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An Investigation to determine the effects of teaching elementary logic to tenth-grade geometry students Hall, William Edward 1968

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AN INVESTIGATION TO DETERMINE THE EFFECTS OF TEACHING ELEMENTARY LOGIC TO TENTH-GRADE GEOMETRY STUDENTS. by WILLIAM EDWARD HALL B.Sc, University of British Columbia, 1964 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of EDUCATION We accept this thesis as conforming to the required standard (Chairman) THE UNIVERSITY OF BRITISH COLUMBIA July, 1968 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may b e g r a n t e d b y t h e H e a d o f my D e p a r t m e n t o r b y h its r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f Education The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a August, 1968 ABSTRACT The purpose of t h i s i n v e s t i g a t i o n was to evaluate the e f f e c t s on achievement i n and a t t i t u d e towards mathematics of teaching c e r t a i n elementary-l o g i c concepts to h igh school mathematics students. To achieve t h i s purpose, four c lasses o f tenth-grade geometry students were se lec ted from a s i n g l e school of the Vancouver School D i s t r i c t . Two of the c lasses served as the experimental group f o r t h i s i n v e s t i g a t i o n . Both the experimental and c o n t r o l groups were taught by the i n v e s t i g a t o r . The program f o r the experimental group i n v o l v e d a one-week i n t r o d u c t i o n to elementary l o g i c concepts fol lowed by a two week study of " S i m i l a r i t y " concepts. The c o n t r o l group's program i n v o l v e d only the two-week study of the " S i m i l a r i t y " concepts. The students were evaluated at the beginning and end of the treatment per iod and again three weeks l a t e r . Most of the instruments administered were developed by the i n v e s t i g a t o r and consequently not s tandardized. The mean t e s t scores obtained were s t a t i s t i c a l l y analyzed f o r s i g n i -f i cance of d i f ferences u s i n g t - s t a t i s t i c s . The n u l l hypothesis was tested at the f i v e percent l e v e l of confidence. A n a l y s i s of the data c o l l e c t e d showed.. . that the n u l l hypothesis i s accepted at the h i g h , medium, and low a b i l i t y l e v e l s . The acceptance of the n u l l hypothesis i m p l i e d that the teaching of l o g i c concepts to tenth-grade geometry students had no s i g n i f i c a n t e f f e c t s on achievement i n mathematics o r a t t i t u d e towards mathematics. W. E . H a l l Approved by: ACKNOVifLEDGMENTS The writer wishes to acknowledge his indebtedness to the teachers and pupils of the classes who participated in the experi-mental study for their interest and friendly helpfulness. Thanks also to the school administrators for permission to work in the school under their supervision. For the interest and encouragement of Dr. Eric MacPherson, Head of the Department of Mathematics Education, University of British Columbia, the writer is truly grateful. Special thanks go to Dr. Roy Yasui for his encouragement, consideration, and supervision; also to Dr. Thomas Howitz and Dr. Soeng Soo Lee for their guidance and helpfulness in super-vising the study to its completion. Finally, to the writer's wife goes an expression of appreciation for her patience and encouragement. W . E . H . i i TABLE OP CONTENTS Page ACKNOWLEDGEMENTS . . i i Chapter I INTRODUCTION I The Problem 2 Statement of the Problem 2 Hypothesis 2 Significance of the Problem 5 Limitations 5 Clarification of Terms 5 Prospectus of the Experimental Design 6 II. REVIEW OF THE RELATED LITERATURE 7 Historical Background 7 Learning Theory 7 Curriculum Studies 8 REVIEW OF THE RELATED RESEARCH 9 Introduction 9 Research at the Elementary School Level 10 Research at the Secondary School Level 11 Research at the University Level 13 Summary and Implications 14 III. DESIGN OF THE STUDY 16 Problem 16 Procedure 16 Subjects 17 Data-Gathering Instruments 18 Treatment and Nontreatment Programs 20 Statistical Procedure 20 i i i TABLE OP CONTENTS—Continued Chapter Page IV. RESULTS OF THE STUDY 22 Introduction 22 Preliminary Test Results 22 Table 1: ; .Means and Standard Deviations . . . 23 Table 2: t-Values and Degrees of Freedom . . 25 Tests of the Hypothesis 21 Hypothesis 1 27 Hypothesis 2 27 Hypothesis 3 28 Hypothesis 4 29 Hypothesis 5 30 Hypothesis 6 31 Hypothesis 7 31 Hypothesis 8 33 V. SUMMARY, CONCLUSIONS, IMPLICATIONS, AND RECOMMENDATIONS 35 Summary:;. 35 Conclusions 35 Implications 36 Recommendations 37 BIBLIOGRAPHY 38 LIST OF APPENDICES 40 Appendix A 41 Appendix B 47 Appendix C 52 Appendix D 54 Appendix E 57 iv CHAPTER I INTRODUCTION In recent years there has been increased concern among mathematics educators to improve the academic calibre of students graduating from our high school mathematics program. This concern has led to many attempts to improve the mathematics curriculum. One of these, the Commission on Mathematics^ was formed to make recommendations for the modernization, modification, and improvement of secondary school mathematics. Among the Commission's recommend-ations was the following: "Understanding of the nature and role of deductive 2 reasoning—in algebra as well as i n geometry". Several other curriculum study groups, which w i l l be reviewed i n Chapter II, have made similar recommendations. An extensive examination of research r e l a t i v e to the effects of teaching logical-deductive reasoning on achievement i n or attitude towards mathematics yields very l i t t l e conclusive evidence. The purpose of this investigation i s to provide experimental evidence to allow evaluation of some of these effects. The tenth-grade modern geometry program has been chosen as the setting f o r this investigation since there exists considerable agreement among mathematics educators that one of the p r i n c i p a l purposes of the program i s to i l l u s t r a t e logical-deductive reasoning. The basis of deductive reasoning i s found i n 1 College Entrance Examination Board, Program for College Preparatory  Mathematics, Report of the Commission on Mathematics (New York, 1959) p.xi. 2 Ibid, p. 39. 2 logic. The nature of logical-deductive reasoning can be summarized as follows: Certain statements are logical consequences of other statements solely on the basis of accepted rules of logic and the logical form of the statements. A cursory survey of available texts for the modern high school geometry program indicates that very few authors have attempted to include logic concepts in their presentations. Many authors inductively illustrate deductive reasoning. The,, s.tu'deftte ist asked to study and attempt a multitude of examples of deductive arguments. From this study the student is expected to learn to reason deductively. Can the high school student be taught to reason deductively? Will teach-ing him logic concepts increase his ability to reason deductively in other settings? What effects will teaching him logic concepts have on his ability to write deductive geometric proofs? It i s the purpose of this study to gather data to allow statistical test-ing of these and other related questions, which are of prime concern.:*o u mathematics educators. THE PROBLEM Statement of the Problem. The present investigation will attempt to answer the following questions: (l) What are the effects on mathematical achievement and attitude toward mathematics of teaching certain elementary logic concepts to tenth-grade geometry students? (2) What are the relationships between these effects and the student's mathematical achievement level? (3) What effect will the passage of time have on the results of learning the element-ary logic concepts? Hypothesis. In order to answer the research questions, the following 3. null hypothesis will be statistically tested: 1. There is no significant difference, between the treatment (receiving instruction on elementary logic concepts) and nontreatment groups, in mean mathematical achievement posttest scores. 2. There are no significant differences, between the treatment and non-treatment groups, in mean mathematical achievement posttest scores at three mathematical ability levels. 3. There are no significant differences, between the treatment and non-treatment groups, in mean posttest scores on geometric proofs at three mathematical ability levels. 4. There is no significant difference, between the treatment and non-treatment groups, in mean posttest scores of attitude towards mathematics. 5. There are no significant differences, between the treatment and non-treatment groups, in mean posttest scores of attitude towards mathematics at three mathematical ability levels. 6. There is no significant difference, between the treatment and non-treatment groups, in mean mathematical achievement posttest scores three weeks after the treatment period. 7. There are no significant differences, between the treatment and non-treatment groups, in mean mathematical achievement posttest scores three weeks after the treatment period at three mathematical ability levels. 8. There are no significant differences, between the treatment and non-treatment groups, in mean posttest scores on geometric proofs three weeks after the treatment period at three mathematical ability levels. Significance of the Problem. Modern mathematics programs require students to understand algebraic and geometric proofs. A familiar example is the present tenth-grade geometry program. In i t modern geometry is taught in a logical-deductive manner. The principles of deductive reasoning are discussed as they arise during the logical-deductive method of presentation. This method of presenting the principles of deductive reasoning and proof, however, is probably not sufficient to illustrate the logical-deductive nature of mathematical 3 reasoning. What is required is a study of materials in which the major emphasis 3 William T. Hale, "The Development and Evaluation of Text Materials Covering Topics from Formal Logic as Related to the Teaching of High School Mathematics" (unpublished Doctoral dissertation, University of Illinois, Urbana, 1964), p. 49-4. i s on the r u l e s of reasoning, ra ther than the " f a c t s " of mathematics. Such m a t e r i a l s are beginning to appear i n modern mathematics t e x t s . 4 C a r l A l l e n d o e r f e r summarizes the need f o r a study of the l o g i c a l -deductive nature of mathematical reasoning by high school students as f o l l o w s : Since the deductive method i s an e s s e n t i a l part of modern mathematical t h i n k i n g , the teacher should use every opportunity to i l l u s t r a t e i t i n every aspect of her work. I l l u s t r a t i o n , however, i s probably not enough to teach the students the e s s e n t i a l s t ruc ture of a deductive system. At some stage i n the h igh school mathematics curr icu lum there should be a ser ious d i s c u s s i o n of deductive systems per se , and l a t e r a p p l i c a t i o n s of t h i s to mathematics and to nonmathematical s i t u a t i o n s should be used to r e i n f o r c e the understanding of students about deductive methods. Perhaps, the tenth grade i s the place f o r t h i s , but no f i r m statement of t h i s k i n d should be made u n t i l more experimental teaching has been c a r r i e d out . In recent h igh school mathematics tex ts attempts are present to inc lude elementary l o g i c concepts i n the mathematics cur r i cu lum. However, there i.-j,; appears^,to.: be;n'-an absence of experimental evidence i n d i c a t i n g that a study of these concepts w i l l f a c i l i t a t e achievement i n mathematics. More d e f i n i t i v e research i n t h i s area i s requ i red . I f i t could be shown that teaching log i c - concepts to h igh school students has a s i g n i f i c a n t e f f e c t on achievement i n mathematics, the i m p l i c a t i o n s would be per t inent to the B r i t i s h Columbia mathematics cur r i cu lum. The present B. C. mathematics cur r icu lum does not requ i re h igh school students to fo rmal l y study the nature o f deductive reasoning. Since modern mathematics programs have been taught f o r s i x years and continue to be taught i n B. C. h igh schools , research i s requ i red to determine the e f f e c t s o f i n c l u d i n g t h i s teaching . 4 C a r l B. A l l e n d o e r f e r , "Deductive Methods i n Mathematics", Ins ights Into  Modern Mathematics, Twenty-Third Yearbook "of the Nat iona l Counci l of Teachers of Mathematics, (Washington, Nat iona l Counci l of Teachers of Mathematics, 1957), p.66. LIMITATIONS 5 In any experimental study of this type in which the treatment period is relatively short i t is extremely difficult to identify and control the numerous factors which could significantly affect the results. Ideally, the students would be handled individually and randomly assigned to treatments. In the high school setting i t is usually not possible to obtain this ideal. Instead, classes must be assigned randomly to treatment classifications and treated intact. Also confounding the results are i n i t i a l student differences in mathematical ability and attitude towards mathematics due to prior teaching. CLARIFICATION OF TERMS By "modern" geometry is intended high school geometry in which the emphasis is on the axiomatic, deductive, and abstract nature. Such geometry programs have been developed for the secondary schools during the last decade. The logic program was developed by the investigator based on materials 5 presented in the sixth chapter of the MacLean, Mumford, et al text. The similarity program was based on materials presented in the fi r s t g half of the twelfth chapter of the Moise and Downs text. At the time of the investigation the students had studied materials from the f i r s t eleven chapters of the text. 5 W. B. MacLean, D.L. Mumford, et al, Secondary School Mathematics: Grade Ten (Toronto: Copp Clark Co., 1964). 6 Edwin E. Moise and Floyd L. Downs, Geometry (Reading, Massachusetts: Addison-Wesley Co., 1964). 6 It lias been assumed that geometric proofs require deductive reasoning. The writing of geometric proofs has been emphasized as an important part of the modern geometry program. Typically, the student i s required to state given and required information followed by the application of one or more mathematical generalizations and the desired conclusion;: PROSPECTUS OP THE EXPERIMENTAL DESIGN In the spring of 1968, four tenth-grade geometry classes were selected from a single school of the Vancouver School System. The classes selected were similar in mathematical ability. The four classes were regularly taught by two mathematics teachers, each teacher teaching two of the classes. The investigation involved about 120 students. The students had studied modern geometry for eight months using the Moise and Downs text. The four classes were randomly assigned to the treatment and nontreatment conditions. Following the administration of pretests on attitude, logic and mathematics, the treatment groups were given a program of study by the invest-igator. The treatment group's program involved one week of instruction on logic concepts followed by two weeks of instruction on similarity concepts. The nontreatment group's program consisted of only the two weeks of instruction on similarity concepts. At the conclusion of the treatment periods, posttests were given to allow evaluation of the student's understanding of the logic concepts, achieve-ment in geometry, and attitude towards mathematics. Three weeks later, another pos.ttest was administered to both groups to measure retention of the similarity concepts. The data was processed according to the procedures described i n Chapter III. CHAPTER I I REVIEW OF RELATED LITERATURE H i s t o r i c a l Background. Since the days of E u c l i d i t has been generally-recognized that mathematics i s i n some way connected with l o g i c . E u c l i d , without e x p l i c i t formulation, u t i l i z e d laws of l o g i c i n many of h i s geometric proofs. For w e l l over 2000 years h i s Elements has served as a model of l o g i c a l reasoning. The Elements i s one of the f i r s t attempts to present geometry i n an organized, l o g i c a l fashion, s t a r t i n g with a few simple assumptions and d e f i n -i t i o n s and developing a complex mathematical system. This development requires the use of logical-deductive reasoning. Learning Theory. We have, f o r centuries, believed i n the a b i l i t y of mathematics to t r a i n our minds f o r l o g i c a l t h i n k i n g i n other areas of knowledge. This statement i s t e s t i f i e d to by the fa c t that mathematics has continued to occupy such an important place i n secondary school and u n i v e r s i t y c u r r i c u l a . The basis of t h i s b e l i e f was the theory of automatic tra n s f e r of t r a i n i n g . About the beginning of the twentieth century experimental evidence mounted against t h i s theory. The reac t i o n to t h i s evidence culminated i n the law of s p e c i f i c i t y of t r a i n i n g . This law rejected mathematics as the irrep l a c e a b l e preparation f o r c l e a r thinking and l o g i c a l reasoning. The t r u t h l i k e l y f a l l s somewhere between these two extreme theories. Bruner^" i n h i s book, The Process of Education, states; V i r t u a l l y a l l of the evidence of t h e l l a s t two decades on the nature of learning and tr a n s f e r has indicated that, while the o r i g i n a l theory of Jerome S. Bruner, The Process of Education (Cambridge: Harvard U n i v e r s i t y Press, I960), p. 6. 7 8 formal discipline was poorly stated in terms of the training of faculties, i t is indeed a fact, massive general transfer can be achieved by appropriate learning even to the degree that learning properly under optimum conditions leads one to "learn how to learn". Curriculum Studies. One of the first major attempts to develop a new mathematics curriculum was begun during the mid-1950's by the University of Illinois. The University of Illinois Curriculum Study in Mathematics (UICSM) Suggested changes in both the content and methodology of secondary school mathematics. The study advocated extensive use of the "discovery" 2 approach to teaching and learning mathematics. Gertrude Hendrix, a UICSM member, asserts that students can acquire an introduction to logical-deductive reasoning through a structural treatment of elementary algebra or an axiomatic study of goemetry. A second attempt to reorganize the secondary school mathematic curriculum was made by the Commission on Mathematics (1955) of the College Entrance Examination Board. The Commission reviewed the existing mathematics curriculum and made recommendations for its modernization, modification and improvement. A study of the logical-deductive nature of mathematical 3 reasoning was recommended by the Commission. The Ball State Teacher's College Experimental Program (1954) developed a new geometry program based on a simplified version of the Hilbert Postulates. Elementary logic was included in this program as an aid to understanding of the logical-deductive development of the program. The School Mathematics Study Group (1958) developed a new mathematics program containing several topics which were new to the school or grade curriculum. In addition SMSG completely reorganized their approach to _ Gertrude Hendrix, "The Psychological Appeal of Deductive Proof," The Mathematics Teacher, 54:518, November, 1961 ^College Entrance Examination Board, op. cit., p. 22. 9. t r a d i t i o n a l topics of mathematics. Their approach to curriculum organization involves strong emphasis on the principles of mathematics. The basic princ-i p l e s of mathematics are used as the l o g i c a l framework within which mathematical facts and s k i l l s can be developed. Their geometric materials emphasize the need for precise definitions of terms and statements of theorems as well as a logical-deductive development of geometry. 3 The Accelerated Learning of Logic Project (i960) of Yale University was developed to provide learning materials for elementary school students. I t was hoped that the ALL materials would help develop a favorable attitude towards symbol-manipulating a c t i v i t i e s and provide practice i n abstract thinking. This objective was approached through games designed to teach some mathematical lo g i c . As a result of the ALL Project a set o f programmed materials and a series of games requiring applications of the programmed materials were developed. REVIEW OF THE RELATED RESEARCH Introduction. During the l a s t half-century there have been numerous appeals for the inclusion of the concepts of elementary logic i n the high 4 school curriculum. Nathan Lazar published an extensive paper i n 1938 suggesting the importance of logic laws and concepts i n the study of geometry. Since then there have been t n u m e r o u s s journal a r t i c l e s written i n support of an e x p l i c i t study of the nature of deductive reasoning i n mathematics and 3 L. E. Allen and others, "The ALL Project (Accelerated Learning of Logic)", American Mathematical Monthly. 68:497, May, 1961. 4 Nathan Lazar, "The Importance of Certain Concepts and Laws of Logic for the Study and Teaching of Geometry", The Mathematics Teacher. 31, March, A p r i l , May, 1938. 10. particularly in geometry. However, there have been very few investigations which report the effects of teaching the nature of deductive reasoning. Research at the Elementary School Level. In the early-1960's Suppes 5 and Binford of Stanford University reported pertinent research findings at the elementary level. They developed an experimental program of elementary logic to be taught to school children of ages 10, 11, and 12. The purpose of the 6 program was described as follows: ... to deepen and extend the mathematical experiences of the able elementary school child at the broadest level of mathematics, the level of methodology and the theory of proof. The approach is. through a study of modern mathematical logic, in particular that portion of i t which is concerned with the theory of logical inference of the theory of deduction. The specific objective of the research project was to determine the effects of teaching mathematical logic to academically talented fifth:- and sixth-grade children. Of primary interest was the evaluation of the capacity of children in this age group to handle deductive arguments. Also of interest was the evalu-ation of their ability to transfer acquired skills of analysis and logical reasoning to other areas of knowledge. On the basis of the experimental 7 evidence obtained from this study, Suppes and Binford concluded that: (l) The upper quartile of elementary school students can achieve a significant conceptual and technical mastry of elementary math-ematical logic. The level of mastery is 85 to 90 percent of that achieved by comparable university students. 5 P. Suppes and P. Binford, "Experimental Teaching of Mathematical Logic in the Elementary School", The Arithmetic Teacher, 12:187 - 195, March, 1965. 6 Patrick Suppes, "Mathematical Logic for the Schools", The Arithmetic  Teacher, 9:396, November, 1962. 7 P. Suppes and P. Binford, op. cit . , p. 194. 11. (2) This mastery of the subject matter by elementary school students can be accomplished in an amount of study time comparable to that needed by college students i f study is allocated over a longer period of time and i f the students receive considerable more direct teacher supervision. (3) The more dedicated and able elementary school teachers can be adequately trained in five or six semester hours to teach classes in elementary mathematical logic. It i s probably essential that this teacher-training program be very closely geared to the actual program of instruction the teacher will follow in the classroom. ( 4 ) Anecdotal evidence from teachers suggests that there is some carryover in c r i t i c a l thinking and attitude into other fields, especially arithmetic, reading, and English. More explicit behavioral data on carry-over in c r i t i c a l thinking would be desirable. (5) The work with the special summer class in 1963 indicates that able elementary school students who have received prior training in mathe-matical logic can make rapid progress in other parts of modern mathe-matics organized on a deductive basis. 8 Smith i n commenting on the Suppes-Binford study agreed with Suppes as to the importance of teaching elementary logic-concepts but disagreed with the conclusions drawn by Suppes and Binford. Smith enumerated three reasons for objection to Suppes' and Binford's conclusions: (l) Suppes generalized his results to the upper quartile of the elementary school population from a specially chosen sample, (2) the composition of the control group used did not allow valid comparisons, and (3) Smith found unacceptable Suppes' definitions of "equal time" and "equal content". 9 Research at the Secondary School Level. In 1959 > Corley designed a study to evaluate the ability of students in grades six through ten to study successfully an introduction to modern geometry. Ih•additiony;:hef s'tudywas 8 G. R. Smith, "Commentary upon Suppes-Binford Report of Teaching Mathe-matical Logic to Fifth- and Sixth-Grade Pupils", The Arithmetic Teacher, 13:640 December, 1966. 9 Glyn J. Corley, "An Experiment in Teaching Logical Thinking and Demonstrative Geometry in Grades Six Through Ten" (unpublished Doctoral dissertation, George Peabody College for Teachers, Nashville, 1959). 12. designed to determine the ability of students in this age group to understand the logical-deductive nature of mathematical reasoning. Based on statistical analysis of the collected data, Corley was able to conclude: (1) The ability of pupils to learn geometric terms and concepts is quite well developed at the sixth grade level. This ability improves at a slow, approximately uniform rate as the pupils progress into the .tenth grade. (2) The ability of pupils to understand the three methods of reaching general conclusion and apply them in either geometric or nongeometric situations i s moderately well developed at this level. At the seventh grade level a stage of much slower growth had been entered. This slower rate of increase continues for the next two or three years. (3) The logical structure of geometry and the proof of theorems in geometry are too complex for comprehension by a l l but very few in the sixth grade. The ability to understand this study is much improved at the seventh grade level, after which the rate of improvement is much slower up to the tenth grade. 10 Retzer and Henderson reported an experimental study designed to aneasuiEe> the effects of teaching select elementary-logic concepts on the ability of college-capable junior-high school students to verbalize discovered mathe-matical generalizations. Also studied was the dependence of achievement in the elementary-logic unit on ability level. To this end the treatment group com-pleted a unit on elementary logic, the nontreatment group did not. Both groups were led to discover generalizations about vectors by means of a programmed unit within which each student was asked to verbalize his discoveries. The authors reported the following conclusions; (1) The treatment group did significantly better than the control group in verbalizing generalizations precisely. (2) The gifted students verbalize significantly better than the other college-capable group. (3) The verbalization of the gifted was aided more by studying logic concepts than that of the others. 10 Kenneth A. Retzer and Kenneth B. Henderson, "The Effect of Teaching Certain Concepts of Logic on Verbalization of Discovered Mathematical General-izations," The Mathematics Teacher, 60:709, November, 1967. 13. 11 Research at the University Level. Elton of the University of Kentucky reported a study carried out on sophomores of his university during the f a l l of 1963. The stated purpose of the study was to evaluate the validity of the assumption that logic-training might affect a student's score on a test of reasoning ability. To this end a l l students enrolled in an elementary logic course, an introductory course devoted to the principles underlying logical-12 deductive reasoning, were pretested using the Valentine Reasoning Test. Concomitantly, students enrolled in a sophomore course in applied psychology were given the same test. During the last week of classes (after 16 weeks) the two groups were again tested using the Valentine Reasoning Test. Analysis of the test data indicated that there were no significant differences, in terms of reasoning ability, between the two treatment groups as a result of the logic-training. 13 Jensen reported research designed to determine the effectiveness of presenting a unit on logical-deductive nature of mathematical reasoning to beginning college students, using computer-assisted instruction. That i s , the computer was programmed to "give individual and immediate guidance to a student as he constructs the steps of a simple deductive proof." After comparison of the data obtained from the computer group and the control group the author came to:?, .the conclusion^ that there were no significant differences between the experimental groups on any measure except the mean pretest and posttest scores, 11 Charles F. Elton, "The Effect of Logic Instruction on the Valentine Reasoning Test," British Journal of Educational Psychology, 35:339 - 41. 12 Charles ¥. Valentine, Reasoning Tests for Higher Levels of Intelligence (Edinburgh : Oliver and Boyd, 1954). 13 Rosalie S. Jensen, "The Development and Testing of a Computer Assisted Instructional Unit Designed .to Teach Deductive Reasoning" (unpublished Doctoral dissertation, Florida State University, Tallahassee, 1966). 14. indicating only that a significant amount of learning took place in each group as a result of the instruction. Summary and Implications. The findings of the Suppes experiment indicate that liable elementary school students are capable of mastering logic subject matter. In addition, Suppes reports evidence of some transfer of this logic training to other settings. Corley suggests, as a result of her investigation, that the majority of the secondary school students are capable of studying modern geometry by the ninth or tenth grade. Retzer reports in favour of teaching select elementary-logic concepts to college-capable secondary school students. He found that this teaching increased the ability of college-capable students to verbalize discovered mathematical general-izations. Also significant was his finding that the higher ability students were aided more by the logic study than were the others. In apparent opposition to these studies were Elton's findings. His study reports that logic-training had no significant effect on the reasoning ability of his subjects. Jensen's investigation suggests that teaching the logical-deductive nature of mathematical reasoning to college students did not significantly improve their performance on deductive reasoning. This review of related research indicates that secondary mathematics students are capable of understanding the nature of deductive reasoning i f i t i s appropriately presented. The high school geometry program would appear to be a suitable Earesi in which to carry out this study since i t purposes to be largely deductive in nature. The important issue is whether a study of the nature of deductive reasoning will facilitate understanding and achieve-ment in other areas. There is a definite need for more experimental evidence at the high school level in order to resolve this issue. In particular, the 15. effect of teaching the nature of deductive reasoning on achievement in high school geometry requires further study. It is the purpose of this research to augment the conclusions of the studies reported in this review. CHAPTER III DESIGN OF THE STUDY Problem. The purpose of this study is to determine the effects on mathematical achievement and attitude towards mathematics of teaching logic concepts to tenth-grade geometry students. The relationship between these results and the student's mathematical achievement level and the effect of the passage of time on these results are also studied. Procedure. During the third week of April, 1968, after the tenth grade geometry students had studied modern geometry for approximately eight months, the present study was initiated. Four comparable tenth grade geometry classes were selected as the sample for the study. The classes were randomly assigned to the experimental and control groups and pretested to measure their attitude towards mathematics and their understanding of the logical-deductive nature of mathematical reasoning. An achievement test given to a l l tenth grade geometry students prior to the Easter recess was used as a pretest evaluation of achievement in geometry. Each of the pre-test instruments will be discussed more thoroughly later in this chapter. The treatments administered to the experimental (treatment) and control (nontreatment) groups were developed by the investigator and designed to teach both groups introductory similarity concepts. Preceding their introduction to the similarity concepts, the experimental group was given a one-week program on logical-deductive reasoning. In order to facilitate concurrent testing of the similarity concepts i t was necessary 16 17 to begin the experimental group's treatment before the control group's. The investigator administered a l l treatments. The content of the experi-mental group's one-week logic unit i s outlined in Appendix A. Both the experimental and control groups were then given a two-week program in which introductory concepts of similarity were presented. An outline of the two-week similarity program appears in Appendix B. Great care was exercised to ensure "equal time" and "equal content" conditions between the groups for the similarity program. At the conclusion of the treatment period a l l students were given three posttests. The f i r s t posttest was designed to measure attitude towards mathematics, the second to measure understanding of logic concepts,- and the third to measure achievement in mathematics on the similarity concepts. Three weeks later, a second posttest of achievement in mathematics on the similarity concepts was administered. Adequate time was provided for writing a l l pre- and post-tests. The test scores obtained from these tests were statistically analyzed for significance of differences between means, both within and between groups, using t-statistics. The data was processed on the University of British Columbia's IBM 7044 computer. Sub .jects. The subjects were selected from a single high school of the Vancouver School System based on the recommendations of the chairman of the thesis committee, the principal of the school selected, and the mathe-matics teachers whose classes were to be used. The sample consisted of four classes of tenth grade geometry students most of whom were studying modern geometry for the f i r s t time. A l l classes were heterogeneously grouped 18 according to scholastic ability. Two of the four classes selected had been taught by one teacher and the remaining two by another teacher. One class from each teacher was randomly assigned to the experimental and control classifications. In order to control absence effects, i t was necessary to delete from the study persons missing any part of the logic program or more than one hour of the similarity program. At the conclusion of the investigation, the experimental and control groups each contained exactly 50 useable subjects. Data-Gathering Instruments. Control of i n i t i a l differences and evaluation of final differences requires careful and extensive pre- and post-testing. Seven data-gathering instruments were administered for this investigation. A l l instruments were given to both the experimental and control groups. None of the instruments used vwaa standardized. Information regarding their validity and re l i a b i l i t y was limited. Valid and reliable standardized instruments suitable for the present investigation were not available. The pretest of mathematical achievement used was constructed and administered by the regular mathematics teachers. The content validity of the instrument can be assumed since the test items were chosen by agreement of a l l geometry teachers involved. The purpose of the test was to evaluate the achievement level of a l l tenth-grade geometry students in the school. The material covered by the test consisted of the geometry concepts studied during the f i r s t five months of the school year. As a result of the reluctance of the school to publish this test, a copy of the test cannot be made available. 19 The Aiken and Dreger Opinionnaire']' Attitude Toward Mathematics, was selected as the pre- and post-test of mathematics attitude. A review of available mathematics attitude scales revealed very few instruments appropriate for use in the high school. The co-authors of the scale 2 selected reported a test-retest reliabi l i t y of 0.94. No evaluation of validity is reported. The Aiken and Dreger Opinionnaire consisted of twenty items and required the student to reply: Strongly Disagree (SD), Disagree (D), Undecided (u), Agree (A), or Strongly Agree (SA) to each item. The items were scored by assigning numerical values from 1 to 5 for each item. A numerical evaluation of attitude was obtained by summing the assigned values. A copy of the Opinionnaire appears in Appendix C. The pre- and post-test of logic concepts used was constructed and administered by the investigator. The instrument was designed to measure knowledge of logic concepts required for understanding of the logical-deductive nature of mathematical reasoning. The students were not told after the ~pretesting that they would be posttested on the logic concepts. Any improvement in performance on the posttest as a result of "practice" was assumed identical for the experimental and control groups. The logic pre- and post-test i s included in Appendix D. The posttest evaluations of mathematical achievement on the ' similarity' concepts were developed by the investigator. Two forms of the 1 Lewis R. Aiken and Ralph M. Dreger, "The Effect of Attitudes on Performance in Mathematics, "Journal of Educational Psychology',' 52: 10-24, February, 1961. 2 Lewis R. Aiken and Ralph M. Dreger, op. cit . , p. 20. 20 achievement posttest were developed. The purpose of Form A was to evaluate mathematical achievement on the similarity concepts immediately following the three week treatment period. The purpose of Form B was to evaluate retention on related concepts. Each form of the achievement posttest consisted of two sections. The second section of each form required the student to write three geometric proofs. The f i r s t sections did not require the student to display this competancy. It was not assumed for the purposes of this study that the two forms of the achievement test or their related sections were equivalent. The investigator did attempt to make Form A and Form B parallel. Thirty-five minutes was allowed for completion of each achievement posttest. Form A and Form B of the achievement posttest are included i n Appendix E. Treatment and Nontreatment Programs. The treatment (experimental) group's program consisted of a one.-week (three hour) study of elementary logic concepts followed by a two-week (six hour) study of similarity concepts presented in Chapter 12 of Geometry by Moise and Downs. The non-treatment (control) group's program consisted solely of the two-week (six hour) study of the similarity concepts. Included in Appendix A is a complete outline of the one-week logic program given to the treatment group. Appendix B contains an outline of the two-week program given to both groups. A l l theorem numbers and page references pertain to the related sections of this text. Statistical Procedure. The mean pre- and post-test scores of the treatment and nontreatment groups were statistically tested for significance 21 o f d i f f e r e n c e s . S i n c e the mean p r e t e s t s c o r e s o f the treatment and non-treatment groups were not s i g n i f i c a n t l y d i f f e r e n t a t the f i v e p ercent l e v e l o f c o n f i d e n c e , a l l hypotheses were t e s t e d u s i n g t - t e s t s . The f i v e p ercent l e v e l o f c o n f i d e n c e was r e q u i r e d i n a l l t e s t s o f the n u l l h y p o t h e s i s . F o r hypotheses 2, 3, 5, 7, and 8 i t was ne c e s s a r y to a s s i g n s u b j e c t s to mathematical a b i l i t y l e v e l s on the b a s i s o f t h e i r p r e t e s t s c o r e s o f mathematical achievement. C u t - o f f p o i n t s based on the p r e t e s t s c o r e s o f mathematical achievement were c a l c u l a t e d such t h a t each a b i l i t y c l a s s i f -i c a t i o n would c o n t a i n a p p r o x i m a t e l y t h i r t y - f i v e s t u d e n t s . Comparisons o f the treatment and nontreatment group's mean p o s t t e s t s c o r e s a t t h r e e l e v e l s o f mathematical a b i l i t y were then p o s s i b l e . CHAPTER IV RESULTS OP THE STUDY Introduction. In brief, this chapter presents the results of the statistical procedures outlined in the last section of Chapter III. The format of presentation will be as follows: statement of the hypothesis, statistical analysis of the data collected, and the decision to accept or reject the hypothesis. Tables I and II have been embodied in this chapter providing convenient reference to the collected data. Preliminary Test Results. In order to evaluate init ial "between-group" differences, i t was necessary to statistically analyze the pretest scores of achievement in mathematics, achievement on logic-concepts, and attitude towards mathematics. Table II reports the t-values of the differences between these scores to be -1.41 (94 degrees of freedom), 1.87 (94 degrees of freedom), and -1.06 (96 degrees of freedom), respect-ively. None of these t-values are statistically different at the five per-cent level of significance. Thus, the treatment and nontreatment groups were not statistically different on any of the three pretest measures. Based on these results i t was decided that t-statistics could be used for further analysis of the data. However, a l l analyses were substantiated using analysis of covariance. Analysis of covariance revealed findings identical to those presented in the remainder of this chapter. 22 23 TABLE I MEANS AND STANDARD DEVIATIONS c VARIABLE GROUP N MEAN S.D. Pretest-Mathematics Attitude # N.T. 49 57.35 18.76 ## T. 47 61.09 15.68 .Pretest-Logic Concepts N.T. 49 4.53 3.26 T. 47 3.43 2.46 Pretest-Mathematical Achievement N.T. 48 22.06 5.89 T. 50 23.76 6.01 Posttest-Mathematics Attitude N.T. 50 56.92 18.44 T. 50 62.12 16.81 (i) High Ability Group N.T. 11 64.45 15.78 T. 20 65.75 16.22 ( i i ) Medium Ability Group N.T. 20 61.35 19.52 T. 14 68.36 15.85 ( i i i ) Low Ability Group N.T. 19 47.89 15.60 T. 16 52.13 14.52 Posttest-Logic Concepts N.T. 50 5.38 4.72 T. 48 17.83 8.33 Posttest-Mathematical Achievement N.T. 47 14.87 5.35 T. - 49 15.67 5.17 (i) High Ability Group N.T. 11 19.82 3.68 T. 19 18.26 4.20 ( i i ) Medium Ability Group N.T. 19 15.21 4.55 T. 14 16.50 5.29 ( i i i ) Low Ability Group N.T. 17 11.29 4.45 T. 16 11.88 3.96 Posttest-Geometric Proofs N.T. 47 8.60 4.54 T. 49 9.59 4.15 (i) High Ability Group N.T. 11 12.45 3.01 r"T. 19 11.37 3.06 24 TABLE I (Continued) VARIABLE GROUP N MEAN S .D. ( i i ) Medium Ability Group N.T. 19 8.63 4.09 T. 14 10.64 4.38 ( i i i ) Low Ability Group N.T. 17 6.06 4.21 T. 16 6.56 3.52 Bosttest-Mathematical Achievement N.T. 46 14.04 5.48 (3 weeks after treatment) T. 49 13-45 5.15 (i) High Ability Group N.T. 11 16.82 3.49 T. 19 14.26 5.11 ( i i ) Medium Ability Group N.T. 17 16.24 4.93 T. 14 15.50 4.09 ( i i i ) Low Ability Group N.T. 18 10.28 4.96 T. 16 10.69 5.12 Posttest-Geo>metric Proofs N.T. 46 7.54 4.22 (3 weeks after treatment) T. 49 7.41 4.23 (i) High Ability Group N.T. 11 9.55 2.50 T. 19 7.58 4.34 (i i ) Medium Ability Group N.T. 17 9.06 3.94 T. 14 9.14 3.63 ( i i i ) Low Ability Group N.T. 18 4.89 4.06 T. 16 5.69 4.16 # N.T. = Nontreatment Group ## T. = Treatment Group TABLE II t-VALUES AND DEGREES OP FREEDOM VARIABLE N.T. MEAN T. MEAN t-VALDE D. of Between-Group Analyses: Pretest-Mathematics Attitude 57.35 61.09 -1.06 94 Pretest-Logic Concepts 4.53 3.43 1.87 94 Pretest-Mathematical Achievement 22.06 23.76 -1.41 96 Posttest-Mathematics Attitude 56.92 62.12 -1.47 98 (i) High Ability Group 64.45 65.75 -0.21 29 ( i i ) Medium Ability Group 61.35 68.36 -1.11 32 ( i i i ) Low Ability Group 47.89 52.13 -0.82 33 Posttest-Logic Concepts 5.38 17.83 -9.15 * 96 Posttest-Mathematical Achievement 14.87 15.67 -0.75 94 (i) High Ability Group 19.82 18.26 1.02 28 ( i i ) Medium Ability Group 15.21 16.50 -0.75 31 ( i i i ) Low Ability Group 11.29 11.88 -0.39 31 Posttest-Geometric Proofs 8.60 9.59 -1U2 94 (i) High Ability Group 12.45 11.37 0.94 28 ( i i ) Medium Ability Group 8.63 10.64 -1.36 31 ( i i i ) Low Ability Group 6.06 6.56 -0.37 31 Posttest-Mathematical Achievement 14.04 13.45 0.55 93 (3 weeks after treatment) (i) High Ability Group 16.82 14.26 1.47 28 ( i i ) Medium Ability Group 16.24 15.50 0.45 29 ( i i i ) Low Ability Group 10.28 10.69 -0.24 32 26 TABLE II (continued) VARIABLE N.T. MEAN T. MEAN t-VALUE D. of P. Posttest-Geometric Proofs (3 weeks after treatment) 7.54 7.41 0.16 93 (i) High Ability Group 9.55 7.58 1.37 28 ( i i ) Medium Ability Group 9.06 9.14 - 0 . 0 6 29 ( i i i ) Low Ability Group 4.89 5.69 - 0 . 5 7 32 * Indicates means are significantly different at the five percent level of conf icteice. 27 Tests of the Hypothesis. Hypothesis 1. There i s no significant difference, between the treatment and non-treatment groups, in mean mathematical achievement posttest scores. Table I reports the mean pretest and posttest scores. The mean mathematical achievement posttest scores for the treatment and nontreat-ment groups were 15.67 and 14.87, respectively. Table I i describes the t-values and degrees of freedom obtained by the group analyses. The reported t-value of -0.75 with 94 degrees of freedom does not exceed the cr i t i c a l value (^1.99) for statistical significance of difference at the five percent level of confidence. Therefore, the null hypothesis can be accepted. Interpretation of this result suggests that students who have been given the one-week program on elementary logic concepts do not obtain significantly higher scores on the test of achievement in geometry than do their counterparts who have not been given the logic program. Hypothesis 2. There are no significant differences, between the treatment and nontreatment groups, in mean mathematical achievement posttest scores at three mathematical ability levels. (a) High Ability Group. Table I reports the means of the scores on the mathematical achievement posttest for the high ability groups to be 18.26 for the treatment group and 19.82 for the nontreatment group. Table II gives the t-value to be 1.02 with 28 degrees of freedom. This t-value does not exceed the c r i t i c a l value of -2.05 for the five percent level of confidence. Therefore, the null hypothesis can be accepted at the five percent level of confidence. 28 (b) Medium Ability Group. The reported means of the mathematical achievement posttest scores for the medium ability groups were 16.50 and 15.21 for the treatment and nontreatment groups, respectively. The t-value obtained was -0.75 with 31 degrees of freedom. Again, the null hypothesis can be accepted at the five percent level of confidence since the obtained t-value does not exceed the c r i t i c a l value of ^ 2.04. (c) Low Ability Group. The mean scores obtained by the low ability groups on the posttest of mathematical achievement were 11.88 for the treatment group and 11.29 for the nontreatment group. Table II reports the t-value to be -0.39 with 31 degrees of freedom. The c r i t i c a l value of ^2.04 was not exceeded by this value. Therefore, the null hypothesis can again be accepted at the five percent level. At each ability level the differences in mean posttest scores obtained for mathematical achievement were not statistically significant at the five percent level of confidence. Hypothesis 3« There are no significant differences, between the treatment and nontreatment groups, in mean posttest scores on geometric proofs at three mathematical ability levels. (a) High Ability Group. The mean posttest scores on geometric proofs of the high ability treatment and nontreatment groups are given in Table I to be 11.37 and 12.45, respectively. The t-value of 0.94 with 28 degrees of freedom given in Table II does not exceed the c r i t i c a l value of -2.05 for the five percent level of confidence. Therefore, the null hypothesis can be accepted for the high ability group. (b) Medium Ability Group. The reported means of the geometric 29 proofs posttest scores for the medium ability groups were 10.64 for the treatment group and 8.63 for the nontreatment group. Table II reports the t-value to be -1.36 with 31 degrees of freedom. This t-value does not exceed the c r i t i c a l value of -2.04 for the five percent level of confidence. The null hypothesis can again be accepted. (c) Low Ability Group. The mean scores obtained by the low ability groups on the geometric proofs posttest were 6.56 and 6.06 for the treat-ment and nontreatment groups, respectively. The t-value obtained was -0.37 with 31 degrees of freedom. Again, the null hypothesis must be accepted at the five percent level of confidence since the obtained t-value does not exceed the c r i t i c a l value (^2.04). The difference in mean scores on the geometric proofs posttest between the treatment and nontreatment groups were not statistically significant at any of the three ability levels. Thus, Hypothesis 3 can be accepted at the five percent level of confidence. Hypothesis 4. There is no significant difference, between the treatment and non-treatment groups, in mean posttest scores of attitude towards mathematics. Table I reports the mean posttest scores of attitude towards math-ematics to be 62.12 for the treatment group and 56.92 for the nontreatment group. Table II reports the t-value for this comparison of means to be -1.47 with 98 degrees of freedom. The reported t-value does not exceed the cr i t i c a l value (-^ 1.99) for statistical significance of difference at the five percent level of confidence. The null hypothesis can be accepted. Consequently, students who have been given a one-week program on elementary 30 logic-concepts do not obtain significantly different scores on a test of attitude towards mathematics than do their counterparts who have not been given the logic program. Hypothesis 5. There are no significant differences, between the treatment and nontreatment groups, in mean posttest scores of attitude towards mathematics at three mathematical ability levels. (a) High Ability Group. Table I reports the mean posttest scores of attitude towards mathematics for the high ability treatment and non-treatment groups to be 65.75 and 64.45, respectively. Table II gives a t-value of -0.21 with 29 degrees of freedom. This t-value does not exceed the c r i t i c a l value (^2.05) for the five percent level of confidence. Therefore, the null hypothesis can be accepted for the high ability groups. (b) Medium Ability Group. The mean posttest scores of attitude towards mathematics of the medium ability treatment and nontreatment groups are given in Table I to be 68.36 and 61.35, respectively. The t-value of -1.11 with 32 degrees of freedom reported in Table II does not exceed the c r i t i c a l value (^2.04) for the five percent level of confidence. Hypothesis 5 can be accepted for the medium ability group. (c) Low Ability Group. The mean scores reported for the low ability groups on the posttest of attitude towards mathematics were 52.13 for the treatment group and 47.89 for the nontreatment group. Table II reports the t-value to be -0.82 with 33 degrees of freedom. This t-value does not exceed the c r i t i c a l value of ^2.04 indicating that the posttest means for the low ability treatment and nontreatment groups were not significantly different at the five percent level of confidence. 31 The differences in mean scores on the posttest of attitude towards mathematics obtained by the treatment and nontreatment groups were not statistically significant at the three mathematical ability levels. Thus, Hypothesis 5 can be accepted at the five percent level for the high, medium and low ability groups. Hypothesis 6. There is no significant difference, between the treatment and non-treatment groups, in mean mathematical achievement posttest scores three weeks after the' treatment period. The mean mathematical achievement posttest scores for the treatment and nontreatment groups, three weeks after the treatment period, are re-ported in Table I to be 13.45 and 14.04, respectively. Table II reports the t-value of the difference in means to be 0.55 with 93 degrees of freedom. The reported t-value does not exceed the c r i t i c a l value of ^ 1.99 for statistical significance of difference at the five percent level of confidence. Hypothesis 6 can be accepted. Thus, students who have been given the one-week program on elementary logic concepts show no significant difference in mean mathematical achievement, three weeks after the treatment period, from their counterparts who have not been give the one-^reek program. Hypothesis 7. There are no significant differences, between the treatment and non-treatment groups, in mean mathematical achievement posttest scores three weeks after the treatment period at three mathematical ability levels. (a) High Ability Group. The mean posttest scores for the treatment and nontreatment groups in mathematical achievement, three weeks after the treatment period, are given in Table I to be 14.26 and 16.82, respectively. 32 Table II reports the t-value for the difference between these means to be 1.47 with 28 degrees of freedom. This t-value does not exceed the c r i t i c a l value (-2.05) for significance of difference at the five percent level of confidence. Therefore, Hypothesis 7 can be accepted for the high ability groups at the five percent level. (b) Medium Ability Group. Table I reports the mean posttest scores in mathematical achievement, three weeks after the treatment period, to be 15.50 for the medium ability treatment group and 16.24 for the medium ability nontreatment group. The t-value reported for this difference in means is 0.45 with 29 degrees of freedom. This t-value does not exceed the cr i t i c a l value of ^ 2.05 for the five percent level of confidence. The null hypothesis can be accepted at the five percent level for the medium ability groups. (c) Low Ability Group. The mean scores reported for the low ability groups on the posttest of mathematical achievement, three weeks after the treatment period, were 10.69 for the treatment group and 10.28 for the nontreatment group. Table II reports the t-value for the difference in these means to be -0.24 with 32 degrees of freedom. Since the obtained t-value does not exceed the c r i t i c a l value of ^ 2.04, the null hypothesis can be accepted at the five percent level for the low ability groups. The differences in mean posttest scores of mathematical achievement, three weeks after the treatment period, between the treatment and nontreat-ment groups were not statistically significant at any of the three ability levels. Consequently, the null hypothesis can be accepted at the five percent level of confidence. 33 Hypothesis 8. There are no significant differences, between the treatment and nontreatment groups, in mean posttest scores on geometric proofs three weeks after the treatment period at three mathematical ability levels. (a) High Ability Group. Table I reports the mean posttest scores on geometric proofs of the high ability treatment and nontreatment groups, three weeks after the treatment period, to be (7.58 and 9-55, respectively. The t-value of this difference is reported in Table II to be 1.37. The t-value reported does not exceed the c r i t i c a l value of -2.05 for significance of difference. Thus, the null hypothesis can be accepted at the five per-cent level of confidence for the high ability groups. (b) Medium Ability Group? The mean posttest scores on geometric proofs of the medium ability group, three weeks after the treatment period, are given in Table I. The treatment group mean score was 9.14- The non-treatment group mean score was 9.06. Table II reports the t-value of the difference of these means to be -0.06 with 29 degrees of freedom. This value does not exceed the c r i t i c a l value (-2.05) for significant difference at the five percent level of confidence. The null hypothesis can be accepted for the medium ability groups at the five percent level. (c) Low Ability Group. The low ability groups' mean scores were reported in Table I for the posttest on geometric proofs, three weeks after the treatment period, to be 5.69 for the treatment group and 4.89 for the nontreatment group. The t-value of -0.57 with 32 degrees of freedom reported in Table II does not exceed the c r i t i c a l value of -2.04 required to reject the null hypothesis. Therefore, Hypothesis 8 can be accepted for the low ability groups at the five percent level of confidence. 34 The differences in mean posttest scores obtained on geometric proofs, three weeks after the treatment period, were not statistically significant between the treatment and nontreatment groups at the five percent level of confidence for any of the three ability groups. The null hypothesis can be accepted for a l l ability groups at the five percent level. CHAPTER V SUMMARY, CONCLUSIONS, IMPLICATIONS, AND RECOMMENDATIONS Summary. The purpose of t h i s i n v e s t i g a t i o n was to evaluate the e f f e c t s on achievement i n and a t t i t u d e towards mathematics o f teaching c e r t a i n l o g i c concepts to tenth-grade geometry students. To t h i s end, f o u r c lasses o f grade ten modern geometry students were se lec ted and assigned to the treatment c o n d i t i o n s . The treatment group was g iven a one-week i n t r o d u c t i o n to l o g i c concepts and a two-week study of s i m i l a r i t y concepts. The non-treatment group was given only the two-week s i m i l a r i t y program. Both groups were given three sets o f t e s t s . The pretreatment t e s t i n g involved a d m i n i s t r a t i o n of the math a t t i t u d e opinionnaire and a t e s t of achievement i n mathematics. The f i r s t -posttreatment t e s t i n g consisted of a re-admin-i s t r a t i o n o f the a t t i t u d e measure fol lowed by an achievement t e s t i n math-ematics. The second p o s t t e s t i n g , three weeks a f t e r the f i r s t , involved a d m i n i s t r a t i o n of another achievement t e s t i n mathematics. The c o l l e c t e d date was s t a t i s t i c a l l y analyzed f o r s i g n i f i c a n c e of d i f ferences between means u s i n g t - t e s t s . S t a t i s t i c a l t e s t i n g o f the hypothesis was c a r r i e d out at the f i v e percent l e v e l of confidence. Conclusions. Comparisons of the mean pretest and post test scores of the treatment and nontreatment groups on the t e s t of a t t i t u d e towards mathematics i n d i c a t e d no s i g n i f i c a n t di f ferences at the f i v e percent l e v e l of confidence. Also not s i g n i f i c a n t were the d i f ferences between the mean scores of the treatment and nontreatment groups on the f i r s t and second 35 36 posttests of achievement in mathematics. At the three mathematical ability levels used, no significant differences were found between the treatment and nontreatment groups in achievement in or attitude towards mathematics as a result of the one-week logic program. The analysis of the gathered data suggests that teaching the one-week logic program to tenth-grade geometry students has no significant effects on achievement in or attitude towards mathematics as measured by the instruments used. Also suggested by this analysis i s that the logical-deductive nature of mathematical reasoning can be taught to tenth-grade geometry students. However, such teaching does not seem to increase their abilitjr to reason deductively in geometry as measured by performance on tests of deductive geometric proofs. Implications. Because of the lack of common ground for comparison, the findings of this study may not legitimately be compared with other research findings. However, i f the variations among the studies can be ignored, several comparisons can be made. The findings of this study suggest no advantages for the high ability group over the lower ability groups. This finding opposes the Suppes-Binford and Retzer results. Also in opposition to the Suppes-Binford findings but corroborated by the Elton study were the transfer of logic-training results. No convincing evidence was obtained from this investigation to suggest that logic-training would affect student achievement in geometry. This finding endures even when achievement on deductive geometric proofs only was considered. This may imply that proofs in modern geometry, although deductive in nature, do not require deductive reasoning. The students may be learning a "ritual" from 37 which deductive proofs result. This "ritual" may not be facilitated by logical-deductive reasoning. If this i s true, tenth-grade geometry students should not be required to write geometric proofs. In addition to i t s limited effects on achievement in geometry, the logic-training program appears to have minimally affected general attitude towards mathematics as measured by the Aiken and Dreger instrument. These results may be attributable to the relatively short treatment period and may not be conclusive. Since the logic-training program has exhibited no significant effects on achievement or attitude as measured by the instruments used in this study, there appears to be few benefits to be gained by modern geometry students from such a program. Suggested Studies. Before concluding that a study of the logical-deductive nature of mathematical reasoning offers no benefits to the high school student, further research is required. Research is required to determine the effects on attitude and achievement of a prolonged logic program in which better control of treatment conditions can be obtained. Settings other than the grade ten geometry course should be considered. Individualized programmed and/or computer-assisted treatments should be utilized. Students should be assigned randomly to treatment classifications. Standardized tests should be used. A valid and reliable mathematics attitude scale suitable for use in the high school setting must be developed. The effects of a study of logic-concepts should be determined using a smaller sample with individualized treatments and stringent control of treatment conditions. BIBLIOGRAPHY Aiken, Lewis R. and Ralph H. Dreger. "The Effect of Attitudes on Performance in Mathematics," Journal of Educational Psychology, 52:19-24, February, 1961. Allen, L.E., and others. "The ALL Project (Accelerated Learning of Logic)," American Mathematical Monthly, 68:497-500, May, 1961. Allendoerfer, Carl B. "Deductive Methods in Mathematics," Insights Into Modern Mathematics, pp. 65-99, Twenty-Third Yearbook of the National Council of Teachers of Mathematics, Washington, 1957. Bruner, Jerome S. On Knowing: Essays for the Left Hand. Cambridge: Harvard University Press, 1962. . The Process of Education. Cambridge: Harvard University Press, I960. College Entrance Examination Board. Program for College Preparatory Mathematics. Report of the Commission on Mathematics. New York: College Entrance Examination Board, 1953. Corley, Glyn J. "An Experiment in Teaching Logical Thinking and Demon-strative Geometry in Grades 6 Through 10." Unpublished Doctoral thesis, George Peabody College for Teachers, Nashville, Tennesee, 1959. Dutton, Wilbur H. "Measuring Attitudes Toward Arithmetic," Elementary  School Journal, 55:24-31, September, 1954-Elton, Charles F. "Effect of Logical Instruction on the Valentine Reasoning Test," British Journal of Educational Psychology, 35:339 .9-41, November, 1965. Hale, William T. "The Development and Evaluation of Text Materials Covering Topics from Formal Logic as Related to the Teaching of High School Mathematics." Unpublished Doctoral thesis, University of Illinois, Urbana, 1964. Hendrix, Gertrude. "The Psychological Appeal of Deductive Proof," The Mathematics Teacher, 54:515-20, November, 1961. Jensen, Rosalie S. "The Development and Testing of a Computer Assisted Instructional Unit Designed to Teach Deductive Reasoning." Unpublished Doctoral thesis, Florida State University, Tallahassee, 1966. 38 39 Lazar, Nathan. "The Importance of Certain Concepts and Laws of Logic for the Study and Teaching of Geometry," The Mathematics Teacher, 31:99-113, 156-74, 216-40, March, April, May, 1938. MacLean, W.B., D.L. Mumford, R.W.BjJock, D.N. Hazell, G.A. Kaye, and D.B. DeLury, Secondary School Mathematics: Grade Ten. Toronto: Copp Clark Co., 1964. Moise, Edwin E. and Floyd L. Downs. Geometry. Reading, Massachusetts: Addison-Wesley, 1964. Retzer, Kenneth A. and Kenneth B. Henderson. "The Effect of Teaching Certain Concepts of Logic on Verbalization of Discovered Mathematical Generalizations," The Mathematics Teacher, 60:707-10, November, 1967. Rosskopf, M.F. and R.M. Exner. "Some Concepts of Logic and Their Application in Elementary Mathematics," The Mathematics Teacher, 48:290-98, May, 1955. Schacht, J.F., R.C. McLennan, and A.L. Griswold. Contemporary Geometry. New York : Holt, Reinhart and Winston, 1962. Smith, G.R. "Commentary Upon Suppes-Binford Report of Teaching Mathematical Logic to Fifth- and Sixth-Grade Pupils," The Arithmetic Teacher, 13:640-43, December, 1966. Suppes, Patrick. "Mathematical Logic for the Schools," The Arithmetic  Teacher, 9:396-99, November, 1962. Suppes, P. and F. Binford. "Experimental Teaching of Mathematical Logic in the Elementary School, " The Arithmetic Teacher, 12: 187-95, March, 1965. Valentine, Charles W. Reasoning Tests for Higher Levels of Intelligence. Edinburgh: Oliver and Boyd, 1954. Weeks, A.W. and J.B. Adkins. A Course in Geometry. Boston: Ginn and Company, 1961. LIST OF APPENDICES Appendices Page A. Outline of One Week Logic Unit 41 B. Outline of Two Week Similarity Unit 47 C. Attitude Towards Mathematics Opinionnaire 52 D. Logic Achievement Test 54 E. Mathematics Achievement Tests - Form A and B 57 40 APPENDIX A DAY 1 Short Study of the Nature of Deductive Reasoning Introduction: Two important types of reasoning in mathematics i) inductive reasoning —method of making probable inference or conjecture from an examination of particular cases. ie. ^/l / a study millions of cases and observe j that always: a + b> c °°° i t i s A / c. probably true. b i i ) deductive reasoning —method by which facts in mathematics can be proved from general cases. deductive reasoning on one general case proves * ABD = * ACD, £_B = Z_P> LJ> = Z_4> • • • are definitely true. B 3 —based on simple logic ideas. DAY 2 Logic: defn: A "sentence" in logic i s a statement which i s either T or P, but not both. defn: A "sentential connective" connects two simple sentences to produce a compound sentence. ex. simple sentence connective compound sentence It i s raining. and It is rjajLning and i t i s It is cold. cold. It is raining. or It is raining or i t It is cold. i s cold. 41 42 APPENDIX A—Continued simple sentence connective compound sentence It i s raining. It i s cold. i f , then If i t is raining, then i t is cold. It i s raining. not It i s not raining. The basic forms of compound sentences are: i) i i ) i i i ) iv) (....) and ( ....) (....) or ( ) If ( ), then ( Not ( ) • • • • ) (conjunction of 2 sentences) (disjunction of 2 sentences) (implication of one sentence by another) (negation of a sentence) In logic capital letters are used to represent simple sentences: i) P and Q i i ) P or Q -i i i ) If P, then & (or) P implies Q (or) P—vQ iv) Not P (or) A defn: The process of making necessary conclusions from accepted statements by applying accepted rules of logic is called "deductive reasoning". ex. major premise : If i t i s raining, then i t i s cold. minor premise : It is raining. logical consequent : It is cold. In symbols the argument is represented as: major premise : P —> Q minor premise : P  logical consequent : .% 0. The rule of logic which permits us to make this conclusion from these statements is called the "Law of Detachment". ex If logic i s easy, then he will master i t . Logic i s easy.  «T0 He will master i t . If I pass my examinations, I am very happy. I am very happy.  No conclusion. Classwork: Day 2 Exercises - to be handed in during the hour and corrected by the instructor . 43 APPENDIX A—Continued  DAY 2 Exercises 1. Which of the following are sentences in logic? Why are they or are they not sentences? 1. A triangle is a polygon. 2. 3 + 2 =T 3. If x + 3 = 7, then x = 5. 4. An isosceles triangle has two sides having the same length. 5. A quadrilateral i s a square. 2. Classify each of the following compound sentences in terms of conjunction, disjunction, implication, or negation (or some combination). 1. If m is parallel to n and n is perpendicular to 1, then m is perpendicular to n. 2. If a triangle is not equilateral, i t is not equiangular. 3. Dick i s sick or Mel i s not swell. 4. If O ABCD has opposite sides parallel or congruent, then a ABCD is a parallelogram. 5. x + 3 ^  6. Using deductive reasoning state the logical consequent, i f there i s one, for each of the following sets of premises: 1. i ) If x - 3 = 0, then x = 3 i i ) x - 3 = 0 2. i) (AD J. BC) — • (m /_ADC = 90) i i ) (AD J. BC) 3. i ) .[AD =_KB and DE // BC) —>- (AE = EC) i i ) AE * EC 4. i ) (M or N) •—» O i i ) (M or N) 5. i ) A — • B i i ) B — > C i i i ) C D iv) A 44 APPENDIX A—Continued  DAY 3 Deductive Proofs; Most deductive arguments consist of more than one application of the Law of Detachment. ie. Premises: 1. If i t is snowing, then i t i s cold. 2. If i t is cold, then I will stay home. 3. It is snowing. Conclusion: I will stay at home. The formal proof i s written as follows: Hypothesis: P —»• Q Q —> R P Conclusion: R Proof: 1. P —*Q 2. P 3. .".Q 4. Q —*• R 5 • =» R Hypothesis Hypothesis Law of Detachment (l,2) Hypothesis Law of Detachment (3,4) ex. Symbolize the following sentences and write a proof to justify the conclusion. Hypothesis: 1. If humid air rises, i t cools. 2. When i t cools, clouds form. 3. Humid air rises. Conclusion: Clouds form. Hypothesis: A —*• B, B —*• C, A Conclusion: C Proof: 1. A —>B 2. A 3. ?° B 4. B —»-C 5. t>° C Hypothesis Hypothesis Law of Detachment (l,2) Hypothesis Law of detachment (3,4) APPENDIX A—Continued 45 Syllogisms; defn: A "syllogism" i s a form of logical argument involving: i) Major Premise: a general implication i i ) Minor Premise: a particular case of i i i i ) Conclusion: the logical consequent of the general and particular statements ex. Major Premise: A l l birds f l y . Minor Premise: Polly i s a bird. Conclusion: Polly f l i e s . ex. Major Premise: A l l right £_*s are =. Minor Premise: / A and / B are =. Conclusion: No Conclusion. ex. Major Premise: If a man is a policeman, then he must be 5'10" t a l l . Minor Premise: John Law i s a policeman. Conclusion: John Law is 5' 10" t a l l . Classwork: Day 3 Exercises - to be handed in during the hour and corrected by the instructor. 46 APPENDIX A—Continued DAY 3 Exercises 1. Complete the following deductive proofs: i ) Hypothesis: A, A >(B), ( B)—>C Conclusion: C Proof: 1. A 1. 2. A — -(B) 2. 3. (B) 3. 4. (B) —> 0 4. 5. C 5. i i ) Hypothesis: A — > • B, B »-C, A, C >-D, D >-E Conclusion: E i i i ) Hypothesis: P — a n d R), P, (Q and E)-+S Conclusion: S iv) Hypothesis: ( P ) — * Q, Q (R), (P) Conclusion: v) Hypothesis: (a = b and b = c), (a = b and b = c)—»(a— c), (a = c ) — v ( c = a) Conclusion: vi) Hypothesis: (c)—* D F _ C —*- (D), (E) — > * (c), D — B. E — * C, E Conclusion: B APPENDIX B Day 1; 1. Introduction to Similarity Congruent 's must have same size and shape. °» corresponding sides are = , and corresponding angles are = . Similar b.'s must have same shape only corresponding angles are = . Vfliat can we say about corresponding sides? ft a C AABC~ A A'B'C iff/_A =/_A', /_B = /_B', /_C = /_C* and a b c 2. Proportionality defn: Given two sequences a,b,c, ... and p,q,r, ... of positive numbers. ^ — = — = —... , then the sequences a,b,c, ... and p,q,r, ... are ^ ^ r "proportional". ie. 1,2,~5,Q- ...and 3,6,9,12, ...are proportional sequences since 3 ~ 6 ~ 9 ~ 12" ie. 2,1,8,3 is proportional to 1, 1, 2, 2. since 2 1 8 3 2 ' 4 ' 4 ¥ = } = 2 = ? ' i e . 3,4,5 and 2. 4. 1 are proportional. 5' 5' ie. 2,4,8 and 2,4,8 are proportional, ie. 1, 5 and 5, 25 are proportional. Proportionalities involving only 4 numbers are called "proportions". 47 46 APPENDIX B—Continued What are some of the properties of proportions? i ) If _ b , then a-q = b«p . (ratio test) i i ) If P ~ q (inversion property) a _ VJ , then_p_ £_ . i i i ) If P ~ q a ~ b a _ Jb , then a _ _p_ . iv) If P ~ q b " a (alternation property) a_ b , then a + p b + q-v) If P ~ q P ~ q a b , then a - p b - q v i ) l f P ~ q P ~ q a _ b_ , then b"2* = a«c or b =]&>c and b is called the "geometric b c mean" between a and c. Exercises: Text pp. 324 #2,3,5,6,7,10,11. Day 2: 3. Similarities Between Triangles a'~ bi!" c' then & ABC ^  ^ A'B'C . defn: If corresponding /__' s are = and corresponding sides are proportional, then the correspondence i s a "similarity". The natural question to ask ourselves i s "Is i t necessary to satisfy a l l six conditions of the definition to show triangles are similar? 4. The Basic Proportionality Theorem and Its Converse It looks like &ABC~AADE To prove this requires the following theorem: 49 APPENDIX B—Continued Theorem 12-1: "The Basic Proportionality Theorem" If DE // BC , then AD _ AE . A AB AC Proof: (see the text p. 330) The converse of Theorem 12-1 can now easily be proved. Theorem 12-2: "Converse of B.P. Theorem" If AD _ AE , the DE // BC . AB AC Proof: (see the text p. 331) Exercises: Text pp. 332 #la-c, 2,3,4a,7 Day 3: 5. The Basic Similarity Theorems Theorem 12-3: "The AAA Similarity Theorem" If [_k = /_D, /_B * /_E, and /_C = /_F, then A ABC^ DEF. Proof: (see the text p. 336) Corollary 12-3.1: "The AA Corollary" If /_A = [_p and /_B = /_E, then tx ABC ~ k DEF . APPENDIX B—Continued Corollary 12-3.2: If DE // BC, thervA ABC ~ &ADE. Exercises: Text pp. 338 #1,3,4,6,8 Day 4: Theorem 12-4: "The Transitive Property of Similar Triangles" If A ABC ^  A DEF and kDEFAi &GHT, then ^ ABC^ 6.GHI. Proof: (follows from the defn. of similar triangles) Theorem 12-5: "The SAS Similarity Theorem" If AB AC and/_A^/_D , DE ' DF then b. ABC <-> & DEF . Proof: (see the text p. 342) Theorem 12-6: "The SSS Similarity Theorem" If AB _ BC _ AC , DE EF DF then A ABC" &DEF Proof: (see text p. 343) Exercises: Text pp. 344 #1,2,4 51 APPENDIX B—Continued Day 5: Review and Analysis of Several Proofs i ) Review of Similarity Theorems i i ) Prove that i f 6. ABC = &DEF, then A ABC ~ADEF. i i i ) Prove that i f A-C // BD, then A ACE-' A.BDE. iv) Prove that i f X is the midpoint of PQ. and Y is the midpoint of PR, then &PXY~kPQR. V c v) Mathematics Opinionnaire. vi) Posttest on Logic. Day 6: Posttest of Achievement in Geometry on the Similarity Material Studied during the Treatment Period. APPENDIX C Math Attitude Opinionnaire Directions: Please write you name in the space provided. Each of the statements on this opinionnaire expresses a feeling which a particular person has toward mathematics. You are to express, on a five-point scale, the extent of agreement between the feeling expressed in each statement and your own personal feeling. The five points are: SD Strongly Disagree D Disagree U Undecided A Agree SA Strongly Agree You are to encircle the letter(s) which best indicates how closely you agree or disagree with the feeling expressed in each statement AS IT CONCERNS YOU. 1. I do not like mathematics. I am always under a terrible strain in a math class. SD D U A SA 2. I do not like mathematics, and i t scares me to have to take i t . SD D U A SA 3. Mathematics is very interesting to me. I enjoy math courses. SD D U A SA 4. Mathematics is fascinating and fun. SD D U A SA 5. Mathematics makes me feel secure, and at the same time i t is stimulating. SD D U A SA 6. I do not like mathematics. My mind goes blank, and I am unable to think clearly when working math. SD D U A SA 7. I feel a sense of insecurity when attempting mathematics. SD D U A SA 8. Mathematics makes me feel uncomfortable, restless, irritable, and impatient. SD D U A SA 9. The feeling that I have toward mathematics is a good feeling. SD D U A SA 10. Mathematics makes me feel as though I'm lost in a jungle of numbers and can't find my way out. SD D U A SA 52 53 APPENDIX C—Continued 11. Mathematics is something which I enjoy a great deal. SD D U A SA 12. When I hear the word math, I have a feeling of dislike. SD D U A SA 13. I approach math with a feeling of hesitation—hesitation resulting from a fear of not being'able to do math. SD D U A SA 14. I really like mathematics. SD D U A SA 15. Mathematics is a course in school which I have always liked and enjoyed studying. SD D U A SA 16. I don't like mathematics. It makes me nervous to even think about having to do a math problem. SD ^  D U A SA 17. I have never liked math, and i t is my most dreaded subject. SD. D U A SA 18. I love mathematics. I am happier in a math class than in any other class. SD D U A SA 19. I feel at ease in mathematics, and I like i t very much. SD D U A SA 20. I feel a definite positive reaction to mathematics; it ' s enjoyable. SD D U A SA APPENDIX D Logic Achievement Test Classify the following sentences as involving conjunction (c), disjunction (D), implication (i) or negation (N) or any combination of C, D, I, and N. 1. If he is certain, then he will not answer. 2 2. If x =4 and x represents a natural number, then x = 2. 3. A person who doesn't understand logic will not understand geometry. 4. If I think logically, I can solve this problem. 5. It will not rain when skies are clear. Rewrite the following sentences using capital letters to represent simple sentences. ie. P —»• Q 1. Logic i s easy. 2. If x = 3, then 3x - 2 = 7 3. If skies are clear, i t will not rain. 4. If Phil i s a p i l l and Mary is a fairy, then Jane is a pain. 5. If today is Saturday or Sunday, there i s no school. Write the logical consequent, i f there i s one, for each of the following: 1. Major Premise: (S or R)—»-(T or U) Minor Premise: R  Consequent: 2. Major Premise: (° ABCD is a II ogram) (AB // DC and BC // AD) Minor Premise: ° ABCD is a //ogram Consequent: 3. Major Premise: ml A + m / B = 90—"/ A and /_B are complementary Minor Premise: m/ A = m / B Consequent: . 54 APPENDIX D—Continued 4. Major Premise: Ik and / B form a linear paic—»m / A + / B  Minor Premise: mZlA + m/B = 180 Consequent: 5. Major Premise: / A and / B form a linear paiaiv»m/ A + m/ B = Minor Premise: 7~A and / B do not form a linear pair. Consequent: Write a proof to justify the following logical conclusions: 1. Hypothesis: A—»-B, C—*k, C Conclusions B Proof: 56 APPENDIX D—Continued 2. Hypothesis: C—*D, E —»C, E—>D, D—»B, E Conclusion: B Proof: 1. 2. '  3. 4. 5. 6. 7. 3. Hypothesis: (A or B), (A or B) Conclusion: G or H Proof: •D, (E and F I — o r H), D-*(E and P) 2. 3. 4. 5. 6. 5 APPENDIX E Achievement Test in Geometry - Form A I. Solve each of the following as directed and write your answer i n the appropriate blank to the right. 1. Solve for a : a + 2 = 20 1. a = 3a 30 2. Solve for x : 14 . x = 21 2. x = 3 3 3. Solve for _p_ : 4 . q - 3 . P 3. £ = q 5 7 q II. Circle T i f the statement i s True, F i f the statement i s False. 4. A l l right triangles are similar 4. T F 5. A l l squares are similar 5. T F 6. A l l rectangles are similar. 6. T F 7. If a line intersects two sides of a triangle and cuts off segments proportional to these two sides, then the line i s parallel to the third side of the triangle. 7. T 8. If the measures of two angles of a triangle are 45 and 60 and the measures of two angles of another triangle are 45 and 75, the triangles are similar. 8. T III. Solve each of the following, simplifying answers whenever possible. Given the figure with MK // RS 9. If PH = 7, PK = 5 and HG = 3, what i s KS? 9. KS = 10. If PR = 12, MR = 4 and PK = 6, what is PS? 10. PS = 57 58 APPENDIX E—Continued Complete the following proofs in the space provided: #1. In the trapezoid ABCD, prove that AAEB^ 6. DEC. Proof: 1. 2. 3. 4. 5. #2. Prove that two isosceles triangles are similar i f they have a base angle of the f i r s t equal to a base angle of the second. ie. A 3 Proof: 1. 2. 3. 4. 5. 6. APPENDIX E—Continued #3. Prove that the t r i a n g l e formed by j o i n i n g the midpoints of the sides of L ABC i s s i m i l a r to A ABB. Proof: 1. 2. 3. 4. 5. 6. 7. 8. 60 APPENDIX E- -Cont inued Achievement Test i n Geometry - Form B I. Solve each of the f o l l o w i n g as d i rec ted and w r i t e your answer i n the appropr iate blank to the r i g h t . 1 . Solve f o r x : x + 3 _ 14_ . 1 . x = 4x _ 35 2 . Solve f o r x : 8 x = 20. 2 . x = 5 5 3 . Solve f o r r : 3_s = 4jr_. 3 . r _ s 5 6 s ~ I I . Complete each sentence w i th the cor rect symbol(s) or word(s) . 4 . - 5 . A correspondence between two t r i a n g l e s i s a s i m i l a r i t y i f the corresponding (4) are p ropor t iona l and the corresponding (5) are congruent. 4 . 5 . 6. I f "m" i s the geometric mean between the two p o s i t i v e numbers " r " and " s " , then m = . I I I . Answer the f o l l o w i n g statements w i th "TRUE" i f the statement i s t rue and "FALSE" i f the statement i s f a l s e . 7 . Two t r i a n g l e s are s i m i l a r i f any p a i r of angles of the f i r s t t r i a n g l e i s congruent to any p a i r of angles of the second. 7 . _ 8 . Two r i g h t t r i a n g l e s are s i m i l a r i f an acute angle of the f i r s t t r i a n g l e i s complementary to an acute angle of the second t r i a n g l e . 8 . _ IV. Given the f i g u r e w i th AC // PQ T 10. I f PT = 3 3 , AP = 12, and DB - 8 , what i s TD? 10. APPENDIX E—Continued  COMPLETE THE FOLLOWING PROOFS IN THE SPACE PROVIDED: 1. 2. 3. Given the parallelogram LMNO with diagonals LN and MO Prove that kKLM^ &KNO. Proof: 1. 2. 3. 4. 5. 6. 7. Given isosceles triangles, &ABC and &XYZ with their vertex angles congruent i . e . /_A = /_X Prove that &ABC ~ kXYZ. Proof: 1. 2. 3-4. 5. 6. 7. 8. 9. Given the quadrilateral ABCD with midpoints L,M,N, and 0 of the sides, Prove that AOLM^AMNO. Proof: 1. 2. 3. 4. 5. 6. 7. 8. 9. 

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