w r i t i n g , and about 25% o f t h e s t u d e n t s have no competence i n w r i t i n g p r o o f s . Thus, t h e r e seems t o be a d i s c r e p a n c y between t h e i n t e n t i o n s o f t h e geometry c u r r i c u l u m i n h i g h s c h o o l and what s t u d e n t s a c t u a l l y learn. C r a i n e (1985), i n an attempt t o improve t h e geometry c o u r s e , made s e v e r a l assumptions. H i s f i r s t assumption, "Students e n t e r i n g t h i s c o u r s e have n o t n e c e s s a r i l y h a d t h e i n f o r m a l g e o m e t r i c - 5 - experiences that should ideally occur in the middle grades." (p. 120) His second assumption, "Many students entering this course are below the third van Hiele level, the minimum level at which one can fully appreciate definitions and relations of class inclusion. Students who have not reached this level cannot be expected to succeed in writing proofs." (p. 120) "According to the van Hieles, the learner, assisted by appropriate instruction, passes through five levels of thinking. The learner cannot achieve one level without having passed through the previous levels." (Fuys, 1985, p. 449) The following i s a brief description of the van Hiele levels: Level 0 (recognition) - Students recognize figures by appearance alone. They can say triangle, square, etc., but cannot identify properties of the figures. Level 1 (analysis) - Students reason about geometric properties of figures, i e . diagonals of a rectangle are equal, but do not interrelate the figures or properties. Level 2 (abstraction or informal deduction) - Students relate figures and their properties, i e . every square i s a rectangle, but do not understand the role and significance of deduction. Level 3 (formal deduction) - Students reason formally, can construct proofs, understand the role of axioms, postulates, theorems, and definitions. Level 4 (rigor) - Students can compare systems based on different axioms and can study various geometries in the - 6 - absence o f c o n c r e t e models. or reach t h i s l e v e l . " "Few s t u d e n t s a r e exposed t o , (Crowley, 1987, p. 2) Other s t u d i e s have found t h a t s t u d e n t s were i l l p r e p a r e d (had low van H i e l e l e v e l s o f g e o m e t r i c t h i n k i n g ) f o r t h e i r geometry 10 c o u r s e . Were t h e s t u d e n t s i n t h i s s t u d y i n t h e same p o s i t i o n ? "The v a n H i e l e model r e v e a l s an a l a r m i n g l a c k o f harmony i n t h e t e a c h i n g and l e a r n i n g o f mathematics." ( H o f f e r , 1983, p. 218) I n an attempt t o " b r i d g e t h e gap" t h i s i n v e s t i g a t i o n endeavoured t o answer t h e f o l l o w i n g q u e s t i o n s : 1) W i l l t h e s t u d e n t s who use t h e Geometric Supposer s o f t w a r e be b e t t e r a b l e t o w r i t e f o r m a l p r o o f s t h a n s t u d e n t s who a r e t a u g h t by more t r a d i t i o n a l methods? 2) What changes i n t h e s t u d e n t s ' v a n H i e l e l e v e l s t a k e p l a c e a f t e r a semester o f geometry? 3) W i l l t h e s t u d e n t s who r e c e i v e t h e t r e a t m e n t have a more p o s i t i v e a t t i t u d e towards geometry t h a n t h e s t u d e n t s i n t h e t r a d i t i o n a l group? S i g n i f i c a n c e o f t h e Study The p r i m a r y s i g n i f i c a n c e o f t h i s s t u d y was t o i n t e g r a t e t h e Geometric Supposer s o f t w a r e i n t o t h e geometry c u r r i c u l u m t o p r o v i d e a b r i d g e between t h e s p a t i a l - v i s u a l a s p e c t s o f geometry and t h e deductive aspects i n order t o increase students' a b i l i t y t o w r i t e proofs. Second, t h e change i n s t u d e n t s ' v a n H i e l e l e v e l s between t h e b e g i n n i n g o f t h e semester and t h e end was measured. Third, this study presented i n f o r m a t i o n regarding the a t t i t u d e s o f students two groups towards geometry a t t h e end o f t h e c o u r s e . - 7 - from One group o f s t u d e n t s u s e d t h e computer s o f t w a r e throughout the course w h i l e the o t h e r group d i d n o t . The r e s u l t s o f t h i s s t u d y p r o v i d e some h e l p f u l s u g g e s t i o n s f o r t h e t e a c h i n g o f g e o m e t r i c p r o o f s t o grade t e n s t u d e n t s . - 8 - CHAPTER 2 REVIEW OF THE LITERATURE This chapter contains a review of the l i t e r a t u r e describing high school geometry and i t s apparent shortcomings, the van Hiele theories, and the Geometric Supposer computer programs. Literature Relating to High School Geometry Geometry i n the high school has been a very controversial topic with opinions ranging from Dieudonne's famous slogan, "Down with E u c l i d . " (Grunbaum, 1981, p. 235) to "Teach them a rigorous Euclidean geometry." The Euclidean camp has dominated despite numerous suggestions f o r changes i n the high school geometry course. Proofs are central Proof i s the cornerstone for teaching Euclidean geometry as, " I t enables us to test the implication of" ideas thus establishing the relationship of the ideas and leading to the discovery of new knowledge." (Smith & Henderson, 1959, p. 178) The purpose then of teaching proof i s to move students from a subjective point of view to an objective one. What i s t h i s term, proof? According to Smith and Henderson: Proof i s a common word i n our vocabularies with various shades of meaning i n i t s d a i l y usage, but i t has a very special and precise meaning i n mathematics. As a mature concept, proof i n mathematics i s a sequence of related statements directed toward establishing the v a l i d i t y of a conclusion (p. I l l ) . - 9- In high school geometry each statement or step i n the proof must be j u s t i f i e d . The j u s t i f i c a t i o n s can be drawn either from given information, d e f i n i t i o n s , postulates, or previously proven theorems. Most often the geometric proof i s written i n two-column form with statements on the l e f t and a reason for each statement on the r i g h t . In order to put the subject of proof i n perspective i t i s necessary to look at the h i s t o r i c a l development. The synthetic methods of E u c l i d existed from approximately 325 B. C. u n t i l the seventeenth century when Descartes f i r s t used numbers i n the study of geometry. This new approach became known as a n a l y t i c geometry. Some flaws were noted i n Euclid's axioms but were corrected. As these corrections were beyond the comprehension secondary of the average school student, the geometric postulates were modified to make them more understandable. Thus, t h i s modified form of Euclidean geometry became the basis of the current geometry course. Other non-Euclidean geometries have been developed, some of which are " a f f i n e , projective, hyperbolic, e l l i p t i c , combinatorial, absolute, analytic, d i f f e r e n t i a l , algebraic, Minkowskian, i n t e g r a l , transformation, vector, linear, topological, conformal, r e l a t i v i s t i c , o p t i c a l , and so forth" (Fehr, 1972, p. 152). Despite these additions, the high school geometry course i s s t i l l mainly Euclidean. "The treatment of geometry i n the high schools today i s remarkably s i m i l a r to the Euclidean model set down more than twenty-three centuries ago." (Eccles, 1972, p. 103) Brumfiel (1973) concurred with t h i s statement but he gave the reason being that "no one has found better proofs" (p. 95). - 10 - Many students enrolled i n geometry courses have had d i f f i c u l t y with the concept of proof and have ended up d i s l i k i n g geometry as a whole. In the eyes of the student "geometry" has become synonymous with "proof" which i s understandable when students spend so much time i n grade ten geometry doing two-column proofs, many of which are self-evident. Because of the preoccupation with r i g o r , students are forced to write down every step along with an associated reason. average student gets l o s t i n the myriad of symbols and steps. example of such a textbook proof appears i n Table 2.1 p. An (Usiskin, 421). Table 2.1 The Complementary Angle Theorem Complements of congruent angles are congruent. GIVEN: [1 i s a complement of L2; /3 i s a complement of Z.4; 12 = Z4 . PROVE: ll PROOF: Reasons Statements 1. 2. 3. 4. 5. 6. 7. 8. £L i s a complement of L2; 13 i s a complement of Z4. mil + ml2 mL3 + ml4 mil + mL2 12 = 14 mL2 = mZ4 . m£L + mLZ mZl = m^3 n =13 = 90; = 90 = m{3 + mlA = m/3 + m/2 i. Given 2. Def. of comp. angles 3. 4. Substitution p r i n c . Given [1] [2] 5. Def. of = angles [4] 6. Substitution p r i n c . [3,5] 7. Add. prop, of equality [6] 8. Def. of = angles [7] - 11 - The 1980, Students i n a secondary school geometry class have to be able "to hypothesize, reason deductively, understand the role of mathematical models, and understand the difference between defining and deducing" (Farrell, 1987, p. 239) . These cognitive a b i l i t i e s are c h a r a c t e r i s t i c of Piaget's formal operational stage. However, the r e s u l t s from various tests measuring cognitive development indicate that a minimum of 30% of these students reason at the concrete operational l e v e l while another 30 - 40% of the students are t r a n s i t i o n a l reasoners ( F a r r e l l , 1987) . Carpenter, Lindquist, Matthews, & S i l v e r (1983) found i n t h e i r report of the r e s u l t s from the t h i r d mathematics assessment of the National Assessment of Educational Progress f o r 13 and 17-year olds, that students are f a i l i n g when mathematical understanding are required. reasoning and "The problem i s p a r t i c u l a r l y c r i t i c a l i n high school geometry, where success depends on propositional thinking and deductive reasoning about geometric properties and r e l a t i o n s . " (Olive & Lankenau, 1986, p. 78) Throughout geometry courses, learning to write proofs has been an important objective of the curriculum. However, proof writing has been perceived to be one of the most d i f f i c u l t topics f o r students to learn. Usiskin (1980) suggested that the amount of time spent on proofs be reduced and that many theorems of lesser importance deleted. These suggestions have been ignored. be "The concept of proof i n mathematics w i l l always be important whatever may be the nature of the curriculum." (Lovell, 1971, p. 66) Given that t h i s p r e d i c t i o n i s true, the basic problem of how to increase student mastery of writing geometric proofs remains c r i t i c a l . - 12 - Proofs become less rigorous The amount of d e t a i l i n the proof i l l u s t r a t e d i n Table 2.1 tended to overwhelm the majority of students who then 'turned o f f . Textbook authors are attempting to keep symbols and technical vocabulary to a minimum. A s i m i l a r proof but from a more recent textbook (Jurgensen, Brown, & Jurgensen, 1985, p. 41) appears i n Table 2.2. Table 2.2 The Supplementary Angle Theorem (a) It two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent. GIVEN: £1 and 12 are supplementary; Z3 and Z.4 are supplementary; 12 = C4 (or m/2 = m^4) PROVE: l l = [3 (or mil = m/3) PROOF: Statements 1. 0- and Z3 and 2. m/l + + 3. m i l + 4. 5. m/l mZ3 Ll are £4 are m/2 = m/4 = m/2 = m/2 = = Reasons supplementary; supplementary. 180; 180 ml3 + mU mZ4 m/3 - 13 - 1. Given 2. Def. of supp. angles 3. Substitution prop. 4. Given 5. Subtraction prop, of = Table 2.3 contains an even more recent textbook Alexander, & Atkinson, 1987, p. 354) example. (Kelly, The examples i n Tables 2.2 and 2.3 show proofs of exactly the same theorem. Table 2.3 The Supplementary Angle Theorem (b) I f two angles are equal, t h e i r supplements are equal. Suppose there are two doors i n the room. Suppose also that each door i s opened the same amount, that i s / @ /0^ Then, we might = predict t h a t a n d £ @ a r e equal. explain why/0 ={0- We can use deductive reasoning to Since l_Q and /@ are supplementary: LO £(D + (2) Since 180 = 0 [1] 180° - / © £(5)and j(Qare supplementary: LO /0 + [0 180 0 180° -[(3) [2] Comparing [1] and [2], we see that the expressions on the right side, are equal, since i t i s given that£(p/0)' - 14 - = /|3= The re fore ,£2) In spite of proofs becoming less rigorous, as can be seen from Tables 2.1, 2.2, and 2.3, students s t i l l have d i f f i c u l t y grasping the concept of deducing a chain of steps. Euclidean geometry questioned Various educators have questioned the value of t r a d i t i o n a l Euclidean based geometry. Fehr (1972) f e l t that Euclid's geometry played a very minor role i n accomplishing the goals of geometric instruction. of spaces" He advocated that geometry "be conceived of as a study (p. 152) integrated into the curriculum and taught every year from grade seven to grade twelve. In 1973 Brumfiel reported on a study he d i d i n 1954 again i n 1973. He was curious about students' understanding of the axiomatic structure a f t e r they l e f t high school. who and repeated "Students of 1954 studied an old-fashioned hodgepodge geometry had no conception of geometric structure. Students of today who have studied a t i g h t axiomatic treatment also have no conception of geometric structure." (p. 102) Is the emphasis on axioms i n school geometry a waste of time? Usiskin (1980) also noted that the reason given f o r studying Euclidean geometry was that i t "provides an example of a mathematical system. It i s the place where the student i s asked to do what mathematicians presumably do, that i s , prove theorems." (p. 419) But mathematicians do a f a i r amount of exploration p r i o r to t h e i r w r i t i n g of a proof. "In contrast, geometry students seldom explore and almost always are t o l d what they should prove." (p. 420) Hoffer (1981) c r i t i c i z e d the high school geometry course f o r putting too great an emphasis on developing the s k i l l of w r i t i n g - 15 - proofs. "When t h i s occurs, precious class time i s taken from providing students with experiences i n other, possibly more p r a c t i c a l , s k i l l s of a geometric nature." (p. 14) Grunbaum (1981) f e l t that there was only a pretense to teach the " c l a s s i c a l " Euclidean geometry when, i n fact, the geometry being taught was "rather misleading" (p. 235). According to D r i s c o l l (1982) " . . . proof has been touted as a means to d i s c i p l i n e the mind, to think i n an orderly fashion, as a vehicle f o r improving l o g i c a l thinking, and as a stimulus toward the kind of responsible, c r i t i c a l and r e f l e c t i v e thinking that should be the mainstay of democratic l i f e . " (p. 155) But does proof r e a l l y promote deductive thinking? Senk (1985) questioned the value of teaching the t r a d i t i o n a l geometry course. Is i t preparing high school students to meet the challenges of the future? In an attempt to use geometry as the vehicle to i l l u s t r a t e mathematics as an axiomatic system, students come to the conclusion that axioms, theorems, and proofs solely belong to t h i s area. Geometry textbooks contain l i s t s of postulates and theorems. ones are even being created, for example, Pasch's axiom. New Niven (1987) posed the questions, "Are we not i n danger that the students w i l l see geometry as just so much nitpicking? Why should the f i r s t course i n geometry carry the special burden of i l l u s t r a t i n g and exemplifying the foundations of mathematics?" (p. 39). Teachers recommend that E u c l i d stays In 1973 Gearhart surveyed a random sample of 999 secondary school mathematics teachers from across the United States to f i n d out - 16 - t h e i r thoughts on the geometry course. Over h a l f of the teachers disagreed that the course should be more informal and less rigorous; 76% agreed that learning to write proofs was important f o r high school students; and over h a l f agreed that the course should be based on Euclid's development as found i n standard textbooks. Thus, t h i s survey indicated support f o r the status quo i n the geometry course. Similarly, i n 1981 the National Council of Teachers of Mathematics also conducted a survey. The results indicated that geometry should be taught for the following reasons: - to develop l o g i c a l thinking a b i l i t i e s ; - to develop s p a t i a l i n t u i t i o n s about the r e a l world; - to impart the knowledge needed to study more mathematics; and - to teach the reading and interpretation of mathematical arguments (Suydam, 1985, p. 481). Some e x p l a n a t i o n s f o r the d i f f i c u l t y w i t h p r o o f s and s u g g e s t i o n s f o r overcoming them Lester (1975) was convinced of the importance f o r students to develop an a b i l i t y to write proofs c o r r e c t l y . I f students were to be properly prepared for t h i s task, Lester f e l t that they should be introduced to various aspects of proof as early as possible. In an attempt to determine the appropriate time for students to be introduced to proofs, Lester (1975) conducted h i s study. In h i s research of the l i t e r a t u r e he found inconsistent evidence regarding developing the a b i l i t y to perform certain formal operations. On the one hand, theories seem to support the suggestion that there i s no relationship between age and l o g i c a l reasoning and on the other hand, - 17 - that l o g i c a l reasoning improves with age. For h i s study Lester selected four groups of subjects, 20 i n each group. consisted of students from grades 1-3, 4-6, The groups 7-9, and 10-12. His subjects a l l interacted with a computer terminal i n a game setting where they were asked to supply proofs of "theorems". The r e s u l t i n g data from Lester's research indicated that "certain aspects of mathematical proof can be understood by children nine years o l d or younger. Perhaps children may be able to deal with formal operations at an e a r l i e r age than proposed by Piaget." (p. 23) F i r s t - y e a r students at the University of Oregon are asked to l i s t t h e i r favorite and least favorite high school subjects. Hoffer (1981) reported that "the subject that was almost u n i v e r s a l l y d i s l i k e d was geometry" (p. 11). He suggested that formal proofs should not be introduced early i n the course as students may reached the formal operational l e v e l of development. spending a good portion of time "exploring geometric informally, without requiring proofs. not have He recommended concepts This enables students to study what they c a l l 'fun things' while preparing for more formal aspects i n the second h a l f of the course.", (p. 18) P r i o r to 1980 research had been l i m i t e d i n t h i s area, consequently l i t t l e was known about the actual nature of the d i f f i c u l t i e s that students experienced i n writing proofs. Thus, i n 1981 the Cognitive Development and Achievement i n Secondary School Geometry (CDASSG) Project commenced to organize research i n t h i s area. The project was designed to address a variety of questions but Senk (1985) reported s p e c i f i c a l l y on the question, "To what extent do secondary school geometry students i n the United States write proofs - 18 - s i m i l a r to the theorems or exercises i n commonly used geometry texts?" (p. 448) . A t o t a l of seventy-four geometry classes from eleven schools i n f i v e states were involved. A proof test consisting of s i x items was administered during the regular class period. The conclusions reached from the data were: - about 70% of students can do simple proofs requiring only one deduction. - achievement i s considerably lower on proofs requiring a u x i l i a r y l i n e s or more than one deduction. - only 30% master proofs s i m i l a r to the theorems and exercises i n standard textbooks. These data indicate a low l e v e l of achievement i n w r i t i n g proofs but perhaps the reason f o r t h i s i s lack of p r a c t i c e . Most mathematical s k i l l s that are acquired have been p r a c t i s e d f o r a length of time at various grade l e v e l s . "In contrast, the t y p i c a l high school mathematics program provides v i r t u a l l y no opportunity f o r students to practice w r i t i n g proofs i n any context outside the geometry course." (Senk, 1985, p. 454) To overcome t h i s weakness, Senk proposed that more e f f e c t i v e ways must be i d e n t i f i e d f o r teaching proof. She specifically recommended: - more attention be given to teaching students how to s t a r t a chain of deductive reasoning. - greater emphasis be placed oh the meaning of proof. - the need to teach students how, why, and when they can transform a diagram i n a proof (p. 455). - 19 - Brown (1982) noted that students entering geometry f i n d the subject quite d i f f e r e n t from other mathematics courses which they have taken. "There are no elaborate arithmetic problems, no polynomials to factor, and few equations to solve. And most d i f f e r e n t of a l l i s geometric proof, where the solution i s not a neat number or algebraic expression that can be underlined and labeled 'answer' " (p. 442). At the same time the student i s having to adjust to these differences; s/he i s expected to "invent a chain of deductions" (p. 442) i n order to a r r i v e at some conclusion f o r which the student can see no purpose. Brown suggested that students should be encouraged to experiment, guess, generalize, and deduce the various formulas and theorems themselves. C r i t i c i s m has been l e v e l l e d at the elementary school f o r not teaching s u f f i c i e n t informal geometry to better prepare students f o r w r i t i n g proofs. The CDASSG Project "confirms the need f o r systematic geometry i n s t r u c t i o n before high school i f we desire greater geometry knowledge and proof-writing success among our students" (Usiskin, 1982, p. 89). However, the mathematics curriculum has not been changed and i n 1987 Usiskin s t i l l bemoaned the fact that there was no geometry curriculum i n the elementary school to prepare students entering high school f o r Euclidean geometry. To combat some of the d i f f i c u l t i e s encountered by students i n the high school geometry course, Prevost (1985) suggested that geometry i n the junior high should be an i n t e g r a l part of mathematics rather than a single chapter i n a whole year's study. He also championed the cause f o r a manipulative approach to geometry. - 20 - He c r i t i c i z e d teachers for using too few devices that allow students to do geometry rather than merely watching i t . Craine (1985) admitted to a preference of a u n i f i e d approach to secondary mathematics where geometry, algebra, and analysis would be integrated throughout the e n t i r e curriculum. However, f a i l i n g t h i s integration, he proposed to reorganize the geometry course. He recommended using informal methods to introduce the basic concepts of geometry followed by an inductive discovery of the properties. Deductive reasoning would gradually be introduced. Similar to Hoffer, Craine saw the l a s t part of the course being devoted to writing proofs. The l o g i c a l arguments that form the basis of Euclidean geometry cause students d i f f i c u l t y . Students are unable to organize t h e i r thoughts to construct a deductive sequence of steps. "To deal d i r e c t l y and e x p l i c i t l y with the organization of students' thought patterns and t h e i r construction of l o g i c a l arguments" (p. 47), Dreyfus and Hadas (1987) developed a set of methods and curriculum materials. twenty-two experimental new To test the effectiveness of t h i s course, classes from f i f t e e n d i f f e r e n t schools ten control classes from other schools were selected. indicated that the students using t h i s new The geometry course and results increased t h e i r a b i l i t y to reason l o g i c a l l y within a geometric context somewhat more than the students using the t r a d i t i o n a l approach. Because of a h i s t o r y of poor achievement, only about one-half of the secondary population e n r o l l s i n the geometry course and of these only about one-third r e a l l y understand i t (Usiskin, 1987) . Consequently, approximately one-sixth of high school - 21 - students are p r o f i c i e n t i n writing proofs. Various suggestions have been given about how to improve the s i t u a t i o n , including the factor of readiness. "In fact, research has suggested that many students at the age when formal geometry i s usually studied are incapable of the formal and abstract thinking required. As a result, they stumble through the yearlong course by mimicking the teacher's two-column proofs, and they emerge at the end with a few facts, a vague sense of the difference between axioms, theorems, and d e f i n i t i o n s . " (Fey, 1984, p. 32) This memorization has prevented students from achieving the major objectives of the geometry course: to develop the a b i l i t y to reason deductively and to appreciate the role of deduction i n mathematics. In summary, the l i t e r a t u r e contains two major c r i t i c i s m s of the current way that geometry i s taught and organized: 1) students are not "ready" f o r geometry and, 2) the method of i n s t r u c t i o n with i t s heavy dependency on writing proofs does not allow students to discover geometric relationships upon which to base t h e i r deductive reasoning. In the present study, the researcher depended upon the van Hiele theories to assess readiness and progress i n geometric thinking, and the Geometric Supposer software to o f f e r students some discovery experiences i n geometry. Literature Relating to the van Hiele Theories Background Two Dutch mathematics teachers, Pierre van Hiele and h i s late wife, Dina van Hiele-Geldof, became troubled about t h e i r students' - 22 - d i f f i c u l t i e s i n learning geometry. developed a theory i n 1957 From t h e i r concerns they involving l e v e l s of thinking i n geometry. They surmised that these l e v e l s could be used to explain why have d i f f i c u l t i e s i n geometry. "They believed that high students school geometry involves thinking at a r e l a t i v e l y 'high' l e v e l and that many students have not had s u f f i c i e n t experiences i n thinking at p r e r e q u i s i t e 'lower' l e v e l s . " (Fuys, 1985, Between 1960 and 1964 p. 448) the Soviet Academy of Pedagogical Sciences v e r i f i e d the v a l i d i t y of the theory of the van Hieles and revised t h e i r own mathematics curriculum accordingly. Freudenthal wrote about Dina van Hiele-Geldof's In 1973 Professor Hans experiments. Wirszup, an American, became acquainted with the work of the Hieles and the way the Soviets had applied i t . Wirszup was van the first to introduce the van Hiele theory to the United States i n 1974 when he presented a paper at the Annual NCTM (National Council of Teachers of Mathematics) Meeting. Despite t h i s early introduction, i t i s only recently that the theory has gained more popularity, possibly because English t r a n s l a t i o n s of t h e i r o r i g i n a l work are now appearing. Description The van Hiele model a c t u a l l y consists of three components: the thought l e v e l s , the properties of the l e v e l s , and the phases of learning. Table 2.4 i l l u s t r a t e s how these components are interrelated. The thought levels Five l e v e l s of geometric thinking were proposed by the Hieles. process. van Each l e v e l describes c e r t a i n c h a r a c t e r i s t i c s of the thinking "These l e v e l s are inherent i n the development of the - 23 - thought processes. The development which leads to a higher geometric l e v e l proceeds b a s i c a l l y under the influence of learning and therefore depends on the content and methods of i n s t r u c t i o n . However, no method not even a perfect one, allows the skipping of levels." (Wirszup, 1976, p. 79) The van Hieles began with the basic l e v e l , l e v e l 0, and ended with l e v e l 4. may be found i n the l i t e r a t u r e . ) (Different numbering systems "According to the van Hieles, two major factors that determine a student's l e v e l are a b i l i t y and p r i o r geometry experiences." (Fuys, Geddes, & T i s c h l e r , 1988, p. 12) Table 2.4 The van Hiele Model PROPERTIES OF THOUGHT PHASES OF THE LEVELS LEVELS LEARNING* Sequential Recognition Inquiry Advancement Analysis Directed Orientation Adjacency Informal Deduction Explanation Linguistics - Formal Deduction Free Orientation Mismatch - Rigor Integration Note: van Hiele (1984) suggested that students move through the phases each time they advance a l e v e l . - 24 - The following i s a description of each of the thought l e v e l s : Level 0 (recognition). At t h i s l e v e l students perceive the geometric figure i n i t s t o t a l i t y . They "do not see the parts of the figure, nor do they perceive the relationship among components of the figure and among the figures themselves." (Wirszup, 1976, p. 77) For example, squares and rectangles would be recognized as d i f f e r e n t kinds of figures. Level 1 (analysis). the properties On t h i s l e v e l students "become aware of of geometric figures by a v a r i e t y of a c t i v i t i e s such as observation, measuring, cutting, and f o l d i n g " (Mayberry, 1981, p. 4 ) . An example would be that the diagonals of a rectangle are equal. "Relationships between properties, however, cannot yet be explained by students at t h i s l e v e l , i n t e r r e l a t i o n s h i p s between figures are s t i l l not seen, and d e f i n i t i o n s are not yet understood." (Crowley, 1987, p. 2) Level 2 (abstraction or informal deduction). l e v e l can " e s t a b l i s h r e l a t i o n s among the properties among the figures themselves. ordering of the properties Students at t h i s of a figure and At t h i s l e v e l there occurs a l o g i c a l of a figure and of classes of figures. The p u p i l i s now able to discern the p o s s i b i l i t y of one property following from another, and the role of d e f i n i t i o n i s c l a r i f i e d . " (Wirszup, 1976, p. 78) Proof i s not understood at t h i s l e v e l . Level 3 (formal deduction). "Students develop sequences of statements to deduce one statement from another, such as showing how the p a r a l l e l postulate implies that the angle sum o f a t r i a n g l e i s - 25 - equal to 180 . However, they do not recognize the need for r i g o r nor do they understand relationships between other deductive systems" (Hoffer, 1983, p. 207). Level 4 (rigor). "Students grasp the significance of deduction as a means of constructing and developing a l l geometric theory." (Wirszup, 1976, p.. 78) The properties of the levels An important aspect of the l i t e r a t u r e related to the van Hiele work i s the properties of the system of l e v e l s . These properties not only describe the relationships between levels but also how a student i s affected by his/her placement and movement i n the l e v e l s . Teachers can use t h i s information to actually plan lessons. Property 1 (sequential). In order to understand geometry, the student must progress through the levels i n order. "A student cannot be at van Hiele l e v e l n without having gone through l e v e l n-1." (Usiskin, 1982, p. 5) Property 2 (advancements . The content and the i n s t r u c t i o n a l methods.can affect the progress of a student from l e v e l to l e v e l . "No method of i n s t r u c t i o n allows a student to skip a l e v e l , some methods enhance progress, whereas others retard or even prevent movement between l e v e l s . " (Crowley, 1987, p. 4) Property 3 (adjacency). At each l e v e l what appears as e x t r i n s i c had become i n t r i n s i c i n the preceding l e v e l . In other words, a student at the recognition l e v e l perceives figures as i s regardless of t h e i r properties. - 26 - Property 4 ( l i n g u i s t i c s ) . "Each l e v e l has i t s own language, i t s own set of symbols and i t s own network of r e l a t i o n s u n i t i n g these symbols." (Wirszup, 1976, p. 82) For example, a student at l e v e l 1 does not r e a l i z e that a figure can have more than one name—a rectangle i s a parallelogram. Property 5 (mismatch). " I f the student i s at one l e v e l and i n s t r u c t i o n i s at a d i f f e r e n t l e v e l , the desired learning and progress may not occur." (Crowley, 1987, p. 4) The phases of learning The van Hieles (1984) stated that the method and organization of i n s t r u c t i o n influenced the progress (or lack of) of a student from l e v e l to l e v e l . They have i d e n t i f i e d a f i v e phase cycle which they consider as a necessary sequence f o r students as they progress through the l e v e l s . Age or maturation are viewed as minor factors. The f i v e phases of learning are described below: Phase 1 (inquiry). Here the teacher and students discuss the objects of study f o r t h i s l e v e l . The teacher discovers what the students already know about the topic and the students become acquainted with the topic to be studied. Phase 2 (directed orientation) . The material, consisting of short tasks where manipulation i s prominent, i s c a r e f u l l y sequenced by the teacher. The teacher i s looking for s p e c i f i c responses from the students. Phase 3 (explanation). At t h i s phase, the students express the r e s u l t s of t h e i r manipulations i n words. The figures take on geometric properties and the role of the teacher i s to introduce the correct terminology. - 27 - Phase 4 (free o r i e n t a t i o n ) . "The complex tasks, to f i n d his/her own way student learns, by doing more i n the network of r e l a t i o n s (e.g. knowing properties of one kind of shape, investigates these properties for a new shape, such as k i t e s ) . " Phase 5 (integration). The students now acquired knowledge and form an overview. are u n i f i e d and i n t e r n a l i z e d into a new 1983, p. "The (Fuys et a l . 1988, p. 7) take t h e i r newly objects and r e l a t i o n s domain of thought." (Hoffer, 208) When the f i f t h phase has been completed, students have reached a new l e v e l of thought. "The new domain of thinking replaces the old, and students are ready to repeat the phases of learning at the next level." (Crowley, 1987 p. 6) At f i r s t glance i t may appear that the van Hiele model simply states the obvious—students need to learn i n an progression. organized However, the U.S.S.R. d i d make major changes i n t h e i r mathematics curriculum based on t h i s work. Research based on the van Hiele theories Despite i t s wide acceptance by the U.S.S.R. i n the 1960's, i n North America only a l i m i t e d amount of research on the van Hiele model has been done. a r t i c l e i n 1976 It was Wirszup's speech i n 1974 followed by his that prompted the American educators to do some investigations of the van Hiele l e v e l s . Three major projects have received U. S. federal funding to carry out research on the model. d e s c r i p t i o n of each one along with t h e i r r e s u l t s follows. - 28 - A The Chicago p r o j e c t : v a n H i e l e l e v e l s and achievement i n secondary s c h o o l geometry The purpose o f t h i s t h r e e y e a r p r o j e c t (1979-82) was t o a d d r e s s v a r i o u s q u e s t i o n s about s t u d e n t achievement i n grade t e n geometry and how t h i s r e l a t e s t o t h e van H i e l e t h e o r y . A p p r o x i m a t e l y 2700 s t u d e n t s i n h i g h s c h o o l geometry c o u r s e s from f i v e d i f f e r e n t s t a t e s were i n c l u d e d i n t h e s t u d y . (Senk, 1985; U s i s k i n , 1982) were a d m i n i s t e r e d n e a r t h e b e g i n n i n g o f t h e s c h o o l y e a r 1980) . Two t e s t s (September, One o f t h e s e t e s t s c o n s i s t e d o f m u l t i p l e - c h o i c e q u e s t i o n s d e a l i n g w i t h p r e r e q u i s i t e geometry knowledge. The second t e s t , a l s o m u l t i p l e - c h o i c e , was i n t e n d e d t o i n d i c a t e t h e v a n H i e l e l e v e l o f each student. Near t h e end o f t h e s c h o o l y e a r (May, 1981) t h e s e sat the van H i e l e t e s t s again. students They a l s o t o o k a s t a n d a r d i z e d m u l t i p l e - c h o i c e t e s t t h a t measured geometry achievement p l u s a t h i r d test, dealing with their proof-writing a b i l i t y . The r e s u l t s show t h a t : - a v a n H i e l e l e v e l c a n be a s s i g n e d t o most s t u d e n t s . - t h e s e l e v e l s a r e good i n d i c a t o r s o f s t u d e n t performance b o t h i n p r o o f - w r i t i n g and s t a n d a r d geometry c o n t e n t . - a p p l i c a t i o n o f t h e v a n H i e l e t h e o r y " e x p l a i n s why many s t u d e n t s have t r o u b l e l e a r n i n g and p e r f o r m i n g i n t h e geometry c l a s s r o o m " ( U s i s k i n , 1982, p. 8 9 ) . The v a n H i e l e l e v e l s were low f o r many s t u d e n t s e n t e r i n g grade t e n geometry. - o v e r h a l f o f t h e s t u d e n t s who e n r o l l i n geometry c o u r s e s w h i c h emphasize p r o o f , e x p e r i e n c e l i t t l e o r no s u c c e s s i n - 29 - w r i t i n g proofs. - one-third of the students rose one l e v e l , o n e - t h i r d rose two or more l e v e l s , and one-third stayed at the same level. - "The geometry course i s not working for large numbers of students. At the end of t h e i r year of study of geometry many students do not possess even t r i v i a l information regarding geometry terminology and measurement." (p. 89) The Brooklyn project: the van Hiele model of thinking i n geometry among adolescents This three year research project (1980-83) focused on four objectives (Fuys, 1985; Fuys et a l . 1988): - to translate the van Hiele writings into English, then develop and implement working modules based on the l e v e l and phases of the van H i e l e s . - to determine whether the van Hiele model describes how s i x t h and ninth graders learn geometry. - to determine i f teachers of these grades can be t r a i n e d to i d e n t i f y the van Hiele l e v e l s of students and of geometry curriculum materials. - to analyze l e v e l s of thinking of the geometric content of several major textbook s e r i e s . Three modules were developed based on the experiments i n Dina van Hiele-Geldof's t h e s i s . These modules were used i n the c l i n i c a l interviews i n v o l v i n g 16 s i x t h graders and 16 ninth graders and were intended to f a c i l i t a t e the students' movement through the lower levels. The students were videotaped as they i n d i v i d u a l l y worked - 30 - through the modules i n s i x to eight 45-minute sessions. This one-on-one contact provided the researchers with information changes i n a student's thinking within a l e v e l or to a higher on level. The r e s u l t s of t h i s study f i r s t v e r i f i e d the existence of each of the properties of the van Hiele model. Next, t h e i r r e s u l t s indicated that a range i n l e v e l s of thinking existed among the s i x t h and ninth graders (level 0 to l e v e l 2). geometry was "Findings i n t h i s study show that a neglected part of the school mathematics experiences of many students, and what was taught was required minimal student explanation." The researchers often taught r o t e l y or (Fuys et a l . 1988, p. 188) found that students i n these grades do have the p o t e n t i a l for l e v e l 1 and l e v e l 2 thinking. However, various factors were found which l i m i t e d a student's progress within a l e v e l or to a higher l e v e l . These factors included: - lack of prerequisite knowledge - poor vocabulary/lack of p r e c i s i o n of language - unresponsiveness to d i r e c t i v e s and given signals - lack of r e a l i z a t i o n of what was - lack of experience i n expected of them reasoning/explaining - i n s u f f i c i e n t time to assimilate new concepts and experiences - rote learning attitude - not r e f l e c t i v e about t h e i r own thinking (p. 139). This study also found that "preservice and inservice teachers learn to i d e n t i f y van Hiele l e v e l s of thinking i n student responses and i n text materials" (p. 154) and that such t r a i n i n g should be included i n teacher preparation programs. - 31 - can In t h e i r analysis of current K-8 mathematics textbooks, the investigators found them to be written at l e v e l 0. "Students w i l l presumably encounter d i f f i c u l t y with a secondary school geometry course at l e v e l 2 i f they can successfully complete grade 8 with l e v e l 0 thinking." (p. 169) The Oregon project: Assessing children's I n t e l l e c t u a l Growth i n Geometry The project was sponsored from September, 1979 to February, 1982. "The purpose of the study was to investigate the extent to which van Hiele levels serve as a model to access student understanding of geometry." (Hoffer, 1983, p. 212) Forty-five students from grades K-12 and college mathematics majors were selected from three states. The candidates were audio-taped during two 45-minute interviews involving tasks with quadrilaterals and t r i a n g l e s . The tasks were designed to r e f l e c t the van Hiele levels and to combine some ideas from Dina van Hiele-Geldof's research with her students. The findings from t h i s project were: - the h i e r a r c h i c a l nature of the van Hiele l e v e l s was confirmed. - the d i f f i c u l t y of assigning some students a van Hiele level. These students may be i n t r a n s i t i o n from one l e v e l to the next. - the movement from one l e v e l to the next i s not discrete. "Students may move back and forth between levels quite a few times while they are i n t r a n s i t i o n from one l e v e l to the next." (Burger & Shaughnessy, 1986, p. 45) - 32 - - that the use of formal deduction (level 3 thinking) among secondary and post-secondary students was nearly absent. - the teachers and students, be at different while t r y i n g to communicate, may levels. - that students may be at a geometric l e v e l quite different from what t h e i r teacher assumes they are. The three projects described above helped to raise the l e v e l of the van Hiele theory. awareness This i n turn began to answer some of the questions about poor performance i n w r i t i n g proofs. The interest s t i r r e d by these projects has resulted i n several a r t i c l e s and dissertations. Mayberry's (1981) d i s s e r t a t i o n centered on preparing .tasks which would be used to place preservice elementary teachers on the van Hiele scale. The Chicago project had developed a 25 question multiple-choice test for t h i s purpose. She prepared 62 tasks which were used while interviewing 19 preservice teachers. Mayberry's r e s u l t s showed that "the general van Hiele l e v e l of the preservice elementary teachers i n the study was rather low" (Mayberry, 1983, p. 102). Using Mayberry's tasks, Denis (1987) assessed Puerto Rican high school students who had already taken the Euclidean geometry course. He found that nearly three-quarters of the high school students were not at a l e v e l s u f f i c i e n t to deal with a t r a d i t i o n a l Euclidean geometry course. Mayberry also tested the h i e r a r c h i c a l nature of the van Hiele levels. Her results v e r i f i e d that a student at l e v e l n could answer a l l questions at and below l e v e l n but none of the questions above that l e v e l . Denis also concurred with the h i e r a r c h i c a l structure. - 33 - Senk's (1983) d i s s e r t a t i o n used the same data as the CDASSG project. One of the issues that she addressed was readiness—were students prepared for the proof w r i t i n g course? She found that the higher the student's van Hiele l e v e l was at the beginning of the geometry course, the greater the prospect for success i n w r i t i n g proofs. However, a high van Hiele l e v e l does not guarantee success i n w r i t i n g proofs. "Instruction plays a large part i n determining which of the students with the basic prerequisite knowledge w i l l eventually be successful on a given t o p i c . teachers, For t h i s reason, curriculum developers, and researchers need to share materials and methods found to be effective i n teaching proofs." (Senk, 1985, p. 455) Following on t h i s theme of i n s t r u c t i o n , Prevost (1985) wrote appealing to teachers to integrate t h e i r geometry curriculum into the van Hiele model. Also, i n keeping with the theory, he urged teachers i n junior high to develop the geometric ideas over time rather than i n a concentrated u n i t . To provide a more effective learning experience for h i s students i n grade ten geometry, Craine (1985) developed an alternative course based on the van Hiele model. He used an informal approach to introduce the basic concepts of geometry gradually b u i l d i n g i n the appropriate vocabulary. Proofs were developed near the end of the second semester. In response to the charge that students do not have the necessary prerequisite experiences to succeed i n w r i t i n g proofs, S c a l l y (1987) proposed a LOGO learning environment as a means to provide these experiences at the grade nine l e v e l . - 34 - Students' van Hiele l e v e l s were obtained by using interview items and tasks based on those by Burger and Shaughnessy. developed A group of ninth grade LOGO students and a group of ninth grade non-LOGO students were interviewed i n d i v i d u a l l y at the beginning and end of each semester. "The vast majority of student responses on both pre- and post-interviews were at the f i r s t and second van Hiele l e v e l s . " (Scally, 1987, p. 51) Overall, the LOGO students made more gains i n performing the various tasks at the end of the semester. Yet to be tested i s whether the LOGO experience w i l l i n fact enhance the students' thought processes i n grade ten geometry. S i m i l a r l y , B a t t i s t a and Clements (1988) recommended the introduction of LOGO into the junior and senior high school geometry classes as they believed "that the Logo environments can be used to help students progress within t h i s hierarchy" (p. 166). They cautioned teachers not to "expect that merely working with Logo automatically moves students into high l e v e l s of geometric (p. 167). thought" There needs to be correlations between LOGO a c t i v i t i e s and curriculum content. They summarized by s t a t i n g that, " I t i s imperative, therefore, that we help students a t t a i n high l e v e l s of geometric thought before they begin a proof-oriented study of geometry." (p. 166) A source of support for teachers to create a "discovery" atmosphere i s the microcomputer. "In p a r t i c u l a r , the microcomputer could prove to be the best bridge yet between the s p a t i a l - v i s u a l aspects of geometry and the logico-deductive aspects" ( D r i s c o l l , 1982, p. 149). The computer language, LOGO, has been used by others as a vehicle to prepare students for geometry and as an i n s t r u c t i o n a l - 35 - a i d while teaching the subject. Rather than LOGO, the investigator used the Geometric Supposer program for t h i s purpose i n the present study. Literature Relating to the Geometric Supposer Software The Geometric Supposer i s a series o f educational software programs, published by Sunburst Communications i n 1985, which allow the user to carry out many d i f f e r e n t geometric constructions and measurements. A more d e t a i l e d description from the manual, The Geometric Supposer: Triangles, follows: The GEOMETRIC SUPPOSER i s a microcomputer program that allows the user to carry out with ease constructions that are possible using and compass. straightedge These include the construction of t r i a n g l e s as well as the drawing of segments, medians, a l t i t u d e s , p a r a l l e l s , perpendiculars, perpendicular bisectors, angle bisectors, and i n s c r i b e d and circumscribed c i r c l e s . In addition, the user can measure lengths, angles, areas and distances as well as arithmetic combinations of these measures, such as the sum of two angles, the product of two lengths, or the r a t i o of two areas (p. 2) . "Part of the rationale behind the SUPPOSER was to provide a t o o l that could help students understand that a picture i s a s p e c i a l case and that examining one picture i s part of a larger process that includes viewing many s p e c i a l cases and not one s t a t i c example." (Yerushalmy & Chazan, 1987, p. 58) One of the problems i n teaching proof w r i t i n g i s that students view the diagrams as fixed, immobile - 36 - objects. "The Supposers provide an exploratory environment where students can experience and develop an i n t u i t i v e understanding of geometric concepts." (Mathis, 1986, p. 45) The researcher's survey has found very few a r t i c l e s and only one study involving the Geometric Supposer software. Reference was first made to t h i s software i n Aieta's (1985) a r t i c l e . He referred to the Geometric Supposer as being a "powerful and accessible" (p. 473) package that teachers could consider as a new approach to geometry. A review of these programs appeared i n The Computing Teacher i n June, 1986. "The software encourages the higher l e v e l thinking s k i l l s involved i n formulating and t e s t i n g hypotheses." (p. 45) The use of the Geometric Supposer was p a r a l l e l e d to that of a science class where the students c o l l e c t data, conjecture, and generalize. According to Yerushalmy and Houde (1986), using the Geometric Supposer "encourages students to behave l i k e geometers because i t o f f e r s a wealth of v i s u a l and numerical data and because conjectures about relationships observed within the data can be quickly tested" (p. 418). For any conjecture that the student makes, t h i s software allows the experiment to be repeated on s i m i l a r figures very quickly. Students would not have the time nor the i n c l i n a t i o n to manually construct counter examples. "Our experience demonstrates that students brought a high degree of enthusiasm to t h e i r work and demonstrated an a b i l i t y to create geometry that we never thought possible." (p. 422) Two of the Geometric Supposer programs were chosen as being i n the top s i x f o r 1987 by Classroom Computer Learning magazine. program i s t r u l y a discovery tool that helps the user become an - 37 - "This active p a r t i c i p a n t i n the quest f o r mathematical knowledge." (p. 20) In the summer of 1986 f i f t y expert high school geometry teachers were brought together i n New Jersey for the purpose of looking at new materials and methods related to the f i e l d of geometry. materials were developed f o r d i s t r i b u t i o n . with the Geometric Supposer. Various One of the topics dealt A series of investigations on t r i a n g l e s and q u a d r i l a t e r a l s , based on Polya's model, were produced f o r teacher and student use. The authors noted, "Do not assume that using the Geometric Supposer w i l l allow you to cover the course material any more quickly. This cannot be guaranteed. It w i l l , however, allow you to teach a much richer course i n which students glean a better understanding of what mathematics i s a l l about." (Birt & Koss, 1986, p. 48) Yerushalmy conducted a yearlong research project i n 1984-85 on inductive reasoning i n geometry and the use of the Geometric Supposer. Three geometry classes (83 students) at d i f f e r e n t s i t e s used t h i s computer software. At each s i t e a comparison class was taught mainly from the textbook using the t r a d i t i o n a l approach. The goal of t h i s project was to provide students with "an opportunity to experiment with geometric shapes and elements, to move from the p a r t i c u l a r t o the general, and to make conjectures before grappling with proofs. This approach to geometry i s absent from the 'formal' secondary geometry curriculum." (Yerushalmy, Chazan, & Gordon, 1987, p. 6) The i n s t r u c t i o n a l approach used i s referred to as guided inquiry. This approach emphasizes a combination of laboratory work and class discussion. From t h e i r comments, the teachers involved were not p o s i t i v e - 38 - about the Geometric Supposer. disappointment They expressed misgivings, i n student progress, and concern about the time-consuming nature of laboratory work and follow-up. They d i d note some improvement i n students' a b i l i t y to organize data and f e l t that the Geometric Supposer had p o t e n t i a l as a diagnostic t o o l . They also f e l t "that these students d i d get more out of t h e i r Geometry class than they would have done i n a t r a d i t i o n a l c l a s s " (p. 40) and that most students had achieved an understanding of the need f o r a proof. The students, on the other hand, were generally p o s i t i v e about the computer experience. They indicated that the Geometric Supposer was easy to use, added to t h e i r understanding, and provided enjoyment when they were successful. making conjectures. The negative aspect f o r the students was "Knowing what to conjecture about, discerning patterns and relationships, and generating conjectures were a l l hard work." (p. 42) Yerushalmy et a l . (1987) concluded that students from both groups (Geometric Supposer and comparison) learned equal amounts of geometry. However, the experimental group " s i g n i f i c a n t l y outperformed the comparison group i n t h e i r a b i l i t y to develop generalizations, and they were equal to and/or somewhat better than the comparison group i n t h e i r a b i l i t y to devise informal arguments and t r a d i t i o n a l formal proofs." (p. 68) In summary, the l i t e r a t u r e indicates that geometry and the proof w r i t i n g a c t i v i t y associated with i t w i l l continue to be viewed as an important and c r i t i c a l part of the high school mathematics curriculum. Despite the well documented fact that students d i s l i k e - 39 - geometry and that only a minority gain the kind of understanding t h e i r instructors hope to i n s t i l l , writing proofs i s considered to be a necessary part of t h e i r education. The van Hiele model appears to have p o t e n t i a l as a way of understanding the problems with proof writing and of designing solutions for these problems. The Geometric Supposer, while less well documented, has been i d e n t i f i e d as a s p e c i f i c instrument for allowing a discovery method to make inroads into the t r a d i t i o n a l proof w r i t i n g method used i n teaching geometry. - 40 - CHAPTER 3 PROCEDURES The procedures used to investigate students' a b i l i t y to write proofs using the computer program, Geometric Supposer, are reviewed i n t h i s chapter. A description of the subjects, the steps taken i n the study, and the data c o l l e c t i o n instruments i s given. The Subjects The community i n which t h i s study was carried out i s a r e l a t i v e l y isolated, northern Canadian town of approximately 4000 residents. The nearest major center i s 240 kilometres (150 miles) to the south. However, as t h i s l i n k i s by a well-paved highway, i s o l a t i o n i s not considered to be a c r i t i c a l factor i n the study. The community i s a service center for a large portion of the northern part of the province. as well as a hospital, administrative o f f i c e s . Three small a i r l i n e s are based there a community college, and some government The community services the tourism industry and more recently, considerable mining exploration. On the other hand, some residents, p a r t i c u l a r l y Treaty Indians, s t i l l earn t h e i r l i v i n g by trapping and f i s h i n g . Thus, despite i t s small size, the community has a wide range of socioeconomic levels. The educational system i n the community consists of three schools: a K-5 elementary school, a K-8 Treaty school, and a grade 6-12 school. The l a t t e r has a population of 500 students and was s i t e of the study. the The population of t h i s school r e f l e c t s the s o c i a l - 41 - makeup of the community which i s roughly 55% Native persons and 45% Europeans. As an overview, the mathematics curriculum i n t h i s province i s prescribed u n t i l grade nine. In grade nine, generally, students have a choice of regular mathematics or general mathematics. students opt for the regular mathematics. Most To graduate from high school, students must have a mathematics credit at the grade ten level. Students can obtain t h i s credit by taking one or more of algebra, geometry, mathematics, or general mathematics. i n grade eleven the same choices are available. Similarly, In grade twelve the mathematics courses offered'are algebra, geometry, and mathematics. In the school where t h i s study took place, the students i n grade nine had to choose between algebra or general mathematics. was not an option. Geometry Algebra, geometry, and general mathematics were offered i n grade ten. In grade eleven an algebra course and a geometry course were included i n the timetable choices. also the case for grade twelve. This was Thus, a f t e r grade ten, a student could take from zero to four senior mathematics classes. The subjects i n t h i s study had opted to take the grade ten geometry course which was scheduled i n the second semester of the school year 1987-88 from February u n t i l June. Of the 62 grade ten students i n the school, 41 (66%) chose to take the geometry course. Three students from grade 11 also elected to take geometry. The 44 students were divided into two classes by the school administration on an ad hoc basis with the researcher having no input. The class assigned to the researcher used the Geometric Supposer software throughout the course and w i l l be referred to as the computer group - 42 - i n t h i s study. In the computer group there were 12 males and 10 females, 9 students were of native ancestry. The second class d i d not use the computer and w i l l be referred to as the t r a d i t i o n a l group. The t r a d i t i o n a l group consisted of 13 males and 9 females, 8 students were of native ancestry. The ages of a l l the students ranged from 15 to 18 with the majority being either 15 or 16 years old. As described above, the students i n t h i s study had had no geometry exposure since grade eight when they studied i t as one of the chapters i n t h e i r textbook. During the semester one student from the computer group withdrew from school and one student was added, thus t h i s group remained at 22 students. In the t r a d i t i o n a l group three students withdrew from school and two others discontinued the geometry course. This l e f t 17 students i n the t r a d i t i o n a l group. Permission was obtained from the parents f o r the p a r t i c i p a t i o n of t h e i r children i n the study (Appendix A). One student i n the t r a d i t i o n a l group was not given permission to p a r t i c i p a t e i n the study. This student obtained the highest f i n a l mark i n that class, thus her absence could have affected the balance of the two groups. Design of the Study A quasi-experimental design was used. The subjects were assigned to two groups based on t h e i r preassigned homerooms. Formalized random selection was not possible i n the school setting. Classical pretest, posttest, and experimental group, control group methodology was followed. In t h i s section a description i s given of the design - 43 - and t h e t e s t s used. A d e s c r i p t i o n i s a l s o i n c l u d e d o f t h e open-ended c l i n i c a l i n t e r v i e w s t h a t were c a r r i e d o u t . Quasi-experiment T h i s s t u d y was a q u a s i - e x p e r i m e n t a l i n v e s t i g a t i o n . The r e s e a r c h e r g a t h e r e d d a t a from two groups o f s t u d e n t s who were e n r o l l e d i n t h e grade 10 geometry c o u r s e . The c u r r i c u l u m c o n t e n t , as p r e s c r i b e d by t h e p r o v i n c i a l Department o f E d u c a t i o n , was t h e same f o r b o t h groups. class The d i f f e r e n c e between t h e two groups was t h a t one (computer group) u s e d a computer program, t h e Geometric Supposer, throughout t h e course w h i l e t h e second c l a s s (traditional group) d i d n o t use t h e computer. The geometry c l a s s e s were b o t h s c h e d u l e d f i r s t p e r i o d (9:00 a.m. - 10:00 a.m.) d a i l y . As t h e a d m i n i s t r a t i o n was unable t o r e - s c h e d u l e , i t was i m p o s s i b l e f o r t h e r e s e a r c h e r t o t e a c h b o t h The r e s e a r c h e r t a u g h t t h e computer group w h i l e a n o t h e r member t h e mathematics department t a u g h t t h e t r a d i t i o n a l group. r e s e a r c h e r had a B.A. degree, a P r o f e s s i o n a l A T e a c h i n g and 22 y e a r s t e a c h i n g e x p e r i e n c e . groups. from The Certificate, The t e a c h e r o f t h e t r a d i t i o n a l c l a s s h a d a B.Ed, degree, a B.Sc. (Honors i n Geology) degree, e i g h t y e a r s e x p e r i e n c e i n e x p l o r a t i o n and m i n i n g p l u s f o u r y e a r s t e a c h i n g experience. Both t e a c h e r s had taught t h e geometry c o u r s e p r e v i o u s l y and kept c o n s t a n t c o n t a c t throughout t h e course r e g a r d i n g t h e c u r r i c u l u m c o n t e n t and e x p e c t a t i o n s o f t h e s t u d e n t s . The Geometric Supposer The computer program, t h e Geometric treatment i n t h i s study. The Geometric Supposer, was u s e d as t h e Supposer i s a s e r i e s o f s o f t w a r e programs e s p e c i a l l y d e s i g n e d t o p r o v i d e a " p l a y g r o u n d " i n - 44 - which students can experiment with geometric figures and form conjectures. The two Supposer programs used i n t h i s study were Triangles and Quadrilaterals. Description Each program i s contained on a 5 1/4" disk and takes approximately 30 seconds to load into a 64K Apple computer. When the Triangle program has been loaded into memory, the screen w i l l look as shown i n Figure 1. Press M t o besin 2 Label 3 Erase Figure 1. K S K N Heasure S c a l e change Repeat Mey trian«le The opening screen of the Geometric Supposer Triangle program Each screen of the Supposer i s divided into three p a r t s — t h e l e f t column i s f o r data recording, the right side i s the area f o r constructions, and the region below the horizontal l i n e i s f o r menus and prompts. After.N (New triangle) i s pressed the user i s presented with a new menu-: 1 Right 4 Isosceles 2 Acute 5 Equilateral 3 Obtuse 6 Your own Depending on the type of triangle the user wishes to experiment on, - 45 - s/he would select accordingly. I f #1 was chosen the screen as shown i n Figure 2 would appear. Figure 2. The Right Triangle The t r i a n g l e s do not appear exactly the same each time selected. Assuming a student had been assigned to investigate a l t i t u d e s i n t r i a n g l e s , s/he would have made menu selections to display such diagrams as i l l u s t r a t e d i n Figure 3 or Figure 4. Figure 3. An acute triangle with a l t i t u d e s . Figure 4. An obtuse triangle with a l t i t u d e s . Then using the measurement function (M), the program offers a choice of: - 46 - 1 Length 5 Distance Point-line 2 Perimeter 6 Distance Line-line 3 Area 7 A d j u s t a b l e element(s) 4 Angle At t h i s p o i n t , t h e u s e r may d e c i d e t o f i n d out what e x i s t s between a n g l e s and. a l t i t u d e s . t h e s c r e e n t o be measured. was O p t i o n #4 a l l o w s any a n g l e on The program r e q u e s t s t h e name o f t h e angle, u s i n g three l e t t e r s . ( F i g u r e 3 ) , i f ADB relationship I n t h e case o f t h e a c u t e triangle e n t e r e d , t h e program would respond w i t h ZADB = 90 as shown i n F i g u r e 5. The u s e r would c o n t i n u e t o "measure" u n t i l satisfied. Data £ A D B = 90 n Heasure S S c a l e change ft Repeat M Heai triaogle 1 Draw | Label 3 Erase Figure 5. Information recorded on angle measurement. The above description provides an overview of some of c a p a b i l i t i e s of t h i s program. The the constructions and measurements are performed quickly by the program giving almost instantaneous feedback to the user. Use i n the The study researcher used the Supposer software as a method of developing geometric concepts. The program was f i r s t demonstrated to the class by using i t as an electronic chalkboard to develop d e f i n i t i o n s for the different types of t r i a n g l e s . Following t h i s group exposure, the students worked i n pairs on s p e c i f i c assignments. In a t y p i c a l period i n which the Geometric Supposer was used, the i n s t r u c t i o n a l period would be divided into four sections. objective would be defined. F i r s t , the For example, the task might be to explore the relationship among the i n t e r i o r angles i n d i f f e r e n t types of t r i a n g l e s . Secondly, the students would carry out a p e n c i l , paper, protractor exercise on t h i s topic. Thirdly, i n pairs, the students would work on the computers using the software to further explore i n t e r i o r angles i n a l l types of t r i a n g l e s . assignment, As part of t h i s they would record t h e i r observations and a f t e r discussions with t h e i r partner, write out t h e i r conjectures. Fourthly, the class would recongregate and discuss t h e i r findings. Depending on the nature of the objectives, a theorem or d e f i n i t i o n would emerge. Without t h i s software, students would have had to use paper and p e n c i l constructions exclusively to explore the various t r i a n g l e s . This process would be tedious, time-consuming, and result i n f r u s t r a t i o n and/or boredom for the students. The class used the Geometric Supposer software i n the way described above approximately twice a week. The average session on the computer was about 15 minutes. Pretests Both groups were given two tests on consecutive days within the f i r s t week of the semester. The tests were an introductory geometry - 48 - t e s t and a test to measure geometric thought levels of students—the van Hiele geometry t e s t . Introductory geometry test (Note: The province i n which t h i s study was to i t s geometry course as "geo t r i g . " c a r r i e d out refers This "geo t r i g " course contains the same content and uses s i m i l a r textbooks as other provinces and states i n t h e i r f i r s t year geometry courses. Hence, the test r e f e r r e d to i n t h i s chapter as the introductory geometry t e s t , appears i n Appendix B under the t i t l e , Introductory Geo T r i g 10 Test.) The f i r s t test given, the introductory geometry test B), was created by the researcher. (Appendix The test was based on the geometry chapter from the grade eight textbook (Fleenor, Eicholz, O'Daffer, 1974) which these students had previously used. As assessment of face v a l i d i t y , the introductory geometry test examined by a grade eight mathematics teacher who reasonableness of the content. lapse of nearly two years & an was approved the However, he f e l t that due to the (three years for the grade elevens) since they had studied t h i s material, the students would not do w e l l . A f t e r a preliminary analysis, one question was removed from the test as only one of the forty students answered i t c o r r e c t l y . reduced the test to 24 questions from the o r i g i n a l 25. r e l i a b i l i t y of t h i s test i s discussed l a t e r i n t h i s van H i e l e geometry This The chapter, test Based on writings of the van Hieles, Usiskin (1982), i n the Cognitive Development and Achievement i n Secondary School Geometry (CDASSG) Project, developed, p i l o t e d , and modified,test items to be - 49 - used to determine geometric thought l e v e l s of students. These l e v e l s were referred to as the van Hiele l e v e l s of the students i n the CDASSG Project. Permission was granted by Professor Zalman Usiskin, Department of Education, University of Chicago, to use the van Hiele geometry test f o r t h i s study (Appendix C). Posttests Three weeks p r i o r to the end of the f i v e month semester, two tests, the van Hiele geometry test and a proof test, were given to both geometry classes on consecutive days at the same time. One week l a t e r an attitude test was administered to both groups, again, at the same time. van H i e l e geometry test This test was i d e n t i c a l to the one given at the beginning of the semester. Proof test Sharon Senk of the CDASSG Project developed three tests on proof. Some of the items on these tests were not suitable f o r the present study as they were either unfamiliar to the students or were i d e n t i c a l to what had been covered i n c l a s s . However, the researcher followed the format of Senk's tests to create her own proof test (Appendix D). Attitude test S i m i l a r l y , Aiken (1963, 1974) developed attitude tests to mathematics. related Since Aiken's test items d i d not deal s p e c i f i c a l l y with geometry, a new test (Appendix E) had to be designed by the researcher. - 50 - Interviews The researcher c a r r i e d out individual interviews with students from the computer group. of four students Eight interviews were scheduled with each (two male students and two female students). Permission (Appendix F) was obtained from the students involved. Each interview lasted 20-30 minutes and took place i n the morning before school started, at lunch hour or after school, depending on which time was convenient for the student. Two students from the t r a d i t i o n a l group (one male, one female) were each interviewed twice i n the l a t t e r h a l f of the semester. Experimental Controls In an attempt to reduce uncontrolled factors i n t h i s study, a l l tests were administered at exactly the same time to both groups. This was done to avoid students obtaining any p r i o r knowledge of the test questions. The teacher of the t r a d i t i o n a l group was very supportive of the study. He was keen to cooperate and was kept informed of the study's progress. He acknowledged the need to keep the kinds of questions assigned, the theorems to be emphasized, and the frequency of quizzes and tests generally equal. The two classes were taught i n rooms some distance from each other. The t r a d i t i o n a l group was thus unaware of when the computer group was actually using the computer l a b . In t h i s study the teacher of the t r a d i t i o n a l group had taught the majority of his students the previous semester i n algebra. Thus, he was aware of t h e i r mathematical strengths, weaknesses, and personality t r a i t s . The researcher had been away from t h i s school on secondment for two years and did not know the students. - 51 - Therefore, the researcher was not biased by previous knowledge about the computer group. O r i g i n a l l y , the researcher had proposed to make formal observations during class time using comment cards, checklists, and/or rating scales. The researcher d i d t r y t h i s on several occasions but found i t was not feasible to combine these a c t i v i t i e s with teaching r e s p o n s i b i l i t i e s . Data C o l l e c t i o n This section provides detailed descriptions of each of the tests employed to obtain data about the subjects' geometric knowledge at the beginning and end of the course. A detailed description of the interviews i s also provided. Tests A t o t a l of f i v e tests were administered to both groups of geometry students—two the end. at the beginning of the semester and three at One of the tests, the van Hiele geometry test, was given both at the beginning and at the end. Introductory geometry test (Appendix B) The primary purpose of the introductory geometry test was to measure the geometry knowledge of the students entering the grade ten course. The students' previous contact with geometry had been i n grade eight. The questions for t h i s test were based on the geometric material they covered i n t h e i r grade eight textbook. The researcher developed the test to cover only the main ideas with l i t t l e emphasis on d e t a i l s . A second purpose for the test was to establish whether any - 52 - s i g n i f i c a n t difference existed between the mean scores of the two groups. The test o r i g i n a l l y consisted of 25 multiple-choice questions but was subsequently reduced to 24 questions when only 3% of the students were able to answer question #7. maximum score of 24 on t h i s t e s t . Thus, a student could achieve a T h i r t y - f i v e minutes were allocated for the test but the majority of students i n both classes were f i n i s h e d within h a l f an hour. The r e s u l t s of an item analysis of the introductory geometry test can be found i n Appendix B. The r e l i a b i l i t y measure was calculated using the answers from a l l students who wrote the t e s t . The Hoyt estimate of r e l i a b i l i t y was .73 (SPSSX package). van Hiele geometry rest (Appendix C) The purpose for giving the van Hiele geometry test was to measure the geometric thought levels of the students at the beginning of the course and again at the end. A comparison of the results should enable the following questions to be answered: - What changes i n the students' van Hiele levels take place a f t e r a semester of geometry? - Did the change i n the van Hiele pre and postlevels of the computer group d i f f e r s i g n i f i c a n t l y from those of the t r a d i t i o n a l group? This test, developed by the CDASSG Project, consists of 25 multiple-choice questions divided into five levels, with f i v e questions at each l e v e l . To test for r e l i a b i l i t y , the CDASSG Project used the Kuder-Richardson formula 20 (.77) and Horst's modification (.79) (Usiskin, 1982, p. 29). - 53 - The CDASSG P r o j e c t developed two methods f o r c a l c u l a t i n g a s t u d e n t ' s van H i e l e l e v e l : criterion. 3 out o f 5 c r i t e r i o n and 4 out o f 5 I n t h i s s t u d y t h e 3 out o f 5 c r i t e r i o n was used. This r e q u i r e s a t l e a s t 3 out o f t h e 5 q u e s t i o n s c o r r e c t a t a l e v e l i n o r d e r t o be a s s i g n e d t h a t l e v e l . F o r example, i f a s t u d e n t had a t l e a s t 3 out o f t h e 5 f i r s t q u e s t i o n s (items 1-5) 3 out o f t h e 5 q u e s t i o n s (items 6-10) c o r r e c t and a t l e a s t c o r r e c t and l e s s t h a n 3 q u e s t i o n s c o r r e c t i n each o f t h e r e m a i n i n g c a t e g o r i e s , s/he would be a s s i g n e d a van H i e l e l e v e l 2 ( i n f o r m a l d e d u c t i o n ) . hand, a s t u d e n t had met the c r i t e r i o n c o r r e c t ) f o r items 1-5, 6-10, a s s i g n e d a van H i e l e l e v e l 2. I f , on t h e o t h e r (a minimum o f 3 out o f 5 and 21-25, t h i s s t u d e n t would a l s o be The reason f o r t h e same van H i e l e l e v e l i s t h a t a c c o r d i n g t o P r o p e r t y 1 o f t h e van H i e l e l e v e l s , s t u d e n t a t l e v e l n must have met not above n. t h e c r i t e r i o n a t l e v e l s below n but I n t h i s case, t h e s t u d e n t has s a t i s f i e d t h e c r i t e r i o n f o r l e v e l s 1 and 2, t h e n jumped t o 5. may a A l t h o u g h t h e van H i e l e t e s t be "a r a t h e r crude d e v i c e f o r c l a s s i f y i n g s t u d e n t s " (p. 30) levels, i t " s t i l l may be u s e f u l f o r a n a l y z i n g b e h a v i o r and students" into treating (p. 3 4 ) . I n o r d e r t o a s s i g n more s t u d e n t s a van H i e l e l e v e l , t h e CDASSG P r o j e c t d e v e l o p e d a schematic d e s c r i p t i o n o f t h e 32 p o s s i b i l i t i e s which c o u l d e x i s t t h a t do not comply w i t h P r o p e r t y 1. p o s s i b i l i t i e s a r e each g i v e n a f o r c e d van H i e l e l e v e l . The v a r i o u s Forced van H i e l e l e v e l s were used i n t h i s s t u d y i n o r d e r t o make comparisons movements o f s t u d e n t s from one l e v e l t o a n o t h e r . The 4 out o f 5 c r i t e r i o n was not u s e d because a g r e a t e r percentage o f s t u d e n t s would have been p l a c e d a t l e v e l 0 (22% as - 54 - in opposed to 6% on the 3 out of 5 c r i t e r i o n ) . One of the purposes of t h i s study was to observe the students' movement between l e v e l s . " I f weaker mastery, say 80%, i s expected of a student operating at a given l e v e l , then i t i s absolutely necessary to use the 3 of 5 c r i t e r i o n , f o r Type II errors with the s t r i c t e r c r i t e r i o n are much too frequent." (p. 24) The CDASSG Project also used two theories i n assigning l e v e l s : the c l a s s i c a l theory and the modified theory. Here again, the c l a s s i c a l theory employs a more r i g i d method of assigning l e v e l s . In the example above, the student s a t i s f y i n g the c r i t e r i o n f o r l e v e l s 1, 2, and 5 would have been assigned a van Hiele l e v e l of zero. The researcher used the modified theory since i t resulted i n more of the students being assigned a van Hiele l e v e l greater than zero. For a more d e t a i l e d description of assigning van Hiele l e v e l s , the reader should r e f e r to Usiskin, 1982. The CDASSG Project found that the van Hiele geometry test could be used to measure changes i n students a f t e r a year of geometry. They also found that the "van Hiele l e v e l s are a very good indicator when i t comes to p r e d i c t i n g success on proof" (p. 51). Proof test (Appendix D) The purpose f o r giving the proof test was to measure the students' a b i l i t y to write proofs and to compare the achievement of the two groups i n t h i s area. The test contained s i x questions resembling those found i n the students' geometry textbook (Jurgensen et a l . 1985) and used a s i m i l a r format to that developed by Sharon Senk of the CDASSG Project. The f i r s t question required the students to f i l l i n the - 55 - missing statements or reasons of the proof. The second, fourth, f i f t h , and s i x t h questions required the students to write f u l l proofs. The t h i r d question required a t r a n s l a t i o n from an English statement to an appropriate "given," "to prove," and a "diagram." The students were then required to write a proof. The teacher of the t r a d i t i o n a l group examined the test to ensure that none of the problems had previously been attempted i n c l a s s . T h i r t y - f i v e minutes (the same time as the CDASSG Project) were a l l o c a t e d for the t e s t . The questions on the test were each scored on a 0-to-4 scale. The c r i t e r i a used for grading the proofs was as follows: 0 - Student writes nothing, writes only the "given," or writes only i n v a l i d or useless deductions. 1 - Student writes at least one v a l i d deduction and gives reason. 2 - Student shows evidence of using a chain of reasoning, either by deducing about h a l f the proof and stopping, or by writing a sequence of statements that i s i n v a l i d because i t i s based on faulty reasoning early i n the steps. 3 - Student writes a proof i n which a l l steps follow l o g i c a l l y but i n which errors occur i n notation, vocabulary, or names of theorems. 4 - Student writes a v a l i d proof with at most one error i n notation (Senk, 1985, p. 449). The researcher marked a l l the proofs using t h i s scale. a cover sheet with the student's name. - 56 - Each test had P r i o r to any marking taking place, both groups of tests had t h e i r cover sheets turned. t h i s , the tests were marked. After Hence, a l l marking took place while the researcher was unaware of whose test she was actually marking. In summary, the proof tests were b l i n d l y scored. Since questions #1 and #3 were of a different type than the other four questions, two measures of o v e r a l l achievement are given. Hoyt estimate of r e l i a b i l i t y on a l l s i x questions was analysis appears i n Appendix D. The .80 and an item The maximum possible score was 24. The second measure pertains to the four questions (2, 4, 5, 6) which were s t r i c t l y proof writing exercises and the maximum score was The Hoyt estimate of r e l i a b i l i t y on t h i s section was 16. .74. A s i m i l a r six-item proof test was given to 1520 students i n the CDASSG Project and formed the basis of conclusions about students' readiness f o r geometry. Attitude test (Appendix E) The attitude test was given to measure the a f f e c t i v e objectives of mathematics i n s t r u c t i o n such as attitude, value, and enjoyment. The purpose f o r giving t h i s test was to analyze any s i g n i f i c a n t difference i n attitude between the two groups at the end of the semester. This test (28 questions) was constructed using Likert's method of summated ratings. Half of the items on the test are worded i n the d i r e c t i o n of a favorable attitude and the remaining h a l f are i n the d i r e c t i o n of an unfavorable attitude. The test can be divided into three sections: (1) questions 1 - 1 0 relate s p e c i f i c a l l y to attitudes towards geometry. - 57 - (2) questions 11 - 20 relate to the enjoyment of mathematics. (3) questions 21 - 28 relate to the value or importance of mathematics. To encourage students to be as honest as possible no names were required on t h i s t e s t . The second part of the test requested written " l i k e s " and " d i s l i k e s " of the geometry course. The purpose of these general questions was to obtain "gut" reactions about what stood out i n the students' minds as p o s i t i v e and negative factors. On the f i r s t 28 questions the students were asked to respond with "Strongly Agree" (SA), "Agree" (A), "Undecided" (U), "Disagree" (D), or "Strongly Disagree (SD). For the attitude items which were stated p o s i t i v e l y , the responses to each item were coded as 5, 4, 3, 2, or 1 respectively. For the attitude items stated negatively, the responses were coded as 1, 2, 3, 4, or 5 respectively. An item analysis f o r each group i s i n Appendix E. The Hoyt estimate of r e l i a b i l i t y f o r the 28 questions was .93 with the estimates f o r each of the three sections being .91, .94, and .74. Interviews The purpose f o r carrying out interviews was to increase the researcher's understanding and appreciation of the d i f f i c u l t i e s students were experiencing with proof writing. The interviews also provided additional insights to the researcher about misconceptions held by students that were not apparent i n t h e i r classroom work. Four students were selected from the computer group. interviewees consisted of two males and two females. The One male (Scml) had a van Hiele l e v e l 1, the second male (Scm2) was a l e v e l 3, one - 58 - female ( S c f l ) was l e v e l 0, and t h e second female (Scf2) was l e v e l 1. The r e s e a r c h e r had a l s o d e r i v e d t h e van H i e l e l e v e l s u s i n g t h e 4 out o f 5 c r i t e r i o n . their levels f o r each These l e v e l s were compared w i t h from t h e 3 out o f 5 c r i t e r i o n . The l e v e l s f o r Scm2 were 3 on t h e l a t t e r c r i t e r i o n and 0 on t h e s t r i c t e r c r i t e r i o n . levels student The f o r t h i s student v a r i e d t h e most i n b o t h groups (computer and traditional) and hence, t h e r e s e a r c h e r s e l e c t e d him f o r t h e i n t e r v i e w s i n o r d e r t o r e s o l v e t h i s quandary. The r e s e a r c h e r met i n d i v i d u a l l y w i t h each student f o r approximately 20-30 minutes on a b i w e e k l y b a s i s . h e l d i n an o f f i c e which p r o v i d e d p r i v a c y . audio-taped and l a t e r t r a n s c r i b e d . nervous w i t h t h e t a p e r e c o r d e r . w i t h homework problems. The s e s s i o n s were Each s e s s i o n was A t f i r s t , t h e s t u d e n t s were The b e g i n n i n g s e s s i o n s m a i n l y I n t h e l a t e r s e s s i o n s each student dealt solved the same geometry p r o o f s . One o f t h e male s t u d e n t s the s e s s i o n s , c o n s e q u e n t l y , (Scml) was not c o n s i s t e n t i n a t t e n d i n g o n l y s i x i n t e r v i e w s were h e l d w i t h h i m b e f o r e t h e end o f semester. A f t e r f o u r s e s s i o n s i t was o b v i o u s t h a t Scm2 had a sound grasp o f t h e g e o m e t r i c c o n c e p t s i n v o l v e d and was a b l e t o q u i c k l y a n a l y z e and write a proof. of t h e study. Hence, s e s s i o n s w i t h h i m were u n p r o d u c t i v e A t t h i s p o i n t , a n o t h e r male student H i e l e l e v e l 1 was s e l e c t e d , and he a t t e n d e d i n terms (Scm3) w i t h a van four interviews. To o b t a i n some i n - d e p t h i n f o r m a t i o n r e g a r d i n g p r o o f w r i t i n g from t h e t r a d i t i o n a l group, two s t u d e n t s were s e l e c t e d — a female (Stf3) w i t h van H i e l e l e v e l 0 and a male (Stm4) w i t h l e v e l 1. These two s t u d e n t s were each i n t e r v i e w e d t w i c e and worked t h r o u g h some o f t h e same - 59 - proofs that the computer group had had. A summary of the number of interviews held with each student follows: Scml - 6 sessions S c f l - 8 sessions Scm2 - 4 sessions Scf2 - 8 sessions Scm3 - 4 sessions Stf3 - 2 sessions Stm4 - 2 sessions These interviews were a l l held out of class time. Given the control of the various extraneous factors i d e n t i f i e d i n t h i s chapter, the quasi-experimental model does appear to be appropriate for t h i s investigation. - 60 - CHAPTER 4 DATA ANALYSIS The r e s u l t s o b t a i n e d from t h e f i v e t e s t s and t h e i n t e r v i e w s are presented i n t h i s chapter. 1) The f o l l o w i n g q u e s t i o n s were W i l l t h e s t u d e n t s who use t h e Geometric raised: Supposer s o f t w a r e be b e t t e r a b l e t o w r i t e f o r m a l p r o o f s t h a n s t u d e n t s who 2) are t a u g h t by more t r a d i t i o n a l methods? What changes i n t h e s t u d e n t s ' van H i e l e l e v e l s take p l a c e a f t e r a semester o f geometry? 3) W i l l t h e s t u d e n t s who positive attitude r e c e i v e t h e t r e a t m e n t have a more towards geometry t h a n t h e s t u d e n t s i n t h e t r a d i t i o n a l group? Assessment o f t h e Groups D i d t h e g e o m e t r i c knowledge o f t h e two groups (computer and t r a d i t i o n a l ) d i f f e r a t t h e b e g i n n i n g o f t h i s study? The instrument u s e d t o answer t h i s q u e s t i o n was t h e i n t r o d u c t o r y geometry p r e t e s t (Appendix B ) . involving An independent t h e two independent t - t e s t was used t o a n a l y z e t h e d a t a groups. The t e s t i s a v a i l a b l e SPSSX package on t h e U. B. C. computer The findings, system. as shown i n Table 4.1, s t a t i s t i c a l l y significant difference s t u d e n t s who The independent indicate t h a t t h e r e was no between t h e two group means a t t h e .05 l e v e l o f s t a t i s t i c a l s i g n i f i c a n c e knowledge. i n the i n terms o f geometric t - t e s t i n v o l v e d t h e d a t a from a l l f o r t y o r i g i n a l l y wrote t h e i n t r o d u c t o r y geometry t e s t . - 61 - The mean scores were 14.95 and 13.16 for the computer and t r a d i t i o n a l groups respectively. 24. The marks on t h i s 24 item test ranged from 5 to An item analysis of t h i s test for each group i s presented i n Appendix B. During the semester four students l e f t the t r a d i t i o n a l group, thus a second t-test was run without the scores of those four students. Again, the result was similar i n that no s t a t i s t i c a l l y s i g n i f i c a n t difference was revealed (t = 1.51; p = .141). Thus, the two groups were accepted as being from the same population and appropriate for t h i s study. Table 4.1 Means, Standard Deviations, and S t a t i s t i c a l Comparison of Groups: Introductory Geometry Pretest Groups n Mean SD Computer 21 14.95 4.73 Traditional 19 13.16 3.10 t-value 1.40 p .169 Geometric Thought Levels P r i o r to Treatment While the f i r s t data analysis was concerned with the actual performance i n geometry of the two groups, the second data analysis was concerned with the geometric thought levels of the subjects. An ordinal scale was used to represent the van Hiele levels (geometric thought levels) of the students. cannot be used as scores. These levels represent rankings and Therefore, the median test, a - 62 - nonparametric test, was appropriate. The median test was used to test the hypothesis that the two groups came from populations that have the same median. This test was applied to the data from the van Hiele pretest. Again, the two groups appear to be homogeneous as can be seen i n Table 4.2. A significance l e v e l of .05 was used. Table 4.2 Median Test: van Hiele Pretest Computer Group Traditional Group Combined Number greater than median 11 4 15 Less than or equal to median 1Q. 11 21 Total 21 15 36 (p = .230) Changes i n Geometric Thought Levels What changes i n the students' van Hiele levels take place a f t e r a semester of geometry? The instruments used to answer t h i s question were the pre and post van Hiele geometry t e s t s . The sign test, another nonparametric test which i s used to measure the same sample on two d i f f e r e n t occasions when i t i s suspected that changes are taking place, was applied to the data. The results, as shown i n Table 4.3, indicate that there was a s t a t i s t i c a l l y s i g n i f i c a n t difference at the .05 l e v e l between the pre and post van Hiele levels for students i n both groups. Therefore, the n u l l hypothesis, there was no s i g n i f i c a n t difference i n the rankings of the geometric thought levels between the pre and post test results within each - 63 - group, was r e j e c t e d . The g e o m e t r i c thought l e v e l s f o r b o t h groups d i d improve a f t e r a semester o f geometry. t o t h e d a t a from t h e van H i e l e p o s t t e s t . A median t e s t was a p p l i e d The r e s u l t o f t h i s t e s t was t h a t no s i g n i f i c a n t d i f f e r e n c e i n t h e r a n k i n g s between t h e computer group and t h e t r a d i t i o n a l group e x i s t e d a t t h e .05 l e v e l o f significance (p = .569). Table 4.3 Sign Test: P r e and P o s t van H i e l e Test Data Computer Traditional Group £r&u£ No change i n van H i e l e l e v e l 6 6 Decrease i n van H i e l e l e v e l 1 1 I n c r e a s e i n van H i e l e l e v e l 14. Total 21 a (p = .0010) Combined 12 2 22 36 15 (p = .0391) I n t h e C o g n i t i v e Development and Achievement i n Secondary S c h o o l Geometry (CDASSG) P r o j e c t i t was found t h a t a f t e r a y e a r ' s s t u d y o f geometry a p p r o x i m a t e l y , "a t h i r d go up one l e v e l ; a t h i r d e x h i b i t 'great growth', i n c r e a s i n g two o r more l e v e l s ; t h e f i n a l e x h i b i t s 'no growth', s t a y i n g t h e same o r d e c r e a s i n g third their level." ( U s i s k i n , 1982, p. 38) The researcher found a s i m i l a r p a t t e r n . A f t e r a semester o f geometry, i n t h e computer group o n e - t h i r d s t a y e d t h e same o r d e c r e a s e d a l e v e l , 43% i n c r e a s e d one l e v e l , and 24% i n c r e a s e d two levels. I n t h e t r a d i t i o n a l group 47% s t a y e d t h e same o r d e c r e a s e d a l e v e l , 20% i n c r e a s e d one l e v e l , and o n e ^ t h i r d i n c r e a s e d two o r more - 64 - levels. Hence, these findings are generally consistent with the CDASSG experience. Written Proofs The written proof test was analyzed using the independent t test. There was a s i g n i f i c a n t difference, at the .05 l e v e l , between the means of the two groups as reported i n Table 4.4. A second t-test was done involving only the questions which required writing p r o o f s — f o u r questions (#2, 4, 5, 6). Again, there was a s i g n i f i c a n t difference between the means of the two groups (Table 4.4). Therefore, the n u l l hypothesis, there was no s i g n i f i c a n t difference between the means of the written proof tests of students i n the computer group and those i n the t r a d i t i o n a l group, was rejected. Table 4.4 Means, Standard Deviations, and S t a t i s t i c a l Comparison of Groups: Proof Test Groups n Mean SD t -value p PART A*: Computer 22 Traditional PART B 11.00 5.46 16 6.00 4.97 22 7.73 4.04 2.89 .006 3.84 < .001 (Proof Writing): Computer Traditional 2.94 16 PART A: 6 questions (24 marks) PART B: 4 questions (16 marks) - 65 - 3.44 Out of a possible 24 marks on the s i x questions, the computer group's results ranged from 2 to 23 while the t r a d i t i o n a l group ranged from 1 to 20. On the f i l l - i n question (Appendix D, question #1) 43% of the computer group and 31% of the t r a d i t i o n a l group could i d e n t i f y the alternate i n t e r i o r angle. On the diagram drawing question (Appendix D, question #3) 57% of the computer group and 38% of the t r a d i t i o n a l group could draw the diagram but were unable to proceed any further. An item analysis f o r both groups on the proof writing test i s i n Appendix D. Half of the computer group and o n e - f i f t h of the t r a d i t i o n a l group were able to apply the converse of the Isosceles Triangle Theorem i n question #2. question #4. Two deductions were required i n Forty-one percent of the computer group and 12% of the t r a d i t i o n a l group achieved h a l f or more of the marks on t h i s question. segment. Question #5 required the addition of an a u x i l i a r y l i n e This question was answered correctly by 59% of the computer group and by 19% of the t r a d i t i o n a l group. required more than two deductions. correctly. The l a s t question No student completed t h i s proof However, 41% of the computer group and 25% of the t r a d i t i o n a l group got h a l f or more of the required answer. Considering the four questions that required proof writing c o l l e c t i v e l y , one student (5%) i n the computer group received no marks while s i x students (31%) i n the t r a d i t i o n a l group had the same result. - 66 - Attitudes The attitude test was analyzed using the independent t - t e s t . The s t a t i s t i c s , as reported i n Table 4.5, support the n u l l hypothesis at the .05 l e v e l of significance. Thus, the n u l l hypothesis of no s i g n i f i c a n t difference i n the means between the attitudes of the two groups was accepted. This result i s consistent whether the whole test i s used or i f the subsets (geometry, mathematics i n general, or the value of mathematics) are used. An item analysis f o r each group along with the test items are included i n Appendix E. Also i n Appendix E i s a summary comparing the mean score of each item between the computer group, the t r a d i t i o n a l group, and the t o t a l sample. Table 4.5 . Means, Standard Deviations, and S t a t i s t i c a l Comparison of Groups: Attitude Test Test A. Geometry: Computer Mean SD B. Mathematics: Mean SD C. Group Group (n=21) (n=17) 31.62 29.06 7.68 9.46 34.00 32.35 8.50 9.41 t-value p. 0.92 ,363 0.57 ,575 -.42 ,679 0.64 ,525 Value of Mathematics: Mean SD D. Traditional Total Test: Mean SD 32.67 4.27 33.24 4.06 98.29 94.65 17.34 17.42 - 67 - The s t u d e n t s were asked f o r t h e i r w r i t t e n comments about t h e course. No p r o m p t i n g was g i v e n . T h e i r spontaneous remarks a r e summarized i n T a b l e 4.6. Table 4.6 W r i t t e n Comments Computer group T r a d i t i o n a l group - Computer use 33% n/a - Constructions 29% 29% - Proofs 10% 18% - Project 10% 6% 5% 6% - Memorizing 5% 35% - Computer 5% n/a - Constructions 5% 0 - Proofs 24% 35% - P y t h a g o r e a n Theorem 19% 0 Likes: - Trigonometry Dislikes: Student s u g g e s t i o n s : - Computer group - " I t would be f a s t e r and e a s i e r t o j u s t t e l l us." - "More computer u s e . " - T r a d i t i o n a l group - " I hope t h a t computers a r e u s e d more and more i n t h e c l a s s r o o m . " - two s t u d e n t s made t h i s comment. The m a j o r i t y o f t h e w r i t t e n comments were c o n c e r n e d w i t h c l a s s r o o m management f u n c t i o n s and added n o t h i n g t o t h i s s t u d y . - 68 - Interview Data This section contains a condensed summary of the a c t i v i t i e s c a r r i e d out during the interview sessions with the students. Responses from the students are also included. Sessions 1-3 The f i r s t three sessions with students from the computer group were used to establish rapport and to set the stage for the sessions focussed on solving proofs. A variety of geometrical concepts, d e f i n i t i o n s , postulates, and theorems were reviewed along with class work and t e s t s . In reporting the findings of the student interviews the researcher used a code to i d e n t i f y each student. of four characters. The code consisted The f i r s t two were either Sc or St to denote the computer group or the t r a d i t i o n a l group respectively. The t h i r d character indicated gender (M or F ) . The l a s t character i s numerical to d i f f e r e n t i a t e among the students. The following are some unexpected student responses from the f i r s t three sessions. After three weeks i n the geometry class S c f l asked, "Does a t r i a n g l e have three sides?" This student was also unable to describe p a r a l l e l l i n e s and had d i f f i c u l t y with the concept of straight angle. In each of the three sessions the exercise i n Figure 6 was reviewed. She- approached the problem i n the same way each time—with complete naivety. She had no idea that a l i n e represents an angle of 180 degrees even though we had used the protractor to measure i t . also never occurred to her to use the protractor herself. - 69 - It Find x. Figure 6. Supplementary angles Scml had d i f f i c u l t y with any question involving equations. In Figure 7 he could i d e n t i f y that £1 = Ll but was unable to proceed to the next step. Given: a//b, Ll= 2x + 5 and Ll = 12x Find: Ll Figure 7. P a r a l l e l l i n e s with alternate i n t e r i o r angles. Scm2 l i k e d questions with numbers. Figure 8 he replied, When given the example i n "You can't do i t i f i t has l e t t e r s . " Figure 8. Relationship between the exterior angle and the remote i n t e r i o r angles. - 70 - Session 4 The fourth session was spent reviewing proof exercises from the classroom. The two g i r l s were unable to make any deductions. would write the "given" and come to a s t a n d s t i l l . They With a constant flow of directed questions from the researcher, they would eventually solve the proof. When looking back and reviewing the steps, S c f l would say, " I t a l l makes sense but I could never do that on my own." Perhaps she was operating i n the zone of proximal development, a theory proposed by Vygotsky. "the The zone of proximal development i s distance between the actual developmental l e v e l as determined by independent problem solving and the l e v e l of p o t e n t i a l development as determined through problem solving under adult guidance" (Vygotsky, 1978, p. 86) . Scml could make simple deductions. Scm2 was selected f o r the interviews because of h i s inconsistent van Hiele l e v e l . He was able to quickly analyze the problem and move forward to writing up the proof. He sometimes assumed a problem to be the same as a previous one such as i n Figure 9. / / J Given: i l = 12, 13 = LA Prove: AB = AD Figure 9. Proving two segments congruent. - 71 - He immediately ABAC ^ AC A ABC AB = wrote down the "given" followed by: LDAC because each t r i a n g l e has 180° AC because of the reflexive property = A ADC because of ASA *= AD because corresponding parts of congruent triangles are congruent. When he was asked to check the "given" he quickly r e a l i z e d h i s mistake and rewrote the proof. The researcher f e l t that t h i s student had a van Hiele l e v e l much closer to 3 than to 0 and having solved the quandary ceased interviewing him. Another male (Scm3, van Hiele l e v e l 1) who was having d i f f i c u l t y with proofs i n class was selected from the computer group to replace Scm2. Session 5 The four students from the computer group and two additional students from the t r a d i t i o n a l group were each given the same two proofs to complete during t h i s interview session. The f i r s t proof i s i n Figure 10 followed by a summary of the students' responses. The second exercise i s shown i n Figure 11 and a s i m i l a r summary of the students' r e p l i e s follows. C Given: CX J _ AB, AC ~ Prove: AACX = ABCX Figure 10. Proof #1 - 72 - BC A l l s i x students immediately wrote the "given" without reading the whole question. what was given. Three students each marked t h e i r diagrams with Two students were able to make a correct deduction from the fact that the segments were perpendicular but the other four (Scfl, Scml, Stf3 and Stm4) had d i f f i c u l t y with t h i s concept. None of the s i x students were able to make a deduction from the fact that two sides of the t r i a n g l e were congruent. They also had d i f f i c u l t y stating why the triangles were congruent. When given pairs of premarked figures such as students were able to give the correct reasons f o r these triangles being congruent. However, i n a proof writing s i t u a t i o n they were unable to relate the written work to t h e i r diagram and then draw a conclusion. Given: 4 ADP with AB = Prove: A APB ~ ADPC DC, PB = PC Figure 11. Proof #2 In proof #2, one student (Scml) correctly used the Isosceles Triangle Theorem to obtain L3 = Z.4. The other students made no connection with Proof #1 which contained similar information. c o r r e c t l y i d e n t i f i e d the supplementary quickly completed the proof. Scm3 angle relationships and Here again, the majority of these students required many probing questions i n order f o r them to make deductions from what the question had given and to achieve what was to be proved. - 73 - Session fi The same s i x students were each given two additional proofs to complete during t h i s session. Proof #4 appears i n Figure 13. Proof #3 appears i n Figure 12 and A summary of the students' responses follows each figure. L p T P Given: TP J_ 71, LP 'Bisects FI = LIBL Prove: Z.FPL F i g u r e 12. Proof #3 On proof #3 Scml successfully completed t h i s proof with no assistance. One Three students said that LF student s a i d that /FLP ~ Ll because LP b i s e c t s FI. = ZlLP because LP b i s e c t s FI. students a l l defined bisect as meaning "to cut i n h a l f . " was adamant about LF being congruent to LL u n t i l she, These four One student without suggestion from the interviewer, turned the page around. Stm4, despite having s u f f i c i e n t information to prove the t r i a n g l e s congruent, kept coming back to LF and LI. He was determined that they should be equal. S c f l asked i f there was a short way to write "bisects" (like a symbol f o r perpendicular). She was unable to write any steps a f t e r the "given" without being s p e c i f i c a l l y directed. - 74 - 5 L I Given Prove LA bisects SI and SA = UAL A SAL = IA F i g u r e 13. Proof #4 On proof #4 a l l s i x students assumed iALS IkLI because LA bisects SI. Scml reread the "given" and erased h i s markings on the diagram, then proceeded to correctly complete the proof. Once the questions, "What does bisect mean?" and "What i s being bisected?" were asked, the five students then corrected t h e i r work. Three students used SAS (Side-angle-side) to prove the triangles congruent and the other h a l f used SSS (Side-side-side). In summary, the students a l l understood that proving the triangles congruent was a c r i t i c a l step i n proofs #3 and #4 the l a s t statement. diagrams. next step. p r i o r to The students had to be encouraged to mark t h e i r Once they did t h i s , they found i t easier to determine the There were no further interviews with the two students from the t r a d i t i o n a l group. Sessions 7 & 8 The four students i n the computer group were interviewed on two more occasions. class. Their responses generally mirrored t h e i r progress i n In the interview situation they experienced the same kind of d i f f i c u l t y as they d i d i n class. As an example of t h i s , the following section contains a report of an interview with each of the - 75 - students that focussed on the concept of overlapping triangles as i l l u s t r a t e d i n Figure 14. \ Y T Given: Prove: ^RTP IPSO RT ZXPS, PT ZTPO XP ~ SP, Figure 14. Overlapping Triangle Proof The students methodically recorded what was given then to mark the diagram. £PSO, Z.TPO, attempted Scm3 was confused between the angles Z.XPS, and ZSPO. Once he had the angles sorted out, he was able to quickly v i s u a l i z e the triangles being congruent by ASA (Angle-side-angle). They a l l had d i f f i c u l t y matching the appropriate vertices of the triangle for the congruence statement. None of them attempted to redraw the diagram. The interviews provided the researcher with an opportunity to c o l l e c t data regarding student approaches to proof writing and student misconceptions about geometric concepts. Additional Data In order to investigate the effects of the Supposer programs on proof writing, the students' geometric thought levels were analyzed. Table 4.7 indicates the percent of a l l the geometry 10 students with van Hiele l e v e l s at the beginning of the course and again at the end. This data was obtained from the pre and post van Hiele geometry - 76 - tests. The results are similar to the findings of U s i s k i n s 1982 study. He found that at the beginning of the geometry course "over 1 h a l f of students c l a s s i f i a b l e into a van Hiele l e v e l are at l e v e l s 0 or 1." (p. 81) He found that at the end of h i s study when the geometry course was completed, more students were at l e v e l 3 than at any other l e v e l . A c h i square test of association showed that the two variables (the van Hiele l e v e l s and the tests) were related at the .05 l e v e l . Table 4.7 Students' van Hiele Levels Percentage of students Level 0 Pretest (n = 36) 6% . Posttest (n = 36) 2 3% . . 1 19 17% . . 6 1 53% . . 2 22% . 8 2.8%. . . 10 3 14% . 5 39% . . 14 4 6% . 2 14% . . 5 The pre and post van Hiele levels f o r each of the students interviewed are shown i n Table 4.8. The results from the proof writing test (possible t o t a l = 16) are also shown i n the table. - 77 - Table 4.8 Interviewees 1 Pre and Post van Hiele Levels and t h e i r Proof Test Scores van Hiele Levels Proof Test Scores Student Pre Post Scfl 0 1 0 Scf2 1 2 11 Scml 1 2 14 Scm2 3 3 11 Scm3 1 2 10 Stf3 0 2 1 Stm4 1 2 5 Data Summary In t h i s chapter the results from the f i v e tests given to the students have been presented. Appropriate s t a t i s t i c a l tests were used to analyze the data gathered with each of these instruments. The findings indicate rejection of the f i r s t n u l l hypothesis (There was no s i g n i f i c a n t difference between the means of the written proof tests of students i n the computer group and those i n the t r a d i t i o n a l group.) and rejection of the second n u l l hypothesis (There was no s i g n i f i c a n t difference i n the rankings of the geometric thought levels between the pre and post van Hiele test r e s u l t s . ) . findings indicate The support for the t h i r d n u l l hypothesis (There was no s i g n i f i c a n t difference i n the means between the attitudes of students i n the computer group and those i n the t r a d i t i o n a l group,). indicates that teaching^geometry, The data with or without computer programs, does improve students' van Hiele l e v e l s . - 78 - CHAPTER 5 SUMMARY AND DISCUSSION The purpose of t h i s study was to investigate how the computer program, Geometric Supposer, would affect a grade 10 geometry class's a b i l i t y to write proofs. As part of t h i s investigation, tests were administered to determine the geometric thought l e v e l s , the geometric knowledge, and the attitudes of the students. Data was c o l l e c t e d from two groups of students—those using microcomputers and those learning geometry the t r a d i t i o n a l way. Summary of the Problem, Methodology, and Results Many high school students who take the grade 10 geometry course experience d i f f i c u l t y with the section on writing geometric proofs. This study was an attempt to investigate the effectiveness of a computer program, the Geometric Supposer, i n increasing the performance l e v e l of students i n writing proofs. The subjects i n t h i s study were a l l the students enrolled i n the grade 10 geometry course i n one p a r t i c u l a r high school. One class of these students used the computer program and the other class d i d not. The van Hiele test, which measures geometric thought levels, was administered at the beginning and the end of the geometry course. An introductory geometry test was also administered at the beginning of the course to measure the geometric knowledge of the students p r i o r to the study. At the end of the course the students wrote a proof test as well as an attitude t e s t . - 79 - A series of interviews were c a r r i e d out with five students from the computer group and two students from the noncomputer group. This was done i n order to gain some insight into the methods the students were actually using to write geometric proofs and to i d e n t i f y changes in their approaches. The van Hiele test was developed and tested i n Chicago by the Cognitive Development and Achievement i n Secondary School Geometry (CDASSG) Project. The Hoyt measure of i n t e r n a l consistency was used as an estimate of r e l i a b i l i t y for the other tests, which were developed by the researcher. The results showed that there was no s t a t i s t i c a l l y significant mean difference i n the geometric knowledge of the two groups (computer and t r a d i t i o n a l ) at the beginning of the study. As was shown i n Table 4.7 the van Hiele levels of the two groups both improved a f t e r one semester of geometry. There was no s i g n i f i c a n t mean difference between the attitudes of the two groups towards geometry at the end of the course. However, the computer group scored s i g n i f i c a n t l y higher than the t r a d i t i o n a l group on the f i n a l proof t e s t . Interpretation of the Findings The introductory geometry test The findings indicate that there was no s i g n i f i c a n t difference i n the geometric knowledge of the two groups at the beginning of the study. The students i n t h i s study had received some geometry i n s t r u c t i o n i n grade 8 but no geometry content i n grade 9. - 80 - The study covers the geometry they took i n the second semester of t h e i r grade 10 year. The introductory geometry test was given at the beginning of t h i s semester to assess the l e v e l of geometry knowledge of the students entering the grade 10 course. The averages on t h i s test were 62% f o r the computer group and 55% for the t r a d i t i o n a l group (Table 4.1). Further, the results of t h i s test are comparable to the results of the entering geometry test given i n the CDASSG Project. The mean percentage correct i n t h e i r study was 54% (Usiskin, 1982, p. 68) . The students i n both groups performed poorly on the questions involving the i d e n t i f i c a t i o n of obtuse angles (23% correct) and the c a l c u l a t i o n of the area of an obtuse triangle (25% c o r r e c t ) . However, 85% of the students could calculate the area of a rectangle, 83% could define an equilateral triangle, 93% could i d e n t i f y a r e f l e c t i o n point, and 90% could f i n d the volume of a rectangular s o l i d (Appendix B). The van Hiele geometry pretest The van Hiele geometry test was given to assess the geometric thought l evels of the students entering the grade 10 geometry course. The results of t h i s test not only supported the hypothesis that there was no s t a t i s t i c a l l y s i g n i f i c a n t difference between the rankings of the geometric thought levels of the students i n the two groups (Table 4.2), but also shed some l i g h t on the o v e r a l l poor performance of both groups on the introductory geometry t e s t . The results from t h i s test indicate that 59% of the students entering the geometry course were at l e v e l 0 or 1 (Table 4.7) . of the CDASSG Project. This data i s consistent with data Using the same c r i t e r i o n , t h e i r results indicated 54% (p. 100) of the students entering grade 10 geometry - 81 - were at van Hiele levels 0 or 1. As indicated e a r l i e r , various researchers (Battista & Clements, 1988; Senk, 1983; Usiskin, 1982; Wirszup, Craine, 1985; Scally, 1987; 1976) have a l l discussed the need for students to be at l e v e l 3 i n order to cope successfully with the abstract concepts of proof writing. In other words, over h a l f the students entering the grade 10 geometry course had not achieved a s u f f i c i e n t geometric thought to deal with the section on writing proofs. level More simply, they were not ready. The van Hiele geometry posttest At the end of the semester the students again wrote the van Hiele geometry t e s t . The results of t h i s test indicated that there was a s i g n i f i c a n t difference i n scores between the beginning of the semester and the end of the semester (Table 4.3 and Table 4.7) and thus the n u l l hypothesis, there was no s i g n i f i c a n t difference i n the rankings of the geometric thought levels between the pre and post test results within each group, was rejected. One would hope that students would perform better on a test of geometric thought a f t e r a semester of geometry. levels Nevertheless, one t h i r d of the students exhibited no change i n t h e i r geometric thought l e v e l . i s also consistent with the CDASSG Project r e s u l t s . This Even a f t e r a semester of geometry 20% of the students i n t h i s study were s t i l l at a van Hiele l e v e l of 0 or 1 (Table 4.7). Were the geometric fundamentals of these students so weak that there was l i t t l e to b u i l d on? Was the content of the geometry 10 course inappropriate for these students and thus were they denied the opportunity to improve t h e i r thought levels? - 82 - Were the teaching methods inappropriate f o r these students? Probably a l l three factors contributed to the students' lack of growth during t h i s semester. Perhaps, more i n d i v i d u a l i z e d instruction needs to be incorporated into the classroom directed at students with van Hiele l e v e l s of 0 or 1. More attention also must be directed at geometry i n s t r u c t i o n i n the elementary and middle year's curriculum. The proof test The n u l l hypothesis, there was no s i g n i f i c a n t difference between the means of the written proof tests of students i n the computer group and those i n the t r a d i t i o n a l group, was rejected. The computer group performed s i g n i f i c a n t l y better than the t r a d i t i o n a l group. The proof test, written at the end of the semester, o v e r a l l averages of 46% — T a b l e 4.4. (computer group) and 25% resulted i n ( t r a d i t i o n a l group) However, i f the analysis i s l i m i t e d to only those questions on the test involving proof writing, the spread between the groups increases. The computer group had an average of 48% while the t r a d i t i o n a l group had an average of 18%. The use of the Geometric Supposer computer programs appears to have contributed to the students' consolidation and understanding of geometric concepts. Using these programs throughout the semester seemed to make i t easier f o r students to make deductions. They had become accustomed to looking for relationships, t e s t i n g t h e i r ideas, and making conjectures. software. The students looked forward to using the On the whole, the students were able to write simple proofs more e a s i l y than previous classes the researcher has taught. However, when the proofs became more complex the students experienced the kind of d i f f i c u l t y t y p i c a l of grade 10 classes. - 83 - Usiskin (1982) s t a t e d t h a t "a s t u d e n t who e n t e r s geometry a t van H i e l e l e v e l s 0 o r 1 has an almost even chance o f f a i l u r e a t p r o o f " (p. 5 7 ) . S i x t y - f i v e p e r c e n t o f t h e s t u d e n t s i n t h i s s t u d y who u n s u c c e s s f u l ( o b t a i n e d below 50%) were i n t h e p r o o f t e s t had b e g i n n i n g van H i e l e l e v e l s o f 0 o r 1 w h i l e 40% o f t h e s u c c e s s f u l ( o b t a i n e d 50% o r more) s t u d e n t s had l e v e l s o f 0 o r 1. I n t h e CDASSG P r o j e c t 71% o f s t u d e n t s u n s u c c e s s f u l a t p r o o f s had b e g i n n i n g van H i e l e l e v e l s o f 0 o r 1 as compared t o 37% who were s u c c e s s f u l . "Thus s t u d e n t s u n s u c c e s s f u l a t p r o o f a r e about t w i c e as l i k e l y as t h e more s u c c e s s f u l o t h e r s t o have t h e s e low van H i e l e l e v e l s . " (p. 61) The above r e s u l t s were i n t h e same d i r e c t i o n as t h e r e s u l t s from t h e CDASSG P r o j e c t . However, a c h i square g o o d n e s s - o f - f i t t e s t l e v e l ) i n d i c a t e d t h a t t h e r e s e a r c h e r ' s r e s u l t s were not significant. T h i s may (.05 statistically have o c c u r r e d because o f t h e r e l a t i v e l y s m a l l sample s i z e . As l o n g as more t h a n 50% o f t h e s t u d e n t s e n t e r i n g t h e grade 10 geometry c o u r s e have van H i e l e l e v e l s o f 0 o r 1, e i t h e r t h e p r o o f s e c t i o n o f t h e course s h o u l d be removed o r more emphasis s h o u l d be p l a c e d on t h e t y p e o f i n s t r u c t i o n employed i n t h e c o u r s e . The attitude test The r e s u l t s were o b t a i n e d from an a t t i t u d e t e s t g i v e n t o b o t h groups a t t h e end o f t h e semester. O v e r a l l b o t h groups had positive a t t i t u d e s towards t h e s t u d y o f geometry i n p a r t i c u l a r , t h e s t u d y o f mathematics i n g e n e r a l , and t h e v a l u e o f mathematics as a whole. D e s p i t e t h e r e l a t i v e ease w i t h which t h e computer group approached p r o o f s , 24% (Table 4.6) s p e c i f i c a l l y mentioned a d i s l i k e f o r p r o o f w r i t i n g as compared t o 35% i n t h e t r a d i t i o n a l group. - 84 - Proof writing may have had such a negative reputation that the p o s i t i v e experience with the computer software was i n s u f f i c i e n t to overcome t h i s negative The valence. interviews During the interviews the researcher discovered why many students were experiencing d i f f i c u l t y i n geometry. basics. The fact that a student can get a l l the way to grade 10 without understanding occur They lacked the (Scfl). what a triangle i s seems i n c r e d i b l e but does Without the constant one-on-one s i t u a t i o n of the interview, the gap i n t h i s student's knowledge may not have been discovered. It was also s u r p r i s i n g to discover the general confusion that existed regarding a segment b i s e c t i n g another segment i n a t r i a n g l e . Despite the fact that "bisect" was used frequently i n class without any apparent d i f f i c u l t y , i t was only during the interviews that the researcher r e a l i z e d these students had f a i l e d to understand the generalized concept of "bisect." The researcher benefited from the contact during the interviews and f e l t that the students d i d likewise, especially Scml, Scm3, and Scf2. Scml approached proofs i n a methodical f a s h i o n — c a r e f u l l y reading, formulating a plan, and carrying out the plan. he enjoyed solving the proofs. to step, reason to reason. He said that At f i r s t , Scm3 would " f l i t " from step Once he was focussed, he could analyze the s i t u a t i o n and foresee what had to be done. It was sheer hard work for Scf2 as she t r i e d to f i n d a rule for every situation, rather than analyzing and dealing with what was given i n the problem. - 85 - These three students a l l achieved well on the proof test (Table 4.8). The researcher recommends one-on-one contact with a l l geometry students who have van Hiele levels 0 or 1. This one-on-one contact should begin as near to the beginning of the course as possible. Obvious weaknesses could be worked on so that the student i s better prepared to cope with geometry concepts as they are presented i n the general classroom. This type of remedial help could also be provided through appropriately designed Computer Assisted Instruction. Limitations of the Study The main limitations of the study were the length of the treatment period, the different instructors, and the effects of the interviews. Limited time for exploration The treatment period of t h i s study was one semester. Hence, the opportunity for students to assimilate computer experiences t h e i r repertoire of problem solving s k i l l s was l i m i t e d . into Also, was a prescribed curriculum to be covered within t h i s time. there This l i m i t e d the amount of time for computer exploration. Different instructors The computer group and the t r a d i t i o n a l group were taught by two different teachers. However, both teachers were q u a l i f i e d mathematics teachers and both had taught the grade 10 geometry course previously. The teachers communicated frequently as to the content being covered and various approaches used. They undertook to s i m i l a r standards for class performance and homework. set The fact that there was no significant difference between the attitudes of the two - 86 - groups towards geometry suggests that the effects of d i f f e r e n t instructors may have been minimal. E f f e c t s of the interviews The interview sessions tended to be t u t o r i a l — c o r r e c t i n g misconceptions and reviewing concepts which had not been understood in class. The same proofs were given to a l l the students and the same format of questioning was followed. Through t h i s interview experience, some students (Scf2, Scml, Scm2, Scm3) could have gained additional geometrical knowledge and, thus obtained higher scores on the proof test than i f they had not been interviewed. These students may also have been influenced by the Hawthorne e f f e c t . In retrospect, the same number of students,, with s i m i l a r van Hiele l e v e l s , from the t r a d i t i o n a l group should have been interviewed for the same length of time as from the computer group. Suggestions for Further Research Given that t h i s study indicates the Geometric Supposer software has value i n the geometry class, further research studying i t s effect over a longer period of time appears warranted. Another possible research suggestion would be the use of other computer programs (i.e., LOGO, LOGOWRITER) i n the grade 10 geometry course and t h e i r effect on proof writing. If proof writing i s to remain i n the grade 10 geometry course then research should be undertaken with respect to methods of incorporating geometric content into the elementary and middle grade levels. Research could center on the development and/or use of manipulatives and computer software at those l e v e l s . - 87 - The current practice i n the province i n which t h i s study took place i s to have proof writing i n the grade 10 geometry course. An alternative would be to include i t i n the grade 11 geometry course. Hence, more time could be spent on informal geometry at the grade 10 level. Research i n t h i s area could be valuable for designing future curriculum. Designing appropriate experiences to help students achieve at least a van Hiele l e v e l 2 p r i o r to undertaking the writing of proofs could also prove f r u i t f u l . Implications Process rather than product One of the current trends i n education i s to give students opportunities to be a c t i v e l y involved i n knowledge construction. Geometric Supposer programs provide such an opportunity. This software i s designed to promote experimentation—the process. no product requirements b u i l t i n . The It has The usefulness and power of the program l i e s i n the context of the task or problem given to the student. The teacher i s the key factor i n d i r e c t i n g t h i s process. defines the student-software interaction. S/he The role of the teacher s h i f t s from the t r a d i t i o n a l one of being the sole source of knowledge to one of supporting and integrating student inquiry. The teacher needs to teach and model such s k i l l s as c o l l e c t i n g data, analyzing, making conjectures, testing, and generalizing. With t h i s software, lesson planning w i l l consist mainly of defining tasks and developing objectives without giving away the outcomes. - 88 - The teacher must encourage students to take d i v e r s i f i e d approaches to solving problems. in pairs. Students should work on the computer Class discussions should follow hands-on sessions. More time involved The use of the Geometric Supposer programs involves a greater investment of time than noncomputer i n s t r u c t i o n . The teacher must be f l e x i b l e i n order to respond to unexpected discoveries by students i n the c l a s s . Similarly, assessment strategies and techniques w i l l need to be adaptable and f l e x i b l e . Hardware accessibility Access to the hardware i s the main factor. Having the computers right i n the room i s the most appropriate arrangement as i t provides opportunities f o r spontaneous investigation. Ideally, students should have access to the program during free periods or a f t e r school. Geometry p r i o r to proof writing The o v e r a l l averages of the two groups (computer and t r a d i t i o n a l ) on the introductory geometry test were r e l a t i v e l y low. This result can be a t t r i b u t e d i n part to the one year gap i n geometric instruction. In order for students to be ready f o r proof writing, they should be exposed to geometry i n the previous grade. Hopefully, studying geometry i n the previous grade would help students to gain at least a van Hiele l e v e l 2 before attempting to write proofs. - 89 - Conclusions The van Hiele geometry test i s a useful a i d for grade 10 geometry teachers to better i d e n t i f y and appreciate the thought levels of t h e i r students. geometric It allows teachers to plan, prepare, and have r e a l i s t i c expectations when teaching the process of writing proofs. The Geometric Supposer computer programs have p o t e n t i a l as an i n s t r u c t i o n a l a i d i n the geometry classroom. require preparation by and guidance They do, however, from the teacher. This study has shown that using the Geometric Supposer software can a s s i s t students i n being able to better write geometric proofs at the grade 10 l e v e l . - 90 - REFERENCES Aieta, J . F. (1985, September). Microworlds: Options for learning and teaching geometry. Mathematics Teacher. IS(6), 473-489. Aiken, L . R . , J r . (1963, May-June). Personality correlates of attitude toward mathematics. The Journal of Educational Research/ 5 £ ( 9 ) - , 476-480. Aiken, L . R . , J r . mathematics. 5(3), 67-71. (1974, March). Two scales of attitude toward Journal for Research in Mathematics Education. Allendoerfer, C. B. (1969, March). The dilemma i n geometry. Mathematics Teacher. £ 2 ( 3 ) , 165-169. The B a t t i s t a . M. T. & Clements, D. A. (1988, March). Using Logo pseudoprimitives for geometric investigations. Mathematics leachar, s i ( 3 ) , 166-174. B i r t , J . 0. & Koss, R. (1986). Geometry and the computer. Geometry Sampler. The Woodrow Wilson National Fellowship Foundation, 45-86. Brady, H. (Ed.). (1987, February). The 1987 classroom computer learning software awards. Classroom Computer Learning. 1(5), 19-24. Brown, R. G. (1982, September). Making geometry a personal and inventive experience. Mathematics Teacher. 15.(6), 442-446. Brumfiel, C. (1973). Conventional approaches using synthetic Euclidean geometry. Geometry i n the Mathematics Curriculum. 36th Yearbook of the National Council of Teachers of Mathematics. Reston, VA: The Council, 95-115. Burger, W. F . & Shaughnessy, J . M. (1986, January). Characterizing the van Hiele levels of development i n geometry. Journal for Research in Mathematics Education. 11(1), 31-48. - 91 - Carpenter, T. P., Lindquist, M. M., Matthews, W., & S i l v e r , E. A. (1983, December). Results of the t h i r d NAEP mathematics assessment: Secondary school. Mathematics Teacher 26.(9), 652-659. r Craine, T. (1985). Integrating geometry into the secondary mathematics curriculum. The Secondary School Mathematics Curriculum, Yearbook of the National Council of Teachers of Mathematics, Reston, VA: The Council, 119-133. Crowley, M. L. (1987). The van Hiele model of the development of geometric thought. Learning and Teaching Geometry, K-12, Yearbook of the National Council of Teachers of Mathematics, Reston, VA: The Council, 1-16. Denis, L. (1987). Relationships between stages of cognitive development and van Hiele levels of geometric thought among Puerto Rican adolescents. (Doctoral dissertation, Fordham University, 1987). Dissertation Abstracts International. 4&, 859A. Dreyfus, T. & taught. National Council, Hadas, N. (1987). Euclid may stay - and even be Learning and Teaching Geometry. K-12. Yearbook of the Council of Teachers of Mathematics. Reston, VA: The 47-58. Driscoll, M. (1982). Research Within Reach: Secondary School Mathematics. Reston, VA: The National Council of Teachers of Mathematics. Eccles, F. M. (1972, February). Transformations in high school geometry. Mathematics Teacher, £5.(2), 103, 165-169. Farrell, M. A. (1987) . Geometry for secondary school teachers. Learning and Teaching Geometry, K-12, Yearbook of the National Council of Teachers of Mathematics, Reston, VA: The Council, 236-250. Fehr, H. F. (1972, February). The present year-long course in Euclidean geometry must go. Mathematics Teacher, £5.(2), 102, 151-154. Fey, J . T. (Ed.). (1984). Computing and Mathematics. National Council of Teachers of Mathematics. - 92 - Reston, VA: Fleenor, C. R., Eicholz, R. E., O'Daffer, P. (1974). School Mathematics 2. Toronto, ON: Addison-Wesley (Canada) Limited. Fuys, D. (1985, A u g u s t ) . Van H i e l e l e v e l s o f t h i n k i n g i n geometry. E d u c a t i o n and Urban S o c i e t y , 11(4), 447-462. Fuys, D., Geddes, D., Tischler, R. (1988). The van Hiele Model of Thinking i n Geometry Among Adolescents. Reston, VA: The National Council of Teachers of Mathematics. Gearhart, G. (1975, October). What do mathematics teachers think about the high school geometry controversy? Mathematics Teacher, Grunbaum, B. Two £fi(6), 486-493. (1981, September). Shouldn't we teach geometry? The Year C o l l e g e Mathematics J o u r n a l , 12(4), 232-238. (1981, January). Geometry i s more Mathematics Teacher/ l i d ) , n-18. Hoffer, A. than proof. H o f f e r , A. (1983). Van H i e l e - b a s e d r e s e a r c h . A c q u i s i t i o n o f Mathematics Concepts and P r o c e s s e s . New York, NY: Academic P r e s s , Inc., 205-227. Howson, G. & Wilson, B. (1986). School Mathematics i n t h e 1990s. Press Syndicate of the University of Cambridge. Jurgensen, R. C , Brown, R. G., Jurgensen, J. W. Boston, MA: Houghton M i f f l i n Company. (1985). Geometry. K e l l y , B., A l e x a n d e r , B., A t k i n s o n , P. (1987). Mathematics 10. Don M i l l s , ON: Addison-Wesley P u b l i s h e r s L t d . L e s t e r , F. K. (1975, J a n u a r y ) . Developmental a s p e c t s o f c h i l d r e n ' s a b i l i t y t o u n d e r s t a n d mathematical p r o o f . J o u r n a l f o r Research i n Mathematics E d u c a t i o n . £(1), 14-25. L o v e l l , K. (1971). The development of the concept of mathematical proof i n abler p u p i l s . Piagetian CognitiveDevelopment Research and Mathematical Education. Proceedings of a conference conducted at Columbia University, 66-80. - 93 - Mathis, J . (1986, June). The Geometric Supposers: preSupposer, t r i a n g l e s , q u a d r i l a t e r a l s . The Computing Teacher, 13(9), 43-45. Mayberry, J . (1981). An investigation of the van Hiele l e v e l s of geometric thought in undergraduate preservice teachers. Doctor of Education t h e s i s , Athens: University of Georgia. Mayberry, J . (1983, January). The van Hiele l e v e l s of geometric thought i n undergraduate preservice teachers. Journal for Reasearch in Mathematics Education. 14.(1), 58-69. Morris, R. (Ed.). (1986). Studies in Mathematics Education: Teaching of Geometry, 5.. United Nations, Educational, S c i e n t i f i c and C u l t u r a l Organization, P a r i s . Niven, I. (1987). Can geometry survive i n the secondary curriculum? Learning and Teaching Geometry K-12, Yearbook of the National Council of Teachers of Mathematics. Reston, VA: The Council, 37-46. r O l i v e , J . & Lankenau, C. A. (1986). Teaching and understanding geometric relationships through Logo. Proceedings of the Second International Conference for Logo and Mathematics Education. • London: University of London I n s t i t u t e of Education, 78-85. Prevost, F . J . (1985, September). Geometry i n the junior high school. Mathematics Teacher. 18.(6), 411-416. S c a l l y , S. P. (1987, J u l y ) . The effects of learning Logo on ninth grade students' understanding of geometric r e l a t i o n s . Psychology of Mathematics Education,. Proceedings of the eleventh International Conference, Montreal, 2, 46-52. Schwartz, J . L . , Yerushalmy, M . , Gordon, M. (1985). The Geometric Supposer: Triangles. P l e a s a n t v i l l e , NY: Sunburst Communications, Inc. Schwartz, J . L . , Yerushalmy, M . , Gordon, M. (1985). The Geometric Supposer: Quadrilaterals. P l e a s a n t v i l l e , NY: Sunburst Communications, Inc. - 94 - Senk, S. L. (1983). Proof-writing achievement and van Hiele l e v e l s among secondary school geometry students. (Doctoral d i s s e r t a t i o n , The University of Chicago, 1983) . D i s s e r t a t i o n Abstracts International, 44, 417A. Senk, S. L. (1985, September). How well do students write geometry proofs? Mathematics Teacher, 2£(6), 448-456. Shaughnessy, J . M. & Burger, W. F. (1985, September). Spadework p r i o r to deduction i n geometry. Mathematics Teacher. 1 & ( 6 ) , 419-427. Smith, E. P. & Henderson, K. B. (1959). Proof. The Growth o f M a t h e m a t i c a l Tdeas Grades K-12. 24th Yearbook of the National Council of Teachers of Mathematics. Washington, D. C : The Council, 111-181. Suydam, M. N. (1985, September). The shape of i n s t r u c t i o n i n geometry: Some highlights from research. Mathematics Teacher, 13.(6), 481-486. U s i s k i n , Z. (1980, September). What s h o u l d n o t be i n t h e a l g e b r a and geometry c u r r i c u l a o f average c o l l e g e - b o u n d s t u d e n t s ? Mathematics Teacher. 1 2 ( 6 ) , 413-424. U s i s k i n , Z. (1982). S c h o o l Geometry. Van H i e l e L e v e l s and Achievement Chicago: U n i v e r s i t y o f Chicago. i n Secondary U s i s k i n , Z. (1987). R e s o l v i n g t h e c o n t i n u i n g dilemmas i n s c h o o l geometry. L e a r n i n g and T e a c h i n g Geometry, K-12. Yearbook o f t h e N a t i o n a l C o u n c i l o f Teachers o f M a t h e m a t i c s . R e s t o n , VA: The C o u n c i l , 17-31. (1959). Development and learning process; A study of some aspects of Fiaget's psychology in relation with the van Hiele, P. M. d i d a c t i c s of mathematics. Utrecht: The University of Utrecht. van H i e l e , P. M. (1984). A c h i l d ' s thought and geometry. English T r a n s l a t i o n o f S e l e c t e d W r i t i n g s o f D i n a van H i e l e - G e l d o f and P i e r r e M. v a n H i e l e . E d u c a t i o n a l Resources I n f o r m a t i o n C e n t e r (ERIC, number ED 287 697), 247-256. - 95 - van Hiele-Geldof, D. (1984). The didactics of geometry i n the lowest class of secondary school. English Translation of Selected Writings of Dina van Hiele-Geldof and Pierre M. van Hiele. Educational Resources Information Center (ERIC, number ED 287 697), 1-246. Vygotsky, L. S. (1978). University Press. Mind i n Society. Cambridge, MA: Harvard Wirszup, I. (1976, August). Breakthroughs i n the psychology of learning and teaching geometry. Space & Geometry, Papers from a Research Workshop, ERIC Center f o r Science, Mathematics, and Environmental Education, 75-97. Yerushalmy, M. & Houde, R. (1986, September). The Geometric Supposer: Promoting thinking and learning. Mathematics Teacher. 22(6), 418-422. Yerushalmy, M., Chazan, D., Gordon, M. (1987, January). G_uid£d, Inquiry and Technology: A Year Long Study of Children and Teachers Using the Geometric Supposer. Technical Report f o r the Educational Technolgy Center, Newton, MA. Yerushalmy, M. & Chazan, D. (1987, J u l y ) . E f f e c t i v e problem posing i n an inquiry environment: A case study using the Geometric Supposer. Psychology of Mathematics Education. Proceedings of the eleventh International Conference, Montreal, 2, 53-59. - 96 - Appendix A PERMISSION LETTER SENT TO PARENTS/GUARDIANS - 97 - February 2, 1988 Dear Parent/Guardian of Geo T r i g 10 Students: Your son or daughter i s enrolled i n one of our Geo T r i g 10 classes. These two classes are p a r t i c i p a t i n g i n a study which I w i l l be doing under the supervision of the University of B r i t i s h Columbia. The purpose of the study i s to determine i f computers can be used to improve the way students learn to write proofs. Over the years I have found that most students have d i f f i c u l t y with writing geometric proofs. One class, which I w i l l be teaching, w i l l use computers and computer software as part of t h e i r course. The other class w i l l use the standard method of learning geometry. I w i l l need to test both classes at the beginning of the semester and again a f t e r the proof writing section. approximately 45 minutes to complete. Each test w i l l require The s p e c i f i c tests to be administered are: • - a geometry test based on the work covered i n grade 8. - a test to determine the geometric thought l e v e l s of the students. This test w i l l be administered twice - beginning of semester and a f t e r proof writing. - a f i n a l proof t e s t . - an attitude t e s t . In addition, I want to carry out regular interviews with four students which I w i l l select from my class. These interviews w i l l be done to f i n d out how the students l i n k the computer work to proof writing. The eight interviews w i l l be one-half hour each, one every two weeks. A l l information c o l l e c t e d i n t h i s project i s for research purposes only. To assure c o n f i d e n t i a l i t y , no family names w i l l be used i n any - 98 - report or release of the information. No personal, family or other sensitive information i s being sought. The parent or student may withdraw from t h i s project at any time by a statement o r a l l y or i n writing. Refusal to cooperate w i l l have no consequences for the student. I w i l l appreciate very much the cooperation of the parents and students i n t h i s project. I w i l l be happy to answer any questions you have regarding the project. office. I can be contacted through the school Please return the form at the bottom. Thank you. Jo Worster I have read the above description of the research project e n t i t l e d AN INVESTIGATION TO DETERMINE THE EFFECTS OF THE GEOMETRIC SUPPOSER SOFTWARE ON GEOMETRIC PROOF WRITING AT THE GRADE 10 LEVEL to be c a r r i e d out by Mrs. [ ] Worster. I consent [ ] I do not consent to my c h i l d writing the written tests of the project. [ ] I consent [ ] I do not consent to my c h i l d being involved i n the individual interviews to be conducted by Mrs. Signature Worster. (Parent/Guardian) Student's name Appendix B INTRODUCTORY GEOMETRY TEST AND ITEM ANALYSIS - 100 - INTRODUCTORY GEO TRIG 10 TEST Directions Do not open t h i s test u n t i l your are t o l d to do so. Please write your name on the l i n e below. This test contains 25 questions. It i s not expected that you w i l l remember everything on t h i s t e s t . When you are t o l d to begin: 1) Read each question c a r e f u l l y . 2) There i s only one correct answer to each question. Print neatly the l e t t e r of your choice on the l i n e to the right of each question. 3) You w i l l have 35 minutes for t h i s t e s t . - 101 - INTRODUCTORY GEO TRIG 10 TEST The area of a rectangle with length 4 cm and width 11 cm i s : 2. 3. 4. a) 30 sq. cm b) 19 sq. cm c) 44 sq. cm d) 15 sq. cm e) 26 sq. cm How many l i n e s of symmetry does a square have? a) 2 only b) 4 only c) 6 only d) 8 only e) infinite The measure of an acute angle i s : a) 90° b) between 45° and 90° c) less than 90° d) between 90° and 180° e) more than 180^ Perpendicular l i n e s : a) do not intersect. b) are two intersecting l i n e s that form right angles. c) intersect to form three acute angles and one obtuse angle. d) intersect to form four acute angles. e) none of the above. - 102 - An e q u i l a t e r a l t r i a n g l e has: a) a l l sides the same length. b) a l l sides with different lengths. c) two sides only with the same length. d) a l l angles with different measures. e) two acute angles and one obtuse angle. If ABAD i s similar to ARSV, then L A i s congruent to which angle i n A RSV? a) R only b) S only c) V only d) SRV only e) none of these Given right a) SC A ABC, d) AC b) A£ BC C) A£ AB e) s i n A equals: m AC EC AB If P i s the center of the c i r c l e , segment PC i s c a l l e d the: a) chord of the c i r c l e . b) diameter of the c i r c l e . c) segment of the c i r c l e . d) radius of the c i r c l e . e) minor arc of the c i r c l e . - 103 - 9. What i s the reason that the two triangles below are congruent? 10. a) AAA (Angle-Angle-Angle Theorem) b) AAS (Angle-Angle-Side Theorem) c) SAS (Side-Angle-Side Theorem) d) ASA (Angle-Side-Angle Theorem) e) SSS (Side-Side-Side Theorem) In every c i r c l e , what i s the r a t i o of the circumference to diameter? a) b) 7/22 err) (1/TT ) c) (22/7) 2 d) (7/22) 2 e) 11. 22/7 there i s no constant r a t i o I f A DEF i s the r e f l e c t i o n of A ABC i n l i n e x, what i s the image of point B? - 104 - a) point D b) point E c) point F d) no image point e) point B 12. 13. The measure of the t h i r d angle i n the triangle below i s : a) ,0 5(T b) 130 c) 20° d) 40' e) 60 0 0 The length of the t h i r d side i n the right t r i a n g l e below i s : a) 8 cm b) 10 cm c) 12 cm d) 14 cm e) 16 cm 8 14. C.r*\ The volume of the box shown i s : 3 a) 63 cm b) 126 cm c) d) 162 cm 1134 cm~ e) 2268 cm~ 3 3 1 cm 15. The horizontal a) 12 b) 14 c) cannot be d) 6 e) 7 lines are p a r a l l e l . < calculated - 105 - / The length of x i s : \ > 16. 17. The p e r i m e t e r o f t h e p a r a l l e l o g r a m below i s : a) 48 cm b) 56 cm c) 30 cm d) 29 cm e) 28 cm g cm la*/ ZABC i s a r i g h t a n g l e . Z.DBC measures 15^. The measure o f Z.ABD is: a) cannot be c a l c u l a t e d b) 105° c) 165 d) 75° 65 18. 19. 0 0 A cube has how many edges? a) 4 b) 8 c) 12 d) 16 e) 20 L i n e s a and b a r e p a r a l l e l . 0 a) 100 b) >0 80 c) cannot be c a l c u l a t e d d) ,0 90' e) 70 The measure o f angle y i s : < 0 - 106 - A* >a 20. Given the number l i n e below, which statement i s true? a) 21. 22. US = TE b) TG = TS U _ l _ C) HT ~ UP d) LG ~ GU e) HG=UH 1 1 1 S 1 «- 1 1 1 1 -f -J -A -/ O I A 3 & 1 +—i -t> T H P In the figure shown, the obtuse angles are: a) ZADC, ZDCB, ZDEC, ZAEB b) ZADC, ZDEB, ZDCB, Z.DEC C) ZAEB, ZABC, ZBAD, ZDEC d) ZAEB, Z.EAB, ZEBA, ZDEC e) ZAED, ZBEC, ZEDC, ZECD A In parallelogram ABCD, point 0 i s the midpoint on AC. Using a 180^ rotation (1/2 turn) around point 0, the rotation image of AB i s : 23. a) BC b) CD C) AD d) AO e) AB ' » A The area of the t r i a n g l e below i s : a) 120 cm b) 108 cm c) 60 cm d) 54 cm e) 31 cm 2 2 . .H- J - 107 - c > The f o r m u l a t o f i n d t h e 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 i s V = 2 1T r h. a) 605 1T cm b) 55 V c) 275 TT cm d) 27.5 If cm 3 e) 3025 TT cm 3 The volume cm o f the f i g u r e below i s : 3 3 II CrVV 3 / V f v fen* A n g l e s a and b a r e : a) interior b) exterior c) vertical d) complementary e) supplementary - 108 - I n t r o d u c t o r y Geometry T e s t Item Analysis Computer Group Item Percentage with choice Humbex £ 1 14 2 14 81 3 24 14 4 5 5 6 81 5 0 86 0 71 Blank % Correct £ D. £ 86 0 0 0 86 0 0 5 0 81 57 0 5 0 57 0 10 0 0 86 0 5 14 0 81 5 0 19 0 71 10 90 0 0 90 19 48 5 0 48 0 52 5 0 90 7 8 0 9 5 10 11 12 13 14 0 23 52 10 5 0 0 90 0 5 48 14 14 19 5 0 48 43 10 33 10 0 43 90 0 0 90 33 10 57 0 57 19 0 0 57 5 33 10 0 15 0 0 16 10 14 57 17 19 0 0 67 14 0 67 0 5 0 48 5 0 33 18 19 20 0 5 43 48 24 33 29 10 10 19 62 5 62 19 14 29 0 29 0 5 21 29 10 22 10 62 23 24 24 5 25 5 5 10 5 10 62 19 10 38 10 0 38 19 57 5 10 5 57 5 24 57 10 57 0 - 109 - Introductory geometry Test Item Analysis Traditional Group Item Number Percentage with choice Blank A fi £ 1 16 0 84 0 0 0 84 2 21 47 0 11 21 0 47 3 26 11 47 11 5 0 47 4 26 47 0 0 26 0 47 5 84 0 11 0 5 0 84 6 0 84 5 11 0 0 84 . Q £ % Correct 7 8 5 5 27 63 0 0 63 9 0 16 32 42 11 0 42 10 42 16 16 0 26 0 42 11 0 95 0 5 0 0 95 12 42 21 0 26 11 0 42 13 0 37 26 32 5 0 37 14 0 5 5 90 0 0 90 15 0 11 21 0 68 0 68 16 11 21 47 11 11 0 47 17 0 11 5 84 0 0 84 18 5 42 47 5 0 0 47 19 21 32 26 0 21 0 32 20 5 0 0 21 74 0 74 21 16 11 42 11 16 5 16 22 11 53 26 5 5 0 53 23 15 .32 32 11 11 0 11 24 26 11 47 11 5 0 47 25 0 21 11 42 26 0 26 - no - Appendix C PERMISSION LETTER AND VAN HIELE GEOMETRY TEST - Ill - VAN HIELE GEOMETRY TEST Directions Do not open this test until your are told to do so. Please write your name on the line below. This test contains 25 questions. It i s not expected that you know everything on this test. When you are told to begin: 1) Read each question carefully. 2) There is only one correct answer to each question. Place the letter of your choice on the line to the right of each question. 3) You will have 35 minutes for this test. This test i s based on the work of P.M. van Hiele. 113 Permission has been granted by-Professor Zalman Usiskin, Director of the CDASSG Project at the University of Chicago to use this test. VAN 1. 2. 3. HIELE GEOMETRY TEST Which of these are squares? a) K only b) L only c) M only d) L and M only e) A l l are squares. Which of these are triangles? a) None of these are t r i a n g l e s . b) V only c) W only d) W and X only e) V and W only Which of these are rectangles? s a) S only b) T only c) S and T only d) S and U only e) A l l are rectangles. - 114 - Which of these are squares? a) None of these are squares. b) G only c) F and G only d) G and I only e) A l l are squares. Which of these are parallelograms? b) L only c) J and M only d) None of these are parallelograms. e) A l l are parallelograms. ? PQRS i s a square. J Which relationship i s true i n a l l squares? a) PR and RS have the same length. b) QS and P"E. are perpendicular. c) PS and Q*R are perpendicular. d) PS and QS have the same length. e) Angle Q i s larger than angle R. | In a rectangle GHJK, GJ and HK are the diagonals. Which of (a) to (d) i s not true i n every rectangl a) There are four right angles. b) There are four sides. c) The diagonals have the same length. d) The opposite sides have the same length. e) A l l of (a) to (d) are true i n every rectangle - 115 - A rhombus i s a 4-sided figure with a l l sides of the same length. Here are three examples. Which of (a) to (d) i s not true i n every rhombus? a) The two diagonals have the same length. b) Each diagonal bisects two angles of the rhombus. c) The two diagonals are perpendicular. d) The opposite angles have the same measure. e) A l l of (a) to (d) are true i n every rhombus. An isosceles t r i a n g l e i s a triangle with two sides of equal length. Here are three examples. A Which of (a) to (d) i s true i n every isosceles triangle? a) The three sides must have the same length. b) One side must have twice the length of another side. c) There must be at least two angles with the same measure. d) The three angles must have the same measure. e) None of (a) to (d) i s true i n every isosceles t r i a n g l e . - 116 - Two c i r c l e s with centers P and Q i n t e r s e c t at R and S t o form a 4-sided f i g u r e PRQS. Here are two examples. Which of (a) t o (d) i s not always true? a) PRQS w i l l have two p a i r s of sides of equal length. b) PRQS w i l l have at l e a s t two angles of equal measure. c) The l i n e s PQ and RS w i l l be perpendicular. d) Angles P and Q w i l l have the same measure. e) A l l of (a) t o (d) are true. Here are two statements. Statement 1: Figure .F i s a rectangle. Statement 2: Figure F i s a t r i a n g l e . Which i s correct? a) I f 1 i s true, then 2 i s true. b) I f 1 i s f a l s e , then 2 i s true. c) 1 and 2 cannot both be true. d) 1 and 2 cannot both be f a l s e . e) None of (a) t o (d) i s c o r r e c t . Here are two statements. Statement S: A ABC has three sides of the same length. Statement T: In A ABC, ZB and Z.C have the same measure. Which i s correct? a) Statements S and T cannot both be true. b) I f S i s true, then T i s true. c) I f T i s true, then S i s true. d) I f S i s f a l s e , then T i s f a l s e . e) None of (a) t o (d) i s c o r r e c t . - 117 - 13. Which of these can be c a l l e d rectangles? P 14. a) A l l can. b) Q only. c) R only. d) P and Q only. e) Q and R only. Which i s true? a) A l l properties of rectangles are properties of a l l squares. b) A l l properties of squares are properties of a l l rectangles. c) A l l properties of rectangles are properties of a l l parallelograms. d) A l l properties of squares are properties of a l l parallelograms. e) 15. None of (a) to (d) i s true. What do a l l rectangles have that some parallelograms do not have? a) opposite sides equal b) diagonals c) opposite sides p a r a l l e l d) opposite angles e) none of (a) to (d) equal equal - 118 - 16. Here i s a right t r i a n g l e ABC. E q u i l a t e r a l triangles ACE, ABF, and BCD have been constructed on the sides of ABC. 3> From t h i s information, one can prove that AD, BE, and CF have a point i n common. a) What would t h i s proof t e l l you? Only i n t h i s t r i a n g l e drawn can we be sure that AD, BE, and CF have a point i n common. b) In some but not a l l right triangles, AD, BE and CF have a point i n common. c) In any right triangle, AD, BE and CF have a point i n common. d) In any t r i a n g l e , AD, BE and CF have a point i n common. e) In any e q u i l a t e r a l t r i a n g l e , AD, BE and CF have a point i n common. 17. Here are three properties of a figure. Property D: It has diagonals of equal length. Property S: It i s a square. Property R: It i s a rectangle. Which i s true? a) D implies S which implies R. b) D implies R which implies S. c) S implies R which implies D. d) R implies D which implies S. e) R implies S which implies D. - 119 - Here are two statements. I. I f a figure i s a rectangle, then i t s diagonals bisect each other. II. I f the diagonals of a figure bisect each other, the figure i s a rectangle. Which i s correct? a) To prove I i s true, i t i s enough to prove that II i s true. b) To prove II i s true, i t i s enough to prove that I i s true. c) To prove I I i s true, i t i s enough to f i n d one rectangle whose diagonals bisect each other. d) To prove II i s false, i t i s enough to f i n d one non-rectangle whose diagonals bisect each other. e) None of (a) to (d) i s correct. In geometry: a) Every term can be defined and every true statement can be proved true. b) Every term can be defined but i t i s necessary to assume that c e r t a i n statements are true. c) Some terms must be l e f t undefined but every true statement can be proved true. d) Some terms must be l e f t undefined and i t i s necessary to have some statements which are assumed true. e) None of (a) to (d) i s correct. - 120 - Examine these three sentences. i) Two l i n e s perpendicular to the same l i n e are p a r a l l e l . ii) A l i n e that i s perpendicular to one of two p a r a l l e l l i n e s i s perpendicular to the other. iii) I f two l i n e s are equidistant, then they are p a r a l l e l . In the figure below, i t i s given that l i n e s m and p are perpendicular and l i n e s n and p are perpendicular. Which of the above sentences could be the reason that l i n e m i s fP p a r a l l e l to l i n e n? a) (i) only b) ( i i ) only c) ( i i i ) only d) E i t h e r (i) or ( i i ) e) E i t h e r ( i i ) or ( i i i ) < P 1 P » < n In F-geometry, one that i s different from the one you are used to, there are exactly four points and s i x l i n e s . contains exactly two points. Every line I f the points are P, Q, R, and S, the l i n e s are {P,Q}, {P,R}, {P,S}, {Q,R}, {Q,S}, and {R,S}. . P a. •s Here are how the words "intersect" F-geometry. and " p a r a l l e l " are used i n The l i n e s {P,Q} and {P,R} intersect at P because {P,Q} and {P,R} have P i n common. The l i n e s {P,Q} and {R,S} are p a r a l l e l because they have no points i n common. From t h i s information, which i s correct? a) {P,R} and {Q,S} intersect. b) {P,R} and {Q,S} are p a r a l l e l . c) {Q,R} and {R,S} are p a r a l l e l . d) {P,S} and {Q,R} e) None of (a) to (d) i s correct. intersect. - 121 - 22. To t r i s e c t an angle means to divide i t into three parts of equal measure. In 1847, P.L. Wantzel proved that, i n general, i t I s impossible to t r i s e c t angles using only a compass and an unmarked r u l e r . a) From h i s proof, what can you conclude? In general, i t i s impossible to bisect angles using only a compass and an unmarked r u l e r . b) In general, i t i s impossible to t r i s e c t angles using only a compass and a marked r u l e r . c) In general, i t i s impossible to t r i s e c t angles using any drawing instruments. d) I t i s s t i l l possible that i n the future someone may f i n d a general way to t r i s e c t angles using only a compass and an unmarked r u l e r . e) No one w i l l ever be able to f i n d a general method f o r t r i s e c t i n g angles using only a compass and an unmarked ruler. 23. There i s a geometry invented by a mathematician J i n which the . following i s true: The sum of the measures of the angles o f a t r i a n g l e i s less than 180°. Which i s correct? a) J made a mistake i n measuring the angles of the t r i a n g l e . b) J made a mistake i n l o g i c a l reasoning. c) J has a wrong idea of what i s meant by "true." d) J started with different assumptions than those i n the usual geometry. e) None of (a) to (d) i s correct. - 122 - 24. Two geometry books define the word rectangle i n d i f f e r e n t ways. Which i s true? a) One of the books has an error. b) One of the d e f i n i t i o n s i s wrong. There cannot be two d i f f e r e n t d e f i n i t i o n s f o r rectangle. c) The rectangles i n one of the books must have d i f f e r e n t properties from those i n the other book. d) The rectangles i n one of the books must have the same properties as those i n the other book. e) The properties of rectangles i n the two books might be different. 25. Suppose you have proved statements I and I I . I. I f P/ then q. I I . I f s, then not q. Which statement follows from statements I and II? a) I f p, then s. b) I f not p, then not q. c) I f p or q, then s. d) I f s, then not p. e) I f not s, then p. - 123 - Appendix D PROOF TEST AND ITEM ANALYSIS - 124 - GEO TRIG 10 PROOF TEST Name. Date today. Your birthdate , Month , Day Year Directions: - You w i l l have 35 minutes to complete t h i s t e s t . - A l l answers should be written on these pages. - P a r t i a l credit w i l l be given so do the best you can on each question. - 125 - GEO TRIG 10 PROOF TEST Statements a) WZIj RftSSOnS XY, XL = ZK, ZZKW = &LY ; b) I f p a r a l l e l l i n e s , then alternate i n t e r i o r angles congruent. c) AWKZ = £YLX d) 8 PROOF: - 126 - C T> 3. I f the diagonals of a parallelogram are perpendicular, then the parallelogram i s a rhombus. To prove the above statement: a) Draw and label a diagram. b) Write what i s given and what i s to be proved i n terms of your diagram. c) Write the proof. - 127 - 5. Write t h i s proof below: GIVEN: Quadrilateral SNOW with SW = WO, SN = NO W PROVE: iS = LO PROOF: - 128 - Write t h i s proof i n the space provided below: GIVEN: Quadrilateral SRIG with SR = GI, SG = I N bisects SI at M. . PROVE: PM = MN PROOF: - 129 - Proof Test Item Analysis Computer Group Question Numbex Percent of students receiving t h i s score • . 1 2 2 4 1 0 14 45 5 36 2 32 14 9 14 32 3 68 14 9 5 5 4 41 18 9 9 23 5 5 9 18 9 59 6 36 23 36 5 0 Item Analysis Traditional Group Question Numbej: Percent of students receiving t h i s score H 1 2 2 4 1 0 6 63 13 19 2 63 25 6 0 6 3 56 38 0 0 6 4 88 0 6 6 0 5 63 0 13 6 19 6 50 25 19 6 0 - 130 - Appendix E ATTITUDE TEST, ITEM ANALYSIS AND SUMMARY OF ITEM STATISTICS - 131 - GEO TRIG 10 June 1988 Draw a c i r c l e around the letter(s) that show(s) how closely you agree with each statement: SD (Strongly Disagree), D (Disagree), U (Undecided), A (Agree), SA (Strongly Agree). 1. I am always under a t e r r i b l e s t r a i n i n GeoTrig. SD D U A SD D U A SD D U A SD D U A SD D U A SD D U A SD D U A SD D U A jungle of information and can't f i n d my way out. SD D U A D U A 2. Geo T r i g i s very interesting to me, and I enjoyed t h i s course. 3. Geo T r i g i s fascinating and fun. 4. Geo T r i g makes me f e e l secure, and at the same time i t i s stimulating. 5. My mind goes blank, and I am unable to think c l e a r l y when working i n Geo T r i g . 6. I f e e l a sense of insecurity when attempting Geo T r i g . 7. Geo T r i g makes me f e e l uncomfortable, restless, i r r i t a b l e , and impatient. 8. The f e e l i n g that I have toward Geo T r i g i s a good feeling. • 9. Geo T r i g makes me f e e l as though I'm lost i n a 10. Geo T r i g i s something which I enjoy a great deal. SD - 132 - 11. When I hear the word math, I have a f e e l i n g of dislike. SD D U A SA SD D U A SA SD D U A SA SD D U A SA SD D U A SA SD D U A SA SD D U A SA D U A SA SD D U A SA SD D U A SA SD D U A SA SD D U A SA 12. I approach math with a f e e l i n g of hesitation, r e s u l t i n g from a fear of not being able to do math. 13. I r e a l l y l i k e mathematics. 14. Mathematics i s a course i n school which I have always enjoyed studying. 15. It makes me nervous to even think about having to do a math problem. 16. I have never l i k e d math, and i t i s my most dreaded subject. 17. I am happier i n a math class than i n any other class. 18. I f e e l at ease i n mathematics, and I l i k e i t very much. SD 19. I f e e l a d e f i n i t e p o s i t i v e reaction to mathematics; i t ' s enjoyable. 20. I do not l i k e mathematics, and i t scares me to have to take i t . 21. Mathematics has contributed greatly to science and other f i e l d s of knowledge. 22. Mathematics i s less important to people than art or l i t e r a t u r e . - 133 - 23. Mathematics i s not important for the advance of c i v i l i z a t i o n and society. SD D U A SA SD D U A SA SD D U A SA SD D U A SA SD D U A SA SD D U A SA 24. Mathematics i s a very worthwile and necessary subject. 25. Mathematics i s not important i n everyday l i f e . 26. Mathematics i s needed i n designing p r a c t i c a l l y everything. 27. Mathematics i s needed i n order to keep the world running. 28. There i s nothing creative about mathematics; i t ' s just memorizing formulas and things. Please write or print your reactions to the following questions: 1. What I l i k e d most about t h i s course was: - 134 - What I d i s l i k e d most about t h i s course was: I would l i k e to make the following suggestions - 135 - Attitude Test Item Analysis Computer Group Item Percentage with choice 2H n II A 1 10 52 29 10 0 2 5 24 14 43 14 3 10 19 43 24 5 4 10 48 19 19 5 5 14 38 19 24 5 6 14 38 19 14 14 7 10 29 29 24 10 8 0 14 52 29 5 9 0 57 5 33 5 10 5 33 19 38 5 11 14 52 10 10 14 12 19 43 19 14 5 13 0 14 38 33 14 14 14 19 24 33 10 15 14 71 5 10 0 16 38 43 10 5 5 17 19 29 29 19 5 18 10 33 29 24 5 19 0 24 33 38 5 20 24 52 10 14 0 21 0 0 10 43 48 22 10 48 14 19 5 23 52 38 0 5 5 24 0 0 0 57 43 25 43 48 0 5 5 26 0 5 0 52 43 27 0 0 10 57 33 28 19 43 14 10 14 nber. - 136 - 2A Attitude Test Item Analysis Traditional Group Item Percentage with choice 2D. n II & 2A l 6 29 29 29 6 2 24 24 12 35 6 3 18 18 29 29 6 4 18 29 41 12 0 5 12 35 6 41 6 6 12 41 18 18 12 7 12 18 18 47 6 8 24 24 18 29 6 9 18 35 12 18 18 10 12 29 18 35 6 11 12 59 6 6 18 12 12 59 0 12 18 13 18 6 24 47 6 14 12 18 24 35 12 15 6 65 24 6 0 16 12 65 6 12 6 17 24 41 35 0 0 18 18 18 29 29 6 19 24 18 12 41 6 20 35 41 12 12 0 21 0 0 24 24 53 22 29 35 18 18 0 23 53 47 0 0 0 24 0 0 18 35 47 25 47 47 6 0 0 26 0 0 6 59 35 27 0 0 18 47 35 28 18 41 18 18 6 - 137 - Attitude Test Summary of Item S t a t i s t i c s Item Numbej: Total Mean Computer Group SD Mean SD. T r a d i t i o n a l Group Mean SD. 1 3.34 0.97 3.62 0.81 3.00 1.06 2 3.11 1.27 3.38 1.16 2.77 1.35 3 2.92 1.10 2.95 1.02 2.88 1.22 4 2.55 1.01 2.62 1.07 2.47 0.94 5 3.21 1.19 3.33 1.16 3.06 1.25 6 3.24 1.26 3.24 1.30 3.24 1.25 7 2.95 1.16 3.05 1.16 2.82 1.19 8 3.00 1.07 3.24 0.77 2.71 1.31 9 3.16 1.22 3.14 1.06 3.18 1.43 10 3.00 1.12 3.05 1.07 2.94 1.20 11 3.42 1.29 3.43 1.29 3.41 1.33 12 3.47 1.22 3.57 1.12 3.35 1.37 13 3.34 1.07 3.48 0.93 3.18 1.24 14 3.11 1.23 3.05 1.24 3.18 1.24 15 3.82 0.73 3.91 0.77 3.71 0.69 16 3.87 1.07 4.05 1.07 3.65 1.06 17 2.40 1.03 2.62 1.16 2.12 0.78 18 2.84 1.13 2.81 1.08 2.88 1.22 19 3.08 1.12 3.24 0.89 2.88 1.36 20 3.92 0.97 3.86 0.96 4.00 1.00 21 4.34 0.75 4.38 0.67 4.29 0.85 22 3.55 1.08 3.38 1.07 3.77 1.09 23 4.40 0.86 4.29 1.06 4.53 0.51 24 4.37 0.63 4.43 0.51 4.29 0.77 25 4.29 0.87 4.19 1.03 4.41 0.62 26 4.32 0.66 4.33 0.73 4.29 0.59 27 4.21 0.66 4.24 0.63 4.18 0.73 28 3.45 1.25 3.43 1.33 3.47 1.18 - 138 - Appendix F PERMISSION LETTER FOR INTERVIEWS - 139 - February 2, 1988 Dear Geo T r i g 10 Student: During t h i s semester I w i l l be conducting a study i n Geo T r i g 10 under the supervision of the University of B r i t i s h Columbia. The purpose of the study i s to determine i f computers can be used to improve the way students learn to write proofs. Over the years I have found that most students have d i f f i c u l t y with writing geometric proofs. Besides administering certain tests to the class, I would l i k e to interview you every two weeks for one-half hour. t o t a l of eight interviews. There w i l l be a The purpose of these interviews i s to f i n d out more s p e c i f i c a l l y how you relate the computer work to proof writing. The interviews w i l l be tape-recorded so that I may analyze them further at a l a t e r time. A l l information c o l l e c t e d i n t h i s project i s for research purposes only. To assure c o n f i d e n t i a l i t y , no family names w i l l be used i n any report or release of the information. sensitive information i s being sought. No personal, family or other You may withdraw from t h i s project at any time by a statement o r a l l y or i n writing. Refusal to cooperate w i l l have no consequences for you. I f you wish any further information please ask me and I w i l l be happy to answer any questions you have regarding the project. I f you agree to being interviewed, please check the most convenient time for your interview on the form on the next page and return to me. Thank you. Mrs. J . Worster - 140 - I have read the above description of the research project e n t i t l e d AN INVESTIGATION TO DETERMINE THE EFFECTS OF THE GEOMETRIC SUPPOSER SOFTWARE ON GEOMETRIC PROOF WRITING AT THE GRADE 10 LEVEL to be c a r r i e d out by Mrs. Worster. [ ] I consent [ ] I do not consent to being interviewed every two weeks f o r one-half hour during second semester. The best time f o r my interview i s : [ ] 8:15 A.M. - 8:45 A.M. [ ] 3:45 P.M. - 4:15 P.M. [ ] Signature - 141 - 12:15 P.M. - 12:45 P.M. (Student)
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- An investigation of the effects of the geometric supposer...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
An investigation of the effects of the geometric supposer software on geometric proof writing at the… Worster, Josephine Regina 1989
pdf
Page Metadata
Item Metadata
Title | An investigation of the effects of the geometric supposer software on geometric proof writing at the grade 10 level |
Creator |
Worster, Josephine Regina |
Publisher | University of British Columbia |
Date Issued | 1989 |
Description | The purpose of this study was to determine if the use of a computer program called the Geometric Supposer would result in improved proof writing by grade 10 geometry students. The researcher studied 44 students enrolled in a grade 10 geometry course. The students were divided into two classes; one class used the Geometric Supposer computer program while the other class did not. Both classes were taught at the same time every day and both classes covered the same content. The researcher kept in close contact with the teacher of the noncomputer group regarding the content, the assignments, and the overall progress of the students. Both classes were given two tests (an introductory geometry test and the van Hiele geometry test) at the beginning of the course. At the end of the course (one semester in length) three tests were given to both classes—the same van Hiele geometry test (measures geometric thought levels), a proof test, and an attitude test. Weekly interviews were conducted with each of five students from the computer group. Two students from the noncomputer group were each interviewed twice near the end of the course. These students were chosen based on their van Hiele levels. The interviews provided the researcher with a better understanding of how some students approach and write geometric proofs. The data gathered from the introductory geometry test, the proof test, and the attitude test were each analyzed using the independent t-test. The median test was applied to the pre van Hiele geometry test results and to the post van Hiele test results. The sign test was used to analyze the pre and post van Hiele data. A chi square test of association was also applied to the van Hiele levels and tests. A .05 level of significance was used in each of these tests. The results indicate that the group of students using the computer program, Geometric Supposer, performed significantly better on the proof test than the group of students who did not use the computers. The pre van Hiele geometry test results indicate that more than 50% of students entering the grade 10 geometry course are at a 0 or 1 level. This level is too low to begin the study of geometric proof writing. The post van Hiele geometry test results indicate that, after a semester of geometry, students do move up in the van Hiele levels, with or without the use of computer programs like the Geometric Supposer. The results from the attitude test indicate that there was no difference between the two groups of students. Both classes value the study of mathematics in general, and geometry in particular. In summary, the computer, with appropriate software and teacher commitment, can contribute to reducing the difficulty generally experienced by students in mastering the writing of geometric proofs. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-09-17 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0302150 |
URI | http://hdl.handle.net/2429/28566 |
Degree |
Master of Arts - MA |
Program |
Mathematics Education |
Affiliation |
Education, Faculty of Curriculum and Pedagogy (EDCP), Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-UBC_1989_A8 W67.pdf [ 5.6MB ]
- Metadata
- JSON: 831-1.0302150.json
- JSON-LD: 831-1.0302150-ld.json
- RDF/XML (Pretty): 831-1.0302150-rdf.xml
- RDF/JSON: 831-1.0302150-rdf.json
- Turtle: 831-1.0302150-turtle.txt
- N-Triples: 831-1.0302150-rdf-ntriples.txt
- Original Record: 831-1.0302150-source.json
- Full Text
- 831-1.0302150-fulltext.txt
- Citation
- 831-1.0302150.ris
Full Text
Cite
Citation Scheme:
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>
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.831.1-0302150/manifest