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An investigation of the effects of the geometric supposer software on geometric proof writing at the… Worster, Josephine Regina 1989

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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 British 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 this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July, 1989 ©Josephine Regina Worster, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mat.hpmatirs anrf SPIPTIPP Education The University of British Columbia Vancouver, Canada DE-6 (2/88) ABSTRACT The purpose of this study was to determine i f 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 - i i -t e s t r e s u l t s and t o the post 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 used t o analyze the pre and post van H i e l e data. A c h i square t e s t of 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 the van H i e l e l e v e l s and t e s t s . A .05 l e v e l of s i g n i f i c a n c e was used i n each of these t e s t s . The r e s u l t s i n d i c a t e that the group of students u s i n g the computer program, Geometric Supposer, performed s i g n i f i c a n t l y b e t t e r on the proof t e s t than the group of students who d i d not use the computers. The pre van 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 than 50% of students e n t e r i n g the grade 10 geometry course are at a 0 or 1 l e v e l . This l e v e l i s too low t o begin the study of geometric proof w r i t i n g . The post van 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 , a f t e r a semester of geometry, students do move up i n the van H i e l e l e v e l s , w i t h or without the use of computer programs l i k e the Geometric Supposer. The r e s u l t s from the a t t i t u d e t e s t i n d i c a t e t h a t there was no d i f f e r e n c e between the two groups of students. Both c l a s s e s value the study of mathematics i n general, and geometry i n p a r t i c u l a r . In summary, the computer, w i t h appropriate software and teacher commitment, can c o n t r i b u t e t o reducing the d i f f i c u l t y g e n e r a l l y experienced by students i n mastering the w r i t i n g of geometric p r o o f s . - i i i -TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS v i i i C h a p t e r I. INTRODUCTION TO THE PROBLEM 1 Statement of the Problem 5 S i g n i f i c a n c e of the Study . 7 I I . REVIEW OF THE LITERATURE 9 L i t e r a t u r e R e l a t i n g t o High School Geometry . 9 L i t e r a t u r e R e l a t i n g t o the van H i e l e Theories. 22 L i t e r a t u r e R e l a t i n g t o the Geometric Supposer. 36 I I I . PROCEDURES 41 The Subjects 41 Design of the Study 43 Data C o l l e c t i o n 52 IV. DATA ANALYSIS 61 Assessment of the Groups 61 Geometric Thought Levels P r i o r t o Treatment. . 62 Changes i n Geometric Thought Levels 63 Wri t t e n Proofs 65 A t t i t u d e s 67 Interview Data 69 A d d i t i o n a l Data 76 Data Summary 78 - i v -Chapter Page, V. SUMMARY AND DISCUSSION 79 Summary of the Problem, Methodology, and R e s u l t s . 79 I n t e r p r e t a t i o n of the Findings 80 L i m i t a t i o n s of the Study 86 Suggestions f o r Further Research 87 Im p l i c a t i o n s 88 Conclusions . . . . 90 References 91 Appendices A Permission L e t t e r sent t o Parents/Guardians 97 B Introductory Geometry Test 100 Item A n a l y s i s 109 C Permission 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 D Proof Test 124 Item A n a l y s i s 130 E A t t i t u d e Test 131 Item A n a l y s i s 136 Summary of Item S t a t i s t i c s 138 F Permission L e t t e r f o r Interviews 139 - v -LIST OF TABLES labia Page 1.1. Comparison between LOGO and the Geometric Supposer. 4 2.1. The Complementary Angle Theorem 11 2.2. The Supplementary Angle Theorem (a) 13 2.3. The Supplementary Angle Theorem (b) 14 2.4. The van H i e l e Model 24 4.1. Means, Standard D e v i a t i o n s , and S t a t i s t i c a l Comparison of Groups: Introductory Geometry P r e t e s t . 62 4.2. Median Test: van H i e l e P r e t e s t 63 4.3. Sign Test: Pre and Post van H i e l e Test Data . . . . 64 4.4. Means, Standard D e v i a t i o n s , and S t a t i s t i c a l Comparison of Groups: Proof Test 65 4.5. Means, Standard D e v i a t i o n s , and S t a t i s t i c a l Comparison of Groups: A t t i t u d e Test 67 4.6. W r i t t e n Comments 68 4.7. Students' van H i e l e Levels 77 4.8. Interviewees' Pre and Post van H i e l e Levels and t h e i r Proof Test Scores 78 - v i -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 . . . . 47 6. Supplementary Angles 70 7. Parallel Lines with Alternate Interior Angles . . 70 8. Relationship Between the Exterior Angle and Remote Interior Angles 70 9. Proving Two Segments Congruent 71 10. Proof #1 72 11. Proof #2 . . . . ' 73 12. Proof #3 74 13. Proof #4 75 14. Overlapping Triangle Proof 76 - v i i -ACKNOWLEDGEMENTS I would like to thank the members of my thesis committee, Dr. Marv Westrom, Dr. Doug Owens, and Dr. Harold Ratzlaff for their guidance and assistance. Secondly, I would like to express my appreciation to the grade ten students and the mathematics teacher who participated in this study. Lastly, I thank my husband, B i l l , for his encouragement, patience, and constant support. j - v i i i -CHAPTER 1 INTRODUCTION TO THE PROBLEM I t is the glory of geometry that from so few principles, fetched from without, it is able to accomplish so much. Newton Despite Newton's admiration f o r t h i s branch of mathematics, the r o l e of geometry i n the mathematics c u r r i c u l u m has been the subject of 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 . " I t i s 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 sound advice on how t o remedy these d i f f i c u l t i e s i s hard 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 , "In geometry . . . there i s not even agreement as t o what the subject i s about." (p. 165) In 1970 the United States Comprehensive School Mathematics P r o j e c t (CSMP) sponsored a conference on the t e a c h i n g of geometry and reported, "Of a l l the 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 respect t o choice of content, u s u a l l y the 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." (Morris, 1986, p. 9) Geometry and the van H i e l e s The Soviet Union had not only 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 of t h e i r mathematics curriculum. This r e v i s i o n was based on a theory developed by a Dutch husband and wife team who were mathematics t e a c h e r s — t h e van H i e l e s . Although not widely known by teachers i n North America today, the work of the van H i e l e s p l a y s a major pa r t i n t h i s i n v e s t i g a t i o n . - 1 -The I n t e r n a t i o n a l Commission on Mathematical 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 i n 1986. Their d i s c u s s i o n s centered on the mathematical c u r r i c u l u m f o r the 1990's and again, geometry-appeared as the 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 area w i t h i n the school c u r r i c u l u m arouses so much concern amongst mathematicians as does geometry, . . . " ( H o w s o n & Wilson, 1986, p. 58) . The l i t e r a t u r e contains 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 the mathematics c u r r i c u l u m . There i s , however, some general debate as t o whether geometry should be i n t e g r a t e d throughout the mathematics c u r r i c u l u m or should be taught as a one year course, u s u a l l y at the grade ten 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 the r o l e of proofs and deductions. Many students i n high school geometry have d i f f i c u l t y w i t h deduction and proof. "They don't understand the r o l e or meaning of an axiomatic system. Despite our best e f f o r t s t o teach them, even the most capable algebra students may s t r u g g l e and get through geometry by sheer willpower and memorization but w i t h l i t t l e understanding of the l o g i c a l system we have been developing a l l year." (Shaughnessy & Burger, 1985, p. 419) We should t h e r e f o r e not be s u r p r i s e d by the f a c t that many students tend t o d i s l i k e g e o m e t r y — i n p a r t i c u l a r , w r i t i n g proofs ( F a r r e l l , 1986; Hoffer, 1981; Senk, 1985; U s i s k i n , 1980). The van H i e l e s a l s o experienced f r u s t r a t i o n s w h i l e teaching geometry. They were f a m i l i a r w i t h the work of Piaget and from t h i s , P i e r r e van H i e l e developed h i s system of thought l e v e l s i n geometry. To help students r a i s e t h e i r thought l e v e l s , the van H i e l e system - 2 -specified a sequence of phases that moved from very direct instruction to the students becoming independent from their teachers (Hoffer, 1983). Mayberry (1981) summarized two consequences of these levels: - a student cannot function adequately on a given level unless he has passed through and learned to think intu i t ive ly on each preceding level . - If instruction, that i s , the language of the instructor, problems in the textbook, or pedagogical techniques assume the student to be on one level while in fact the student is on a lower level , there w i l l be serious communication problems between the instructor and the student because their geometric knowledge is organized differently (p. 6). Further, van Hiele in his 1959 ar t i c l e , stated, "The bad results of the teaching of geometry must almost entirely be attributed to the disregard of the levels. The learning process in geometry, as we have seen, covers many levels, but appreciation of this has s t i l l so l i t t l e penetrated into the teaching world that one even encounters teaching methods in which beginners are confronted with modes of reasoning based on symbols of the th ird (formal deduction) level ." (p. 21) Wirszup (1976) also stated, "The majority of our high school students are at the f i r s t level of development in geometry, while the course they take demands the fourth level of thought. It is no wonder that high school graduates have hardly any knowledge of geometry, and that this irreparable deficiency haunts them continually later on." (p. 96) Geometry and the Microcomputer Geometry is c learly a visual subject, yet much of the student's time is spent writing. If students are to have an opportunity to - 3 -t h i n k i n t u i t i v e l y they need a f a s t e r , l e s s cumbersome medium i n which to experience i t . The microcomputer has these c h a r a c t e r i s t i c s but has only been used i n a l i m i t e d way i n North American geometry c l a s s e s . Can the microcomputer be used t o reduce the d i f f i c u l t i e s t h a t students have w i t h proofs? This w i l l be p o s s i b l e only i f appropriate software i s used. The researcher i n i t i a l l y considered u s i n g the programming language LOGO t o teach geometric concepts. This n o t i o n was di s c a r d e d when the Geometric Supposer software was brought t o the researcher's a t t e n t i o n . The advantages of the Geometric Supposer over LOGO are l i s t e d i n Table '1.1. Table 1.1 Comparison between LOGO and the Geometric Supposer LOGO GEOMETRIC SUPPOSER 1. Need t o l e a r n language before studying geometric concepts. Easy-to-use menu d r i v e n programs. 2. Need t o develop experiments/ e x e r c i s e s that demonstrate geometric concepts. B u i l t on geometric shapes and r e l a t i o n s h i p s . 3. Need t o develop a l l support m a t e r i a l s . Some teaching and student l e a r n i n g m a t e r i a l a v a i l a b l e . This study i n v e s t i g a t e d the e f f e c t s on a grade ten geometry c l a s s of i n c l u d i n g the use of microcomputers and the Geometric Supposer software i n the course. Further, the c l a s s was compared t o a second c l a s s which was taught the same course content i n the t r a d i t i o n a l , proof w r i t i n g manner. - 4 -Statement of the Problem Because of c r i t i c i s m of high school geometry, Gearhart (1975) conducted a nationwide survey of secondary school mathematics teachers. "Proof was regarded as an important t o p i c by n e a r l y a l l teachers." (p. 490) However, the teachers " a l s o i n d i c a t e d t h a t many students do i n f a c t have t r o u b l e w i t h the m a t e r i a l and do not l i k e i t " (p. 490). More r e c e n t l y , Suydam's (1985) report on the NCTM (1981) survey i n d i c a t e d that mathematics teachers p r e f e r r e d the geometry c u r r i c u l u m be kept i n t a c t w i t h the focus on Eucl i d e a n geometry. When students were asked what they d i s l i k e d most about t h e i r geometry course, "There i s only one strong answer: proof." ( U s i s k i n , 1980, p. 419) Despite the emphasis on proofs i n t e n t h grade geometry, students seem t o emerge from t h i s course w i t h only a l i m i t e d a b i l i t y t o generate proofs and not much understanding about the nature of proof. Senk (1985) reported the r e s u l t s of the Co g n i t i v e Development and Achievement i n Secondary School Geometry P r o j e c t — a p p r o x i m a t e l y 30% of the students i n geometry courses i n which proof i s taught reach a 75% mastery l e v e l i n proof w r i t i n g , and about 25% of the students have no competence i n w r i t i n g p r o o f s . Thus, there seems t o be a discrepancy between the i n t e n t i o n s of the geometry c u r r i c u l u m i n high school and what students a c t u a l l y l e a r n . Craine (1985), i n an attempt t o improve the geometry course, made s e v e r a l assumptions. His f i r s t assumption, "Students e n t e r i n g t h i s course have not n e c e s s a r i l y had the i n f o r m a l geometric experiences - 5 -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 is 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, ie. 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, ie. every square is 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 of concrete models. "Few students are exposed t o , or reach t h i s l e v e l . " (Crowley, 1987, p. 2) Other s t u d i e s have found that students were i l l prepared (had low van H i e l e l e v e l s of geometric t h i n k i n g ) f o r t h e i r geometry 10 course. Were the students i n t h i s study i n the same p o s i t i o n ? "The van H i e l e model reveals an alarming l a c k of harmony i n the te a c h i n g and l e a r n i n g of mathematics." (Hoffer, 1983, p. 218) In an attempt t o "bridge the gap" t h i s i n v e s t i g a t i o n endeavoured t o answer the f o l l o w i n g questions: 1) W i l l the students who use the Geometric Supposer software be b e t t e r able t o w r i t e formal proofs than students who are taught by more t r a d i t i o n a l methods? 2) What changes i n the students' 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 of geometry? 3) W i l l the students who rec e i v e the treatment have a more p o s i t i v e a t t i t u d e towards geometry than the students i n the t r a d i t i o n a l group? S i g n i f i c a n c e of the Study The primary s i g n i f i c a n c e of t h i s study was t o i n t e g r a t e the Geometric Supposer software i n t o the geometry c u r r i c u l u m t o provide a bridge between the s p a t i a l - v i s u a l aspects of geometry and the deductive aspects i n order t o increase students' a b i l i t y t o w r i t e p r o o f s . Second, the change i n students' van H i e l e l e v e l s between the beginning of the semester and the end was measured. T h i r d , t h i s study presented i n f o r m a t i o n regarding the a t t i t u d e s of students from two groups towards geometry at the end of the course. One group of - 7 -students used the computer software throughout the course while the other group d i d not. The r e s u l t s of t h i s study provide some h e l p f u l suggestions f o r the t e a c h i n g of geometric proofs to grade ten students. - 8 -CHAPTER 2 REVIEW OF THE LITERATURE This chapter contains a review of the literature 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 in the high school has been a very controversial topic with opinions ranging from Dieudonne's famous slogan, "Down with Euclid." (Grunbaum, 1981, p. 235) to "Teach them a rigorous Euclidean geometry." The Euclidean camp has dominated despite numerous suggestions for changes in the high school geometry course. Proofs are central Proof i s the cornerstone for teaching Euclidean geometry as, "It 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 is this term, proof? According to Smith and Henderson: Proof is a common word in our vocabularies with various shades of meaning in i t s daily usage, but i t has a very special and precise meaning in mathematics. As a mature concept, proof in mathematics is a sequence of related statements directed toward establishing the validity of a conclusion (p. I l l ) . - 9 -In high school geometry each statement or step in the proof must be ju s t i f i e d . The justifications can be drawn either from given information, definitions, postulates, or previously proven theorems. Most often the geometric proof i s written in two-column form with statements on the l e f t and a reason for each statement on the right. In order to put the subject of proof in perspective i t i s necessary to look at the historical development. The synthetic methods of Euclid existed from approximately 325 B. C. u n t i l the seventeenth century when Descartes f i r s t used numbers in the study of geometry. This new approach became known as analytic geometry. Some flaws were noted in Euclid's axioms but were corrected. As these corrections were beyond the comprehension of the average secondary school student, the geometric postulates were modified to make them more understandable. Thus, this modified form of Euclidean geometry became the basis of the current geometry course. Other non-Euclidean geometries have been developed, some of which are "affine, projective, hyperbolic, e l l i p t i c , combinatorial, absolute, analytic, differential, algebraic, Minkowskian, integral, transformation, vector, linear, topological, conformal, r e l a t i v i s t i c , optical, and so forth" (Fehr, 1972, p. 152). Despite these additions, the high school geometry course is s t i l l mainly Euclidean. "The treatment of geometry in the high schools today i s remarkably similar to the Euclidean model set down more than twenty-three centuries ago." (Eccles, 1972, p. 103) Brumfiel (1973) concurred with this statement but he gave the reason being that "no one has found better proofs" (p. 95). - 10 -Many students enrolled in geometry courses have had d i f f i cu l ty with the concept of proof and have ended up d i s l ik ing geometry as a whole. In the eyes of the student "geometry" has become synonymous with "proof" which is understandable when students spend so much time in grade ten geometry doing two-column proofs, many of which are self-evident. Because of the preoccupation with rigor, students are forced to write down every step along with an associated reason. The average student gets lost in the myriad of symbols and steps. An example of such a textbook proof appears in Table 2.1 (Usiskin, 1980, p. 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 = Z.4 PROVE: ll PROOF: Statements Reasons 1. £L i s a complement of L2; 13 i s a complement of Z4. i . Given 2. mil + ml2 = 90; mL3 + ml4 = 90 2. Def. of comp. angles [1] 3. mil + mL2 = m{3 + mlA 3. Substitution princ . [2] 4. 12 = 14 4. Given 5. mL2 = mZ.4 5. Def. of = angles [4] 6. m£L + mLZ = m/3 + m/2 6. Substitution princ . [3,5] 7. mZl = m^ 3 7. Add. prop, of equality [6] Def. of = angles [7] 8. n =13 8. - 11 -Students in 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 characteristic of Piaget's formal operational stage. However, the results from various tests measuring cognitive development indicate that a minimum of 30% of these students reason at the concrete operational level while another 30 - 40% of the students are transitional reasoners (Farrell, 1987) . Carpenter, Lindquist, Matthews, & Silver (1983) found in their report of the results from the third mathematics assessment of the National Assessment of Educational Progress for 13 and 17-year olds, that students are f a i l i n g when mathematical reasoning and understanding are required. "The problem is particularly c r i t i c a l in high school geometry, where success depends on propositional thinking and deductive reasoning about geometric properties and relations." (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 for students to learn. Usiskin (1980) suggested that the amount of time spent on proofs be reduced and that many theorems of lesser importance be deleted. These suggestions have been ignored. "The concept of proof in mathematics w i l l always be important whatever may be the nature of the curriculum." (Lovell, 1971, p. 66) Given that this prediction 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 detail in the proof illustrated in 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 similar proof but from a more recent textbook (Jurgensen, Brown, & Jurgensen, 1985, p. 41) appears in Table 2.2. It two angles are supplements of congruent angles (or of the same Table 2.2 The Supplementary Angle Theorem (a) 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 Reasons 1. 0- and Ll are supplementary; 1. Given Z3 and £4 are supplementary. 2. m/l + m/2 = 180; 2. Def. of supp. angles mZ3 + m/4 = 180 3. mil + m/2 = ml3 + mU 4. m/2 = mZ4 5. m/l = m/3 3. Substitution prop. 4. Given 5. Subtraction prop, of = - 13 -Table 2.3 contains an even more recent textbook (Kelly, Alexander, & Atkinson, 1987, p. 354) example. The examples in Tables 2.2 and 2.3 show proofs of exactly the same theorem. Table 2.3 The Supplementary Angle Theorem (b) If two angles are equal, their supplements are equal. Suppose there are two doors in 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. We can use deductive reasoning to explain why/0 ={0-Since l_Q and /@ are supplementary: LO +£(D 180 0 (2) = 180° - / © Since £(5)and j(Qare supplementary: LO +/0 [0 180 0 180° -[(3) [1] [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)' The re fore ,£2) =/|3-- 14 -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 traditional Euclidean based geometry. Fehr (1972) f e l t that Euclid's geometry played a very minor role in accomplishing the goals of geometric instruction. He advocated that geometry "be conceived of as a study of spaces" (p. 152) integrated into the curriculum and taught every year from grade seven to grade twelve. In 1973 Brumfiel reported on a study he did in 1954 and repeated again in 1973. He was curious about students' understanding of the axiomatic structure after they l e f t high school. "Students of 1954 who studied an old-fashioned hodgepodge geometry had no conception of geometric structure. Students of today who have studied a tight axiomatic treatment also have no conception of geometric structure." (p. 102) Is the emphasis on axioms in school geometry a waste of time? Usiskin (1980) also noted that the reason given for 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 prior to their writing of a proof. "In contrast, geometry students seldom explore and almost always are told what they should prove." (p. 420) Hoffer (1981) c r i t i c i z e d the high school geometry course for putting too great an emphasis on developing the s k i l l of writing - 15 -proofs. "When this occurs, precious class time is taken from providing students with experiences in other, possibly more practical, s k i l l s of a geometric nature." (p. 14) Grunbaum (1981) fel t that there was only a pretense to teach the "classical" Euclidean geometry when, in fact, the geometry being taught was "rather misleading" (p. 235). According to Driscoll (1982) " . . . proof has been touted as a means to discipline the mind, to think in an orderly fashion, as a vehicle for improving logical thinking, and as a stimulus toward the kind of responsible, c r i t i c a l and reflective thinking that should be the mainstay of democratic l i f e . " (p. 155) But does proof really promote deductive thinking? Senk (1985) questioned the value of teaching the traditional 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 illustrate mathematics as an axiomatic system, students come to the conclusion that axioms, theorems, and proofs solely belong to this area. Geometry textbooks contain l i s t s of postulates and theorems. New ones are even being created, for example, Pasch's axiom. Niven (1987) posed the questions, "Are we not in danger that the students w i l l see geometry as just so much nitpicking? Why should the f i r s t course in 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 Euclid stays In 1973 Gearhart surveyed a random sample of 999 secondary school mathematics teachers from across the United States to find out - 16 -their thoughts on the geometry course. Over half of the teachers disagreed that the course should be more informal and less rigorous; 76% agreed that learning to write proofs was important for high school students; and over half agreed that the course should be based on Euclid's development as found in standard textbooks. Thus, this survey indicated support for the status quo in the geometry course. Similarly, in 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 logical thinking a b i l i t i e s ; - to develop spatial intuitions about the real 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 explanations for the d i f f i c u l t y with proofs and suggestions for overcoming them Lester (1975) was convinced of the importance for students to develop an a b i l i t y to write proofs correctly. If students were to be properly prepared for this task, Lester fel 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 his study. In his research of the literature 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 is no relationship between age and logical reasoning and on the other hand, - 17 -that logical reasoning improves with age. For his study Lester selected four groups of subjects, 20 in each group. The groups consisted of students from grades 1-3, 4-6, 7-9, and 10-12. His subjects a l l interacted with a computer terminal in a game setting where they were asked to supply proofs of "theorems". The resulting data from Lester's research indicated that "certain aspects of mathematical proof can be understood by children nine years old or younger. Perhaps children may be able to deal with formal operations at an earlier age than proposed by Piaget." (p. 23) First-year students at the University of Oregon are asked to l i s t their favorite and least favorite high school subjects. Hoffer (1981) reported that "the subject that was almost universally disliked was geometry" (p. 11). He suggested that formal proofs should not be introduced early in the course as students may not have reached the formal operational level of development. He recommended spending a good portion of time "exploring geometric concepts informally, without requiring proofs. This enables students to study what they c a l l 'fun things' while preparing for more formal aspects in the second half of the course.", (p. 18) Prior to 1980 research had been limited in this 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 in writing proofs. Thus, in 1981 the Cognitive Development and Achievement in Secondary School Geometry (CDASSG) Project commenced to organize research in this area. The project was designed to address a variety of questions but Senk (1985) reported specifically on the question, "To what extent do secondary school geometry students in the United States write proofs - 18 -similar to the theorems or exercises in commonly used geometry texts?" (p. 448) . A total of seventy-four geometry classes from eleven schools in five states were involved. A proof test consisting of six 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 auxiliary lines or more than one deduction. - only 30% master proofs similar to the theorems and exercises in standard textbooks. These data indicate a low level of achievement in writing proofs but perhaps the reason for this i s lack of practice. Most mathematical s k i l l s that are acquired have been practised for a length of time at various grade levels. "In contrast, the typical high school mathematics program provides v i r t u a l l y no opportunity for students to practice writing proofs in any context outside the geometry course." (Senk, 1985, p. 454) To overcome this weakness, Senk proposed that more effective ways must be identified for teaching proof. She specifically recommended: - more attention be given to teaching students how to start 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 in a proof (p. 455). - 19 -Brown (1982) noted that students entering geometry find the subject quite different 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 different 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) in order to arrive at some conclusion for 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. Criticism has been levelled at the elementary school for not teaching sufficient informal geometry to better prepare students for writing proofs. The CDASSG Project "confirms the need for systematic geometry instruction 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 in 1987 Usiskin s t i l l bemoaned the fact that there was no geometry curriculum in the elementary school to prepare students entering high school for Euclidean geometry. To combat some of the d i f f i c u l t i e s encountered by students in the high school geometry course, Prevost (1985) suggested that geometry in the junior high should be an integral part of mathematics rather than a single chapter in a whole year's study. He also championed the cause for a manipulative approach to geometry. He - 20 -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 unified approach to secondary mathematics where geometry, algebra, and analysis would be integrated throughout the entire curriculum. However, f a i l i n g this 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 last part of the course being devoted to writing proofs. The logical 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 their thoughts to construct a deductive sequence of steps. "To deal directly and e x p l i c i t l y with the organization of students' thought patterns and their construction of logical arguments" (p. 47), Dreyfus and Hadas (1987) developed a set of methods and new curriculum materials. To test the effectiveness of this course, twenty-two experimental classes from fifteen different schools and ten control classes from other schools were selected. The results indicated that the students using this new geometry course increased their a b i l i t y to reason logically within a geometric context somewhat more than the students using the traditional approach. Because of a history of poor achievement, only about one-half of the secondary population enrolls in the geometry course and of these only about one-third really understand i t (Usiskin, 1987) . Consequently, approximately one-sixth of high school students - 21 -are proficient in writing proofs. Various suggestions have been given about how to improve the situation, 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 definitions." (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 in mathematics. In summary, the literature contains two major criticisms of the current way that geometry i s taught and organized: 1) students are not "ready" for geometry and, 2) the method of instruction with i t s heavy dependency on writing proofs does not allow students to discover geometric relationships upon which to base their deductive reasoning. In the present study, the researcher depended upon the van Hiele theories to assess readiness and progress in geometric thinking, and the Geometric Supposer software to offer students some discovery experiences in geometry. Literature Relating to the van Hiele Theories Background Two Dutch mathematics teachers, Pierre van Hiele and his late wife, Dina van Hiele-Geldof, became troubled about their students' - 22 -d i f f i c u l t i e s in learning geometry. From their concerns they developed a theory in 1957 involving levels of thinking in geometry. They surmised that these levels could be used to explain why students have d i f f i c u l t i e s in geometry. "They believed that high school geometry involves thinking at a relatively 'high' level and that many students have not had sufficient experiences in thinking at prerequisite 'lower' levels." (Fuys, 1985, p. 448) Between 1960 and 1964 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 their own mathematics curriculum accordingly. In 1973 Professor Hans Freudenthal wrote about Dina van Hiele-Geldof's experiments. Wirszup, an American, became acquainted with the work of the van Hieles and the way the Soviets had applied i t . Wirszup was the f i r s t to introduce the van Hiele theory to the United States in 1974 when he presented a paper at the Annual NCTM (National Council of Teachers of Mathematics) Meeting. Despite this early introduction, i t i s only recently that the theory has gained more popularity, possibly because English translations of their original work are now appearing. Description The van Hiele model actually consists of three components: the thought levels, the properties of the levels, and the phases of learning. Table 2.4 illustrates how these components are interrelated. The t h o u g h t l e v e l s Five levels of geometric thinking were proposed by the van Hieles. Each level describes certain characteristics of the thinking process. "These levels are inherent in the development of the - 23 -thought processes. The development which leads to a higher geometric level proceeds basically under the influence of learning and therefore depends on the content and methods of instruction. However, no method not even a perfect one, allows the skipping of levels." (Wirszup, 1976, p. 79) The van Hieles began with the basic level, level 0, and ended with level 4. (Different numbering systems may be found in the literature.) "According to the van Hieles, two major factors that determine a student's level are a b i l i t y and prior geometry experiences." (Fuys, Geddes, & Tischler, 1988, p. 12) Table 2.4 The van Hiele Model PROPERTIES OF THE LEVELS Sequential Advancement Adjacency Linguistics Mismatch THOUGHT LEVELS Recognition Analysis Informal Deduction - Formal Deduction - Rigor PHASES OF LEARNING* Inquiry Directed Orientation Explanation Free Orientation Integration Note: van Hiele (1984) suggested that students move through the phases each time they advance a level. - 24 -The following i s a description of each of the thought levels: Level 0 (recognition). At this level students perceive the geometric figure in 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 different kinds of figures. Level 1 (analysis). On this level students "become aware of the properties of geometric figures by a variety of ac t i v i t i e s such as observation, measuring, cutting, and folding" (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 this level, interrelationships between figures are s t i l l not seen, and definitions are not yet understood." (Crowley, 1987, p. 2) Level 2 (abstraction or informal deduction). Students at this level can "establish relations among the properties of a figure and among the figures themselves. At this level there occurs a logical ordering of the properties of a figure and of classes of figures. The pupil i s now able to discern the possi b i l i t y of one property following from another, and the role of definition i s c l a r i f i e d . " (Wirszup, 1976, p. 78) Proof i s not understood at this level. 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 of a triangle i s - 25 -equal to 180 . However, they do not recognize the need for rigor 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 literature related to the van Hiele work is the properties of the system of levels. These properties not only describe the relationships between levels but also how a student is affected by his/her placement and movement in the levels. Teachers can use this information to actually plan lessons. Property 1 (sequential). In order to understand geometry, the student must progress through the levels in order. "A student cannot be at van Hiele level n without having gone through level n-1." (Usiskin, 1982, p. 5) Property 2 (advancements . The content and the instructional methods.can affect the progress of a student from level to level. "No method of instruction allows a student to skip a level, some methods enhance progress, whereas others retard or even prevent movement between levels." (Crowley, 1987, p. 4) Property 3 (adjacency). At each level what appears as extrinsic had become intrinsic in the preceding level. In other words, a student at the recognition level perceives figures as is regardless of their properties. - 26 -Property 4 (linguistics). "Each level has i t s own language, i t s own set of symbols and i t s own network of relations uniting these symbols." (Wirszup, 1976, p. 82) For example, a student at level 1 does not realize that a figure can have more than one name—a rectangle i s a parallelogram. Property 5 (mismatch). "If the student i s at one level and instruction i s at a different level, 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 instruction influenced the progress (or lack of) of a student from level to level. They have identified a five phase cycle which they consider as a necessary sequence for students as they progress through the levels. Age or maturation are viewed as minor factors. The five phases of learning are described below: Phase 1 (inquiry). Here the teacher and students discuss the objects of study for this level. 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 is prominent, i s carefully sequenced by the teacher. The teacher i s looking for specific responses from the students. Phase 3 (explanation). At this phase, the students express the results of their manipulations in words. The figures take on geometric properties and the role of the teacher i s to introduce the correct terminology. - 27 -Phase 4 (free orientation). "The student learns, by doing more complex tasks, to find his/her own way in the network of relations (e.g. knowing properties of one kind of shape, investigates these properties for a new shape, such as kites)." (Fuys et a l . 1988, p. 7) Phase 5 (integration). The students now take their newly acquired knowledge and form an overview. "The objects and relations are unified and internalized into a new domain of thought." (Hoffer, 1983, p. 208) When the f i f t h phase has been completed, students have reached a new level 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 in an organized progression. However, the U.S.S.R. did make major changes in their mathematics curriculum based on this work. Research based on the van Hiele theories Despite i t s wide acceptance by the U.S.S.R. in the 1960's, in North America only a limited amount of research on the van Hiele model has been done. It was Wirszup's speech in 1974 followed by his a r t i c l e in 1976 that prompted the American educators to do some investigations of the van Hiele levels. Three major projects have received U. S. federal funding to carry out research on the model. A description of each one along with their results follows. - 28 -The Chicago p r o j e c t : van H i e l e l e v e l s and achievement i n secondary school geometry The purpose o f t h i s three year p r o j e c t (1979-82) was t o address v a r i o u s questions about student achievement i n grade t e n geometry and how t h i s r e l a t e s t o the van H i e l e theory. Approximately 2700 students i n high school geometry courses 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 the study. (Senk, 1985; U s i s k i n , 1982) Two t e s t s were administered near the beginning of the school year (September, 1980) . One of these t e s t s c o n s i s t e d of m u l t i p l e - c h o i c e questions 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 intended t o i n d i c a t e the van H i e l e l e v e l of each student. Near the end of the school year (May, 1981) these students sat the van H i e l e t e s t s again. They a l s o took a st 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 t e s t , d e a l i n g w i t h t h e i r p r o o f - w r i t i n g a b i l i t y . The r e s u l t s show t h a t : - a van H i e l e l e v e l can be assigned t o most students. - these l e v e l s are good i n d i c a t o r s o f student performance both i n p r o o f - w r i t i n g and standard geometry content. - a p p l i c a t i o n of the van H i e l e theory " e x p l a i n s why many students have t r o u b l e l e a r n i n g and performing i n the geometry classroom" ( U s i s k i n , 1982, p. 89). The van H i e l e l e v e l s were low f o r many students e n t e r i n g grade t e n geometry. - over h a l f of the students who e n r o l l i n geometry courses which emphasize proof, experience l i t t l e or no success i n - 29 -writing proofs. - one-third of the students rose one level , one-third rose two or more levels, and one-third stayed at the same leve l . - "The geometry course i s not working for large numbers of students. At the end of their 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 in 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 level and phases of the van Hieles. - to determine whether the van Hiele model describes how sixth and ninth graders learn geometry. - to determine i f teachers of these grades can be trained to identify the van Hiele levels of students and of geometry curriculum materials. - to analyze levels of thinking of the geometric content of several major textbook series. Three modules were developed based on the experiments in Dina van Hiele-Geldof's thesis. These modules were used in the c l i n i c a l interviews involving 16 sixth graders and 16 ninth graders and were intended to fac i l i ta te the students' movement through the lower levels . The students were videotaped as they individual ly worked - 30 -through the modules in six to eight 45-minute sessions. This one-on-one contact provided the researchers with information on changes in a student's thinking within a level or to a higher level. The results of this study f i r s t v e r i fied the existence of each of the properties of the van Hiele model. Next, their results indicated that a range in levels of thinking existed among the sixth and ninth graders (level 0 to level 2). "Findings in this study show that geometry was a neglected part of the school mathematics experiences of many students, and what was taught was often taught rotely or required minimal student explanation." (Fuys et a l . 1988, p. 188) The researchers found that students in these grades do have the potential for level 1 and level 2 thinking. However, various factors were found which limited a student's progress within a level or to a higher level. These factors included: - lack of prerequisite knowledge - poor vocabulary/lack of precision of language - unresponsiveness to directives and given signals - lack of realization of what was expected of them - lack of experience in reasoning/explaining - insufficient time to assimilate new concepts and experiences - rote learning attitude - not reflective about their own thinking (p. 139). This study also found that "preservice and inservice teachers can learn to identify van Hiele levels of thinking in student responses and in text materials" (p. 154) and that such training should be included in teacher preparation programs. - 31 -In their analysis of current K-8 mathematics textbooks, the investigators found them to be written at level 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 level 2 i f they can successfully complete grade 8 with level 0 thinking." (p. 169) The Oregon project: Assessing children's Intellectual Growth in 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 triangles. The tasks were designed to reflect the van Hiele levels and to combine some ideas from Dina van Hiele-Geldof's research with her students. The findings from this project were: - the hierarchical nature of the van Hiele levels 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 in transition from one level to the next. - the movement from one level to the next i s not discrete. "Students may move back and forth between levels quite a few times while they are in transition from one level 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, while trying to communicate, may be at different levels . - that students may be at a geometric level quite different from what their teacher assumes they are. The three projects described above helped to raise the awareness level of the van Hiele theory. This in turn began to answer some of the questions about poor performance in writing proofs. The interest s t i rred by these projects has resulted in several art ic les and dissertations. Mayberry's (1981) dissertation 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 this purpose. She prepared 62 tasks which were used while interviewing 19 preservice teachers. Mayberry's results showed that "the general van Hiele level of the preservice elementary teachers in 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 level sufficient to deal with a tradi t ional Euclidean geometry course. Mayberry also tested the hierarchical nature of the van Hiele levels . Her results ver i f i ed that a student at level n could answer a l l questions at and below level n but none of the questions above that l eve l . Denis also concurred with the hierarchical structure. - 33 -Senk's (1983) dissertation used the same data as the CDASSG project. One of the issues that she addressed was readiness—were students prepared for the proof writing course? She found that the higher the student's van Hiele level was at the beginning of the geometry course, the greater the prospect for success in writing proofs. However, a high van Hiele level does not guarantee success in writing proofs. "Instruction plays a large part in determining which of the students with the basic prerequisite knowledge w i l l eventually be successful on a given topic. For this reason, teachers, curriculum developers, and researchers need to share materials and methods found to be effective in teaching proofs." (Senk, 1985, p. 455) Following on this theme of instruction, Prevost (1985) wrote appealing to teachers to integrate their geometry curriculum into the van Hiele model. Also, in keeping with the theory, he urged teachers in junior high to develop the geometric ideas over time rather than in a concentrated unit . To provide a more effective learning experience for his students in 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 building in 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 in writing proofs, Scally (1987) proposed a LOGO learning environment as a means to provide these experiences at the grade nine level . Students' van Hiele levels were - 34 -obtained by using interview items and tasks based on those developed by Burger and Shaughnessy. A group of ninth grade LOGO students and a group of ninth grade non-LOGO students were interviewed individually 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 levels." (Scally, 1987, p. 51) Overall, the LOGO students made more gains in performing the various tasks at the end of the semester. Yet to be tested is whether the LOGO experience w i l l in fact enhance the students' thought processes in grade ten geometry. Similarly, Battista 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 this hierarchy" (p. 166). They cautioned teachers not to "expect that merely working with Logo automatically moves students into high levels of geometric thought" (p. 167). There needs to be correlations between LOGO act i v i t i e s and curriculum content. They summarized by stating that, "It i s imperative, therefore, that we help students attain high levels 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 particular, the microcomputer could prove to be the best bridge yet between the spatial-visual aspects of geometry and the logico-deductive aspects" (Driscoll, 1982, p. 149). The computer language, LOGO, has been used by others as a vehicle to prepare students for geometry and as an instructional - 35 -aid while teaching the subject. Rather than LOGO, the investigator used the Geometric Supposer program for this purpose in the present study. Literature Relating to the Geometric Supposer Software The Geometric Supposer i s a series of educational software programs, published by Sunburst Communications in 1985, which allow the user to carry out many different geometric constructions and measurements. A more detailed 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 straightedge and compass. These include the construction of triangles as well as the drawing of segments, medians, altitudes, parallels, perpendiculars, perpendicular bisectors, angle bisectors, and inscribed 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 ratio of two areas (p. 2) . "Part of the rationale behind the SUPPOSER was to provide a tool that could help students understand that a picture i s a special case and that examining one picture i s part of a larger process that includes viewing many special cases and not one static example." (Yerushalmy & Chazan, 1987, p. 58) One of the problems in teaching proof writing 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 intuitive understanding of geometric concepts." (Mathis, 1986, p. 45) The researcher's survey has found very few articles and only one study involving the Geometric Supposer software. Reference was f i r s t made to this software in 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 in The Computing Teacher in June, 1986. "The software encourages the higher level thinking s k i l l s involved in formulating and testing hypotheses." (p. 45) The use of the Geometric Supposer was paralleled to that of a science class where the students collect data, conjecture, and generalize. According to Yerushalmy and Houde (1986), using the Geometric Supposer "encourages students to behave like geometers because i t offers a wealth of visual 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, this software allows the experiment to be repeated on similar figures very quickly. Students would not have the time nor the inclination to manually construct counter examples. "Our experience demonstrates that students brought a high degree of enthusiasm to their 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 in the top six for 1987 by Classroom Computer Learning magazine. "This program i s truly a discovery tool that helps the user become an - 37 -active participant in the quest for mathematical knowledge." (p. 20) In the summer of 1986 f i f t y expert high school geometry teachers were brought together in New Jersey for the purpose of looking at new materials and methods related to the f i e l d of geometry. Various materials were developed for distribution. One of the topics dealt with the Geometric Supposer. A series of investigations on triangles and quadrilaterals, based on Polya's model, were produced for 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 in 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 in 1984-85 on inductive reasoning in geometry and the use of the Geometric Supposer. Three geometry classes (83 students) at different sites used this computer software. At each site a comparison class was taught mainly from the textbook using the traditional approach. The goal of this project was to provide students with "an opportunity to experiment with geometric shapes and elements, to move from the particular to 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 instructional approach used i s referred to as guided inquiry. This approach emphasizes a combination of laboratory work and class discussion. From their comments, the teachers involved were not positive - 38 -about the Geometric Supposer. They expressed misgivings, disappointment in student progress, and concern about the time-consuming nature of laboratory work and follow-up. They did note some improvement in students' a b i l i t y to organize data and f e l t that the Geometric Supposer had potential as a diagnostic tool. They also f e l t "that these students did get more out of their Geometry class than they would have done in a traditional class" (p. 40) and that most students had achieved an understanding of the need for a proof. The students, on the other hand, were generally positive about the computer experience. They indicated that the Geometric Supposer was easy to use, added to their understanding, and provided enjoyment when they were successful. The negative aspect for the students was making conjectures. "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 "significantly outperformed the comparison group in their a b i l i t y to develop generalizations, and they were equal to and/or somewhat better than the comparison group in their a b i l i t y to devise informal arguments and traditional formal proofs." (p. 68) In summary, the literature indicates that geometry and the proof writing activity 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 dislike - 39 -geometry and that only a minority gain the kind of understanding their instructors hope to i n s t i l l , writing proofs is considered to be a necessary part of their education. The van Hiele model appears to have potential 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 identified as a specific instrument for allowing a discovery method to make inroads into the traditional proof writing method used in 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 in this chapter. A description of the subjects, the steps taken in the study, and the data collection instruments i s given. The Subjects The community in which this study was carried out is a relatively isolated, northern Canadian town of approximately 4000 residents. The nearest major center is 240 kilometres (150 miles) to the south. However, as this link i s by a well-paved highway, isolation i s not considered to be a c r i t i c a l factor in the study. The community is a service center for a large portion of the northern part of the province. Three small airlines are based there as well as a hospital, a community college, and some government administrative offices. The community services the tourism industry and more recently, considerable mining exploration. On the other hand, some residents, particularly Treaty Indians, s t i l l earn their l i v i n g by trapping and fishing. Thus, despite i t s small size, the community has a wide range of socioeconomic levels. The educational system in the community consists of three schools: a K-5 elementary school, a K-8 Treaty school, and a grade 6-12 school. The latter has a population of 500 students and was the site of the study. The population of this school reflects the social - 41 -makeup of the community which is roughly 55% Native persons and 45% Europeans. As an overview, the mathematics curriculum in this province i s prescribed un t i l grade nine. In grade nine, generally, students have a choice of regular mathematics or general mathematics. Most students opt for the regular mathematics. To graduate from high school, students must have a mathematics credit at the grade ten level. Students can obtain this credit by taking one or more of algebra, geometry, mathematics, or general mathematics. Similarly, in grade eleven the same choices are available. In grade twelve the mathematics courses offered'are algebra, geometry, and mathematics. In the school where this study took place, the students in grade nine had to choose between algebra or general mathematics. Geometry was not an option. Algebra, geometry, and general mathematics were offered in grade ten. In grade eleven an algebra course and a geometry course were included in the timetable choices. This was also the case for grade twelve. Thus, after grade ten, a student could take from zero to four senior mathematics classes. The subjects in this study had opted to take the grade ten geometry course which was scheduled in the second semester of the school year 1987-88 from February unt i l June. Of the 62 grade ten students in 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 -in this study. In the computer group there were 12 males and 10 females, 9 students were of native ancestry. The second class did not use the computer and w i l l be referred to as the traditional group. The traditional 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 in this study had had no geometry exposure since grade eight when they studied i t as one of the chapters in their textbook. During the semester one student from the computer group withdrew from school and one student was added, thus this group remained at 22 students. In the traditional group three students withdrew from school and two others discontinued the geometry course. This l e f t 17 students in the traditional group. Permission was obtained from the parents for the participation of their children in the study (Appendix A). One student in the traditional group was not given permission to participate in the study. This student obtained the highest fi n a l mark in 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 their preassigned homerooms. Formalized random selection was not possible in the school setting. Classical pretest, posttest, and experimental group, control group methodology was followed. In this section a description i s given of the design - 43 -and the 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 of the open-ended c l i n i c a l i n t e r v i e w s that were c a r r i e d out. Quasi-experiment This study was a quasi-experimental i n v e s t i g a t i o n . The researcher gathered data from two groups of students who were e n r o l l e d i n the grade 10 geometry course. The c u r r i c u l u m content, as p r e s c r i b e d by the p r o v i n c i a l Department of Education, was the same f o r both groups. The d i f f e r e n c e between the two groups was that one c l a s s (computer group) used a computer program, the Geometric Supposer, throughout the course while the second c l a s s ( t r a d i t i o n a l group) d i d not use the computer. The geometry c l a s s e s were both scheduled 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 the a d m i n i s t r a t i o n was unable t o r e -schedule, i t was impossible f o r the researcher t o teach both groups. The researcher taught the computer group while another member from the mathematics department taught the t r a d i t i o n a l group. The researcher had a B.A. degree, a P r o f e s s i o n a l A Teaching C e r t i f i c a t e , and 22 years t e a c h i n g experience. The teacher of the t r a d i t i o n a l c l a s s had a B.Ed, degree, a B.Sc. (Honors i n Geology) degree, eight years experience i n e x p l o r a t i o n and mining p l u s four years teaching experience. Both teachers had taught the geometry course p r e v i o u s l y and kept constant contact throughout the course regarding the cu r r i c u l u m content and expectations of the students. The Geometric Supposer The computer program, the Geometric Supposer, was used as the treatment i n t h i s study. The Geometric Supposer i s a s e r i e s of software programs e s p e c i a l l y designed t o provide a "playground" i n - 44 -which students can experiment with geometric figures and form conjectures. The two Supposer programs used in this 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 in Figure 1. Press M to besin 2 Label 3 Erase K Heasure S Scale change K Repeat N Mey trian«le Figure 1. The opening screen of the Geometric Supposer Triangle program Each screen of the Supposer is divided into three parts—the l e f t column i s for data recording, the right side i s the area for constructions, and the region below the horizontal line i s for menus and prompts. After.N (New triangle) i s pressed the user i s presented with a new menu-: 1 Right 2 Acute 3 Obtuse 4 Isosceles 5 Equilateral 6 Your own Depending on the type of triangle the user wishes to experiment on, - 45 -s/he would select accordingly. If #1 was chosen the screen as shown in Figure 2 would appear. Figure 2. The Right Triangle The triangles do not appear exactly the same each time selected. Assuming a student had been assigned to investigate altitudes in triangles, s/he would have made menu selections to display such diagrams as illustrated in Figure 3 or Figure 4. Figure 3. An acute triangle Figure 4. An obtuse triangle with altitudes. with altitudes. Then using the measurement function (M), the program offers a choice of: - 46 -1 Length 5 Distance P o i n t - l i n e 2 Perimeter 6 Distance L i n e - l i n e 3 Area 7 Adjustable element(s) 4 Angle At t h i s p o i n t , the user may decide t o f i n d out what r e l a t i o n s h i p e x i s t s between angles and. a l t i t u d e s . Option #4 allows any angle on the screen t o be measured. The program requests the name of the angle, u s i n g three l e t t e r s . In the case of the acute t r i a n g l e (Figure 3), i f ADB was entered, the program would respond w i t h ZADB = 90 as shown i n Figure 5. The user would continue t o "measure" u n t i l s a t i s f i e d . Data £ADB = 90 1 Draw | Label 3 Erase n Heasure S Scale change ft Repeat M Heai triaogle Figure 5. Information recorded on angle measurement. The above description provides an overview of some of the capabilities of this program. The constructions and measurements are performed quickly by the program giving almost instantaneous feedback to the user. Use in the study The 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 definitions for the different types of triangles. Following this group exposure, the students worked in pairs on specific assignments. In a typical period in which the Geometric Supposer was used, the instructional period would be divided into four sections. F i r s t , the objective would be defined. For example, the task might be to explore the relationship among the interior angles in different types of triangles. Secondly, the students would carry out a pencil, paper, protractor exercise on this topic. Thirdly, in pairs, the students would work on the computers using the software to further explore interior angles in a l l types of triangles. As part of this assignment, they would record their observations and after discussions with their partner, write out their conjectures. Fourthly, the class would recongregate and discuss their findings. Depending on the nature of the objectives, a theorem or definition would emerge. Without this software, students would have had to use paper and pencil constructions exclusively to explore the various triangles. This process would be tedious, time-consuming, and result in frustration and/or boredom for the students. The class used the Geometric Supposer software in 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 -test and a test to measure geometric thought levels of students—the van Hiele geometry test. Introductory geometry test (Note: The province in which this study was carried out refers to i t s geometry course as "geo t r i g . " This "geo t r i g " course contains the same content and uses similar textbooks as other provinces and states in their f i r s t year geometry courses. Hence, the test referred to in this chapter as the introductory geometry test, appears in Appendix B under the t i t l e , Introductory Geo Trig 10 Test.) The f i r s t test given, the introductory geometry test (Appendix B), was created by the researcher. 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 an assessment of face validity, the introductory geometry test was examined by a grade eight mathematics teacher who approved the reasonableness of the content. However, he f e l t that due to the lapse of nearly two years (three years for the grade elevens) since they had studied this material, the students would not do well. After a preliminary analysis, one question was removed from the test as only one of the forty students answered i t correctly. This reduced the test to 24 questions from the original 25. The r e l i a b i l i t y of this test i s discussed later in this chapter, v a n H i e l e g e o m e t r y t e s t Based on writings of the van Hieles, Usiskin (1982), in the Cognitive Development and Achievement in Secondary School Geometry (CDASSG) Project, developed, piloted, and modified,test items to be - 49 -used to determine geometric thought levels of students. These levels were referred to as the van Hiele levels of the students in the CDASSG Project. Permission was granted by Professor Zalman Usiskin, Department of Education, University of Chicago, to use the van Hiele geometry test for this study (Appendix C). Posttests Three weeks prior to the end of the five 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 later an attitude test was administered to both groups, again, at the same time. v a n H i e l e g e o m e t r y t e s t This test was identical 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 for the present study as they were either unfamiliar to the students or were identical to what had been covered in class. However, the researcher followed the format of Senk's tests to create her own proof test (Appendix D). Attitude test Similarly, Aiken (1963, 1974) developed attitude tests related to mathematics. Since Aiken's test items did not deal specifically with geometry, a new test (Appendix E) had to be designed by the researcher. - 50 -Interviews The researcher carried out individual interviews with students from the computer group. Eight interviews were scheduled with each of four students (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 in the morning before school started, at lunch hour or after school, depending on which time was convenient for the student. Two students from the tradit ional group (one male, one female) were each interviewed twice in the latter half of the semester. Experimental Controls In an attempt to reduce uncontrolled factors in this study, a l l tests were administered at exactly the same time to both groups. This was done to avoid students obtaining any prior knowledge of the test questions. The teacher of the tradit ional 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 in rooms some distance from each other. The tradit ional group was thus unaware of when the computer group was actually using the computer lab. In this study the teacher of the tradit ional group had taught the majority of his students the previous semester in algebra. Thus, he was aware of their mathematical strengths, weaknesses, and personality t r a i t s . The researcher had been away from this school on secondment for two years and did not know the students. Therefore, - 51 -the researcher was not biased by previous knowledge about the computer group. Originally, the researcher had proposed to make formal observations during class time using comment cards, checklists, and/or rating scales. The researcher did try this on several occasions but found i t was not feasible to combine these activities with teaching responsibilities. Data Collection 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 total of five tests were administered to both groups of geometry students—two at the beginning of the semester and three at the end. 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 in grade eight. The questions for this test were based on the geometric material they covered in their grade eight textbook. The researcher developed the test to cover only the main ideas with l i t t l e emphasis on details. A second purpose for the test was to establish whether any - 52 -significant difference existed between the mean scores of the two groups. The test originally 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. Thus, a student could achieve a maximum score of 24 on this test. Thirty-five minutes were allocated for the test but the majority of students in both classes were finished within half an hour. The results of an item analysis of the introductory geometry test can be found in 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 test. 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 in the students' van Hiele levels take place after a semester of geometry? - Did the change in the van Hiele pre and postlevels of the computer group differ significantly from those of the traditional group? This test, developed by the CDASSG Project, consists of 25 multiple-choice questions divided into five levels, with five questions at each level. 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 student's van H i e l e l e v e l : 3 out of 5 c r i t e r i o n and 4 out of 5 c r i t e r i o n . In t h i s study the 3 out of 5 c r i t e r i o n was used. This r e q u i r e s at l e a s t 3 out of the 5 questions c o r r e c t at a l e v e l i n order t o be assigned t h a t l e v e l . For example, i f a student had at l e a s t 3 out of the 5 f i r s t questions (items 1-5) c o r r e c t and at l e a s t 3 out of the 5 questions (items 6-10) c o r r e c t and l e s s than 3 questions c o r r e c t i n each of the remaining c a t e g o r i e s , s/he would be assigned a van H i e l e l e v e l 2 (informal deduction). I f , on the other hand, a student had met the c r i t e r i o n (a minimum of 3 out of 5 c o r r e c t ) f o r items 1-5, 6-10, and 21-25, t h i s student would a l s o be assigned a van H i e l e l e v e l 2. The reason f o r the same van H i e l e l e v e l i s t h a t according to Property 1 of the van H i e l e l e v e l s , a student at l e v e l n must have met the c r i t e r i o n at l e v e l s below n but not above n. In t h i s case, the student has s a t i s f i e d the c r i t e r i o n f o r l e v e l s 1 and 2, then jumped t o 5. Although the van H i e l e t e s t may be "a r a t h e r crude device f o r c l a s s i f y i n g students" (p. 30) i n t o l e v e l s , 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 behavior and t r e a t i n g students" (p. 34). In order t o a s s i g n more students a van H i e l e l e v e l , the CDASSG P r o j e c t developed a schematic d e s c r i p t i o n of the 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 that do not comply w i t h Property 1. The v a r i o u s p o s s i b i l i t i e s are each given a f o r c e d van H i e l e l e v e l . Forced van H i e l e l e v e l s were used i n t h i s study i n order t o make comparisons i n movements of students from one l e v e l t o another. The 4 out of 5 c r i t e r i o n was not used because a g r e a t e r percentage of students would have been pla c e d at l e v e l 0 (22% as - 54 -opposed to 6% on the 3 out of 5 criterion). One of the purposes of this study was to observe the students' movement between levels. "If weaker mastery, say 80%, i s expected of a student operating at a given level, then i t i s absolutely necessary to use the 3 of 5 criterion, for Type II errors with the stricter criterion are much too frequent." (p. 24) The CDASSG Project also used two theories in assigning levels: the classical theory and the modified theory. Here again, the classical theory employs a more r i g i d method of assigning levels. In the example above, the student satisfying the criterion for levels 1, 2, and 5 would have been assigned a van Hiele level of zero. The researcher used the modified theory since i t resulted in more of the students being assigned a van Hiele level greater than zero. For a more detailed description of assigning van Hiele levels, the reader should refer to Usiskin, 1982. The CDASSG Project found that the van Hiele geometry test could be used to measure changes in students after a year of geometry. They also found that the "van Hiele levels are a very good indicator when i t comes to predicting success on proof" (p. 51). Proof test (Appendix D) The purpose for 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 in this area. The test contained six questions resembling those found in the students' geometry textbook (Jurgensen et a l . 1985) and used a similar 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 in the - 55 -missing statements or reasons of the proof. The second, fourth, f i f t h , and sixth questions required the students to write f u l l proofs. The third question required a translation 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 traditional group examined the test to ensure that none of the problems had previously been attempted in class. Thirty-five minutes (the same time as the CDASSG Project) were allocated for the test. 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 invalid or useless deductions. 1 - Student writes at least one valid deduction and gives reason. 2 - Student shows evidence of using a chain of reasoning, either by deducing about half the proof and stopping, or by writing a sequence of statements that i s invalid because i t is based on faulty reasoning early in the steps. 3 - Student writes a proof in which a l l steps follow logically but in which errors occur in notation, vocabulary, or names of theorems. 4 - Student writes a valid proof with at most one error in notation (Senk, 1985, p. 449). The researcher marked a l l the proofs using this scale. Each test had a cover sheet with the student's name. Prior to any marking taking - 56 -place, both groups of tests had their cover sheets turned. After this, the tests were marked. 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 blindly scored. Since questions #1 and #3 were of a different type than the other four questions, two measures of overall achievement are given. The Hoyt estimate of r e l i a b i l i t y on a l l six questions was .80 and an item analysis appears in Appendix D. 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 16. The Hoyt estimate of r e l i a b i l i t y on this section was .74. A similar six-item proof test was given to 1520 students in the CDASSG Project and formed the basis of conclusions about students' readiness for geometry. Attitude test (Appendix E) The attitude test was given to measure the affective objectives of mathematics instruction such as attitude, value, and enjoyment. The purpose for giving this test was to analyze any significant difference in 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 in the direction of a favorable attitude and the remaining half are in the direction of an unfavorable attitude. The test can be divided into three sections: (1) questions 1-10 relate specifically 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 this test. The second part of the test requested written "likes" and "dislikes" of the geometry course. The purpose of these general questions was to obtain "gut" reactions about what stood out in the students' minds as positive 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 positively, 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 for each group i s in Appendix E. The Hoyt estimate of r e l i a b i l i t y for the 28 questions was .93 with the estimates for each of the three sections being .91, .94, and .74. Interviews The purpose for 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 in their classroom work. Four students were selected from the computer group. The interviewees consisted of two males and two females. One male (Scml) had a van Hiele level 1, the second male (Scm2) was a level 3, one - 58 -female ( S c f l ) was l e v e l 0, and the second female (Scf2) was l e v e l 1. The researcher had a l s o d e r i v e d the van H i e l e l e v e l s f o r each student u s i n g the 4 out of 5 c r i t e r i o n . These l e v e l s were compared w i t h t h e i r l e v e l s from the 3 out of 5 c r i t e r i o n . The l e v e l s f o r Scm2 were 3 on the l a t t e r c r i t e r i o n and 0 on the s t r i c t e r c r i t e r i o n . The l e v e l s f o r t h i s student v a r i e d the most i n both groups (computer and t r a d i t i o n a l ) and hence, the researcher s e l e c t e d him f o r the in t e r v i e w s i n order t o reso l v e t h i s quandary. The researcher 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 biweekly b a s i s . The sessions were h e l d i n an o f f i c e which provided p r i v a c y . Each s e s s i o n was audio-taped and l a t e r t r a n s c r i b e d . At f i r s t , the students were nervous w i t h the tape recorder. The beginning sessions mainly d e a l t w i t h homework problems. In the l a t e r sessions each student s o l v e d the same geometry pr o o f s . One of the male students (Scml) was not c o n s i s t e n t i n at t e n d i n g the s e s s i o n s , consequently, only s i x i n t e r v i e w s were h e l d w i t h him before the end of semester. A f t e r four sessions i t was obvious that Scm2 had a sound grasp of the geometric concepts i n v o l v e d and was able t o q u i c k l y analyze and w r i t e a proof. Hence, sessions w i t h him were unproductive i n terms of the study. At t h i s p o i n t , another male student (Scm3) with a van H i e l e l e v e l 1 was s e l e c t e d , and he attended four i n t e r v i e w s . To ob t a i n some in-depth i n f o r m a t i o n regarding proof w r i t i n g from the t r a d i t i o n a l group, two students 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) with l e v e l 1. These two students were each i n t e r v i e w e d twice and worked through some of the same - 59 -proofs that the computer group had had. A summary of the number of interviews held with each student follows: Scml - 6 sessions Scfl - 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 identified in this chapter, the quasi-experimental model does appear to be appropriate for this investigation. - 60 -CHAPTER 4 DATA ANALYSIS The r e s u l t s obtained from the f i v e t e s t s and the i n t e r v i e w s are presented i n t h i s chapter. The f o l l o w i n g questions were r a i s e d : 1) W i l l the students who use the Geometric Supposer software be b e t t e r able to w r i t e formal proofs than students who are taught by more t r a d i t i o n a l methods? 2) What changes i n the students' van H i e l e l e v e l s take place a f t e r a semester of geometry? 3) W i l l the students who receive the treatment have a more p o s i t i v e a t t i t u d e towards geometry than the students i n the t r a d i t i o n a l group? Assessment of the Groups Did the geometric knowledge of the two groups (computer and t r a d i t i o n a l ) d i f f e r at the beginning of t h i s study? The instrument used t o answer t h i s question was the i n t r o d u c t o r y geometry p r e t e s t (Appendix B). An independent t - t e s t was used t o analyze the data i n v o l v i n g the two independent groups. The t e s t i s a v a i l a b l e i n the SPSSX package on the U. B. C. computer system. The f i n d i n g s , as shown i n Table 4.1, i n d i c a t e 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 d i f f e r e n c e between the two group means at the .05 l e v e l of s t a t i s t i c a l s i g n i f i c a n c e i n terms of geometric knowledge. The independent t - t e s t i n v o l v e d the data from a l l f o r t y students who o r i g i n a l l y wrote the i n t r o d u c t o r y geometry t e s t . The - 61 -mean scores were 14.95 and 13.16 for the computer and traditional groups respectively. The marks on this 24 item test ranged from 5 to 24. An item analysis of this test for each group i s presented in Appendix B. During the semester four students l e f t the traditional group, thus a second t-test was run without the scores of those four students. Again, the result was similar in that no s t a t i s t i c a l l y significant difference was revealed (t = 1.51; p = .141). Thus, the two groups were accepted as being from the same population and appropriate for this study. Table 4.1 Means, Standard Deviations, and Statistical Comparison of Groups: Introductory Geometry Pretest Groups n Mean SD t-value p Computer 21 14.95 4.73 1.40 .169 Traditional 19 13.16 3.10 Geometric Thought Levels Prior to Treatment While the f i r s t data analysis was concerned with the actual performance in 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. These levels represent rankings and cannot be used as scores. 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 in Table 4.2. A significance level of .05 was used. Table 4.2 Median Test: van Hiele Pretest Computer Traditional Combined Group Group Number greater than median 11 4 15 Less than or equal to median 1Q. 11 21 Total 21 15 36 (p = .230) Changes in Geometric Thought Levels What changes in the students' van Hiele levels take place after a semester of geometry? The instruments used to answer this question were the pre and post van Hiele geometry tests. The sign test, another nonparametric test which is used to measure the same sample on two different occasions when i t i s suspected that changes are taking place, was applied to the data. The results, as shown in Table 4.3, indicate that there was a s t a t i s t i c a l l y significant difference at the .05 level between the pre and post van Hiele levels for students in both groups. Therefore, the null hypothesis, there was no significant difference in 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 geometric thought l e v e l s f o r both groups d i d improve a f t e r a semester of geometry. A median t e s t was a p p l i e d t o the data from the van H i e l e p o s t t e s t . The r e s u l t of t h i s t e s t was that no s i g n i f i c a n t d i f f e r e n c e i n the rankings between the computer group and the t r a d i t i o n a l group e x i s t e d at the .05 l e v e l of s i g n i f i c a n c e (p = .569). Table 4.3 Sign Test: Pre and Post van H i e l e Test Data Computer Traditional Combined Group £r&u£ No change i n van H i e l e l e v e l 6 6 12 Decrease i n van H i e l e l e v e l 1 1 2 Increase i n van H i e l e l e v e l 14. a 22 T o t a l 21 15 36 (p = .0010) (p = .0391) In the Cog n i t i v e Development and Achievement i n Secondary School Geometry (CDASSG) P r o j e c t i t was found that a f t e r a year's study of geometry approximately, "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 or more l e v e l s ; the f i n a l t h i r d e x h i b i t s 'no growth', s t a y i n g the same or decreasing t h e i r l e v e l . " ( 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 of geometry, i n the computer group o n e - t h i r d stayed the same or decreased a l e v e l , 43% increased one l e v e l , and 24% increased two l e v e l s . In the t r a d i t i o n a l group 47% stayed the same or decreased a l e v e l , 20% increased one l e v e l , and o n e ^ t h i r d increased two or 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 significant difference, at the .05 level, between the means of the two groups as reported in Table 4.4. A second t-test was done involving only the questions which required writing proofs—four questions (#2, 4, 5, 6). Again, there was a significant difference between the means of the two groups (Table 4.4). Therefore, the null hypothesis, there was no significant difference between the means of the written proof tests of students in the computer group and those in the traditional group, was rejected. Table 4.4 Means, Standard Deviations, and Statistical Comparison of Groups: Proof Test Groups n Mean SD t -value p PART A*: Computer 22 11.00 5.46 2.89 .006 Traditional 16 6.00 4.97 PART B (Proof Writing): Computer 22 Traditional 16 PART A: 6 questions (24 marks) PART B: 4 questions (16 marks) 7.73 4.04 3.84 < .001 2.94 3.44 - 65 -Out of a possible 24 marks on the six questions, the computer group's results ranged from 2 to 23 while the traditional 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 traditional group could identify the alternate interior angle. On the diagram drawing question (Appendix D, question #3) 57% of the computer group and 38% of the traditional group could draw the diagram but were unable to proceed any further. An item analysis for both groups on the proof writing test i s in Appendix D. Half of the computer group and one-fifth of the traditional group were able to apply the converse of the Isosceles Triangle Theorem in question #2. Two deductions were required in question #4. Forty-one percent of the computer group and 12% of the traditional group achieved half or more of the marks on this question. Question #5 required the addition of an auxiliary line segment. This question was answered correctly by 59% of the computer group and by 19% of the traditional group. The last question required more than two deductions. No student completed this proof correctly. However, 41% of the computer group and 25% of the traditional group got half or more of the required answer. Considering the four questions that required proof writing collectively, one student (5%) in the computer group received no marks while six students (31%) in the traditional group had the same result. - 66 -Attitudes The attitude test was analyzed using the independent t-test. The stati s t i c s , as reported in Table 4.5, support the null hypothesis at the .05 level of significance. Thus, the null hypothesis of no significant difference in 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 in general, or the value of mathematics) are used. An item analysis for each group along with the test items are included in Appendix E. Also in Appendix E i s a summary comparing the mean score of each item between the computer group, the traditional group, and the total sample. Table 4.5 . Means, Standard Deviations, and Statistical Comparison of Groups: Attitude Test Test A. Geometry: Mean SD Computer Group (n=21) 31.62 7.68 Traditional Group (n=17) 29.06 9.46 t-value p. 0.92 ,363 B. Mathematics: Mean 34.00 SD 8.50 32.35 9.41 0.57 ,575 C. Value of Mathematics: Mean 32.67 SD 4.27 33.24 4.06 -.42 ,679 D. Total Test: Mean 98.29 SD 17.34 94.65 17.42 0.64 ,525 - 67 -The students were asked f o r t h e i r w r i t t e n comments about the course. No prompting was given. Their spontaneous remarks are summarized i n Table 4.6. Table 4.6 Wr i t t e n Comments Computer group T r a d i t i o n a l group L i k e s : - Computer use 33% n/a - Constructions 29% 29% - Proofs 10% 18% - P r o j e c t 10% 6% - Trigonometry 5% 6% D i s l i k e s : - Memorizing 5% 35% - Computer 5% n/a - Constructions 5% 0 - Proofs 24% 35% - Pythagorean Theorem 19% 0 Student suggestions: - Computer group - " I t would be f a s t e r and e a s i e r to j u s t t e l l us." - "More computer use." - T r a d i t i o n a l group - " I hope that computers are used more and more i n the classroom." - two students made t h i s comment. The m a j o r i t y of the w r i t t e n comments were concerned w i t h classroom management funct i o n s and added nothing t o t h i s study. - 68 -Interview Data This section contains a condensed summary of the activities carried 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, definitions, postulates, and theorems were reviewed along with class work and tests. In reporting the findings of the student interviews the researcher used a code to identify each student. The code consisted of four characters. The f i r s t two were either Sc or St to denote the computer group or the traditional group respectively. The third character indicated gender (M or F). The last character is numerical to differentiate among the students. The following are some unexpected student responses from the f i r s t three sessions. After three weeks in the geometry class Scfl asked, "Does a triangle have three sides?" This student was also unable to describe parallel lines 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 in Figure 6 was reviewed. She- approached the problem in the same way each time—with complete naivety. She had no idea that a line represents an angle of 180 degrees even though we had used the protractor to measure i t . It also never occurred to her to use the protractor herself. - 69 -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 identify that £1 = Ll but was unable to proceed to the next step. Given: a//b, Ll Find: Ll = 2x + 5 and Ll = 12x Figure 7. Parallel lines with alternate interior angles. Scm2 liked questions with numbers. When given the example in Figure 8 he replied, "You can't do i t i f i t has letters." Figure 8. Relationship between the exterior angle and the remote interior 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. They would write the "given" and come to a standstill. With a constant flow of directed questions from the researcher, they would eventually solve the proof. When looking back and reviewing the steps, Scfl would say, "It a l l makes sense but I could never do that on my own." Perhaps she was operating in the zone of proximal development, a theory proposed by Vygotsky. The zone of proximal development i s "the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance" (Vygotsky, 1978, p. 86) . Scml could make simple deductions. Scm2 was selected for the interviews because of his inconsistent van Hiele level. 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 in Figure 9. / / J Given: i l = 12, 13 = LA Prove: AB = AD Figure 9. Proving two segments congruent. - 71 -He immediately wrote down the "given" followed by: ABAC ^ LDAC because each triangle has 180° AC = AC because of the reflexive property A ABC = A ADC because of ASA AB *= AD because corresponding parts of congruent triangles are congruent. When he was asked to check the "given" he quickly realized his mistake and rewrote the proof. The researcher fe l t that this student had a van Hiele level much closer to 3 than to 0 and having solved the quandary ceased interviewing him. Another male (Scm3, van Hiele level 1) who was having d i f f i c u l t y with proofs in 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 traditional group were each given the same two proofs to complete during this interview session. The f i r s t proof i s in Figure 10 followed by a summary of the students' responses. The second exercise i s shown in Figure 11 and a similar summary of the students' replies follows. C Given: CX J _ AB, AC ~ BC Prove: AACX = ABCX Figure 10. Proof #1 - 72 -A l l six students immediately wrote the "given" without reading the whole question. Three students each marked their diagrams with what was given. 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 this concept. None of the six students were able to make a deduction from the fact that two sides of the triangle 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 were able to give the correct reasons for these triangles being congruent. However, in a proof writing situation they were unable to relate the written work to their diagram and then draw a conclusion. 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. Scm3 correctly identified the supplementary angle relationships and quickly completed the proof. Here again, the majority of these students required many probing questions in order for them to make deductions from what the question had given and to achieve what was to be proved. premarked figures such as students Given: 4 ADP with AB = DC, PB = PC Prove: A APB ~ ADPC Figure 11. Proof #2 - 73 -Session fi The same six students were each given two additional proofs to complete during this session. Proof #3 appears in Figure 12 and Proof #4 appears in Figure 13. A summary of the students' responses follows each figure. p L T P Given: TP J_ 71, LP 'Bisects FI Prove: Z.FPL = LIBL F i g u r e 12. Proof #3 On proof #3 Scml successfully completed this proof with no assistance. Three students said that LF ~ Ll because LP bisects FI. One student said that /FLP = ZlLP because LP bisects FI. These four students a l l defined bisect as meaning "to cut in half." One student was adamant about LF being congruent to LL u n t i l she, without suggestion from the interviewer, turned the page around. Stm4, despite having sufficient information to prove the triangles congruent, kept coming back to LF and LI. He was determined that they should be equal. Scfl asked i f there was a short way to write "bisects" (like a symbol for perpendicular). She was unable to write any steps after the "given" without being specifically directed. - 74 -5 L I Given LA bisects SI and SA = IA Prove A SAL = UAL Figure 13. Proof #4 On proof #4 a l l six students assumed iALS IkLI because LA bisects SI. Scml reread the "given" and erased his markings on the diagram, then proceeded to correctly complete the proof. Once the questions, "What does bisect mean?" and "What is being bisected?" were asked, the five students then corrected their work. Three students used SAS (Side-angle-side) to prove the triangles congruent and the other half 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 in proofs #3 and #4 prior to the last statement. The students had to be encouraged to mark their diagrams. Once they did this, they found i t easier to determine the next step. There were no further interviews with the two students from the traditional group. Sessions 7 & 8 The four students in the computer group were interviewed on two more occasions. Their responses generally mirrored their progress in class. In the interview situation they experienced the same kind of di f f i c u l t y as they did in class. As an example of this, the following section contains a report of an interview with each of the - 75 -students that focussed on the concept of overlapping triangles as illustrated in Figure 14. \ Y T Given: ^RTP ZXPS, PT ~ SP, IPSO ZTPO Prove: RT XP Figure 14. Overlapping Triangle Proof The students methodically recorded what was given then attempted to mark the diagram. Scm3 was confused between the angles Z.XPS, £PSO, Z.TPO, and ZSPO. Once he had the angles sorted out, he was able to quickly visualize 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 collect 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 levels 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 Usiskin 1s 1982 study. He found that at the beginning of the geometry course "over half of students classifiable into a van Hiele level are at levels 0 or 1." (p. 81) He found that at the end of his study when the geometry course was completed, more students were at level 3 than at any other level. A chi square test of association showed that the two variables (the van Hiele levels and the tests) were related at the .05 level. Table 4.7 Students' van Hiele Levels Percentage of students Level Pretest (n = 36) Posttest (n = 36) 0 6% . 2 3% . . 1 1 53% . . 19 17% . . 6 2 22% . 8 2.8%. . . 10 3 14% . 5 39% . . 14 4 6% . 2 14% . . 5 The pre and post van Hiele levels for each of the students interviewed are shown in Table 4.8. The results from the proof writing test (possible total = 16) are also shown in the table. - 77 -Table 4.8 Interviewees1 Pre and Post van Hiele Scores Levels and their Proof Test van Hiele Levels Proof Test Student Pre Post Scores 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 this chapter the results from the five 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 null hypothesis (There was no significant difference between the means of the written proof tests of students in the computer group and those in the traditional group.) and rejection of the second null hypothesis (There was no significant difference in the rankings of the geometric thought levels between the pre and post van Hiele test results.). The findings indicate support for the third null hypothesis (There was no significant difference in the means between the attitudes of students in the computer group and those in the traditional group,). The data indicates that teaching^geometry, with or without computer programs, does improve students' van Hiele levels. - 78 -CHAPTER 5 SUMMARY AND DISCUSSION The purpose of this 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 this investigation, tests were administered to determine the geometric thought levels, the geometric knowledge, and the attitudes of the students. Data was collected from two groups of students—those using microcomputers and those learning geometry the traditional 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, in increasing the performance level of students in writing proofs. The subjects in this study were a l l the students enrolled in the grade 10 geometry course in one particular high school. One class of these students used the computer program and the other class did 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 prior to the study. At the end of the course the students wrote a proof test as well as an attitude test. - 79 -A series of interviews were carried out with five students from the computer group and two students from the noncomputer group. This was done in order to gain some insight into the methods the students were actually using to write geometric proofs and to identify changes in their approaches. The van Hiele test was developed and tested in Chicago by the Cognitive Development and Achievement in Secondary School Geometry (CDASSG) Project. The Hoyt measure of internal 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 in the geometric knowledge of the two groups (computer and traditional) at the beginning of the study. As was shown in Table 4.7 the van Hiele levels of the two groups both improved after one semester of geometry. There was no significant mean difference between the attitudes of the two groups towards geometry at the end of the course. However, the computer group scored significantly higher than the traditional group on the fi n a l proof test. Interpretation of the Findings The introductory geometry test The findings indicate that there was no significant difference in the geometric knowledge of the two groups at the beginning of the study. The students in this study had received some geometry instruction in grade 8 but no geometry content in grade 9. The study covers the - 80 -geometry they took in the second semester of their grade 10 year. The introductory geometry test was given at the beginning of this semester to assess the level of geometry knowledge of the students entering the grade 10 course. The averages on this test were 62% for the computer group and 55% for the traditional group (Table 4.1). Further, the results of this test are comparable to the results of the entering geometry test given in the CDASSG Project. The mean percentage correct in their study was 54% (Usiskin, 1982, p. 68) . The students in both groups performed poorly on the questions involving the identification of obtuse angles (23% correct) and the calculation of the area of an obtuse triangle (25% correct). However, 85% of the students could calculate the area of a rectangle, 83% could define an equilateral triangle, 93% could identify a reflection point, and 90% could find the volume of a rectangular solid (Appendix B). The van Hiele geometry pretest The van Hiele geometry test was given to assess the geometric thought levels of the students entering the grade 10 geometry course. The results of this test not only supported the hypothesis that there was no s t a t i s t i c a l l y significant difference between the rankings of the geometric thought levels of the students in the two groups (Table 4.2), but also shed some light on the overall poor performance of both groups on the introductory geometry test. The results from this test indicate that 59% of the students entering the geometry course were at level 0 or 1 (Table 4.7) . This data i s consistent with data of the CDASSG Project. Using the same criterion, their results indicated 54% (p. 100) of the students entering grade 10 geometry - 81 -were at van Hiele levels 0 or 1. As indicated earlier, various researchers (Battista & Clements, 1988; Craine, 1985; Scally, 1987; Senk, 1983; Usiskin, 1982; Wirszup, 1976) have a l l discussed the need for students to be at level 3 in order to cope successfully with the abstract concepts of proof writing. In other words, over half the students entering the grade 10 geometry course had not achieved a sufficient geometric thought level to deal with the section on writing proofs. 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 test. The results of this test indicated that there was a significant difference in scores between the beginning of the semester and the end of the semester (Table 4.3 and Table 4.7) and thus the null hypothesis, there was no significant difference in 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 levels after a semester of geometry. Nevertheless, one third of the students exhibited no change in their geometric thought level. This is also consistent with the CDASSG Project results. Even after a semester of geometry 20% of the students in this study were s t i l l at a van Hiele level 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 build on? Was the content of the geometry 10 course inappropriate for these students and thus were they denied the opportunity to improve their thought levels? Were the teaching - 82 -methods inappropriate for these students? Probably a l l three factors contributed to the students' lack of growth during this semester. Perhaps, more individualized instruction needs to be incorporated into the classroom directed at students with van Hiele levels of 0 or 1. More attention also must be directed at geometry instruction in the elementary and middle year's curriculum. The proof test The null hypothesis, there was no significant difference between the means of the written proof tests of students in the computer group and those in the traditional group, was rejected. The computer group performed significantly better than the traditional group. The proof test, written at the end of the semester, resulted in overall averages of 46% (computer group) and 25% (traditional group) —Table 4.4. However, i f the analysis i s limited 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 traditional 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 for students to make deductions. They had become accustomed to looking for relationships, testing their ideas, and making conjectures. The students looked forward to using the software. On the whole, the students were able to write simple proofs more easily 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 typical of grade 10 classes. - 83 -U s i s k i n (1982) s t a t e d that "a student who enters geometry at van H i e l e l e v e l s 0 or 1 has an almost even chance of f a i l u r e at proof" (p. 57). S i x t y - f i v e percent of the students i n t h i s study who were unsuc c e s s f u l (obtained below 50%) i n the proof t e s t had beginning van H i e l e l e v e l s of 0 or 1 while 40% of the s u c c e s s f u l (obtained 50% or more) students had l e v e l s of 0 or 1. In the CDASSG P r o j e c t 71% of students unsuccessful at proofs had beginning van H i e l e l e v e l s of 0 or 1 as compared t o 37% who were s u c c e s s f u l . "Thus students un s u c c e s s f u l at proof are about twice as l i k e l y as the more s u c c e s s f u l others t o have these 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 the same d i r e c t i o n as the r e s u l t s from the CDASSG P r o j e c t . However, a c h i square goodness-of-fit t e s t (.05 l e v e l ) i n d i c a t e d that the researcher's r e s u l t s were not s t a t i s t i c a l l y s i g n i f i c a n t . This may have occurred because of the r e l a t i v e l y s m a ll sample s i z e . As long as more than 50% of the students e n t e r i n g the grade 10 geometry course have van H i e l e l e v e l s of 0 or 1, e i t h e r the proof s e c t i o n of the course should be removed or more emphasis should be p l a c e d on the type of i n s t r u c t i o n employed i n the course. The a t t i t u d e t e s t The r e s u l t s were obtained from an a t t i t u d e t e s t given t o both groups at the end of the semester. O v e r a l l both groups had p o s i t i v e a t t i t u d e s towards the study of geometry i n p a r t i c u l a r , the study of mathematics i n general, and the value of mathematics as a whole. Despite the r e l a t i v e ease with which the 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 proof w r i t i n g as compared t o 35% i n the t r a d i t i o n a l group. Proof - 84 -writing may have had such a negative reputation that the positive experience with the computer software was insufficient to overcome this negative valence. The interviews During the interviews the researcher discovered why many students were experiencing d i f f i c u l t y in geometry. They lacked the basics. The fact that a student can get a l l the way to grade 10 without understanding what a triangle i s seems incredible but does occur (Scfl). Without the constant one-on-one situation of the interview, the gap in this student's knowledge may not have been discovered. It was also surprising to discover the general confusion that existed regarding a segment bisecting another segment in a triangle. Despite the fact that "bisect" was used frequently in class without any apparent d i f f i c u l t y , i t was only during the interviews that the researcher realized these students had failed to understand the generalized concept of "bisect." The researcher benefited from the contact during the interviews and f e l t that the students did likewise, especially Scml, Scm3, and Scf2. Scml approached proofs in a methodical fashion—carefully reading, formulating a plan, and carrying out the plan. He said that he enjoyed solving the proofs. At f i r s t , Scm3 would " f l i t " from step to step, reason to reason. Once he was focussed, he could analyze the situation and foresee what had to be done. It was sheer hard work for Scf2 as she tried to find a rule for every situation, rather than analyzing and dealing with what was given in the problem. These - 85 -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 is better prepared to cope with geometry concepts as they are presented in 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 this study was one semester. Hence, the opportunity for students to assimilate computer experiences into their repertoire of problem solving s k i l l s was l imited. Also, there was a prescribed curriculum to be covered within this time. This l imited the amount of time for computer exploration. D i f f e r e n t i n s t r u c t o r s The computer group and the tradit ional group were taught by two different teachers. However, both teachers were qualif ied 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 set similar standards for class performance and homework. The fact that there was no significant difference between the attitudes of the two - 86 -groups towards geometry suggests that the effects of different instructors may have been minimal. Effects of the interviews The interview sessions tended to be tutorial—correcting 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 this 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 effect. In retrospect, the same number of students,, with similar van Hiele levels, from the traditional group should have been interviewed for the same length of time as from the computer group. Suggestions for Further Research Given that this study indicates the Geometric Supposer software has value in 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) in the grade 10 geometry course and their effect on proof writing. If proof writing i s to remain in 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 levels. - 87 -The current practice in the province in which this study took place i s to have proof writing in the grade 10 geometry course. An alternative would be to include i t in the grade 11 geometry course. Hence, more time could be spent on informal geometry at the grade 10 level. Research in this area could be valuable for designing future curriculum. Designing appropriate experiences to help students achieve at least a van Hiele level 2 prior 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 in education is to give students opportunities to be actively involved in knowledge construction. The Geometric Supposer programs provide such an opportunity. This software is designed to promote experimentation—the process. It has no product requirements built in. The usefulness and power of the program l i e s in the context of the task or problem given to the student. The teacher is the key factor in directing this process. S/he defines the student-software interaction. The role of the teacher shifts from the traditional 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 collecting data, analyzing, making conjectures, testing, and generalizing. With this 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 diversified approaches to solving problems. Students should work on the computer in pairs. 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 instruction. The teacher must be flexible in order to respond to unexpected discoveries by students in the class. Similarly, assessment strategies and techniques w i l l need to be adaptable and flexible. Hardware a c c e s s i b i l i t y Access to the hardware is the main factor. Having the computers right in the room is the most appropriate arrangement as i t provides opportunities for spontaneous investigation. Ideally, students should have access to the program during free periods or after school. Geometry prior to proof writing The overall averages of the two groups (computer and traditional) on the introductory geometry test were relatively low. This result can be attributed in part to the one year gap in geometric instruction. In order for students to be ready for proof writing, they should be exposed to geometry in the previous grade. Hopefully, studying geometry in the previous grade would help students to gain at least a van Hiele level 2 before attempting to write proofs. - 89 -Conclusions The van Hiele geometry test i s a useful aid for grade 10 geometry teachers to better identify and appreciate the geometric thought levels of their students. 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 potential as an instructional aid in the geometry classroom. They do, however, require preparation by and guidance from the teacher. This study has shown that using the Geometric Supposer software can assist students in being able to better write geometric proofs at the grade 10 level. - 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). 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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. & Hadas, N. (1987). Euclid may stay - and even be taught. Learning and Teaching Geometry. K-12. Yearbook of the National Council of Teachers of Mathematics. Reston, VA: The Council, 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. Reston, VA: National Council of Teachers of Mathematics. - 92 -Fleenor, C. R., Eicholz, R. E., O'Daffer, P. (1974). School Mathematics 2. Toronto, ON: Addison-Wesley (Canada) Limited. Fuys, D. (1985, August). Van Hiele l e v e l s of thi n k i n g i n geometry. Education and Urban Society, 11(4), 447-462. Fuys, D., Geddes, D., Tischler, R. (1988). The van Hiele Model of Thinking in 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, £fi(6), 486-493. Grunbaum, B. (1981, September). Shouldn't we teach geometry? The Two Year College Mathematics Journal, 12(4), 232-238. Hoffer, A. (1981, January). Geometry i s more than proof. Mathematics Teacher/ l id) , n-18. Hoffer, A. (1983). Van Hiele-based research. A c q u i s i t i o n of Mathematics Concepts and Processes. New York, NY: Academic Press, Inc., 205-227. Howson, G. & Wilson, B. (1986). School Mathematics i n the 1990s. Press Syndicate of the University of Cambridge. Jurgensen, R. C , Brown, R. G., Jurgensen, J. W. (1985). Geometry. Boston, MA: Houghton M i f f l i n Company. K e l l y , B., Alexander, B., Atkinson, P. (1987). Mathematics 10. Don M i l l s , ON: Addison-Wesley Publishers Ltd. Lester, F. K. (1975, January). Developmental aspects of children's a b i l i t y to understand mathematical proof. Journal f o r Research i n Mathematics Education. £(1), 14-25. Lovell, K. (1971). The development of the concept of mathematical proof in abler pupils. Piagetian Cognitive- Development Research and Mathematical Education. Proceedings of a conference conducted at Columbia University, 66-80. - 93 -Mathis, J . (1986, June). The Geometric Supposers: preSupposer, triangles, quadrilaterals. The Computing Teacher, 13(9), 43-45. Mayberry, J . (1981). An investigation of the van Hiele levels of geometric thought in undergraduate preservice teachers. Doctor of Education thesis, Athens: University of Georgia. Mayberry, J . (1983, January). The van Hiele levels of geometric thought in 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, Sc ient i f ic and Cultural Organization, Paris . Niven, I. (1987). Can geometry survive in the secondary curriculum? Learning and Teaching Geometryr K-12, Yearbook of the National Council of Teachers of Mathematics. Reston, VA: The Council, 37-46. Olive, 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 Institute of Education, 78-85. Prevost, F . J . (1985, September). Geometry in the junior high school. Mathematics Teacher. 18.(6), 411-416. Scally, S. P. (1987, July) . The effects of learning Logo on ninth grade students' understanding of geometric relations. 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. Pleasantvil le , NY: Sunburst Communications, Inc. Schwartz, J . L . , Yerushalmy, M . , Gordon, M. (1985). The Geometric Supposer: Quadrilaterals. Pleasantvil le , NY: Sunburst Communications, Inc. - 94 -Senk, S. L. (1983). Proof-writing achievement and van Hiele levels among secondary school geometry students. (Doctoral dissertation, The University of Chicago, 1983) . Dissertation 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 prior to deduction in geometry. Mathematics Teacher. 1&(6), 419-427. Smith, E. P. & Henderson, K. B. (1959). Proof. The Growth of Mathematical 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 instruction in geometry: Some highlights from research. Mathematics Teacher, 13.(6), 481-486. U s i s k i n , Z. (1980, September). What should not be i n the algebra and geometry c u r r i c u l a of average college-bound students? Mathematics Teacher. 12(6), 413-424. U s i s k i n , Z. (1982). Van H i e l e Levels and Achievement i n Secondary School Geometry. Chicago: U n i v e r s i t y of Chicago. U s i s k i n , Z. (1987). R e s o l v i n g the c o n t i n u i n g dilemmas i n school geometry. Learning and Teaching Geometry, K-12. Yearbook of the N a t i o n a l C o u n c i l of Teachers of Mathematics. Reston, VA: The Co u n c i l , 17-31. van Hiele, P. M. (1959). Development and learning process; A study of some aspects of Fiaget's psychology in relation with the didactics 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. E n g l i s h T r a n s l a t i o n of Sele c t e d W r i t i n g s of Dina van Hi e l e - G e l d o f and P i e r r e M. van H i e l e . E d u c a t i o n a l Resources Information Center (ERIC, number ED 287 697), 247-256. - 95 -van Hiele-Geldof, D. (1984). The didactics of geometry in 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). Mind in Society. Cambridge, MA: Harvard University Press. Wirszup, I. (1976, August). Breakthroughs in the psychology of learning and teaching geometry. Space & Geometry, Papers from a Research Workshop, ERIC Center for 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 for the Educational Technolgy Center, Newton, MA. Yerushalmy, M. & Chazan, D. (1987, July). Effective problem posing in 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 Trig 10 Students: Your son or daughter is enrolled in one of our Geo Trig 10 classes. These two classes are participating in a study which I w i l l be doing under the supervision of the University of British 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 their 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 after the proof writing section. Each test w i l l require approximately 45 minutes to complete. The specific tests to be administered are: • - a geometry test based on the work covered in grade 8. - a test to determine the geometric thought levels of the students. This test w i l l be administered twice -beginning of semester and after proof writing. - a f i n a l proof test. - an attitude test. 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 find out how the students link 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 collected in this project is for research purposes only. To assure confidentiality, no family names w i l l be used in any - 98 -report or release of the information. No personal, family or other sensitive information is being sought. The parent or student may withdraw from this project at any time by a statement orally or in 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 in this project. I w i l l be happy to answer any questions you have regarding the project. I can be contacted through the school office. Please return the form at the bottom. Thank you. Jo Worster I have read the above description of the research project entitled AN INVESTIGATION TO DETERMINE THE EFFECTS OF THE GEOMETRIC SUPPOSER SOFTWARE ON GEOMETRIC PROOF WRITING AT THE GRADE 10 LEVEL to be carried out by Mrs. Worster. [ ] I consent [ ] I do not consent to my child writing the written tests of the project. [ ] I consent [ ] I do not consent to my child being involved in the individual interviews to be conducted by Mrs. Worster. Signature (Parent/Guardian) Student's name Appendix B INTRODUCTORY GEOMETRY TEST AND ITEM ANALYSIS - 100 -INTRODUCTORY GEO TRIG 10 TEST Directions Do not open this test unt i l your are told to do so. Please write your name on the l ine below. This test contains 25 questions. It is not expected that you w i l l remember everything on this test. When you are to ld to begin: 1) Read each question carefully. 2) There is only one correct answer to each question. Print neatly the letter of your choice on the l ine to the right of each question. 3) You w i l l have 35 minutes for this test. - 101 -INTRODUCTORY GEO TRIG 10 TEST The area of a rectangle with length 4 cm and width 11 cm i s : a) 30 sq. cm b) 19 sq. cm c) 44 sq. cm d) 15 sq. cm e) 26 sq. cm 2. How many lines of symmetry does a square have? a) 2 only b) 4 only c) 6 only d) 8 only e) i n f i n i t e 3. 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^ 4. Perpendicular lines: a) do not intersect. b) are two intersecting lines 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 equilateral triangle 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 is similar to ARSV, then L A is congruent to which angle in A RSV? a) R only b) S only c) V only d) SRV only e) none of these Given right A ABC, sin A equals: a) SC AC b) A£ BC d) m AC e) EC AB C) A£ AB If P i s the center of the circ l e , segment PC is called 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 circ l e . - 103 -9. What is the reason that the two triangles below are congruent? 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) 10. In every cir c l e , what is the ratio of the circumference to diameter? a) 2 2 / 7 err) b) 7/22 (1/TT ) c) ( 2 2 / 7 ) 2 d) ( 7 / 2 2 ) 2 e) there i s no constant ratio 11. If A DEF is the reflection of A ABC in line x, what is the image of point B? a) point D b) point E c) point F d) no image point e) point B - 104 -12. The measure of the third angle in the triangle below i s : ,0 a) 5(T b) 130 c) 20° 0 d) 40' e) 60 0 13. 14. 15. The length of the third side in the right triangle below i s : a) 8 cm b) 10 cm c) 12 cm d) 14 cm e) 16 cm 8 C.r*\ The volume of the box shown i s : 3 a) 63 cm b) 126 cm3 c) 162 cm3 d) 1134 cm~ e) 2268 cm~ 1 cm The horizontal lines are parallel. The length of x i s : a) 12 b) 14 < / \ > c) cannot be calculated d) 6 e) 7 - 105 -16. The perimeter of the p a r a l l e l o g r a m below i s : a) 48 cm g cm b) 56 cm c) 30 cm la*/ d) 29 cm e) 28 cm 17. 18. ZABC i s a r i g h t angle. Z.DBC measures 15^. The measure of Z.ABD i s : a) cannot be c a l c u l a t e d b) 105° c) 165 d) 75° 0 0 65 A cube has how many edges? a) 4 b) 8 c) 12 d) 16 e) 20 19. Lines a and b are p a r a l l e l . The measure of angle y i s : 0 < A * >a a) 100 >0 b) 80 c) cannot be c a l c u l a t e d ,0 d) 90' e) 70 0 - 106 -20. Given the number line below, which statement i s true? a) US = TE b) TG = TS & T H S «- P U l _ _ + — i 1 1 1 1 1 1 1 1 1 > C) HT ~ UP -t> -f -J -A -/ O I A 3 d) LG ~ GU e) HG=UH 21. 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 A e) ZAED, ZBEC, ZEDC, ZECD 22. 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 : a) BC ' » c b) CD C) AD d) AO e) AB A 23. The area of the triangle below i s : a) 120 cm b) 108 cm 2 c) 60 cm 2 d) 54 cm . .H- J e) 31 cm - 107 -The formula t o f i n d the volume of 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. The volume of the f i g u r e below i s : a) 605 1T cm 3 b) 55 V cm 3 c) 275 TT cm 3 II CrVV d) 27.5 If cm 3 f v / e) 3025 TT cm 3 V fen* Angles a and b are: a) i n t e r i o r b) e x t e r i o r c) v e r t i c a l d) complementary e) supplementary - 108 -I n t r o d u c t o r y Geometry Test Item Analysis Computer Group Item Humbex Percentage with choice £ £ D. £ Blank % Correct 1 14 0 86 0 0 0 86 2 14 81 0 0 5 0 81 3 24 14 57 0 5 0 57 4 5 86 0 10 0 0 86 5 81 0 0 5 14 0 81 6 5 71 5 0 19 0 71 7 8 0 0 10 90 0 0 90 9 5 23 19 48 5 0 48 10 52 10 5 0 33 0 52 11 0 90 0 5 5 0 90 12 48 14 14 19 5 0 48 13 5 43 10 33 10 0 43 14 10 0 0 90 0 0 90 15 0 0 33 10 57 0 57 16 10 14 57 19 0 0 57 17 19 0 0 67 14 0 67 18 5 43 48 0 5 0 48 19 24 33 29 10 5 0 33 20 0 5 10 19 62 5 62 21 29 10 19 14 29 0 29 22 10 62 5 10 5 10 62 23 24 19 10 38 10 0 38 24 5 19 57 5 10 5 57 25 5 0 5 24 57 10 57 - 109 -Introductory geometry Test Item Analysis Traditional Group Item Number A Percentage with choice fi £ Q £ Blank % Correct 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 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 - I l l -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 is 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 is based on the work of P.M. van Hiele. Permission has been granted by Professor Zalman Usiskin, Director of the CDASSG Project at the University of Chicago to use this test. - 113 -VAN HIELE GEOMETRY TEST 1. 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. 2. Which of these are triangles? a) None of these are triangles. b) V only c) W only d) W and X only e) V and W only 3. 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 is a square. J Which relationship i s true in 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) is not true in 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 in 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) is not true in 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 in every rhombus. An isosceles triangle i s a triangle with two sides of equal length. Here are three examples. A Which of (a) to (d) is true in 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) is true in every isosceles triangle. - 116 -Two c i r c l e s with centers P and Q intersect at R and S to form a 4-sided figure PRQS. Here are two examples. Which of (a) to (d) i s not always true? a) PRQS w i l l have two pairs of sides of equal length. b) PRQS w i l l have at least two angles of equal measure. c) The lin 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) to (d) are true. Here are two statements. Statement 1: Figure .F i s a rectangle. Statement 2: Figure F i s a tr 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 false, then 2 i s true. c) 1 and 2 cannot both be true. d) 1 and 2 cannot both be false. e) None of (a) to (d) i s correct. 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 false, then T i s false. e) None of (a) to (d) i s correct. - 117 -13. Which of these can be called rectangles? P a) A l l can. b) Q only. c) R only. d) P and Q only. e) Q and R only. 14. 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) None of (a) to (d) is true. 15. What do a l l rectangles have that some parallelograms do not have? a) opposite sides equal b) diagonals equal c) opposite sides parallel d) opposite angles equal e) none of (a) to (d) - 118 -16. Here i s a right triangle ABC. Equilateral triangles ACE, ABF, and BCD have been constructed on the sides of ABC. 3> From this information, one can prove that AD, BE, and CF have a point in common. What would this proof t e l l you? a) Only in this triangle drawn can we be sure that AD, BE, and CF have a point in common. b) In some but not a l l right triangles, AD, BE and CF have a point in common. c) In any right triangle, AD, BE and CF have a point in common. d) In any triangle, AD, BE and CF have a point in common. e) In any equilateral triangle, AD, BE and CF have a point in common. 17. Here are three properties of a figure. Property D: It has diagonals of equal length. Property S: It is a square. Property R: It is a rectangle. Which is 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. If a figure i s a rectangle, then i t s diagonals bisect each other. II. If the diagonals of a figure bisect each other, the figure i s a rectangle. Which i s correct? a) To prove I is true, i t i s enough to prove that II is true. b) To prove II is true, i t i s enough to prove that I i s true. c) To prove II is true, i t i s enough to find one rectangle whose diagonals bisect each other. d) To prove II i s false, i t i s enough to find one non-rectangle whose diagonals bisect each other. e) None of (a) to (d) is 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 certain statements are true. c) Some terms must be le f t undefined but every true statement can be proved true. d) Some terms must be le f t undefined and i t is necessary to have some statements which are assumed true. e) None of (a) to (d) is correct. - 120 -Examine these three sentences. i) Two lines perpendicular to the same line are parallel. i i ) A line that i s perpendicular to one of two parallel lines i s perpendicular to the other. i i i ) If two lines are equidistant, then they are parallel. In the figure below, i t is given that lines m and p are perpendicular and lines n and p are perpendicular. Which of the above sentences could be the reason that line m i s parallel to line n? f P < P » < n a) (i) only b) (ii) only c) ( i i i ) only 1 P d) Either (i) or (ii) e) Either (ii) or ( i i i ) In F-geometry, one that i s different from the one you are used to, there are exactly four points and six lines. Every line contains exactly two points. If the points are P, Q, R, and S, the lines are {P,Q}, {P,R}, {P,S}, {Q,R}, {Q,S}, and {R,S}. . P a. • s Here are how the words "intersect" and "parallel" are used in F-geometry. The lines {P,Q} and {P,R} intersect at P because {P,Q} and {P,R} have P in common. The lines {P,Q} and {R,S} are parallel because they have no points in common. From this information, which is correct? a) {P,R} and {Q,S} intersect. b) {P,R} and {Q,S} are parallel. c) {Q,R} and {R,S} are parallel. d) {P,S} and {Q,R} intersect. e) None of (a) to (d) is correct. - 121 -22. To trisect an angle means to divide i t into three parts of equal measure. In 1847, P.L. Wantzel proved that, in general, i t I s impossible to trisect angles using only a compass and an unmarked ruler. From his proof, what can you conclude? a) In general, i t i s impossible to bisect angles using only a compass and an unmarked ruler. b) In general, i t i s impossible to trisect angles using only a compass and a marked ruler. c) In general, i t i s impossible to trisect angles using any drawing instruments. d) It i s s t i l l possible that in the future someone may find a general way to trisect angles using only a compass and an unmarked ruler. e) No one w i l l ever be able to find a general method for trisecting angles using only a compass and an unmarked ruler. 23. There i s a geometry invented by a mathematician J in which the . following i s true: The sum of the measures of the angles o f a triangle i s less than 180°. Which i s correct? a) J made a mistake in measuring the angles of the triangle. b) J made a mistake in logical reasoning. c) J has a wrong idea of what is meant by "true." d) J started with different assumptions than those in the usual geometry. e) None of (a) to (d) is correct. - 122 -24. Two geometry books define the word rectangle in different ways. Which is true? a) One of the books has an error. b) One of the definitions i s wrong. There cannot be two different definitions for rectangle. c) The rectangles in one of the books must have different properties from those in the other book. d) The rectangles in one of the books must have the same properties as those in the other book. e) The properties of rectangles in the two books might be different. 25. Suppose you have proved statements I and II. I. If P/ then q. II. If s, then not q. Which statement follows from statements I and II? a) If p, then s. b) If not p, then not q. c) If p or q, then s. d) If s, then not p. e) If 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 this test. - A l l answers should be written on these pages. - Partial credit w i l l be given so do the best you can on each question. - 125 -GEO TRIG 10 PROOF T E S T Statements RftSSOnS a) WZIj XY, XL = ZK, ZZKW = &LY ;  b) If parallel lines, then alternate interior angles congruent. c) AWKZ = £YLX d) PROOF: 8 C T> - 126 -3. If 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 is to be proved in terms of your diagram. c) Write the proof. - 127 -5. Write this proof below: GIVEN: Quadrilateral SNOW with SW = WO, SN = NO W PROVE: iS = LO PROOF: - 128 -Write this proof in the space provided below: GIVEN: Quadrilateral SRIG with SR = GI, SG = IN bisects SI at M. . PROVE: PM = MN PROOF: - 129 -Proof Test Item Analysis Computer Group Question Percent of students receiving this score Numbex . • 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 Percent of students receiving this score Numbej: 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 T R I G 10 June 1988 Draw a ci 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 terrible strain in GeoTrig. SD D U A 2. Geo Trig i s very interesting to me, and I enjoyed this course. SD D U A 3. Geo Trig i s fascinating and fun. SD D U A 4. Geo Trig makes me feel secure, and at the same time i t i s stimulating. SD D U A 5. My mind goes blank, and I am unable to think clearly when working in Geo Trig. SD D U A 6. I feel a sense of insecurity when attempting Geo Trig. SD D U A 7. Geo Trig makes me feel uncomfortable, restless, i r r i t a b l e , and impatient. SD D U A 8. The feeling that I have toward Geo Trig is a good feeling. • SD D U A 9. Geo Trig makes me feel as though I'm lost in a jungle of information and can't find my way out. SD D U A 10. Geo Trig is something which I enjoy a great deal. SD D U A - 132 -11. When I hear the word math, I have a feeling of dislike. SD D U A SA 12. I approach math with a feeling of hesitation, resulting from a fear of not being able to do math. SD D U A SA 13. I really like mathematics. SD D U A SA 14. Mathematics is a course in school which I have always enjoyed studying. SD D U A SA 15. It makes me nervous to even think about having to do a math problem. SD D U A SA 16. I have never liked math, and i t is my most dreaded subject. SD D U A SA 17. I am happier in a math class than in any other class. SD D U A SA 18. I feel at ease in mathematics, and I like i t very much. SD D U A SA 19. I feel a definite positive reaction to mathematics; i t ' s enjoyable. SD D U A SA 20. I do not like mathematics, and i t scares me to have to take i t . SD D U A SA 21. Mathematics has contributed greatly to science and other fields of knowledge. SD D U A SA 22. Mathematics is less important to people than art or literature. SD D U A SA - 133 -23. Mathematics is 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 24. Mathematics is a very worthwile and necessary subject. SD D U A SA 25. Mathematics is not important in everyday l i f e . SD D U A SA 26. Mathematics is needed in designing practically everything. SD D U A SA 27. Mathematics is needed in order to keep the world running. SD D U A SA 28. There is nothing creative about mathematics; i t ' s just memorizing formulas and things. SD D U A SA Please write or print your reactions to the following questions: 1. What I liked most about this course was: - 134 -What I disliked most about this course was: I would like to make the following suggestions - 135 -Attitude Test Item Analysis Computer Group Item Percentage with choice nber. 2H n II A 2A 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 - 136 -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 -A t t i t u d e Test Summary of Item Statistics Item Numbej: Total Mean SD Computer Group Mean SD. Traditional 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 Trig 10 Student: During this semester I w i l l be conducting a study in Geo Trig 10 under the supervision of the University of British Columbia. The purpose of the study is 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 like to interview you every two weeks for one-half hour. There w i l l be a total of eight interviews. The purpose of these interviews is to find out more specifically 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 later time. A l l information collected in this project is for research purposes only. To assure confidentiality, no family names w i l l be used in any report or release of the information. No personal, family or other sensitive information is being sought. You may withdraw from this project at any time by a statement orally or in writing. Refusal to cooperate w i l l have no consequences for you. If 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. If 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 entitled AN INVESTIGATION TO DETERMINE THE EFFECTS OF THE GEOMETRIC SUPPOSER SOFTWARE ON GEOMETRIC PROOF WRITING AT THE GRADE 10 LEVEL to be carried out by Mrs. Worster. [ ] I consent [ ] I do not consent to being interviewed every two weeks for one-half hour during second semester. The best time for my interview i s : [ ] 8:15 A.M. - 8:45 A.M. [ ] 3:45 P.M. - 4:15 P.M. [ ] 12:15 P.M. - 12:45 P.M. Signature (Student) - 141 -

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