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MITCHELL, Bruce Alex, 1937- THE EFFECT OF A TEACHER-DEVELOPED UNIT IN HYPERBOLIC GECMETRY ON STRUCTURAL OBJECTIVES IN TENTH GRADE GECMETRY.
The Ohio State University, Ph.D., 1972 Education, general
University Microfilms, AXEROX Company , Ann Arbor, Michigan
THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. THE EFFECT OF A TEACHER-DEVELOPED UNIT IN HYPERBOLIC
GEOMETRY ON STRUCTURAL OBJECTIVES IN
TENTH GRADE GEOMETRY
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By
Bruce Alex Mitchell, B.S., M.S
* * * * *
The Ohio State University 1972
Approved by
A civ is or lece of Education
PLEASE NOTE:
Some pages may have
Indistinct print.
Filmed as received.
University Microfilms, A Xerox Education Company AC KNOWLEDGMENTS
I wish to acknowledge those people who helped make
the completion of this research project possible. To
Dr. F. Joe Crosswhite, my major adviser, and to Dr. Alan
Osborne for their time investment in me and their many
helpful suggestions, my deep appreciation.
A thank you to Ted Scheick of the mathematics department for his help, encouragement, and friendship.
Thanks to Dr. Dan Eustice of the mathematics department for his help in the instrument development stage. I also thank
Dr. Peter Anderson and Dr. Jae Lee for their assistance.
To Ray Tata, Hike Myers, and Gary Smith, of the
Brookhaven mathematics department, thank you so much for your help.
A special thanks to my parents and the Hemingtons
for the help and encouragement they have always given me.
To my wife Judy and children, Kathe, Bob, and Jim, who gave love, understanding, patience and help, I express my deepest appreciation.
The combination of people and events that effected my life during the last three years contributed to what I consider the best three years of my life. VITA
March 26, 19 37 * Born - Chicago, Illinois
1959 . . . . B.S. in Education, Northern Illinois University, DeKalb, Illinois
1959-1962. . . Mathematics Teacher, Zion-Benton Township High School, Zion, Illinois
1962-1963. . . M.S. in Mathematics, University of Tennessee, Knoxville, Tennessee
1963-1966. . . Mathematics Teacher, Niles Township High School - West Division, Skokie, Illinois
1966-1967 Participant, NSF Academic Year and Summer Institute, University of Wisconsin, Madison, Wisconsin
1967-1969. . . Mathematics Instructor,.Lake Forest College, Lake Forest, Illinois
1969-1971. . . Teaching Associate, Department of Mathematics, The Ohio State Uni versity, Columbus, Ohio
1971-1972, Teaching Associate, Department of Science and Mathematics Education, The Ohio State University, Columbus, Ohio
FIELDS OF STUDY
Major Field: Mathematics Education Professor F. Joe Crosswhite Professor Alan Osborne Professor Harold C. Trimble
i i i Minor Fields: Mathematics Professor John T. Scheick
Teacher Education Professor Herbert Coon
iv TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS . i i
VITA...... i l l
LIST OF TABLES v
Chapter
I. THE PROBLEM
Introduction and Background of the Problem Rationale for the Study ...... Method and Procedure ......
Instrumentation
Hypotheses...... 8 Analyses of Data. 8 Delimitations. 9 An Overviev.’ of the Report g
II REVIEW OF THE RELATED LITERATURE .... 11
Goals of Geometry Related to This St.rdy . 11 Dissatisfaction VJith Student Fulfillment of the Goals of Geometry Related to This Study ...... 19 Recommendations for the Geometry Course . 2 3 Non-Euclidean Geometry in the Iligh School Feasibility of Teaching Hyperbolic Non- Euclidean Geometry to High School Geomctr Students...... 32 Summary...... 34
III RESEARCH PROCEDURE. 36
Sample and Design 36 Treatment . 39 Instrumentation . 41 Analysis of Data. 44
v Chapter Page
IV. THE SEMINAR ...... 46
Background...... 46 Preparing the Teachers...... 50 Developing the Unit...... 5 3 Discussing Teaching Strategies ..... 54
V. ANALYSIS OI-* D A T A ...... 59
Initial Difference Hypotheses ...... 59 Analyses Related to Main Hypotheses . . . 64 Analyses of Posttest ...... 67
Chi-Square Analysis ...... 72 Item Analysis of Posttest: Results . . . 76
Achievement Test in Hyperbolic Geometry . 8 3
VI. SUMMARY AND CONCLUSIONS...... 86
Summary...... 86 Conclusions ...... 88 Discussion...... 89 Recommendations for Further Research . . . 90
APPENDIX
A ...... 94
B ...... 103
C ...... 114
D ...... 137
E ...... 139
BIBLIOGRAPHY ...... 347
vi LIST OF TABLES
Table Page
1. Results of the Posttest Pilot ...... 43
2. Distribution of Pre-Student Teachers and Student Teachers...... 47
3. Summary of Results of the Elementary Geometry Facts Test— September 1 9 7 1 ...... 60
4. Summary of the Results of the t-Tests for the September Geometry Facts Test ...... 61
5. Summary of Results of the Pretest...... 62
6. Summary of the Results of the t-Tests Used on the Pretest ...... 6 3
7. Summary of t-Test R e s u l t s ...... 65
8. Blocking on the P r e t e s t ...... 66
9. Summary of Results of 2 X 3 Analysis of Covariance...... 67
10. Summary of Posttest Results. . 68
11. Chi-Square Item by Item Analysis ...... 70
12. Relative Difficulty and Discrimination Indices of Posttest I t e m s ...... 77
13. Response Pattern on Posttest ...... 79
14. Results of an Achievement Tost in hyperbolic Non-Euclidean Geometry...... 84
15. Item by Item Mean Scores on Hyperbolic Geometry Achievement Test...... 85
vii CHAPTER I
THE PROBLEM
Introduction and Background of the Problem
Each year thousands of high school sophomores com plete a course in Plane Geometry. The literature suggests that the understanding of mathematical structure, inference patterns, and patterns of proof are commonly accepted goals for this course. The degree to which these goals are achieved appears to be unknown. To some, the attainment of these goals is questionable. As Brumfiel states:
Students of 1954 who studied an old fashioned hodge podge geometry had no conception of geometric structure. Students of 1971 who have studied a tight axiomatic treatment have no conception of geometric structure (4, 5).
The content that will enable students to emerge with an appreciation of mathematical structure and the ability to recognize valid inference and proof patterns has been a controversial issue among mathematicians and mathematics educators. A source of fuel for this controversy has been supplied by attempts to utilize recent developments in geometry (20) . The affine approach to Euclidean geometry
(24), the transformational approach (13), the vector
1 2 approach (46), and the inclusion of non-Euclidean geometry
(30), were among the suggestions for the geometry course that reflected these developments.
It was the purpose of this research to develop a unit in hyperbolic non-Euclidean geometry as a means of achieving goals related to mathematical structure and inference and proof patterns. Specifically, the study tested the effect of the unit on student understanding of three variables: (1) the components of a mathematical system, (2) recognition of valid inference and proof patterns, and (3) the ability to operate in an unfamiliar mathematical system.
Rationale for the Study
The first of four reasons for choosing hyperbolic non-Euclidean geometry as the treatment was that it challenged the individuals predetermined intuitive notions of the plane. When operating in a system in which many of the results are contrary to one's intuition, several con clusions seem feasible. The role of the undefined terms might be more clearly understood. Because intuition and
"picture-drawing” are no longer reliable, a greater degree of reliance on the axioms, definitions, and theorems of the system should be necessary. It is possible that this additional reliance could produce a deeper awareness of the roles of the components of a mathematical system than is possible by examining only the Euclidean approach.. Students 3
studying hyperbolic non-Euclidean geometry will have been
exposed often to the idea that what can be proved depends
entirely upon what1 was accepted as the axiom set. In
hyperbolic ncn-Euclidean geometry this "rub" against
intuition comes quickly. The characteristic postulate
states: "Through a point, P, not on a given line, 1, there
exists more than one line through P which does not intersect
1." If students are willing to accept this, which may be a
non-trivial accomplishment, they could be beginning to
understand the nature and role of axioms in a mathematical
system. Of course, there is more to axiomatics than the
acceptance of a non-intuitive axiom, but such an acceptance
might be an indication of a more mathematically mature
outlook at axiomatics. In addition, the new ideas such as
"ideal point," "ideal triangle," "Sacchori Quadrilateral,"
and the new interpretation of "parallel" require carefully
formulated definitions.
Secondly, the availability of Euclid's first twenty-
eight propositions gives the students an initial security.
The fact that many of the familiar theorems are useable in
this new geometry enables the students to build upon a
familiar foundation. Use of the first twenty-eight propo
sitions provide students opportunity to question and
explore why these particular propositions are available.
As one would expect, high school students will be using
Euclidean theorems not included in the first twenty-eight 4
propositions. When this occurs, the exploration shifts to
attempts to find out why some theorems are not available.
The task of demonstrating non-availability to tenth graders
is sometimes quite difficult due to the subtle use of the
fifth postulate. At any rate, there is an opportunity to
have first hand experience with the concept of independence of an axiom set.
Thirdly, the extensive use of indirect proof in the
hyperbolic non-Euclidean Geometry unit provides exposure to
this important component of the concept of proof. It is
hard to believe and even harder to justify the apparent
ignoring of indirect proof in the first three years of the
high* school mathematics curriculum. Because of the indirect method is an efficient method of proof to use in many of the
theorems and problems, the hyperbolic geometry provides a
setting in which the students get experience using this
strategy.
The confidence that a unit in hyperbolic non-
Euclidean could be developed and made palatable for high
school sophomores, provided the fourth reason. A unit in hyperbolic geometry had been developed and taught to tenth grade students at the Niles Township High School during the years 1963-1966. The work that Maiers did at Valparaiso
(30), and the Greater Cleveland Mathematics Program's (6)
inclusion of hyperbolic geometry in their program provided evidence that such a program had been tried. The author's 5 personal involvement with the program at Niles West, demonstrated that a program in non-Euclidean Geometry was feasible at the tenth grade level.
Method and Procedure
This study was conducted at the Brookhaven High
School in Columbus, Ohio through the 1971-72 school year.
During this period of time, three activities, related to the study, took place: (1) development of a hyperbolic non-
Euclidean unit, (2) preparing teachers to teach the unit, and (3) implementation of the treatment.
The unit was prepared in a seminar setting with the cooperation of those who were to teach the hyperbolic geometry. The seminars, held at least once a week, were also used to help prepare the teachers to use the unit effectively. The activities of this seminar were a major part of the study and, for that reason, will be discussed in detail in Chapter IV.
Each of three teachers involved in the study had two geometry classes. One of each teacher's classes was randomly assigned to the treatment on May 15. The classes that were not assigned to the treatment comprised the control group.
The control group studied the material in chapters thirteen and fourteen of Dolciani's Modern Geometry (25) while the treatment group studied the hyperbolic non-Euclidean geometry. 6
Two designs, the Static-Group Comparison and the
Nonequivalent Control Group (10), were utilized. In the first design the class was considered to be the unit of analysis and in the second the unit of analysis was the student.
Instrumentation
The Educational Testing Service's Cooperative
Mathematics Test, Geometry Form A, Part 1, was used as a pretest and was administered on May 15, 1972.
One achievement test concerned with the content of hyperbolic non-Euclidean geometry was administered. This test supplied no input into the main hypothesis of the study, but it did supply evidence concerning the level of achievement attained by the students in hyperbolic non-
Euclidean geometry. There will be a discussion of the results of this test in Chapter V.
Since an existing instrument could not be located which tested the three variables to be studied, a test was developed to be used as a posttest and was administered on
June 9, 1972. The posttest contained three subtests. The goals of each of these subtests are listed below.
Subtest A: The Components of a Mathematical System.
Questions in this category were directed at providing information about the following: 7
1. The meaning of "undefined term," "definition," "axiom,"
and "theorem"
2. the function of the components of a mathematical system
within the system
3. recognition of the structure in Euclidean geometry by
distinguishing between the definitions, axioms, and
theorems of that geometry.
Subtest B: Inference Patterns and Patterns of Proof.
Questions in this category were directed at providing
information about the following:
1. The relationship that exists between a theorem its
converse, inverse and contrapositive
2. the use of the law of detachment and the law of the
contrapositive
3. the extent to which "child's logic" is used
4. arriving at valid conclusions from given information
that is assumed to be true
5. alternative forms of writing the implication, "if p,
then q"
6. recognition of the indirect proof pattern.
Subtest C: Ability to Operate Within the Framework of an
Unfamiliar Mathematical System, Questions in this category were directed at providing information about the following:
1. Arriving at accurate deductions from a set of undefined
terms and the axioms of a finite system 8
2. arriving at accurate deductions, of an elementary
nature, within the constraints of elliptic geometry
given the characteristic postulate of elliptic geometry.
Hypotheses
H^: There is no significant difference in achievement
between the control group and the treatment group on
the post-test.
H2: There is no significant difference in achievement
between the control group and the treatment group on
sub-test A of the post-test.
H3: There is no significant difference in achievement
between the control group and the treatment group on
sub-test B of the post-test.
There is no significant difference in achievement
between the control group and the treatment group on
sub-test C of the post-test.
Analyses of Data
Both the t-test and analysis of covariance were used in analyzing the data. The t-test was used to test for significant differences between the means of the control and treatment groups on the post-test. The unit of analysis for the t-test was the class. In the analysis of co- variance, blocking on the covariate was used with the student as the unit of analysis. 9
In addition to the analyses described above, an item by item analysis of the post-test and achievement test in hyperbolic geometry was performed.
Delimitations
The problems related to teacher training and the logistics of effecting a curriculum change in the public school setting were major factors in the decision to restrict this study to one high school. This decision, to work with one high school, helped make the study possible but placed limitations on the generalizability of the results.
Length of treatment and teacher knowledge of hyper bolic geometry were sources of limitations. In allowing only three weeks for the treatment it is conceivable that this was not enough time to affect the concepts about mathematical structure and proof strategies that took ten years for the students to develop. It is also possible that
teacher knowledge of the content of hyperbolic geometry and mathematical structure was a limiting factor. The high degree of cooperation and ambition the teachers demonstrated and the design of the seminar served to minimize this problem.
An Overview of the Report
It was the purpose of Chapter I to provide an under
standing of the problem being studied, its background, and 10 its significance. A summary of the methods and procedures, and a brief explanation of the environment that allowed the study to take place was described.
Chapter II will present a review of the related literature.
The procedures utilized during this study are discussed in detail in Chapter III. This chapter describes the sample of the population used, assignment to treatment, test development, the experimental design of the study, and the statistical methods used.
Chapter IV describes in detail the activities of the Seminar.
Chapter V contains the presentation and analysis of the data.
A summary of the study, conclusions, interpretations, and recommendations for further research are presented in
Chapter VI. CHAPTER II
REVIEW OF THE RELATED LITERATURE
It is the intent of this review to: (1) demonstrate the literature suggests that goals related to mathematical structure and patterns of proof and inference are among the commonly accepted goals for the tenth grade geometry course;
12) provide evidence indicating student achievement of the goals mentioned above might be unsatisfactory; (3) examine recommendations for the geometry course that not only reflect a dissatisfaction with the course as it now exists, but also represent an attempt to increase student under standing of mathematical structure and patterns of proof and inference; (4) focus on a specific recommendation to this study— the inclusion of non-Euclidean geometry in the tenth grade geometry course; and (5) present evidence related to the feasibility of teaching hyperbolic non-Euclidean geometry to high school geometry students.
Goals of Geometry Related to This Study
The search for demonstrating the goals tested in this study are among the commonly accepted goals of the tenth grade geometry course begins with a report published
11 12 in 1892. In their "Special Report on Demonstrative Geome try," the Committee on Secondary Schools Studies (The
Committee of Ten) urged that the role of postulates in the demonstrative geometry course be correctly understood (37).
The importance of the role of one component, the postulate, in a mathematical system gets consideration at this early date. On the place of inference and proof patterns the
Committee had this to say:
The elementary ideas of logic may be introduced early in the course in demonstrative geometry with great advantage . . . to make it easily understood why the converse proposition B is A is not a necessary conse quence of A is B and under what conditions it becomes such a consequence; and why, on the other hand the "contrapositive" not B is not A is the logical equiva lent of A is B and the "obverse" not A is not B of B is A. The contrapositive of a proposition is oftentimes more readily demonstrated than the proposition itself, . . . (3, 139).
In response to an increasing failure rate, decreasing enrollments, recognition of individual differences, and growing criticism of drill and manipulative formalism, the
Mathematical Association of America formed the National
Committee on Mathematical Requirements in 1916. The report of this committee is perhaps best known as the 1923 Report
(3). Contained in the section that stated recommendations for demonstrative geometry was a rather open ended statement of purpose:
The principal purposes of the instruction in this subject are: . . . , to develop understanding and appreciation of a deductive proof and the ability to use this method of reasoning where it is applicable, 13
and to form habits of precise and succinct statement, of the logical organization of ideas, and of logical memory (3, 4IS).
Although it was not stated explicitly, it seems reasonable to assume that in order to develop an under standing and appreciation of a deductive proof, the role of the components of a mathematical system and proof strategies would be important parts of the geometry course.
In 1938 one of the classical works in mathematics education received wide spread exposure via the Thirteenth
Yearbook of the National Council of Teachers of Mathematics.
This book, titled The Nature of Proof, authoried by Harold P.
Fawcett (18) reported a study in which a group of high school geometry students took an active part in the deductive development of the tenth grade geometry. Fawcett summarized pre-1938 conceptions of the high school geometry course when he stated:
Demonstrative geometry has long been justified on the ground that its chief contribution to the general education of the young people in our secondary school is to acquaint them with the nature of deductive thought and to give them an understanding of what it really means to prove something (18, 29).
The relationship between Fawcett's study and this study is made more clear by examining what Fawcett meant by "the nature of a deductive proof."
In Fawcett's mind the pupil understands the nature of a deductive proof when he understands: 14
1. The place end significance cf undefined concepts in proving any conclusion. 2. The necessity for clearly defined terms and their effect on the conclusion. 3. The necessity for assumptions or unproved propo sitions. 4. That no demonstration proves anything that is not implied by the assumptions (18, 38).
Each of the four goals are closely related to the goals of
Sub-test A.
The next major revision related to the goals of mathematics appeared as the Second Report of the Committee on Post-War Plans in 194 5. In the section titled "Mathematics
In Grades Ten to Twelve" Thesis 15 was directed toward what the Committee labeled "functional competence" in mathematics.
The 29 question on the check list for functional competence, added to the original list of 28 in a 1947 publication, was related to mathematical structure: "Can he analyze a statement in a newspaper and determine what is assumed, and whether the suggested conclusions really follow from the given facts or assumptions {3, 636)?"
The College Entrance Examination Board in 1959 listed as one of the three main objectives of geometry in the high school: "... the development of an understanding of the deductive method as a way of thinking, and a reason able skill in applying this method to mathematical situ ations (11, 22)."
The late fifties and early sixties saw the "modern math" movement grow from infancy to a point of domination in the mathematics education curriculum. The College 15
Entrance Examination Board's proposals for the tenth grade geometry, the geometry text of the School Mathematics Study
Group, and the geometry of the University of Illinois
Committee on School Mathematics, all reflected the thinking of many top mathematicians and mathematics educators of this period. A more refined set of axioms, greater emphasis on preciseness of definitions, special attention to deductive
reasoning as logic rather than a characteristic of geometry
alone, and the geometry content were common elements of the
three groups (51). In addition to the groups mentioned, the
Greater Cleveland Mathematics Program was active in the
movement. Charles Buck, who directed the writing of the
geometry text, wanted the tenth grade course to be designed
so that the structure of the subject could be revealed (7).
Mathematical structure and proof appeared to be two concepts
stressed, at least implicitly, by the people associated with
the movement.
In the summer of 1963 a group of twenty-five pro
fessional mathematicians and "mathematics-users" participated
in the Cambridge Conference on School Mathematics. The
purpose of the meeting was to review school mathematics and
establish goals for mathematical education. Although the
Conference reported to be uncertain how thorough a treatment
of logic should be in the geometry course, guidelines for
what they called a "meagre" treatment were suggested in the
section on "Logic and Foundations." That section included: 16
1. Truth tables for simple connectives 2. Common schemes of inference 3. Preliminary recognition of the roles of axioms and theorems in relation to the real number system 4. Simple use of indirect proof (18, 38).
In Appendix C of the report, "An Introduction to Formal
Geometry," a more definite position concerning Geometry was stated:
The scheme of the geometry course would be somewhat different: it would be the students first experience with a formal mathematical structure. For several reasons, we believe that geometry is a good choice for this purpose (18, 77) .
The drafters of this appendix felt that geometry had an educational significance above and beyond its substantive content.
The International Commission . Congress of Mathematicians in Moscow in the summer of 1966. The booklet, The Role of Axiomatics and Problem Solving in Mathematics (12), was a result of this meeting, and it supplied more evidence of the concern that mathematicians and mathematics educators have for the role of axiomatics in the mathematics curriculum. The geometry course has the potential of making significant contributions toward this general goal. For this reason a few chapters in the booklet will be briefly discussed. 17 The chapters in the booklet include those by Allen, Buck, Henkin, Suppes, Blank, and Klein. Allen contended that the axiomatic method helps students acquire a better understanding of the nature of proof. Allen cited examples from classroom situations. Twelve laws of logic are stated that the student must understand and there was a heavy emphasis on flow diagramming. Buck favored the treatment of axiomatics in pre-college courses to be principally at the naive level. Here, axioms are formulated to describe or codify basic properties of an existing system and are then used to derive additional properties of the system (8). Both Henkin and Suppes attested to the importance of the axiomatic method. Henkin began his chapter with the axiom that the axiomatic method is an important element of con temporary mathematics (23). While Suppes stated: "The importance of the axiomatic method in modern mathematics. scarcely needs a general defense (42, 69}." There were of course those who were less enthusiastic in the application of the axiomatic method in the high school (28). The general spirit of the publication was to accept the axiomatic method and structure to be important concepts that need continual development, but at the same time, there was a general caution concerned with the degree of rigor required to do this. More recently, at the CSMP International Conference on the Teaching of Geometry at the Pre-college Level held 18 March 19 through March 28, 1970, a variety of possibilities were presented for the geometry course. The conference recommendations stress the evolution of geometry as an important means of understanding and organizing spatial phenomena. The recommendations mention that deductive structures should arise to the degree to which children are able to analyze situations (14). In a presentation to the second CSMP International Conference on the Teaching of Geometry at the Pre-college Level, Marshall Stone discussed the learning and teaching of axiomatic geometry. Stone believed that all programs in geometry for high school students should be developed to include some discussion of axiomatics. By the end of the tenth grade Stone thought students should possess "... sufficiently clear notions of logic and the nature of proof that he can understand both the possibility and significance of an axiomatic treatment of geometry (41, 18)." The continued concern for the geometry component of the high school mathematics curriculum was demonstrated by the National Council of Teachers of Mathematics and the Mathematics Association of America in their joint publication of a booklet devoted to geometry (38). The near future will see the next yearbook of the National Council of Teachers of Mathematics focus on geometry. From the time of the Report of the Committee of Ten through the present thinking of mathematicians and mathematics 19 educators, mathematical structure and inference and proof patterns have been and continue to be thought of as important goals for the tenth grade geometry course. Dissatisfaction with Student Pullfillment of the Goals of Geometry Re laved to This S tudy Assuming one accepts that structure and patterns of inference and proof are legitimate objectives of the geome try course, it is important to know if these objectives are being attained or to know the degree to which they are being attained. If these goals are being attained by the students in the tenth grade course, then it does not make much sense to administer a treatment that would help in the attainment of these goals. The literature will point out that there is suitable evidence to suspect these goals are not being reached. This evidence will provide a reason to administer some treatment in an attempt to effect a positive change in the level of attainment of the goals tested in this study. Brumfiel expressed his views and experiences related to this problem in this way: It would be interesting to run a series of studies and find out what understandings of axiomatic structure of geometry students retain at the end of their geometry course, later as a college freshman, and again as college seniors (4, 3). In his own search he had this to relate: I started gathering bits of information on this appreci ation for axiomatic structure that students carry away from their high school geometry course in 1954 . I asked a group of 40 freshmen in a college algebra class to 20 list some theorems they had proved in school geometry. Five minutes of thought produced two responses. 1. We proved that if equals are added to equals the results are equal. 2. We proved that an isosceles triangle has 2 equal sides. I asked for a theorem and what did I get? First an axiom, then a definition. These students of 1954 had no conception of the structure of geometry. In a core recent investigation Brumfiel stated: Recently, in the winter of 1971, I posed similar questions to a group of twenty-five university juniors and majors and minors who will themselves soon be teaching high school geometry. Nearly all*had studied as S.M.S.G. influenced geometry based upon the postulate of a distance function, the ruler postulate, the pro tractor postulate, etc. The only axiom they could recall was that two points determine a line. Only three out of the twenty-five classified the two statements below correctly. 1. An isisceles triangle has two congruent sides {a definition, of course). 2. An isosceles triangle has two congruent angles (a theorem they all had seen proved) [4, 4]. Brumfiel's concern with the importance of structure in geometry was apparent when he stated that honesty dictates careful evaluation in at least three attributes■of students who have studied geometry. The third attribute being: "Their overall understanding of the role of the postulates in the development of geometry--how specific axioms were used as the structure of the geometry unfolded (4, 25)." Dodes express his concern in this way: People think of geometry as a well-structured course. In fact, the traditional course has neither structure, logic nor (suitable) method. The very fact that few students know the status of statements like 1. Radii of a circle are equal ("half of a definition). 21 2. Any line segment has one and only one midpoint (usually part of the Birkoff Point-Number Axiom, therefore a postulate). is evidence that the traditional course has no structure (16, 28). A study by O'Brien, Shapiro and Reali (35) involved 180 students of grades 4, 6, 8, and 120 students in grade 10 in the public school system of suburb of St. Louis, Missouri. The study involved the use of "childs logic" (reasoning from the statement to the converse or inverse) and "math logic" (making correct inferences). The results illustrated the predominance of "childs logic" over "math logic" at all levels (35). Apparently the same misgivings related to the geome try course existed long before the modern mathematics movement. Fawcett reported in 19 38 as one of the con clusions of his study: "The usual formal course in demon strative geometry does not improve reflective thinking of pupils (18, 45)." This was a strong statement during a period when geometry assumed more responsibility, within the mathematics curriculum, for the development of critical thinking powers and structural considerations than it does today. Fawcett's judgment of practices further suggested: "Actual classroom practice indicated that the major emphasis is placed on a body of theorems to be learned rather than on the method by which these theories are established (18, 23).” Some of the literature not only implicitly suggested the goals under discussion might not be achieved but also 22 supplied a plausible explanation for the lack of achieve ment . In a study by Buchalter that examined 19 popular series of text books, 45 texts in all, for grades 7-14, it was found that the majority of textbooks evaluated failed to satisfy the criteria for testing their validity with respect to the objective of presenting mathematics through an understanding of its underlying structure. It should be noted that logic and foundations were two of the criteria used {5>. The inconsistency of students learning about proof and structure from those whose knowledge in those areas is weak, was noted by Van Engen. He stated: Most certainly; the concept of structure can not be understood without some idea of the role of proof, since proof is an integral part of the "whole story." Unless teachers understand v:hat we mean by proof in building structure, their teaching efforts will not be in line with the modern movement for mathematics reform (45, 637) . Rettig developed an inventory that measured a person's "view of mathematics." The 253 secondary school mathematics teachers in the state of Pennsylvania who com pleted and returned the inventory produced some interesting results. Rettig reports: "Most of the teachers saw un defined terms as necessary but a considerable number of them saw undefined terms as unnecessary or as manifesting the laziness of mathematicians (39, 88)." Concerning teacher 23 understanding of a mathematical system Rettig had this to say: A fairly large percentage of the teachers do not seem to understand what a mathematical system is. This is indicated by the 17 per cent of the teachers who did not agree that undefined terms are necessary in a mathe matical system, and also, by the 40 per cent who did not agree that a mathematical system has the same properties on earth as it does on the moon and Mars (39, 52). Recommendations for the Geometry Course Most frequently mentioned among the suggestions for a major restructuring and revision of the entire geometry course are: (1) the affine approach to Euclidean Geometry, (2) the transformational approach, and (3) the vector approach. While all three approaches have the general effect of uniting the algebra and geometry, supporters of each approach also feel that the goals related to this study are accomodated within the changes they are suggesting. When a student faces a great many axioms in early encounters with an axiomatic system, Johnson (24) pointed out that the student is likely to be confused. He may feel that too much is being assumed, and much of the axiomatic method's power is not revealed to him. Among other con tributions, Johnson felt the axiomatic structure of the affine approach is more adaptable for inculcating an appreciation for axiomatics in the students. Although the affine approach to Euclidean geometry, at the secondary school level, is probably to be regarded as 24 experimental, it has some strong backers. At the present time a variety of suitable textbooks for high school students is not available. The Wesleyan Coordinate Geometry Group did, however, produce a text for this purpose. Fehr included in his objectives to guide the development of secondary school geometry instruction: Develop an understanding of an axiomatic structure by this sequence of study: the affine line, the affine plane, the affine space, metric space, Euclidean space as a vector space (20, 8). Fehr would eliminate any year-long course in geometry from the curriculum. He would build it into an integrated unified body of secondary school study of mathematics (20). There are individuals who have called for the insertion of transformations into the tenth grade geometry course. The publication of Yaglom's Geometric Transfor mations (52) by SMSG in 1962 helped give prominence to the subject in the minds of many school people. More recently (1971) the emergence of a hardbound text for the tenth grade geometry course, Geometry: A Transformation Approach (13) and its use in approximately thirty schools was an indication of the progress of this approach. Avoiding the whole idea of superposition, the use of function in its most natural form as a geometric transformation, the applications of symmetry in other fields, and the ability to use transfor mations as problem-solving instruments have been cited as advantages of this approach. Although the proponents of 25 this approach made no special claims related to "structural" goals, the increased interest in geometric transformations might easily lead to an examination of the relationship between this approach and achievement of the goals related to this study. Vector geometry represents another alternative to the traditional geometry course. A two year program con cerned with vector geometry in the secondary school was developed by the University of Illinois Committee on School Mathematics. The UICSM course was intended for bright students, but the course was taught with apparent success to students of varied abilities and backgrounds. Troyer advocated a one-semester course in vector geometry in the twelfth grade to provide a closer tie between algebra and geometry (46) , Troyer (4 3) felt that the vector approach was consistant with the development of the deductive method as a way of thinking and the application of this method in other mathematical situations. Adler (1) recommended the inclusion of a vector unit to show more clearly the relation ship between algebra and geometry. In addition to the three revisions discussed above, the insertion of a unit in logic was suggested by some as a means to improve critical thinking ability. A study by Platt measured the effect of the use of mathematical logic in high school geometry was evaluated on four variables. One of these variables was critical thinking. Platt found 26 that a course which included instruction in mathematical logic did not result in a significant difference in its effect on critical thinking ability over the traditional course (36) . There are, of course, those who would contend that formal logic, should be a part of the school mathematics curriculum (17). Martens inferred that the deletion of instructions in the rules of inference will produce students that have developed invalid inference patterns as well as valid ones (34). Miller, after a study that involved 329 tenth-, eleventh-, and twelfth-graders and the ranking of logical fallacies, recommended that formal instruction be given on recognizing the common logical fallacies (33). Fawcett found that when youngsters took an active role in the actual development of the structure of the geometry course it was possible to improve reflective thinking abilities of secondary school pupils (18). Ulmer extended Fawcett's work using more than 1,200 students and 16 teachers. He worked with three groups in an effort to make conclusions about the ability to teach reflective thinking. Two of the groups were composed of students enrolled in geometry classes; in one group appli cations of reflective thinking were stressed, and, in the other, there was no particular emphasis on this type thinking. The third group of people were not enrolled in 27 geometry. The categories tested related to patterns of proof were: (1) "If then or postulational thinking," (2) "inverses and converses," and (3) "indirect proof." Although no significance levels were reported, results seemed to indicate that students specifically taught methods of reflective thinking made clearly higher gains on a reasoning test than those not given special methods (44). Cook reported only "slightly increased" scores on reasoning tests for a group of geometry students that had been taught reasoning and transfer over those geometry students for which there was no special emphasis (15). Lewis on the other hand found that geometry taught with emphasis on critical thinking resulted in increases in ability to think reflectively (29). The above conclusion was duplicated by Massimiano (31) in a study involving 531 sophomore high school students. Although geometry students performed slightly better than non-geometry students on the Watson-Glaser Critical Thinking Test the difference was not significant at the .05 level. His data did not support the hypothesis that success in geometry was closely associated with ability to think critically. It should be noted that many of these studies use the Watson-Glaser Critical Thinking Test to measure critical thinking ability. Regardless of how one defines critical thinking and what fancy notions the definition includes, 28 the instrument actually determines what abilities are being measured. Although the Watson-Glaser instrument is defi nitely related to the goals of sub-test B , it is not very closely related with the other goals of this study. Mathematicians and mathematics educators have expressed a variety of opinions aimed at the high school geometry course. In February 1972 the high school geometry course was the topic under discussion in The Mathematics Teacher (21) in "The Forum." In this issue the question "What Should Become of the High School Geometry Course?" was examined by Fehr, Eccles, and Meserve. Although Fehr was echoing Dieudonne's point of view: "The present year-long course in Euclidean geometry must go," mathematical structure was still an important part of his proposed revisions: Today, geometry must be conceived of as a study of spaces. Each geometry is a (set, structure) where the elements of the set are called points and the structure is a set of axioms, including definitions, which relate the points and their important subsets (21, 152). Meserve, on the other hand, believed that there should be a full year's course primarily concerned with geometry. According to Meserve, students in geometry should be ready for explicit considerations of: . . . Compound statements using "and," "or," "if-then"; truth values of statements; truth tables; Venn diagrams; forms of statements of implication, negation (denial) of a statement; the converse, inverse, and contrapositive of a statement of implication;- biconditional statements; rules of inference; application of rules of inference to obtain proofs; direct proofs; indirect proofs; 29 disproofs (proofs that a statement is false, usually by a counter example); existential quantifiers; and universal quantities (21, 178). Eccles concluded that geometric transformations of the plane should play a major role in the geometry course but that much of the traditional Euclidean approach should be retained (21). Of course there are other opinions concerning the • tenth grade geometry course. Klamkin would encourage more emphasis on significant applications of geometry to other areas (27). Shuster believed students should be exposed' to several approaches (40). The literature related to this section seemed to indicate that there was a growing dissatisfaction with the current Euclidean geometry. This dissatisfaction was easier to detect than a viable substitute to which there was a general agreement. Non-Euclidean Geometry in the High School To say that the inclusion of non-Euclidean geometry in the secondary school geometry curriculum was a unanimous choice among mathematicians and mathematics educators would be misleading. There is, however, enough evidence to con clude that it is thought a worthy contender for a spot in the curriculum. The specific reasons for the use of hyper bolic non-Euclidean geometry in this study were discussed in Chapter I. 30 Perhaps the strongest statement concerning exposure to non-Euclidean geometry came from Kemeny: The neglect of non-Euclidean geometry is one of the cultural crimes of our secondary education. We'd rather not admit to high school students that there is such an animal. Basically, the kids all think they are living in a Euclidean universe, and they ought to be told that this is false. I think this is a revelant piece of information for a child in high school. These Euclidean axioms hold true in a relatively small part of our universe; it is amatter of established scientific fact that the axioms are not true in the universe as a whole. Kids should be let in on this (7, 410). To tell high school students in a meaningful way that they are living in such a world requires that they be taught some non-Euclidean geometry. A few historical statements about Gauss, Labachevsky, Bolyai, Saccheri, and Riemann and a brief message about the importance of the fifth postulate are not sufficient (7). The importance of the non-Euclidean geometries and the necessity of their development beyond mere mention was also stated by Kattsoff: The existence of alternative geometries is no longer a complete secret to school mathematics teachers and students. This, however, is merely to say that they are mentioned as existing--a kind of curiosity. The logical import of these alternatives is lost because usually nothing more is done with them. Only when the alterna tive axioms are stated and their consequences drawn does the existence of non-Euclidean geometries have its full impact on the minds of teachers and students. The deductive character of geometry is then revealed com pletely (26, 559). Meserve devoted a section of an article to a way of helping students accept each of the four distinct geometries: Euclidean, Spherical, Elliptic and Hyperbolic (32) . Also he 31 stated: "We need to recognize that any geometry is a mathe- matical system (32, 7)." According to Meserve: Our acceptance of the existence of many geometries is a necessary step in the obtaining of a sufficient under standing of geometry to apply geometric concepts effectively to mathematical problems and to the teaching of geometry (32, 11). Among the major movements of scientific thought that should influence elementary geometry and the way it is presented, Veblen listed the discovery of non-Euclidean geometries (47). One of Adler’s five significant goals for high school geometry was "Development of Critical Thinking." In ex plaining what he meant by "critical thinking" Adler mentioned that ability to reason deductively from a set of assumptions does not by itself guarantee the ability to think critically. In fact, he states: . . . if the student knows of only one set of con clusions, deduced from the commonly used assumptions, he can easily fall into the trap of thinking that the conclusions are necessarily true rather than being merely consequences of the assumptions (1, 234). Adler suggested that an effective way of demonstrating that conclusions depend on assumptions was to change the axioms of Euclidean geometry to convert them into the axioms of hyperbolic geometry (1). Allendoerfer summarized his opinion of a reasonable curriculum for geometry in the schools in a 1969 article. In the summary was a brief treatment of non-Euclidean geometry in the eleventh or twelfth grade (2). 32 In listing some important pedagogical character istics of his geometry program Fawcett includes the intro duction of other geometries. A considerable extension of this introduction appears in the text book he authored with Cummins (19). In describing the advances made in geometry since 1827, including the development of the non-Euclidean geome try, Fehr believed that it was: "Plainly evident that geometry today has a tremendously different aspect than that which is still prevalent in today's high school program (20, 6)." Fehr further states: "Euclid's geometry is just one of many, and to imply otherwise would be to deny ail that has happened in mathematics and science during the last 100 years (20, 7)." Feasibility of Teaching Hyperbolic Non-Euclidean Geometry to High School Geometry Students Although it is quite likely that hyperbolic non- Euclidean geometry has been introduced, in some form, in many high school geometry classrooms throughout the country, the results of these endeavors have not been reported in the literature. Luckily existence arguments require producing a single example and at this writing three can be produced. Maiers taught a unit in hyperbolic non-Euclidean geometry. He investigated the geometry up to the point of proving that similar triangles, in the Euclidean sense, do not exist. This rather astonishing result leads to the 33 angle-angle-angle congruence theorem of ordinary triangles in hyperbolic geometry. The time spent on the unit varied from six to ten days. Maiers reported that, for the most part, students were excited about the material, class discussion was livelier than usual, and students seemed to be more appreciative of the role of axioms in a mathematical system. Mo statistical evidence was cited (30). The unit taught at the Niles West High School for three years was similar in content to that of Maiers. Again no data was collected, but judging from student and teacher reactions the following could be concluded: (1) students were able to understand the material, at least enough to work achievement tests effectively; (2) the unwillingness, at first, of the students to accept the Characteristic Postulate indicated a lack of understanding of the axiomatic process; and (3) the class was more lively and appeared interested in the material. The Greater Cleveland Mathematics Program's Geometry course (6) included a substantial section on hyperbolic non- Euclidcan Geometry. In booklets one and two the geometry proceeded on the basis that those postulates essential to both Euclidean and non-Euclidean were accepted and the consequences of these were discussed. Booklet three takes the Euclidean route. Absolute geometry is re-examined in booklet four as a lead-in to non-Euclidean geometry. The dependence of theorems on the postulates and other theorems 34 was emphasized as was logical dependence among the propo sitions. The whole course approached geometry as a study of a mathematical system in which there was a point, the fifth postulate, where a decision had to be made. However, by delaying the decision there were still many results that could be reached. It would have been interesting to know how students in this type program would have performed on the post-test used, in this study and how they would answer the question "what is geometry?" Unfortunately( nothing of a statistical nature could be located. Summary This review began with a discussion of the goals related to mathematical structure and patterns of proof and inference claimed for the tenth grade geometry course. Since these were the goals related to this study, the intent was to show that the goals stated above were among the commonly accepted goals for the geometry course. Next, evidence was presented that indicated there was reason to believe these goals were not being achieved. Recommendations for changes in the high school geometry curriculum were presented. These changes reflect a dissatisfaction with the geometry course as it exists. Most of the recommendations included positive indicators related to the goals of this study. The inclusion of non-Euclidean geometry in the high school geometry curriculum was cited as one of the 35 recommendations. Opinions of some mathematicians and mathematics educators, regarding the inclusion of non- Euclidean geometry, were presented. The main body of the chapter concluded with examples that indicated the non-Euclidean geometry used in this study, the hyperbolic type, was feasible at the tenth grade level. CHAPTER III RESEARCH PROCEDURE « Sample and Design This study was conducted at the Brookhaven High School in Columbus, Ohio during the 1971-72 school year. The subjects were those students, mostly sophomores, taking the geometry course at Brookhaven. Because the junior high schools in the area have a program in which mathematically talented students take geometry in the ninth grade, these students could not be included in the sample. No special patterns existed for assigning students to the geometry classes. The class grouping was not consciously homogeneous and special scheduling problems were scattered. A follow-up study of Brookhaven graduates of 1964, 1966, and 1967 showed that 26,70 per cent of the graduates enrolled in schools leading to a bachelors degree and an additional 8.70 per cent enrolled in other schools beyond the secondary school. It was estimated by the curriculum coordinator of the school that about 90 per cent of the plane geometry students at Brookhaven go on to college. To test for initial differences and effects of history and maturation two measures were taken. The first 36 37 measure# a fifteen item test of elementary geometry facts based on junior high school experiences# was administered in early September. The test v/as part of a masters degree candidate's thesis. Two "initial differences" hypotheses were tested at this stage: G^: There is no significant difference in the means of the control class and treatment class of each geometry teacher on the geometry achievement test. G2 : There is no significant difference in the means of the total control group and total treatment group on the geometry achievement test. The results of the achievement test and the tests on the hypotheses will be discussed in Chapter V. The test appears in Appendix A. The second measure was the pretest given on May 15, 1972. The test used for this purpose was the Educational Testing Service's Cooperative Mathematics Test# Geometry Form A, Part 1. In order to examine the groups for differ ences at this point, the following hypotheses were tested: G^: There is no significant difference in the means of the control and treatment class of each geometry teacher on the pretest. 38 G^: There is no significant difference in the means of the total control group and the total treatment group on the pretest. The results of the tests and the tests on the hypotheses will be discussed in Chapter V. The pretest date, May 15, 1972, was also the date that three of the six geometry classes were randomly assigned to the experimental group. This assignment was delayed until this date to avoid the possibility of the experimental classes receiving special treatment during the school year. Each of the three teachers taught one control and one treatment group. It was decided, prior to the treatment, to use two designs in this study. The Static-Group Comparison was the first of these two designs utilized in this study. This design may be diagrammed as follows: X ox Campbell and Stanley (10) classified the static group comparison design as "pre-expcrimental." In this design it is helpful to have data concerning the equality of the groups. This data will be supplied in Chapter V. The data analysis related to this design used the class as the unit * of analysis. 39 The second design used in this study was the Non equivalent Control Groups Design. This quasi-experimental design makes use bf intact groups. The design can be diagrammed in the following way: i This design corrects many of the difficulties that are present in the static group comparison but required, for this study, the unit of analysis to be the student. Treatment The treatment group, the fifth, eighth, and ninth period geometry classes, studied the unit in hyperbolic non- Euclidean geometry. The unit began with a section concerned with the importance of the fifth postulate of Euclid. The undefined terms and postulate set, not including Euclid's fifth postulate, was introduced next. The characteristic postulate was given a special introduction because of its importance. Pasch's Axiom was discussed because it would be used in some of the proofs in the unit. The twenty- eight propositions of Euclid, common to both geometries, were stated, preceded by an explanation justifying their availability in the hyperbolic geometry. The topics of the previous paragraph were written on pages one through five of the unit. There are two 40 comments to be made related to the brief exposure to these introductory ideas'. The seminar members felt that their students interest level would be higher if they could get into the hyperbolic geometry quickly. The second reason was that the introduction could be supplemented by the classroom teachers if this seemed appropriate. The unit continued with a careful description of the three types of lines that would frequently be the center of attention: intersecting lines, non-intersecting lines, and the special case of the non-intersecting lines, the parallels. The main topics of the unit included the following: ideal triangles, Saccheri Quadrilaterals, Lambert Quadrilaterals, and the properties of ordinary triangles which includes the angle sum theorem and the angle-angle-angle congruence theorem. A copy of the unit appears in Appendix C. Although no special effort was made to keep the classes working at the same pace, casual progress checks indicated that there was never more than a two-page differ ence between classes. This might largely be due to the seminar activities. Because of the seminar, each teacher was aware of the pacing of the others. This awareness might possibly have resulted in an effort to maintain an approxi mation of the pace of the others. The control group studied chapters thirteen and fourteen in Dolciani's Modern Geometry (25) and received 41 supplementary problems directly related to the text material. The supplementary problems represented an attempt to equalize the Hawthorne effect between the control and treatment groups. Both control and treatment groups associated something special with their class during the implementation of the treatment. Instrumentation The pretest used for this study was the Educational Testing Service’s Cooperative Mathematics Test, Geometry Form A, Part 1. The reliability reported for this forty item test was .80. The pretest was administered on May 15, 1972 to all geometry students. The mean score on this exam was 22.41. The national high school norms for this test place a score off 22 in the 51-80 percentile band. The same score, 22, on the urban high school norms was in the 51-81 percentile band. A more thorough analysis of the results of the pretest will be presented in Chapter V. Mo existing test could be located that tested mathe matical structure via the role of the components of a mathematical system, operating within the framework of an unfamiliar system, and also strategies of proof and inference in a mathematical setting. It was therefore necessary to develop the posttest for this study. Six stages could be identified with this development: (1) generating items, (2) categorizing items (content validity), (3) th e ‘pilot 42 test, (4) evaluating the pilot and writing the final draft, (5) administering the test, and (6) evaluating test results. The item generation phase took place between August 1, 19 71 and April 10, 1972. The items developed throughout this period were submitted for criticism to four members of the mathematics education department, a member of the mathematics department, and fellow graduate students. The culmination of this activity was a draft containing 101- questions . During the week from April 17, 1972 to April 24, * 1972, the questions were reviewed by two members of the mathematics education faculty, one member of the mathematics department, and one Ph.D. student in mathematics education. The reviewers were asked to decide in v/hich of the three subtests, A, B, or C, each question should be placed. The description of the subtest objectives that appeared in Chapter I was given to the reviewers. In addition there was an opportunity to rate a question as not belonging to any of the categories. The only items that remained after the rating were those items on which there was unanimous agree ment on placement in one subtest area. The content validity of the test was established in this way. On May 1, 1972 a pilot test was given to two geometry classes at the Northland High School in Columbus, Ohio. Northland is located approximately two miles from Brookhaven and serves about the same type population. The pilot test 43 consisted of fifty-five items chosen from those items that received unanimous placement by the reviewer's. The fifty- five items were distributed in the following way: subtest A, 20 questions; subtest B, 17 questions; and subtest C , 18 questions. The results of this test are summarized in Table 1 below. TABLE 1.--Results of the Posttest Pilot. Description of Measurement Results Number of Students Taking Test 53 Number of Items on Test 55 Mean Score 19.38 Kuder-Richardson 20 .728 Kuder-Richardson 21 .657 The test that was given at Northland appears in Appendix B. The next phase in the development was evaluation of the results of the pilot. It was apparent from the analysis of the individual item results that the students did not have time to finish the test. The final draft of the test therefore contained fewer questions. Eliminated were those questions on the pilot taken from the "Watson-Glaser Analysis of Critical Thinking Test" and some nonsense syllogisms since these items were the most removed from the goals of the posttest. Some "true-false" questions on elliptic 44 geometry that were repetitive and were already incorporated in previous questions on the exam were also removed. VThen the evaluation .process was completed, the posttest emerged with thirty-six items. Of these 36 items, 19 were on subtest A, 8 on subtest B, and 9 on subtest C. This test can be found in Appendix E. The posttest was administered to the six geometry classes at Brookhaven on June 9, 1972. The results of this test are discussed in Chapter V. One achievement test on hyperbolic geometry was administered to the treatment classes during the course of the study. This test, generated in the seminar, was intended to provide a casual check on student progress in hyperbolic non-Euclidean geometry. Each teacher graded the exams for his class within the guidelines for determining partial credit suggested in the seminar. The test appears in Appendix D and the results are discussed in Chapter V. Analysis of Data A decision was made, early in May, to analyze the data in two ways. Because it was possible that, by May, the students within each class were acting more like members of that class than they were as individuals whose actions were not affected by the other members of the class, the first analysis used the "class" as the unit of measure. Using the class as the unit of measure it was nece'ssary to use a t-test on the means of the posttest. 45 The second analysis used student as the unit of measure. With the "student" as the unit of measure the analysis of covariance could be used with the pretest as a covariate. Because the correlation between the pretest and posttest was only .34, blocking seemed the appropriate technique to use. Dr. Jae Lee of the statistics department concurred with this decision. The next chapter presents a description of the seminar activities. The seminar provided the setting in which the hyperbolic non-Euclidean geometry could be studied, the unit was developed and discussions related to both the mathematics and teaching of the unit took place. CHAPTER IV THE SEMINAR Background An understanding of the cooperative program between the mathematics faculty of the high school and the mathe matics education faculty of Ohio State University will help provide the reader with a conception of the setting that made the seminar and this research possible. The Brookhaven mathematics department accepted 42 pre-student teachers and 11 student teachers during the 1971-72 school year. The writer taught two mathematics classes at the high school normally taught by the department chairman. In exchange the chairman assisted with the pre-service teachers. Table 2 summarizes the distribution of the pre-student teachers and student teachers among the mathematics teachers at the high school. The pre-student teachers were at the high school each school day from 8:00 a.m. to noon. Typical activities of the pre-student teachers were: tutoring, grading papers, providing help at grading time, leading small group lessons, teaching mini lessons to an entire class, and occasionally taking full responsibility for the lesson. Because the 46 47 TABLE 2.--Distribution of Pre-Student Teachers and Student Teachers. Teacher 1 2 3 4 5 6 Pre-Student Teachers Autumn Quarter First Half 2 2 2 2 2 1 Second half 2 2 2 2 2 1 Winter Quarter- First Half 2 2 2 2 2 0 Second Half 2 2 2 2 2 0 Student Teachers Winter Quarter 1 1 1 1 1 0 Spring Quarter 1 1 1 1 1 1 Total 10 10 10 10 10 3 student teachers had been through a teacher education program with a field experience component that began in the junior year, the student teachers were able to assume a greater amount of responsibility quicker than student teachers of a program that supplies little or no field experiences prior to student teaching. Since the classroom teachers were freed from many of their normal responsibi lities, they were able to devote this released time to providing leadership for the pre-service teachers. As a result of the writer teaching two of the department chairman's classes, the chairman was released two periods a day to assume leadership responsibilities 48 beyond those of <:he other teachers. In addition tc making a large number of classroom observations, having conferences with the pre-service people, and leading a seminar for the pre-student teachers once a week, he acted as a liaison between the high school administration and the university. The chairman did not receive released time for the duties connected with the chairmanship. The design of this program placed the author in the high school from 8:00 a.m. to noon daily and this stay was extended until 3:00 p.m. approximately twice a week. During this time, observations were made, student conferences held, impromtu discussions were generated as problems arose, suggestions to classroom teachers for optimum utilization of the pre-service teachers were made, and seminars were conducted. In addition to those people mentioned, a faculty member from Ohio State made observations, directed seminars and was ultimately responsible for the teacher education activities that took place at Brookhaven. There were three aspects of this program that facilitated the materialization of this study. The first was that the program enabled the author to become a teaching member of the Brookhaven faculty. Because of this situation there was an apparent atmosphere of openness, frankness, friendship and mutual trust. It should be noted that this * atmosphere had to be created since a university associated 49 person could be looked upon as a threat by a high school faculty. This atmosphere, and the friendships that grew out of it, perhaps contributed more to the success of the total program than any single factor. Secondly, because the people involved in the program were all at the school together at least four hours per day, there were many opportunities for interactions between the pre-service teachers, in-service teachers, and the mathematics education representatives from Ohio State. Questions could be answered quickly, techniques could be discussed, and spontaneous discussions concerning the hyperbolic geometry unit frequently developed. The third aspect was the • flexibility of teacher time made possible by the assistance they were receiving from the pre-service teachers. Since each classroom teacher was assigned pre-service teachers throughout the school year, it was possible for the teachers to meet one or two periods per week in a seminar during the school day. The seminars conducted at Brookhaven High School met at least once a week during the period January 3, 1972 through June 12, 1972. Each seminar was conducted during the school day and lasted one to two hours. The participants were the teachers involved in the study and the writer. The seminar activity was directed toward the attainment of three goals: (1) preparing teachers to teach a unit on hyperbolic non-Euclidean geometry, (2) developing a unit on hyperbolic 50 non-Euclidean geometry suitable for high school students to digest in about thrfee weeks, and (4) discussing'teaching strategies both prior to and during the teaching of the unit. Preparing the Teachers Although each of the teachers involved in the study had a conversational knowledge of hyperbolic geometry, none of them had been exposed to formal course work in that subject. The text for the seminar was a rough draft of the unit that would be used in the study. Wolfe's book Non- Euclidean Geometry (50) was used in a supplementary capacity. The unit was looked on as something to be built on and changed. Wolfe's book added depth where it seemed appropri ate . This writer lead the first three seminars while the teachers were adjusting to the new material. After the first three meetings, the leadership responsibility was rotated among the other teachers in the seminar. The seminar leader was responsible for proofs of theorems and solutions of problems designated for the seminar meeting he was directing. The atmosphere was informal and each member made contributions. Prior to the introduction of the characteristic postulate of hyperbolic geometry it became evident that some historical background, an examination of some of the short comings of Euclid's geometry, a close look at the fifth 51 postulate, some discussion of axiomatics, and a review of indirect proof was appropriate. A major portion of the introduction was devoted to those propositions of Euclid (one through twenty-eight) that could be used in hyperbolic non-Euclidean geometry. The emphasis was placed on understanding why they were available. Some of the important consequences of Euclid's fifth postulate (Playfairs- axiom, exterior angle theorem, sum of the interior angles of a triangle equals 180, etc.) were examined to see why they were consequences of this postulate. It was anticipated that students would be using Euclidean propositions that were not available in hyperbolic geometry. The teachers wanted to develop the techniques necessary to explain the inconsistency of using these propositions. Pasch's Axiom, Playfair's Axiom, and a look at indirect proof concluded the introduction to hyperbolic geometry. Pasch's Axiom was necessary to have available for many of the theorems. Playfair's Axiom was emphasized because it was the popular form of Euclid's Fifth Postulate. The teachers were asked to show that Euclid's Fifth Postulate presented necessary and sufficient conditions for Playfair's Axiom. Because indirect proof played a vital role in so many of the proofs of the hyperbolic geometry, the teachers felt the need to review this type .proof strategy. VanEngen's article, "Strategies of Proof" (45) was used in conjunction 52 with examples of proofs. A few indirect type proofs were assigned as homework exercises. There were several times in the seminar when topics were discussed that would not be included in the unit. Among these topics were: Dedekind's postulate, absolute geometry, and a brief look beyond the unit. Two theorems that occur early in the unit were not proved for the high school students. The proofs were difficult and it seemed likely that they would discourage the high school students. These two theorems were: (1) if line 1 ^ is parallel (in the hyperbolic sense) to line lg, then lj is parallel to 1^, and (2) if a straight line is parallel to a given line in a given sense, then it is parallel at each of its points in that same sense. Both theorems were proved in the seminar. At some time in each seminar the following occurred: (1) the seminar leader introduced the new material in any way he saw fit, (2) problems assigned during the previous seminar were solved, (3) problems were assigned for home work, (4) suggestions for the unit were proposed and discussed, and (5) teaching strategies were considered. By this time in the school year the writer and the other seminar members had become good friends. Because of this, discussion was open and activity abounded in these sessions. r>3 Doveloping the Uni t The writing of the unit was the responsibility of each person in the seminar. The writer coordinated this activity bat made a special effort to incorporate the feelings of the teachers, and, many times, their exact words into the unit. It was felt that the teachers might be more enthusiastic teaching a unit that they had developed rather than one that was a finished product of someone else. It was during the first seminar meeting that the strategy for writing the unit was established. On the basis of the discussion that took place in the seminars, it was decided that the author would write a rough draft. The draft was written in installments of two to three pages each. A copy of the installment was given to each seminar member for comments and changes. These comments were used to make changer and prepare the final draft. After the rough drc>ft, the writer assumed the responsibility for the mathematical accuracy of the unit while the high school teachers concentre-tod on form and phrasing. Most of the criticisms were related to simplification of presentation. Theorems that had more than one part were changed so than each part was a separate theorem. Some of the proofs that were only outlined in the draft appeared in mere detail in the final version'. A more careful intro duction to the notion of parallel, in the hyperbolic sense, war suggested and incorporated. Because the teachers know 54 they would be teaching from the unit, they made suggestions that they felt would'make the unit teachable. Discussing Teaching Strategies The teaching problems connected with the unit were partitioned into two classes: (1) problems that were anticipated prior to teaching, and (2) problems that were not anticipated but arose as the unit was being taught. A portion of the seminar time was devoted to the problems in these categories. Because the seminar members worked together most of the. school day, there were many times when teaching and other problems were discussed during the school day outside of the seminar. Some of the problems that were correctly anticipated were: (1) students would use theorems from Euclidean that were not available in hyperbolic, 12) the concept of "parallel" in the hyperbolic sense would be confusing, (3) ideal points might be confused with ordinary points, (4) students ability to make indirect proofs would be low, and (5) students would not be willing to accept the character istic postulate of hyperbolic geometry. In an attempt to solve the first problem those propositions among the first twenty-eight that would be most frequently used were discussed. In addition, the propo sitions that might be tempting to use but were not available, were pointed out. 55 It was decided by the seminar members to define rather than develop the hyperbolic parallel concept. The three classes of lines: intersecting, non-intersecting, and parallel were mentioned to be consequences of the character istic postulate. The theorem that showed any line within the angle formed by the two parallel lines to a given line, 1, through a goven point, P does not intersect 1 was proved. After this proof, the classed of lines were summarized. Students were asked to show: Given a point, P, and a line, 1, with PT intersecting 1 at T, and parallel to 1, that any line within angle TPS intersects 1. This idea was utilized often in proofs, and it reemphasized the significance of "parallel" meaning the first non intersecting line. Rather than stating that parallel lines share an ideal point, it was decided to use the word "approach." The teachers felt that this might help.prevent the students thinking of the ideal point as the point they were familiar with in Euclidean geometry. The high school teachers believed that their students would be able to make indirect type proofs if there was some general pattern of attack. The general patterns that emerged from the seminar took the following form: (1) accept the hypothesis to be true and assume the negation of the conclusion to be true, (2) based on the 56 information in (1) search for a contradiction, {3) when a contradiction, R, was found, state: "but R cannot be correct," and, as a reason, state exactly what was con tradicted, (4) write "assumption incorrect" and as a reason write "contradiction of1' followed by the classification (definition, axiom, theorem, etc.) of what was contradicted, and (5) the negation of the assumption was written justified by the- reason: "if one of two contradictory statements was proved to be incorrect, then the other was correct. The teachers seemed to feel more comfortable with this structure so it was adopted. Because it was the author's experience that students were not willing to accept the characteristic postulate of hyperbolic geometry, feasible explanations were discussed in the seminar. These explanations did not include Poincare's model since the author wanted to force the students to reason strictly from the axioms, and previously proven theorems. When the characteristic postulate was introduced in the classroom, the teachers noticed that lively discussions took place. Students who had not talked very much during the year took an active part in trying to demonstrate the impossibility of this new postulate. It took about, three to five days before most of the students began to feel comfortable with this unusual assumption. After about eight days, a youngster that reverted back to Euclidean 57 ideas not available in hyperbolic received a resounding "boo” from his classmates. Class spirit was high and the students seemed to be enjoying the unit. In general attempted solutions to foreseeable difficulties took the form of more careful development, summaries, discussing different approaches to the concept, and adding problems that fore-shadowed the difficulty. Although no formal measurements were taken to . determine the impact of the seminar on the teachers at Brookhaven, some information, anecdotal in nature can be reported. The chairman of the mathematics department had this to say: "this is the first time in ten years I've had to work hard at mathematics and I'm really enjoying it." The other two members of the seminar had similar sentiments when asked about their feelings concerning the seminar. One of the members emphasized learning more about a non-Euclidean geometry was an important by-product for him. Two of the members mentioned several times that "this is the best year of teaching I've ever had." This statement was a reflection of the total teacher education program at Brookhaven. Many times the members would stop me in the hall with an original proof of a theorem, a counter example, or a question about a proof. The motivation level was high. There was never any dissatisfaction expressed either directly or indirectly related to the seminar activities 58 and each member expressed, in his own way, that he was pleased with the activities of the seminar. The activity of the seminar was an integral part of this dissertation. The seminar provided the teachers of the Brookhaven mathematics department a setting for curriculum development, the learning of new mathematical ideas, and an opportunity to discuss teaching strategies. The seminar discussed in this chapter provided what could be an effective model for inservice teacher education. CHAPTER V ANALYSIS OF DATA Analyses of the data collected during the study are presented in this chapter. The chapter is divided into four sections. The first section reports analyses related to the hypotheses of initial differences. The second section reports analyses related to the main hypotheses of the study. The third section presents an analysis of posttest results and relates some of these results to the results obtained on the pilot test. In the fourth section the data from the results of an achievement test in hyperbolic geometry are presented. Initial Difference Hypotheses Two measures were taken to test for initial differ ences and effects of history and maturation. The first measure, a test of elementary geometry facts based on junior high school experiences, was designed and administered in early September by a masters degree candidate. Table 3 is a summary of the results of this fifty item test. Two initial difference hypotheses, G^ and , were tested. These hypotheses were: 59 60 TABLE 3.— Summary of Results of the Elementary Geometry Facts Test— September 1971. Group Mean Score Standard Deviation Period 1 21.15 6. 55 Period 3 18.60 5.68 Period 5 19.14 5.93 Period 6 18.54 4 .66 Period 8 22.22 5.77 Period 9 21. 52 4 .76 Total Control (1,3,6) 19.43 5.85 Total Treatment (5,8,9) 20.90 5. 70 G^: There is no significant difference in the means of the control class and treatment class of each geometry teacher on the geometry achievement test. : There is no significant difference in the means of the total control group and total treatment group on the geometry achievement test. Hypothesis required three tests for significance: one test for the classes that met periods one and eight (teacher one), a test for the third and fifth period classes (teacher two), and a test for the classes that met sixth and ninth periods (teacher three). The t-test was used on each of 61 the hypotheses. Table 4 summarizes the results of these tests. TABLE 4.— Summary of the Results of the t-Tests for the September Geometry Facts Test. Hypothesis df t-value P Less Than G i Teacher 1 52 .625 .268 Teacher 2 57 . 351 . 373 Teacher 3 47 2.168* .015 160 .054 G2 1.610 *Significant at the .05 level. Hypothesis was rejected at the .05 level for teacher three (sixth and ninth period classes). The t-value for hypothesis Gj approached the .05 level of significance. The second measure was the pretest. The test used was the Educational Testing Service's Cooperative Mathematics Test, Geometry, Form A, Part 1. The pretest was administered immediately prior to treatment on May 15, 19 72. The results of this test appear in Table 5. The two initial difference hypotheses tested at this point were G^ and as stated below: G^i There is no significant difference in the means of the control and treatment class of each geometry teacher on the pretest. 62 TABLE 5.— Summary of Results of the Pretest. Group Mean Standard Deviation Period 1 21.348 6.617 Period 3 22.158 5.047 Period 5 21.476 4.578 Period 6 22.428 3.682 Period 8 22.750 ' 3.669 Period 9 25.480 3.831 Total Control Student as Unit 22.09 5.22 Total Treatment Student as Unit 23.28 4.29 Total Control Class as Unit 21.97 .41 Total Treatment Class as Unit 23.23 1.45 63 G-: There is no significant difference in the means 4 of the total control and total treatment group on the pretest. The hypotheses were tested using the t statistic. A summary of the results appears in Table 6. TABLE 6.--Summary of the Results of the t-Tests Used on the Pretest • Hypothesis df t-Value P Less Than g 3 Teacher 1 41 .822 .206 Teacher 2 40 .437 .333 Teacher 3 44 2.693* .004 G4 Student as Unit 129 1.402 .081 Class as Unit 4 1.029 . 152 ♦Significant at the .05 level. The only hypothesis that could be rejected at the .05 level was for teacher three (periods six and nine). When the total control group was compared to the total treatment group, the equality of means hypothesis could not be rejected at the .05 level. 64 Analyses ReJ a ted to Main Hypotheses This section contains a discussion of the data related to the main hypotheses. The following hypotheses were tested: There is no significant difference in achievement between the control group and the treatment group on the posttest, 1^: There is no significant difference in achievement between the control group and the treatment group on subtest A (the role of the components of a mathematical system) of the posttest. There is. no significant difference in achievement between the control group and the treatment group on subtest B (proof patterns and inference patterns) of the posttest. H^: There is no significant difference in achievement between the control group and the treatment group on subtest C (operating in an unfamilar mathematical system) of the posttest. Since the treatment was administered late in the school year (May 16, 1972 to June 8, 1972), it was possible that the interactions between the students and teacher in each class molded that class into a unit. That is, the students within each class were acting more like numbers of that class than they were as individuals whose actions were 65 not affected by the other members of the class. For this reason, it was appropriate to use the class as the experi mental unit in the analysis. The statistical technique utilized with the class considered to be the experimental unit was the t-test. Table 7 summarizes the analysis. TABLE 7.— Summary of t-Test Results. Test df t-Value P Less Than Posttest Total 4 1.474 .11 Subtest A 4 1. 365 .13 Subtest B 4 1. 308 . 14 Subtest C 4 1.194 .16 None of the hypotheses, or could be rejected at the .05 level. A disadvantage of the class being used as the experimental unit in this study is that it rules out analysis of covariance as an alternative method of analyzing the data. In this research, using the class as the unit of analysis, the number of units was 6. In a 2 X 3 analysis of covariance, which was what this study required, the cell size would have been one. The degrees of freedom associated with the error term is pq(n-l)-l (49). In this study p-2,q=3(2X3 factorial experiment) and n = 1 (cell size). By substituting in the formula given 66 above, the degrees of freedom associated with the error term becomes -1. The results cannot be interpreted with -1 degrees of freedom in the error term. It was therefore decided, prior to treatment, that the data would also be analyzed using analysis of covariance with the student as the experimental unit. It was thought that this additional analysis might provide useful information despite its limitations. The correlation between the pretest and posttest was .34. With a correlation this close to zero, blocking on the covariate seemed to be the appropriate technique. Table 8 summarizes the blocking assignments. Within the blocks, scores were collapsed on the mean score on the posttest. TABLE 8,— Blocking on the Pretest. Raw Score Frequency Cum. Frequency Block Assignment 27-40 25 131 4 24-26 39 106 3 20-23 36 67 2 1-19 31 31 1 A 2 X 3 analysis of covariance was performed to test hypotheses H^, Hj* and . Table 9 shows a summary of the results of the analysis of covariance. 67 TABLE 9,— Summary of Results of 2 X 3 Analysis of Covari ance Test SS df MS FP Less Than Posttest 39.788 1 39.788 2.985 .087 Subtest A 11.948 1 11.948 1. 582 .211 Subtest B 2.196 1 2.196 1.166 .282 Subtest C 1. 875 1 1.875 .888 . 348 None of the hypotheses 11^, or H4 could be rejected at the .05 level. Ana lysis of Posttcst The posttest was administered to the six classes on June 9, 1972. A summary of results of this test appears in Table 10. The next section of this chapter contains a thorough analysis of the posttest. The reliability estimates on the posttest represented a decrease on this measure from those obtained on the pilot test. There are two explanations for this decrease. According to Guilford (22) test length and homogeneity of difficulty of the omitted items are factors that decrease reliability. The pilot test was reduced from fifty-five items to thirty-six in the final version. The decision to make the reduction was based upon the evidence present on the student response sheets that there was not enough time to complete the exam. A more complete discussion .of the reduction of item process was presented in Chapter- III. 68 TABLE 10.— Summary of Posttesc Results. Description of Subtest Subtest Subtest Total Measurement ABC Number of Items on Test 19 8 36 Moan--Control Group 9.5 4 2.32 1.99 13.88 Standard Deviation— Control Group .72 .24 .41 1.22 Mean— Treatment Group 10.34 2.69 2 .40 15.43 Standard Deviation-- Treatment Group .41 .32 .26 .85 Mean Score— Total 10.02 2 .52 2.24 14.78 Standard Deviation— Total 2.79 1. 39 1.56 3.96 Kuder Richardson 20 .563 Kuder Richardson 21 .457 69 Eleven of the nineteen items eliminated had item diffi culties that were within the interval .792 - .642. It does not seem unreasonable to assume that the homogeneity of difficulty criteria would apply to these items. The ratios of correct responses to total responses on the pilot test and posttest were close enough to merit further investigation. The mean for the pilot was 19.38 (fifty-five possible responses), and the mean for the treatment group on the posttest was 15.4 3 (thirty-six possible responses). These represent per cents of 35.23 and 42.86 respectively. A closer investigation revealed that there were thirty-four items common to the pilot test and posttest. Both tests were rescored on the basis of performance on these thirty-four items. The mean score for the group taking the pilot test was 10.75 (standard deviation 3.21) while the mean for the treatment group was 14.42 (standard deviation .81). These two scores represented per cents of 31.62 for the pilot group and 42.41 for the treatment group. Although no significant differences were obtained for the main hypothesis, a closer look at the posttest results seemed desirable. Two post-hoc analyses were performed. The first was a chi-square test. Table 11 summarizes the results of the item by item chi-square tests. The table is followed by a discussion of the results. The second analysis includes many of the measures 70 TABLE 11.— Chi-square Item by Item Analysis. Control Group Treatment Group 2 Item ------■ ■■■ — ------■— X Correct Incorrect Correct Incorrect Subtest A 2 28 36 31 36 .084 7 22 42 19 48 .551 9 30 34 29 38 .171 10 21 43 35 32 5.047* 11 51 13 55 12 .122 12 43 21 38 29 1.521 13 53 11 58 9 .357 14 33 31 37 30 . 176 15 54 10 58 9 . 127 16 28 36 36 31 1.305 17 39 25 47 20 1.232 18 18 46 23 44 . 586 19 31 33 44 23 3.972* 20 50 14 46 21 1. 499 21 45 19 44 23 . 324 22 33 31 37 30 . 176 23 12 52 11 56 .123 35 11 53 22 45 4.253* 36 15 49 16 51 .004 71 TABLE 11.— Continued. Control Group Treatment Group - I t e m ------:------X Correct Incorrect Correct Incorrect Subtest B 1 10 54 21 46 4.477* 3 33 31 38 29 .350 5 12 52 5 62 3. 693 6 26 38 29 38 .095 31 9 55 14 53 1.056 32 8 56 9 58 .025 33 26 38 31 36 .•424 34 27 37 32 35 .411 Subtest C 4 15 49 19 48 .412 8 21 43 16 51 1.288 24 8 56 17 50 3.513 25 23 41 30 37 1.062 26 19 45 14 53 1.343 27 16 48 25 42 2 . 308 28 21 43 14 53 1.837 29 5 59 15 52 5.376* 30 3 61 12 55 5.055* *Signifleant at .05 level. 72 associated with a standard item analysis. The summary of these results appeals in tables and is also foil-owed by a discussion. Chi-Square Analysis Two categories of questions will be discussed: {1) questions that produced significant differences at the .05 level (2) questions for which the chi-square value was less than .1. Chi-Square Significant at .05 Level. Student performance yielded significant differences favoring the experimental group on six of the thirty-six items. Three of these were in subtest A. The questions were: 10. Regardless of axioms used, the sum of the interior angles of a quadrilateral is 360 degrees. {True- False Item) 19. A statement and its negation cannot both be true in a mathematical system. (True-False Item) 35. Mathematicians include undefined terms in a mathe matical system {1) In order to keep the system flexible. (2) Because they don’t want to be bothered defining the obvious. (3) To add difficulty to the system. (4) Because some terms are hard to define. (5) To avoid circularity of definitions. Each of these questions may have been related to topics emphasized in the experimental treatment. The 73 control gro*p had studied only the Euclidean approach while the treatment group had just studied a system.in which the sum of the interior angles in a quadrilateral was less than 360 degrees. The emphasis on indirect proof in the hyper bolic geometry unit required students to frequently use the fact that a statement and its negation cannot both be true. The treatment group students were also forced to broaden their conception of "line," one of the undefined terms with which they were familiar. There was only one question in subtest B which resulted in significant differences at the .05 level. This question was number one, a multiple choice item, which stated: 2 1. In proving; if x is an odd number, then x is an odd number, a valid alternative would be to 2 (1) Prove: if x is not an odd number, then x is not an odd number. 2 (2) Prove: if x is not an odd number, then x is not an odd number. 2 (3) Prove: if x is an odd number, then x is an odd number. (4) Pick 100 odd perfect squares and show their square roots are odd. (5) None of the above are valid alternatives. It is possible that the extensive use of indirect proof had some carry-over to this type question. It should be noted, however, that the contrapositive was not utilized in any of the proofs. The two other questions that produced results * significant at the .05 level appeared in subtest C. Both 74 questions were in the section in which students were asked to assume that any two straight lines intersect. The questions were: 29. The least number of lines that can inclose an area is (1) 0 (2) 1 (3) 2 {4) 3 (5) 4 30. If angle 1 = angle 3, then (1) 11 intersects 12 (2) 1 \ is parallel to 12 1 (3) it is not known if ll is lei to or if paral H 1 inter sects I2 C 4) angle 1 can't = angle 3 (5) both (2) and (4) It seemed reasonable to assume that the experience of making deductions, many of which were not consistant with pre-formed notions of the plane, might transfer to other unfamiliar situations of approximately the same difficulty level. Although questions twenty-nine and thirty seemed to indicate the treatment group was better adapted to making deductions based on an unfamiliar assumption, the overall results of this subtest were not that encouraging. Chi-Square Less Than .1. There were four questions for which the chi-square value was less than .1; that is, there was virtually no difference in the responces of the 75 two groups. Two of the items, two and thirty-six, appeared in subtest A. The items were: 2. In a mathematical system definitions are (1) Statements which are agreements as to how a symbol or term is to be used,and they do require proof. (2) Statements which are agreements as to how a symbol or term is to be used,and they do not require proof. (3) Statements that should be flexible so they can be used in a variety of situations. (4) Self evident truths. (5) Hone of the above. 36. A statement that is a self evident truth, accurately describes (1) An undefined term (2) A definition (3) An axiom (4) A theorem (5) None of the above In each case the results are understandable. Although definitions were present in the hyperbolic unit, there was no special emphasis placed upon the meaning of the term "definition." While the treatment group was defining terms in the unit the control gioup was defining terms from the text. Each group simply made more definitions; and there fore, their perceptions regarding the nature of a definition may have remained unchanged. The results of question thirty-six may reflect exposure to axioms that were presented and discussed as "self-evident truths" (the most popular distractor). Apparently, the .three week unit in hyperbolic geometry did not affect these pre-formed notions. 76 The I'enaining two items for which the chi-square was less than .1 appeared in subtest B. 6. In trying to prove /2 is not rational using the indirect method of proof, one should (1) Assume /?. is rational and show this does not lead to a contradiction of a known fact. (2) Assume /2 is not rational and show this leads to a contradiction of a known fact. (3) Assume /2 is not rational and show this does not lead_to a contradiction of a known fact. (4) Assume /2 is rational and show that this leads to a contradiction of a known fact. (5) None of the above. 32. Accept as being true the statcsment: "If each of two sets, A and B are compact, then the union of these two sets is compact." Given two sets A and B, and the knowledge that the union of A and B is compact, we can conclude (1) A is compact and B is compact. (2) Either A or B is compact. (3) Neither of the two sets in compact. (4) Nothing definite about A or B can be concluded. (5) The question can't be answered unless the terms are understood. The results of question six may question the transfer effect of the emphasis on indirect proof in the treatment group. Item Analysis of Posttest Results This section will present two summary tables and a brief discussion of some of the results highlighted in the tables. Table 12 is a summary of the relative difficulty (per cent of students missing the item) and discrimination indices (degree to which the item'discriminates between the upper and lower 27.5 per cent). Table 13 gives the item by 77 12.— Relative Difficulty and Discrimination Indices of Posttest Items. — -i i I ' ■■ 1 ■ l' 1— ■ i Tl ■ II l ii i i ~1 ■— 1 I I Relative Difficulty Discrimination Index Control Treatment Control Treatment Group Group Group Group .837 .686 2.9 9.6 2 . 560 . 540 53.7 29.9 3 .486 .432 33.0 47.9 4 . 763 .728 31.6 19.0 5 .814 . 623 30. 8 13.9 6 . 590 , 567 42.6 45.2 7 .651 . 721 48.6 31. 3 8 .673 .839 36.2 12. 6 9 . 521 . 567 13.7 44.6 10 .675 .478 33.5 32.2 11 .209 .175 14.2 22.6 12 . 336 .283 32.5 40.1 13 .175 .129 19 .5 31.5 14 .495 .455 28 .6 45.5 15 .160 .127 11.5 15.7 16 .568 .462 27 .8 28.2 17 .396 . 306 18.7 26.5 18 .719 .658 18.7 24.1 19 . 525 .333 18.3 37.7 20 . 221 . 312 36.9 22.9 21 .307 .346 16. 3 37.4 78 12.— Con t i nued. Relative Difficulty Discrimination Index Ite Control Treatment Control Treatment Group Group Group Group 22 488 .440 38.9 51.1 23 842 .833 27.8 18 .9 24 864 .746 25.2 47.8 25 652 .552 44.2 69.4 26 707 .794 42.3 21.3 27 764 . 625 57.2 43.3 28 671 .795 25.5 14 .4 29 921 .778 15.0 25.0 30 953 .815 30.6 15.1 31 863 .801 14.7 14.1 32 871 .862 4.7 5.3 33 593 . 535 38. 6 46.4 34 582 .524 27.8 42.1 35 829 .686 15.8 13.5 36 772 .757 12,8 20.6 79 TABLC 13.— Response Pattern on Posttest. Total Responses Item Correct Correct Control Group Treatment Group Answer Answer (64 Students) (67 Students) 1 2 3 4 5 Blnk 1 2 3 4 5 Blnk 1 15 10 24 3 12 0 2 20 21 17 2 7 0 2 2 16 28 8 7 5 0 2 17 31 4 11 4 0 2 3 1 15 33 3 12 0 3 2 13 38 4 10 0 3 4 8 25 15 16 4 0 3 13 18 19 9 7 1 3 5 41 4 12 5 2 0 3 49 4 5 8 1 0 3 6 11 13 8 26 6 0 4 12 5 8 29 3 0 4 7 1 2 22 31 8 0 3 5 2 19 34 7 0 3 8 12 19 21 7 5 0 3 15 25 16 4 5 2 3 9 30 33 1 0 0 0 3 29 38 0 0 0 0 1 10 41 21 0 1 0 1 2 30 35 0 2 0 0 2 11 12 51 1 0 0 0 2 10 55 2 0 0 0 2 12 43 20 1 0 0 0 1 48 17 0 0 2 0 1 13 10 53 0 0 1 0 2 9 58 0 0 0 0 2 14 33 29 1 1 0 0 1 37 28 0 0 1 1 1 15 54 8 1 1 0 0 1 58 7 2 0 0 0 1 16 33 28 1 0 2 0 2 31 36 0 0 0 0 2 17 39 23 1 1 0 0 1 47 18 2 0 0 0 1 18 43 18 0 2 1 0 2 44 23 0 0 0 0 2 19 31 32 0 0 0 1 1 44 22 0 0 1 0 1 20 6 50 6 1 1 0 2 15 46 6 0 0 .0 2 21 11 7 45 0 0 1 3 20 2 34 0 1 0 3 80 TABLE 13.--Continued. r^, — ■■ J-t , | t _ u . i l T I' !■ 1 T • I f I I T ~ ' 1 T Total Responses Item Control Group - Correct Treatment Group Correct (64 Students) Answer (67 Students) Answer 1 2 3 4 5 Blnk 1 2 3 4 5 Blnk 22 33 4 26 1 0 0 1 37 5 23 2 0 0 1 23 26 12 24 1 1 0 2 25 11 28 3 0 0 2 24 8 36 7 1 11 1 1 17 23 9 6 12 0 1 25 23 32 1 0 a 0 1 30 19 5 1 12 0 1 26 13 7 20 19 5 0 4 15 9 18 14 11 0 4 27 31 16 7 2 8 0 2 17 25 3 11 11 0 2 28 22 4 6 21 9 .2 4 28 6 10 14 8 1 4 29 7 9 5 38 4 1 3 7 9 15 34 2 0 3 30 3 28 10 8 14 1 1 12 24 14 5 2 0 1 31 14 13 9 2 25 1 3 19 10 14 8 16 0 3 32 21 9 8 8 17 1 4 23 9 9 9 16 1 4 33 3 26 7 3 23 2 2 6 31 4 4 22 0 2 34 7 27 9 4 15 2 2 8 32 4 6 16 I 2 35 24 8 4 15 11 2 5 20 2 3 18 22 2 5 36 9 20 11 7 15 2 5 10 20 17 4 16 0 5 81 item response pattern for the control and treatment groups. The total number of students responding to each distractor and the correct answer is listed for each group. Item Difficulty. The posttest overall mean was 14.78 on a thirty-six item test. It is evident, then, that there were many items for which the relative difficulty was high. There wore four items on which the relative diffi culty was greater than .800 for both control and treatment groups. Less than 20 per cent of the students taking the exam were able to answer each of questions twenty-three, thirty, thirty-one, and thirty-two. Question twenty-three asked students to classify the following as "1. axiom, 2, definition, or 3. theorem." 23. Point B is said to lie between points A and C if and only if all three points are distinct points on a line and AB + BC =■ AC. This definition of "betweenness of points" was one that received special emphasis at the beginning of the course. Fifty-one students thought it was an axiom and fifty-two picked it as a theorem. Question thirty was discussed in this chapter (chi- square significant at .05 level). The most popular dis tractor for this item was two (1^ is parallel to 1^)* Fifty-two people marked two as their answer while twenty- four students answered that it is not known if 1^ is parallel to I2 ^r if 1^ intersects Ij. 82 Question thirty-one asked: 31. It is known that "if a function is differentiable, then it is continuous." which of the following follows from this fact? (1} If a function is not differentiable, then it is not continuous. (2) If a function is continuous, then it is differentiable. (3) If a function is not continuous, then it is not differentiable. (4) It is desirable for a function to be differentiable. (5) The question can't be answered unless the terms are understood. Distractors one and five were most popular with thirty-three and forty-one students respectively. It appeared that students had virtually no understanding of the relationship • that exists betv/een a theorem and its contrapositive. Thirty-two was a similar question which was also discussed in the previous section of this chapter. The question related a statement with its converse. Forty-four students reasoned that the converse of a statement {assumed true) could be concluded to be true itself. This time only thirty-three students believed that the question could not be answered unless the terms were understood. There were only two items on which at least 80 per cent of the total group answered correctly. These two items were thirteen and fifteen. Both questions were the true- false type. 83 13. Once a theorem is proved in one mathematical system, it is true in any other mathematical system. 15, A theorem is a statement that must be proved. Not only might one expect, perhaps, even a higher rate of correct response to question fifteen, but also the results of thirteen do not seem consistant with the general results of the posttest. Achievement Test In Hype rboTlc Geometry One achievement test in hyperbolic non-Euclidean geometry, developed in the seminar, was administered on June 2, 1972. The test was given just prior to the intro duction of the Lambert quadrilateral. The test was intended only to provide a rough estimate of student achievement in hyperbolic geometry. The classroom teachers graded the test in which partial credit was given according to a pattern discussed in the seminar. The test appears in Appendix D, and the results are summarized in Table 14. The results of the test indicated that students performed well on items that asked for the statement of Euclid's fifth postulate (Playfair's Axiom being acceptable), stating the characteristic postulate of hyperbolic geometry and explaining tho meaning of the term "parallel" in hyper bolic geometry. Items that required classification of lines as intersecting, non-intersecting or parallel received 84 TABLE 14.— Results of an Achievement Test in Hyperbolic Mon-Euclidean Geometry. Possible Standard Class Points Mean Deviation Fifth Period 70 44.23 10.12 Eighth Period 70 45.81 7.65 Ninth Period 70 47.27 8.65 relatively few incorrect answers. Question five defined the Lambert quadrilateral and asked students to make conjectives about other properties of this quadrilateral. Out of five points the mean score on this item was 2.72. Two proofs were included. Question 6 asked students to prove that if two lines were perpendicular to the seme line, then the lines did not intersect. The mean score on this item was 7.77 out of a possible fifteen points. The second proof involved an elementary property of an "isosceles" ideal triangle. A score of 8.11 out of 15 was the mean on this item. A summary of the mean scores, item by item, appears in Table 15. 85 TABLE 15.--Item by Item Mean Scores on Hyperbolic Geometry Achievement Test. Item Mumber Point Value Moan Score on Item 1 5 4.41 2 5 4.59 3 5 3.82 4a 5 4.45 4b 5 4.62 4c 5 4.54 5 5 2.72 6 15 7.77 7 15 6.11 8 5 2.72 CHAPTER VI SUMMARY AND CONCLUSIONS Summary This study was designed to test the effects of a unit in hyperbolic non-Euclidoan geometry on tenth grade geometry students achievement. Three criterion variables were measured: the role of the components of a mathematical system, proof and inference patterns, and operating in an unfamiliar mathematical system. The unit materialized as part of a curriculum development project in a seminar setting. The seminar met at least once each week from January to June. The investigator and the teachers who were to teach the unit studied hyperbolic geometry, considered teaching strategies, and developed the unit in the seminar. The cooperative teacher education program that existed between the Ohio State University and Brookhavcn High School both facilitated this study and imposed limitations on its design. The study took place at Brookhaven High School in Columbus, Ohio. Six geometry classes were used, three for the control group, and three for the treatment group. Students were assigned to classes within the constrains of 87 their class schedules. There was no conscious effort to group classes homogeneously. Two pretests, one administered in September, 19 71 and one in May, 19 72, were used to test the hypothesis: there is no significant difference in the means of the control group and treatment group. Using the September measure, the hypothesis was rejected. However, the hypothesis was not rejected using the May data. The treatment was conducted between May 16, 1972 and June 8, 1972. The pretest was administered on May 15, 197 2. The treatment group studied the hyperbolic non-Euclidean geometry unit for seventeen days. The control group studied chapters thirteen and fourteen in Dolciani's Modern Geometry (25) and received supplementary problem sheets on the material from those chapters. The posttest, developed by the re searcher, was administered on June 9, 1972. The four hypotheses tested were: H^: There is no significant difference in achievement between the control group and the treatment group on the posttest. There is no significant difference in achievement between the control group and the treatment group on subtest A (role of components of a mathematical system) of the posttest. 88 11^: There is no significant difference in achievement between the control group and the treatment group on subtest B (proof and inference patterns) of the post- tost. H^: There is no significant difference in achievement between the control group and the treatment group on subtest C (operating in an unfamiliar mathematical system) of the posttost. These hypotheses were tested with a t-test and a 2 X 3 analysis of covariance. Conclusions Statistical analyses indicated that.none of the four hypotheses could be rejected at the .05 level of significance. These results were consistent whether the class or the student was used as the unit of analysis. The t-values obtained in testing hypotheses and were 1.474 .16) respectively. The F ratios associated with the tests on the hypotheses , H^* and were 2.985 (p<.087), 1.582 (p<.211), 1.166 (p<.282), and .888 (p<.348) respectively. Discussion One might expect the goals reflected in the posttest would be realized by the end of a year of geometry. With the emphasis on axiomatic structure in the first year 89 algebra, students completing the geometry course would appear to have a foundation for learning about structure and strategies of proof. The results of this research indictated the students in the study could apply little concerning mathematical structure and patterns of inference and proof. If presented an unfamiliar mathematical system, they seemed to ignore the stated axioms of that system and reverted to the familiar axioms and theorems of Euclidean geometry. Does the school mathematics curriculum promote a type of mathematical thought that is in conflict with the objectives of the posttcst? The axioms a student sees are almost always self evident truths and so this is how a student perceives an axiom. The necessity and nature of undefined terms in a mathematical system is not always made clear. The curriculum rarely provides an opportunity for students to explore a mathematical system whose components arc contrary to the students intuition concerning physical space or the real numbers. The reliance on Euclidean notions is so strong that students do not seem willing to make conclusions contrary to those notions, regardless of the axioms. Tenth grade geometry pupils appear to emerge with a collection of "facts" which they can repeat reasonably well on achievement tests. The same students emerge with very little understanding about structure or patterns of proof. Indirect proof seems to be rarely used. It is possible that the total mathematical experiences or perhaps 90 "inexperiences" a youngster receives throughout the cur riculum implants too many misconceptions and too much confusion to change in a three week period. Recommendations for Further Research Using the instrument developed in this study or a refined instrument, it is recommended that the effect of different approaches to tenth grade geometry on the achieve ment variables, identified in this study, be investigated. The different approaches might include the transformational approach, vector approach, and affine approach. Another dimension could be added by comparing, within the approaches, those students who would study a unit in hyperbolic non- Euclidean geometry and those who would not. There are three recommendations for further research related to a hyperbolic geometry unit. In this study it was decided not to present the Poincare model. A similar study could be made in which the model was presented at the beginning of the unit. The second recommendation is related to placement of the unit. The characteristics postulate of hyperbolic geometry could be accepted at that point in the course where some form of Euclid's fifth postulate is first introduced. The unit could be presented at this stage in the geometry course. The effect of this placement on the criterion variables of this study could be determined. The third recommendation involves a change in the sequence of 91 the geometry curriculum. Considering that length of treat ment time might have been a contributing factor for not obtaining significant differences, creating a course similar to that of the Greater Cleveland Mathematics Projects ( ) would help to eliminate time as a constraint. The next three recommendations are related to the classroom teacher. The structural type understandings that students can learn in the geometry class may well be dependent on the teacher's knowledge of mathematical structure. It is therefore recommended that a study of the classroom teacher's understanding of the mathematical foundations significant to this study be conducted. The results of the previous recommendation could be combined with attitude measures on the classroom teacher and the effect of these two variables (attitude and knowledge of structure) on student achievement and attitudes, Ihe results of the posttest seemed to indicate that some of the commonly accepted goals of geometry were not attained by the students in the sample. If this is a generalizable deficiency, a study related to the feasibility of attainment of these goals by high school geometry students might be appropriate. The final recommendation reflects the researcher's experience with the seminar. The pattern of meeting during the school day to develop curricular materials, learn some mathematics, and discuss teaching techniques prior to and 92 during implementation of the unit# presents a feasible model for an in-service teacher training program. The. recommen dation is that the model be replicated with measurements taken to determine the effect of the program on the teachers. APPENDICES A. Sept.eir.ber Geometry Facts Test B. May Pilot Test C. Hyperbolic Non-Euclidean Unit D. Hyperbolic Non-Euclidean Geometry Achievement Test E. Posttest 93 APPENDIX A SEPTEMBER GEOMETRY FACTS TEST 1. Angles 1 and 2 are called (a) adjacent angles (b) complementary angles (c) vertical angles (d) opposite angles (e) interior angles 2, Perpendicular lines (a) meet at right angles (b) form two acute and two obtuse angles when they Intersect (c) do not intersect at all (d) form four acute angles where they meet (e) none of the above The figure shown here is (a) a regular polygon (b) a convex polygon (o) a pentagon (d) an equilateral polygon (e) none of the above The plane figure produced by drawing all points exactly 6 inches from a given point is a (a) circle with a diameter of 6 inches (b) circle with a radius of 6 Inches (c) sphere with a diameter of 6 inches (d) cylinder 6 inches high and 6 Inches wide (e) square with a side of 6 inches 5. The slope of a line passing through the points (5, -2) and (o#8) is (a) 6 (b) 10 (c) 1/10 (d) -10 (e) — 1/6 6. The area of a rectangle with length 3 inohes and width 12 inches is (a) 18 sq in 94 95 (b) 72 sq in (c) 36 sq In . (d) 15 sq in (e) 30 sq in 7* In triangle ABC, side AC has the same^length as aide BC, and angle BAC has a meaaure of 50° • What is the measure of angle BCA? (a) 80" M 90" (b) between and 90 (c) less than 90‘ . # (d) between 90" and 180 (e) more than 180° 11* The measure of angle ABC is (a) 120c (b) 60c (c) 80° (d) 240° 120 (e) need more information A B 12* Parallel lines are lines (a) in the same plane which never meet (b) in different planes which never meet 96 (c) which always form angles of 90 when they meet (d) which have the same length (e) none of the above 13* The longest side of the triangle shown is (a) AB (a) 8 cm B Angle BAG Is a (a) kOO sq cm (b) 31^ sq cm (c) 62.8 sq cm (d) 1256 sq cm (e) 100 sq cm 16. A rectangle (a) has four diagonals (b) can be made by putting any two congruent triangles together (c) is sometimes a parallelogram (d) is always a parallelogram {e ) 17. If 0 is the center of the circle, segment OA Is called the (a) radlU3 of the circle (b) diameter of the circle (c) chord of the circle (d) segment of the circle (e) sector of the circle 18, Angles 1 and 2 are called (a) opposite angles (b) parallel angles (c) alternate interior angles 97 (d) alternate exterior angles (e) corresponding angles 19* The measure of a right angle is (a) less than 90° a (b) between 90 and 180 (c) 45‘ (d) 9 0 \ (e) 180 20. An isoceles triangle has (a) all three angles congruent (b) all three angles with different measures (c) two sides the same length (d) all three sides different lengths (e) none of the above 21, Lines 1 and m are parallel. The measure of angle 1 is (a) 70, (b) 60" (c) 30; 24, Skew lines are lines (a) that are not perfectly straight (b) in the same plane that never meet (c) in different planes which never meet (d) which form at least one acute and one obtuse angle when they meet 98 (e) none of the above 25. The slope of the line shown is (a) 0 (b) greater than 2 (c) non-existentj the line has no slope (d) negative (e) positive 26, Given that the radius of a circle is 5 Inches, its circumference Is about (a) 5 in (b) 50 in 29. Two angles are complementary if (a) the sum of their measures Is 180 the product of their measures isel80c (b (o) the sum of their measures is 360° (d) the product of their measures is 90* (e) the sum of their measures Is 90° 30. If AC bisects angle BAD, then angle ACB has a measure of ^-rsA (a) 35 t (b) 105 99 (c> 70 0 (d) 110 (e) 30° 31. A line perpendicular to the y-axls has (a) a negative slope (b) a slope of 0 (c) a positive slope (d) no slope (e) a very large slope 32. Angle ABC, with its vertex B on the circle, has a measure of 30 • What Is the measure of the aro AC? (a) 2/3 of one inch (b) 30 (c) 330 (d) 15 (e) 60 33* Angles 1 and 2 are called (a) interior (b) vertical (c) supplementary (d) complementary (e) scalene 3^< A3CD is a parallelogram. The measure of angle BCD is A (a) 20 X?0 (b) 110° (o) (d) 70“ (e) need more information 35. If a pair of intersecting lines form two adjacent angles which have equal measure, then the lines are (a) vertical (b) horizontal (c) oblique (d) perpendicular (e) transversals 36. Triangle ABC is equilateral with a side of 5 0“ . The measure of angle ACB is (a) 50 100 (b) 30 (o) 60° (d) 90° (e) need inore lnformtlon 37. The area of this quadrilateral, where AB Is parallel to Cd, Is 96 aq in. What is the length of AB? (a) 9.6 in B (b) 12 in (c) 10.6? in 8 In (d) 1*1 in (e) 5.33 10 In 30. The perimeter of this parallelogram is (a) 25 cm i 15 cm B (b) *12 cm (c) 21 cm 6 cm / 4 cm (d) 8*1 era (a) 8** sq cm (b) *12 sq cm (c) 90 sq cm (d) 2*f sq cm (e) 60 ora * 1 0, A certain triangle has a shortest side of t cm* its longest side is 3 times the length of its shortest side; and the third side is *l cm more than half the longest side. What is tne perimeter of the triangle? (a) 2t + 8 cm (b) 5t + *1 cm (c) lit + 8 2 cm (d) 7t + U cm (e) 15t + ** om *fl. The circle shown has its center at 0 and an area of 36 • The area of triangle OA is (a) 18 (b) 18 (c) 9 (d) 9 (e) 36 101 42. The line given by the equation y * 3x + 2 crosses the y-axis at the point (a) (0,3) (b) (-3.0) (c) (-2 .0 ) (d) (0,1/3) (e) (0 .2 ) 43. If triangle ABC Is congruent to triangle DEF, which of the following Is true? (a) AB is congruent to EF (b) angle ABC Is congruent to angle FED (c) angle ACB Is congruent to angle FDE (d) FE Is congruent to CA (e) DE Is congruent to CB 44. Triangle ABC is similar to triangle DEF. The measure of AB la (a) 15 In (b) 10 In 71 (c) 11 In E 45. Lines a. b, o, and d are parallel, The length of XY la (a) 79 cm (b) 44 cm (e) 72 cm (d) 68 cm (e) 66 cm 46. If arc AB has a measure of 70 and 0 Is the center of the circle, the measure of angle AOB is (a) 120 (b) 140 (c) 70 (d) 35 (e) 110 47. The coordinates of point B are 1*.7> (a) (2.7) (b) (-3.7) _d (c) ('•.2 ) (d) (-3.^) 102 («) (-3.7) W3. The length of segment AB Is (a) 5 , 4 2 .6 ) (b) ^.5 (c) 8 B^<5.2) (d) 3.75 ------fr. (e) 7 U9m Which of the following Is a true statement? (a) tv/o different planes always meet at a line (b) three different points always determine a unique piano (c) If tv/o different lines do not Intersect, they must be parallel (d) if two different points lie on both lines 1 and. m, then line 1 must coincide with line m (e) all of the above are true. 50. The total measure of all the interior angles of the figure shown Is (a) 5^0 I \ (b) 720 \ (c) 360 / X (d) 130 / (e) 120 APPENDIX B TEST INSTRUCTIONS 1* Pleas© print your name In Pencil on the answer sheet and darken the matching grids directly under your name. Continue to use pencil on the exam. 2. On this test a "mathematical system" contains undefined terms, definitions, postulates, theorems, and laws of logic. 3* The term "postulate" will mean the same thing as the term "axiom". k. Some sections of the te3t that do not have 5 multiple choice answers have special Instructions preceding the questions. Be sure to read these special instruc tions carefully so that you are able to mark the answer sheet correctly. 5. You will have to work quickly and carefully. When • you finish the exam be sure your name Is on the answer sheet. At the bottom of the answer sheet in the place for your instructors name, print his name and the name of your school. 6. Please hand in BOTH the test and the answer sheet. 103 One year a particular farmer*a stand of wheat yielded 40 bU3hels per acre. We nay conclude from this statement (1) The farmer's land wao extremely fertile, (2) The farmer has raised wheat on his land, (3) The weather that year was unfavorable for growing wheat. (4) Forty bushels per acre is a high yield, (5) The field would be more suitable for some other crop. In proving: If x2 Is an odd number, then x Is an odd number, a valid alternative would be to (1) Prove: If x2 la not an odd number, then x la not an odd number. (2) Prove: if x Is not an odd number, then xz la not an odd number. (3) Prove: If x Is an odd number, then x2 is an odd number. (4) Pick 100 odd perfect squares and show their square roots are odd. (5) None of the above are valid alternatives. In a mathematical system definitions are (1) Statements which are agreements as to how a symbol or term Is to be used, and they do require proof. (2) Statements which are agreements as to how a symbol or term is to be used, and they do not require proof. (3) Statements that should be flexible so they oan be used In a variety of situations. (4) Self evident truths. (5) None of the above. Life expectancy tables show that one out of each 100 people in the U.S. will live to be more than 95 years old and 54 will live to be at least 68. We may con clude from this statement (1) Less than half of the U.S. population la expected to die before the age of 68. (2) Women have a greater life expectancy than men. (3) host men who live to be 68 will live to be 95« (4) Living to be very old is a result of having a long life expectancy. (5) The average man will not live to be 68. * 105 5* Theorem: If a trLangle la equilateral, then It Is isosceles. An equivalent form of this theorem Is (1) Some triangles are equilateral and some are isosceles.. (2) If a triangle is isosceles, then it is equilateral. (3) It will never happen that a triangle will be equilateral and not isosceles. (^) If a triangle la not equilateral, then it might be isosceles. (5) Both (1) and (*0. 6. Given a mathematical system in which the undefined terms are 11 mugs11 and "chairo" and in which the uned- flned relations "mug on a chair" and "chair on a mug" mean the same thing. The axioms are: Axiom 1. There exists exactly three mugs. Axiom 2. Not all mugs are on the same chair. Axiom 3* Cn any two distinct mugs there Is exactly one chair. Axiom U. On any two distinct chairs there is at least one common mug. The least number of chairs that satisfies this system is? (1) The angles opposite these sides are equal, (2) The angles opposite these sides are not equal. (3) The triangle Is equilateral. 106 (**■) There are at least two equal angles In the triangle. (5) None of the above. 9. In trying to prove /2 Is not rational using the Indirect method of proof, one should (1) Assume 'fZ is rational and show this does not lead to a contradiction of a known faot, (2) Assume ^2 Is not rational and show this leads to a contradiction of a known fact. (3) Assume ^2 is not rational and show this does not lead_to a contradiction of a known fact. (4) Assume /2 la rational and show that this leads to a contradiction of a known fact. (5) Nono of the above. The following questions are to be answered "True” or "False". It is understood that those items that are true only some of the time are to be considered false. Remember that on the answer sheet that 1. is True and 2. is False. 10. An axiom could be a statement that is the opposite to ones basic intuition. 11. Regardless of the axioms used, the sum of the in terior angles of a quadrilateral Is 360 degrees. 12. Undefined terms are not really necessary in geometry. 13. When a mathematician remarked: "Cne must be able to say at all times - instead of points, straight lines and planes - tables, chairs and beer mugs." he was referring to undefined terms. 14. Once a theorem is proved in one mathematical system, it Is true in any other mathematical system. 15. If a statement is too difficult to prove it could be accepted without proof and the mathematical system that contained it would not ohange much* 16. Definitions never have to be proved. 1?. Definitions sometimes depend upon undefined terras for their meaning. 18. A theorem is a statement that must be proved, 19. Some axioms must be proved. 107 20. The definition commonly used for the area of a circle Is Area = TT rf 21. It Is Impossible to define all words in a manner that avoids any confusion about their meanings. Classify the following as 1. axiom (used Interchangeably with "postulate" on this test). 2. definition, or 3. theorem. Determine the correct choice using the same classification that your textbook does. 22. An Isosceles triangle is a triangle In which at least two sides are equal. 23. If two angles of a triangle are equal, then the sides opposite those angles are equal. 24-, Through a point outside a line there Is exactly one parallel to that line. 25. Pclnt B is said to lie between points A and C If and only If all three points are distinct points on a line and AB + DC =» AC. 26. Given a set of undefined terms, definitions, postulates, and laws of logic for proving theorems, then the theorems deduced and proved In this system (1) Would be different on :*ars than on Earth. (2) Would be different on the moon than on Earth, (3) Would be the same on all planets and moons. (£) Eoth (1) and (2) are correct. (5) It depends a lot on the theorems. 27. Given a mathematical system In which the undefined terms are "point" and "line" and In which the unde fined relations "point on a line" and "line on a point" have the same meaning. The axioms are: Axiom 1. Not all points are on the same line. Axiom 2. There exists exactly three points. Axiom 3* Cn any two distinct points there Is exactly one line. Axiom 4. c-n any two distinct lines there Is at least one common point. The least number of lines that satisfies the system is? (1) 1 108 ( 2 )' 2 (3) 3 < * > * (5) The number of lines io unlimited* 28. Mathematicians include undefined terms in a mathe matical system (1) In order to keep the system flexible. (2) Because they don't want bo be bothered defining the obvious. (3) To add difficulty to the system. (4) Because some terms are hard to define. (5) To avoid circularity of definitions. 29. Recently, it has been discovered that It is possible to rid o.n area of starlings If a recording of a starling in distress is played over loud-speakers for several evenings. It can be concluded from this that (1) One starling control technique uses recordings of starlings In distress. (2) When starlings hear the distress call of other starlings, they fleo. (3) Starlings have been multiplying and spreading at an appalling rate. <4-) The starling control problem could be solved if ouch recordings were used In enough places. (5) The best way to oontrol starlings is to frighten them. A postulate about parallel lines states: Given a point A not lying on a line 1, there exists one and only one line through A that is parallel to line 1. In the questions that follow, assume that we have replaced the above postulate with: Any two straight lines always interseot. Everything else, except, of course, those ideas that are a consequence of or follow fron the fifth postulate, remains the saute. It v/ill be helpful to know that the only congruence theorem that is needed ia the SAS theorem. 11 30. If angle 1 = angle 2, then 12 109 (1) li Intersects lg (2) 1- Is parallel to lp (3) It Is not known if intersects I2 or if 1^ is parallel to I2 (4) angle_l can't equal angle 2 (5) both (1) and (*t) 31* Given a line 1 and a point P not on 1, then the number of straight lines that can be drawn through P that do not intersect 1 is (1)0 P . (2 ) 1 (3 ) 2 W 3 (5) more than 3 1 ______ 32. If C A x A C and OB J.AC and BC = AB* then (1) OC * OA (2) 0C_l AC (3 ) C>A and CB can11 both be perpendicular to AC (4) both (1) and (2) (5) neither (1) or (2) 33* Given AC _l AB and BB-uAB* then (1) AC || SB (2) AC and Ep intersect (3) AC and DB can't both be perpendicular to AB (**) impossible to determine what AC and ED do when exton le i------— ------(5) both (2) and (3) A B 34. Given OAJ-AB and AO = AO' and 0B1AB, 0,A,0* colllnear with points O' and B connected, then (1) triangle AOB * triangle AO'B (2) angle O'BA is a right angle AB (3) 0, B, and 0* are on the aarae line (*0 (1)» (2), and (3) are allcorrect (5) (1), and (2) are correct but not (3)* 0* 110 35* The least number of lines that can inclose an area la (1) 0 (2) 1 (3) 2 W 3 (5) If 36. If angle 1 = angle 3* then (1) 1- Intersects I2 (2) lx is parallel to I2 (3) It is not known If lx ls parallel to or if lx Intersects I2 (4) angle 1 can't =» angle 3 (5) both (2) and (^) Still assuming that we have accepted the postulate: "Any two straight lines Intersect", which of the following would be correct and v/hich would not. If the statement is correct, respond with the numeral "I"• If the state ment is not always correct, respond with the numeral "2", 37* Between any two points there exists a straight line. 38. Between any two points there exists one and only one straight line. 39* The exterior angle of a triangle is greater than each opposite interior angle. 4*0. All right angles are equal to. one another. 41. An Isosceles triangle has two sides of equal length. **-2. There exists a pair of lines everywhere equidistant from one another. 43* If two lines are cut by a transversal so that the alternate interior angles are equal, then the lines are parallel. It is known that "if a function is differentiable, then it is continuous." Which of the following follows from this fact? (1) If a function is not differentiable, then it Is differentiable. (2) If a function la continuous, then it is dlfferen- Ill tlablc, (3) If a function la not continuous, then It is not differentcable, (4) It la desirable for a function to be differen tiable, (5) The question can't be answered unless the terras are understood, 45- Accept as belnr: true the statement: nIf each of two seta, A and B are compact, then the union of these two sets la compact.” Given two uets A and B, and the knowledge that the union of a and B is compact, we can conclude (1) A is compact and B is compact. (2 ) Either A or B la compact. (3) Neither of the two sets is compact. (4) Nothing definite about A or B can be concluded. (5) The question can't bo answered unless the tei-ma sore understood. 46. Given a mathematical system in which "town” and "road arc undefined. The undefined relations "town on a road" and "road on a town" mean the same thing. There arc two relations between roads; either they are parallel, have no common town on both, or they Intersect at a certain town. The axioms are: Axiom 1. There is -at least one town on the map. Axiom 2. Every road has exactly two towns on it. Axiom 3* Every town is on exactly two roads, Axiom 4. Every road has exactly three roads parallel to it. Axiom 5* For every two roads, there is a road parallel to one and intersecting the other. How many roads are in the system? (1) exactly 2 (2) exactly 4 (3) exactly 6 (4) exactly 8 (5) none of the above 47. A statement that is a self evident truth, accurately describes (1) An undefined term (2) A definition (3) An axiom .112 (4) A theorem (5) None of the above **8. Accept as being true, "If a number is composite, then it can be expressed an a product of primes," Which of the following statements below implies that 12,243 can be expressed as a product of primes? (1) 12,243 is a prime number, (2) 12,243 is a composite number. (3) 12,243 is a large number, (4) 12,243 laan odd number, (5) In order to answer this question it is necessary to understand what prime and composite numbers are, 49, Accept as being true the statement: "If a nurd is a dingle,then the nurd is a nood." Which of the following implies that Ned a nurd, is also a nood? (1) Ned is a nurd. (2) Med is a dingle. (3) Ned is not a dingle. (4) Ned is not a nurd. (5) Impossible to tell without understanding the terms• Assume the first two statements in the following problems are correct. If the third statement, the conclusion, show s good reasoning, mark the column 1 on the answer sheet. If the conclusion shows poor reasoning, mark the column 2 on your answer sheet, 50. No cats are electrified. All ghosts are electrified. Therefore no ghost is a cat. 51* Some kettles are giraffes. All zebras are kettles. Therefore some giraffes are zebras. 52. All dogs are ink bottles. Some ink bottles are squirrels. Therefore some squirrels are dogs. 53. Some people in our town are not famous. Everyone in our town is rich. Therefore some rich people are not famous, 54. No onions are parsnips. Some parsnips are tangerines. Therefore some tangerines are not onions. 55 Some soldiers who were in the Civil War used green peaches for gunpowder. This soldier uses green 113 peaches for gunpowder. Thereforfc he must have teen In the Civil War* APPENDIX C Hyperbolic Non-Euclidean Geometry When Euclid organized his geometry In the Elements (about 300 B.C.), he began with 5 Common Notions and 5 Postulates (listed on page 2,) It la the fifth postulate of Euclid that has been the center of so much controversy. Although there are many equivalent forma of thia postulate, the particular one that is usually used in place of the rather lengthy original version Is credited to John Playfair (17**8-1819). Playfair1s Arlom: Through a given point can be drawn only one line parallel to a given line. The troublesome part of this particular postulate is that, unlike the others, one must place a certain amount of faith that every other lino through the given point will eventually Intersect the given line. Suppose another line through the given point meets the parallel at an angle of 179*9999999999999 degrees. (See figure 1.) 12 - : 13 179.9999999999999 1 ______ Can we really be sure It intersects the given line? Now, if someone could prove this fifth postulate from the others, it could be elevated to the status of a theorem. Assuming the acceptance of the other postulates. Euclidean geometry would then be the "true11 geometry. Because of a strong belief of the early mathematicians in the correctness of Euclidean geometry there were numerous attempts to prove the fifth postulate. Proolus (**10-**85) tells of Ptolemy*s attempt, and Proclus himself attempted a proof. Some of the following attempts were: Nasiraddin (1201-127*0. Wallis (1616-1703). Saccheri (1667-1733). Lambert (1728-1?77). and Legendre (1852-1833). Generally speaking the attempts all failed because some where in the proof a theorem that was actually an equivalent form of the fifth postulate, or a tacit assumption about parallels was used. Today we know that it is impossible 114 115 to prove the fifth postulate from the others; It Is In dependent. The three men usually credited with the discovery of non-Euclldean geometry are: Carl Gauss (1777-1855)* Johann Bolyai (1802-1860) and Hioolal Lobachevsky (1793- 1856). Although Gauss and Bolyai were probably the first to see the Independence of the fifth postulate, Lobachevsky was the first to publish an organized development of the subject. The geometry that these men were studying is called Hyperbolic non-Euclldean Geometry and Is sometimes referred to as Lohachevaklan Geometry. Since this geometry Is another example of a mathe matical system, some undefined terms will be needed. There will be a set of axioms (postulates), definitions, and theorems will bo proved using the logical atruoture that was developed for Euclidean Geometry, Undefined Terms The same terns that were undefined in Euclidean Geometry, "point", "line", and "plane", will be the undefined terms of hyperbolic non-Euclldean geometry. There will be times in this unit when the reader will have to keep his inlnd from thinking of line as It perceives it In Euclidean space. Postulates Available Prom Euclidean Geometry. Because the fifth postulate of Euclid Is the only one we are refecting, the other 9 assumptions of Euclid become our flr3t 9 postulates. (In this unit the terms "axiom", "postulate", and "assumption" all mean the same thing and can be used Interchangeably.) Since these postulates will be necessary to make proofs. It is important that the student knows what they are and what they mean. 1. Things which are equal to the same thing are also equal to one another 2. If equals are added to equals the wholes are equal. 3. If equals are subtracted from equals, the re mainders are equal. *»■. Things which coincide with one another are equal to one another* 5* The whole is greater than the part. 116 These five assumptions are what Euclid called his COMMON NOTIONS. The following four assumptions, stated essentially as Euclid did, were called POSTULATES by Euclid* It Is noted again that In this unit we will use the terms "axioms", "postulates", and "assumptions" to mean the same thing. 6. A straight line can be drawn from any point to any point. 7. A finite straight line can be produced contin uously in a straight line. 0. A circle can .be drawn with a given center and radius. 9. All right angles arc equal to one another. We are now ready to make a drastic change in the Euclidean postulate set. Although it is not the Intent of this unit to defend any particular geometry as the "true" geometry. It Is interesting to note again that Playfair's form of Euclid's fifth postulate can not be proved from the other postulates. This means that the geometry that we are going to begin to develop might be a more accurate dlscriptlon of our world than Euclidean geometry. The practical consequences oner short distances would not be significant but what about astronomical distances? Remember, we are accepting the next postulate; reguardless of what our intuition and "locked in" ideas about our world being Euclidean are. The main idea is to examine what follows logically from this assumption and how does it compare to the Euclidean approach. THE CHARACTERISTIC POSTULATE OF HYPERBOLIC NCN-EUCLIDEAN GEOMETRY THROUGH A GIVEN POINT NOT CN A GIVEN LINE THERE EXISTS NOPE THAN ONE LINE THAT DOES NOT INTERSECT THE GIVEN LINE. Problem 1: Which la correct: through a point not on a line there exists one and only one line that does not Intersect the given line or there exists more than one that does not intersect the given line? Definitions. The definitions of new terras will be supplied as needed. Before looking at the theorems of hyperbolic' geometry, it should be mentioned that Indirect proof is 117 used in many cases, and, at the beginning, reference is made to Pasch*s Axiom, Put very briefly the spirit of indirect proof involves assuming the negation of the entire statement that is being proved and searching for a contradiction of a known fact. Since the assumption led to a contradiction, it is an incorrect assumption; there fore, the original statement has been proved within the structure of the system, Euclid made a few tacit assump tions on developing his geometry. Moritz Fasch (18^3- 1930) recognized one of these; the result of this was Pasch>s Axiom which roughly states: If a line passes through one side of a triangle, not at a vertex, then it must intersect at least one of the other two sides. As a consequence of this axiom it follows that: If a line enters a triangle at a vertex then it Intersects the opposite side. Both the axiom and it3 consequence are available for our use. Theorems: Although we will be examining theorems that are unique to hyperbolic geometry, there are 28 theorems from Exiclidean geometry that are available for our use. When Euclid wrote his text. The Elcmonto, he proved the first 28 propositions without using the fifth postulate or an equivalent form of it. It is interesting to note that this might be considered evidence that Euclid himself was not satisfied with the parallel postulate. By avoiding its use, he gave us a nice start on the theorems of hyperbolic geometry. In order to enable the students to adjust more easily to this different geometry it la ESSENTIAL that they learn the propositions that are available from Euclidean geometry. For simplicity of achieving the goal stated in the previous sentence, the first 28 propositions of Euclid will be stated. In most cases they are not stated in the original translated form, but they are ciodidled to read more like the theorems in your text book. NOTE: THE 28 PROPOSITIONS THAT FOLLOW CAN EE USED IN THIS UNIT: OTHER THEOREMS FROM EUCLIDEAN GEOMETRY CAN NOT BE USED. 1X8 THE FIRST 28 PROPOSITIONS 1. An equilateral triangle can be constructed on a given line segment* 2. A straight line segment may be constructed equal to a given straight line segment* 3* Given two unequal straight line segments, a straight line segment on the longer can be out off equal to the shorter. If two sides and the Included angle of one triangle are equal respectively to two aides and the included angle of another triangle, then the triangles are congruent, and their corresponding parts are equal. (SAS). 5* In an Isosceles triangle the angles at the base will be equal to one another, and. If the equal straight lines are extended, the angles under the base will be equal to one another. 6. If two angles of a triangle are equal, then the sides opposite these angles are equal. 7* Given a triangle, a triangle congruent to the given triangle can be constructed on a side of the given triangle so that the equal sides share a vertex. 8, If three sides of one triangle are equal respectively to three sides of another triangle, then the triangles are congruent, and their corresponding parts are equal. (SSS) 9. The bisector of an angle may be constructed. 10. The blseotor of a line segment may be constructed. 11. At a given point on a line a perpendicular may be constructed. 12. From a point not on a line a perpendicular may be constructed to the line. 13. If two adjacent angles have their exterior sides in a straight line, then either the two angles are right angles or their sum Is equal to the sum of two right angles. 119 14. If the sum of the degrees In two adjacent angles Is equal to the sum of the degrees of two right angles, the exterior sides of the adjacent angles are in a straight line. 15. If two straight lines intersect the vertical angles are equal. 16. The exterior angle at any angle of a triangle Is greater than either of the interior and opposite angles, 17. In any triangle two interior angles have a degree sum less than that of two right angles, 1.8, In any triangle the greater side Is opposite the greater angle. 19- In any triangle the greater angle is opposite the greater side. 20. In any triangle the sum of the lengths of two sides la greater than the length of the third. 21. If from the extremities of one side of a triangle two straight linea are constructed meeting within the triangle, the sum of the lengths of these con structed segments will bo less than the sun of the lengths of the remaining sides of the given triangle, but they will contain a greater angle. 22. A triangle may be constructed given its three sides. 23. An angle may be constructed equal to a given angle. 24. If two sides of one triangle are equal to two sides of another triangle, then the two sides that contain the greater angle will have the greater third side. 25* If two sides of one triangle are equal to two sides of another, then the triangle with the greater base will have the greater angle opposite it. 26. If two triangles have two angles and the included sides respectively equal, then the triangles are congruent and the corresponding parts are equal. (ASA) 27. If two lines are cut by a transversal so as to form equal alternate interior angles, then the lines are parallel. 120 28. If two lines are cut by a transversal so that the corresponding angles are equal or the interior angles on the same side of the transversal are supplementary* then the lines are parallel. PROBLEM 1. Prove proposition 27 using only the material that preceded it and. of course, the system of logic that was developed for making proofs. Because of the CHARACTERISTIC POSTULATE OF HYPERBOLIC PLANE GEOMETRY we are able to classify three types of lines that can be drawn through a point not on a given line that have a relation to the given line. The three classes of lines mentioned in the above paragraph are pictured below. (1) Those lines through P that intersect 1. 1 <1> (2) The FIRST lines on the left and right that do not intersect 1. (Recall there are at lea t 2 non-intersecting lines from our postulate.) (3) Those lines contained within the angle formed by the lines in (2). 1 (The dotted lines.) (3) The lines in (1) are called INTERSECTING LINES. Those in (2) are called PARALLEL. The lines of (3) are * called NON-INTERSECTING. Since the lines that are called intersecting seem to agree with ones conception of Intersecting lines, there is no need here for further discussion. It would be very easy to go into more detail about the lines called "parallel" by examining some of the work of Dedekind and applying it to our situation. The authar chooses not to do that here. It is Important to note here that the word "PARALLEL" in hyperbolic geometry refers to 121 the vary first non-intersecting lines on the loft and right* Tha lines of (3) will be examined through a theorem, THEOREM 29. Any line within the angle formed by the tvio parallels to a given line, 1, does not lnterseot 1, 0^ 1. Assume k, any line 1. Assumption within Z.GPB intersects V \ 1. \ % 2. Construct PQJ-1. 2 * Why? 1 " ' "j Z J 3. PB intersects 1 3 . A line through the vertex of a triangle intersects the opposite side. k. But PB can not Given (students can write Intersect 1 the given and to prove for this problem, ) 5. Assumption incorrect. 5 . Led to contradiction of knowna fact (the Given) 6. k does not intersect 1. 6. If one of two contradictory statements Is proved false, then the other is correct. At this point wo have established that k is not an intersecting line. In order for it to be classified as a NON-INTERSECTING line it must be shown that it is not a PARALLEL. (Again the term “non-intersecting" will generally refer to the lines that do not intersect and are also not the parallel lines.) Since the parallel lines are the first non-intersecting lines, k could not be parallel since PB is already the first non-intersecting line, and there is only one first non-intersecting line on each side. PROBLEMS 2. Why was theorem 29 called theorem 29 instead of theorem 1? 3. How many non-lntersectlng lines can be drawn, in hyperbolic geometry, through a point not on a line to a given line? (Hint: see theorem 29) Explain what the following terms mean in hyperbolic non-Euclidean geometry: Intersecting lines, parallel lines, non-intersecting lines. 122 SUMMARY The following picture summarizes many of the Intro ductory concepts we have discussed thus far. NOW-1NTER3 ECTTNG CN-IN£SRSECTING LTNES- — LINES LEFT HAND PARALLEL Problem 5, Can you show that any line within angle APB is an intersecting line? (THIS WILL BE AN EXTREMELY USEFUL FACT IN SOME OF THE THEOREMS AND PROBLEMS TO COME) We are now ready to prove an Important theorem about the angles that PARALLEL lines (the FIRST non-intersecting lines) through a point, P, to a line, 1, form with the perpendicular from P to 1. THEOREM 30. If 1 is any line and P Is a point not on 1, then the parallels (left and right hand) to 1 through P form equal acute angles with the perpendicular from P to 1, NOTE: There are two conclusions In this theorem: (1) the angles formed by the parallels and the perpendicular are equal and (11) the angles formed by the parallels and the perpendicular are acute. GIVEN: FQX1* k and k* parallel to 1. k» PROVE: XSPQ = XTPQ 123 1. Assume /SPQ * ^TPQ 1. say /SPQ>/iTPQ 2. Construct ^LPQ = ZTPQ 2. 3. PL intersects 1 at J 3. 4. Construct QR = QJ 4. 5. Draw PH 5. 6. PQ = PQ 6. 7. PQL 1 7. 8. z4 = IS 8. 9. AJFQ ^ A RHQ 9. 10. P. = 12 Problems 6. Complete the proof of Theorem 30, part (1), 7. Give an Informal but convincing argument that demon strates ZSPQ and iTPQ can't be either right angles or obtuse angles. Once this is established, they must be acute. 8. If two lines BA and BC are both parallel to line 1, then the bisector of <£ABC is perpendicular to 1, (Hint: Assume the bisector Is not perpendicular to 1. Construct the perpendicular from B to 1. and use theorem 30 to help find a contradiction. / 1 ______i~ _C— ------DEFINITION: RIGHT AND LEFT HAND. PARALLELS. Given a line 1 and a point P with 0 any point on 1; a line SP is parallel to 1 In a rlght hand sense If each line through P and a point lying* within /.OKi intersects 1. A line RP is parallel to*l In a left hand sense If each line through P and a point lying within 10PR intersects 1, NOTE: The following technique will be helpful In some of the theorems and problems coming up. It will be helpful to examine it here. Suppose in the figure at the right It is known that k and 1 do not Intersect. Futhermore it 13 known that any line through P within OPT Intersects 1. (0 Is any point on 1). This is enough Informa tion to show that k and 1 are PARALLEL. Our basis for this is the definition of 124 k right and left hand parallels, and the '----- 7— - T reason we will use in proofs is "definl- ' tion of parallel." The reader should be/ sure that he understands the relation-/ ship between the definition and the ^ j example just given. 1 ^ ------There are two theorems that are important to the develop ment of this geometry. The proofs of these theorems are quite rigorous and the author does not wish to present the proofs in this unit. The interested student who feels that he would like to see the proofs can find them in Wolfe's Non-Euclidean Geometry on pages 68-70. A more complete reference can be found in the bibliography. THEOREM 31. If a straight line is parallel to a given line in a given sense (left hand or right hand), then it is parallel at each of its points in that same sense. THEOREM 32* If a line is parallel to a second line then the second is parallel to the first. (if k is parallel to 1, then 1 is parallel to k.) THEOREM 33* If two lines are both parallel to a third line they are parallel to each other. (parallel in the same sense). A AJL B GIVEN: AB parallel to CD parallel to PROVE: AB parallel to 1. AB Is parallel to_EF 1. 2. Construct A * H1 XC D 2 . 3. Let R be any point 3* within /.H * A1B 4. A 1R intersects EF at I If. 5. £d Is parallel to EP 5. 6. £F is parallel Jto CD 6. 7. Construct I J X C D 7. 8. A*I extended will 8. Intersect CD _ 9. AB is parallel to CD. Problem 9. It is Important to this proof that A*H' Intersects EP, 125 Although this fact seems obvious, we are sometimes called upon In mathematics to prove these type statements. Prove that A fH' Intersects EF. # (This Is easy but tricky!I) Since the proof of theorem 33 jslven on page 9 depended on the relative positions of AB, CD. and EF, another case must be examined. GIVENi AB parallel to EF CD parallel to EF C ------D PROVE: CI5 parallel to AB E ------F 1. Assume AB is not parallel 1 . to CD 2. There exists a line 2. through K, say KI such that KI la parallel to CD Problems 10. Complete the proof of the above case. Keep In mind that there exists only one parallel to a given line In a given SENSE. 11. Are there any other cases of theorem 33? If yes, how would you prove the desired outcome? NCTE: In this unit UNLESS OTHERWISE STATED when two lines are parallel, they are parallel in the right hand sense and anything proved about these lines In this sense also holds about lines parallel In the left hand sense. The next section of this unit is devoted to a type of triangle that will be unfamiliar to most of you. The triangle is called an IDEAL TRIANGLE, and. In this triangle, two of the sides are parallel. As in the case of ordinary triangles we will be looking at some of the properties that all ideal triangles share, some special ideal triangles. 126 and congruence theorems for Ideal triangles. It is Important that you know the definitions and see the theorems as a logical consequence of what we have done thus far. DEFINITION: IDEAL POINT. Two parallel lines approach an ideal point. It could also be stated that two parallel lines "share” an Ideal point. NOTE: Symbolically the Ideal point will be denoted by. -TL (OMEGA). It is Important that the student thinks of this Ideal point not as he thinks of an ordinary point. We will not represent it with a dot as we represent ordinary points. An ideal point is nothing more than it is defined to be, something given a apeaial name that two parallel lines share. NOTE: Because of the reversibility property of definitions, it is also known that if two lines share the SAME ideal point, then the lines must be PARALLEL, DEFINITION: IDEAL TRIANGLE. Let A and B bo any two parallel lines in the same sense, with A and B ordinary points and-ft is an ideal point. The figure ABA. will be called an ideal triungle. A and B are the ordinary vertices and4L is called the ideal vertex. The Ideal Triangle THEOREM 3^. (Analogue to Paseh*s Axiom for Ordinary Triangles) If a line passes through a vertex of triangle AB and contains a point in the interior of triangle AB then the line intersects the opposite side of the triangle. Since there are two types of verticeis, ordinary and ideal, this theorem must be proved by considering two separate cases. CASE 1. A line through an ordinary vertex. A GIVEN: Ideal triangle A^it AH with H in the Interior of triangle ABIL B PROVE: AH intersects BA, 127 1. 4JIB A- Ideal triangle 1 . 2. AB Intersects B 2 . 3* AiL parallel to B-XL 3. 4. AH Intersects BA. 4. For a line EH the proof Is similar A CASE 2. A line through the Ideal vertex* A. GIVEN: Ideal triangle ABA, B< with P in the Interior of the triangle. PROVE: PIL interseots AB Problems 12. Supply the correct reasons for case 1 of theorem 34. 13* Prove case 2 of theorem 34. THECREj; 35* If * straight line intersects one of the sides of triangle AR but does not pass through a vertex, it will intersect one and only one of the other two sides. Again there are two cases to consider: a line inter secting a side containing , and a line intersecting the ordinary side. CASE 1, The line Intersects A or B GIVEN: Ideal triangle AHilwlth PK and PKf Intersecting Alt at P SL B PROVE: PK* intersects AB PK Intersects BA. Problems 14. Prove this case using the hint to draw PB. 15. Draw the diagram, state the "given" and "to prove1'. 128 and prove case 2. (Let the line intersect AB and show it inuat intersect AiL or B^b . Uniqueness will be discussed by, the author. The only part of this theorem to be proved Is that these lines intersect one and only one of the remaining sides. This is accomplished by assuming they intersect both of the remaining sides and show this leads to the contradiction that two different lines are drawn between two points. Except for this brief outline, this part of the theorem will not be proved In detail. THEOREM 36. The exterior angles of AABAat A and B made by extending AI3 are greater than their respective opposite interior angles. A Jb C fig.'l- -• D fig. 2 GIVEN: Ideal A ABA with AB extended PROVE: ZCBA > /BA It The general plan of the proof will be to construot /CBD at B so that /CUD » /BAA and show that &D is within ZCBil (fig. 1). Bjr_aasuming that Bt) coincides with BH. (fig. 3) or that BD is within ZABIL (fig. 2) we find contra dictions. 1. Assume Sd Intersects AIL at 1. K (fig. 2) 2. ZCBD - ABAll 2. 3. AABK is an ordinary 3. triangle But /CBD can’t equal -^BAJL 5. Assumption Incorrect 5. 129 6. Assume BD coincides with B 6. (flff. 3) 7. Construct M the.mid-point 7. of A B _ 8. Construct NEJ-BD 8. 9. Draw AL * BE 9. 10. /I + /2 = 2 rt, angles 10. 11. /3 + 14 =* 2 rt. angles 11. 12. iX + 12 = 13 + /4 12. 13. 12 =* L 4 13. 14. ix = n 14. 15. BM = KA 15. 16. ANAL « ANBE 16. 17. AMLA is a right angle 17. 18. ILKA - /BME 18. 19. AB is a straight line segment 19. 20. L, K, and E are collinear 20. 21. BUT AMLA CAN NOT BE A RIGHT 21. ANGLE 22. Assumption Incorrect 22. AT THIS FOINT WE KNOW THAT FIGURE 1, THE ONLY REMAINING POSSIBILITY IS CORRECT* - 23* /CBD = *BAO- 23. 24. /CBA^ZCBD 24. 25. ICB--H >/BAfl- 25. The proof for the exterior angle at A Is similar. Problem 16. Fill In the reasons In the above• DEFINITION: CONGRUENT IDEAL TRIANGLES: Two ideal triangles, AB and A'B* 1 are congruent If: (1) /ABB. * ZA'B'JU (2 ) ^BAA = iB'A'4' (3) AB = A* Br Just as you did In the case of the ordinary triangle, we will search for sufficient conditions for the congruence of Ideal triangles. In other words, are there any ways to get the congruence without showing all three of the above conditions must hold? THEOREM 37. In triangles ABJL and A'SWl* If AB * A*B' and /BAA- * AB'A'il*, then AABil « AA* B'Jl* A GIVEN* A ABU, and A A * B */L1 AB = A'B* ABASL« iB'A'-fl-' Jl PROVE: AAB0-* A a 'B'A* B B* 1* Assume zABIL* ZA'BrCL* and 1. that > M ,B ,A.1 2. Construct IAE0 = A A 1Bid* 2. 3. BO intersects AIL at- some 3. point, say D. 4, Construct A'D* = AD 4. 5. AB - A'B* 5 . 6. ABJUl* /B'Ain.* 6 . 7. A A B D ■« AA'B'D* 7. 8. il - I 2 8. 9. But L\ « LZ + {J 9* 10. 12 - 12 + J. 3 10. 11. But /3 > 0 11. 12. Therefore 2 can NOT 12. equal 12 + Z3 13. Assumption Incorrect 13. By a similar method it could easily be AB can*t be less than A'B* '. In this case the roles of AB and A*B' f are Just switched* 14. AB A»B* 14. 15. AB A* B* 15. THEOREM 38. In triangles AHA.and A'B'a ' if IBAd = IB'A'JI and AABIL** AA’Btfl.1, then the two ideal triangles are congruent. A- GIVEN: A ABiland AA'B1-^1 ZAB-d- = AA'B'A* iBAB. a ZB'A'Jl' C Jl Jl PROVE: A ABU. A A1 Blil1 BL B'/ The proof will appear as an excerolse for the reader to enjoy. PROBLEMS 17* Supply the correct reasons for theorem 37, 18, In the given of theorem 37 replace BA ® B'A* 1 with AB - A* B* *. Does the conclusion still follow ( AB a AfBf % )? Demonstrate that your answer is correct. 19. Supply a proof for theorem 38* (Hint: Assume 131 AB =* A*B* and that AB A fBf. Draw AC * A'B* and then draw C4. ) THEOREM 39* In the triangles AfAand A'B'A* If AB * A* B* , ^ A B a » L BAA»and ZA1B'XU* = iB'AtlL1, then AA8A3* AA'B’JU . pt - 0. B* PROBLEMS 20. Prove theorem 39 by assuming the pairs of angles are not « and const ret ^BAC = and^ABE = ZA'BUL1 • 21. Prove that In the figure ABJl, the sum of the angles A B A and B A A is less than 2 right angles, 22. If two lines are cut by a transversal so that the sum of the Interior angles on the same side of the transversal Is equal to two right angles, then the lines cannot meet and are not parallel. They are non-intersecting, 23* Given triangle A B A with angles ABjIand B A A equal. Prove that if M la the mid-point of AD, then iVUL is perpendicular to AB. Prove also that the line perpendicular to A B at M is parallel to AA and BA and that all points on it are eauidistant from AA and BA. What type of triangle in Euclidean grometry is like the one described above? 24. Given figure ABA, prove that if the perpendicular to AB at its mid-point lo parallel to All and BiL, then AK « BM . 25* Let TT(h) denote the angle of parallelism (the angle between the right hand parallel of left hand parallel and the perpendicular) for a given height,,h. Prove that if h>k, then TT(h) perpendiculars drawn to a line and then their end points are connected*) AB is called the BASIS, CD is called the SUMMIT, the angles at A and B are the BASE ANGLES, and the angles at C and D are called the SUMMIT ANGLES. D ______£ THEOREM 40. The line segment Joining the mid-points of the base and summit of a Saccheri quadrilateral Is perpen dicular to the base and summit. PROBLEMS 26. Prove theorem 40. 27. Prove that the summit angles of a Saccheri quadrl lateral are equal. THEOREM 41. The summit angles of a Saccheri quadrilateral are acute. „ - i u -- H ^ o < " i GIVEN: ABCD is a Saccheri 4"-. quadrllateral N s PROVE: Angles C and D are B acute a JL NOTE; A fact used in this proof is that the base and summit of a Saccheri quadrilateral are not parallel. Before continuing with the proof of theorem 4l, you should prove this. Assume they are parallel and see if you can find a contradiction. 1. Draw the right hand 1. parallels CJL and BS l (In order for these lines _not to coincide with DC _lt must be known DC and AB are not parallel.) 2. AD * BC 2. 3. iDAB - LCBB 3. 4. A-DA A. = ACBlt 4. 5. a = Li* 5- 6. 11 >L2 6. 7. /BCE >2ADC 7. 8. M C D - /BCE 8. 9. /BCE >/ BCD 9. 10. /BCD is acute 10. 133 11. /ADC is also acute I 11. DEFINITION: LAMBERT QTJADRILARERAL. A quadrilateral with three of Its angles right angles la called a Lambert quadrilateral* THEOREM. In a Lambert quadrilateral one of the four Interior angles is acute* GIVEN: ABCD a Lambert quadrilateral with right angles at At Bf and D. PROVE: Angle C Is acute 1. Extend BA so that BA * AE 1 2. Construct F E x E B and make 2 EF = BC 3. Draw FD _ 3 4. Draw AF and AC 4 5. A AFE = AACB 5 6. /DAE - /.DAB 6 7. /FAE =* / DAC 7 8. DA =■ DA 8 9. FA * AC 9 10. A FAD b ACAD 10 11. 11 =* JL2 = 90 11 12. FDC is a straight line 12 13. EBCF is a Saccheri 13 quadrilateral 14. Angle C is acute 14 THEOREM 43. If in a quadrilateral ABCD the angles at two consecutive vertices are right angles, say at A and B, then the angle at C Is larger or smaller than the angle at D, depending on AD being greater than or less than BC. GIVEN: AD > BC lA = ZB » 90 PROVE: Z.BCD> ZD 1* Construct on AD 1 . AE * BC 2* ZBCD > /BCE 2. 3. ABCE is a Saccheri 3. quadrilateral 4. Z1 *= ^2 4. 5- ZAEC^/ADC 5- 6. 12 > 4ADC 6. 7. L BCD > < D 7. In order to complete the proof It can easily be demon strated In a similar manner that If AD BC, then BCD D. PROBLEM 28* Given a quadrilateral ABCD with right angles at A and B and the angles at C and D are equal, prove the figure is a Saccheri quadrilateral* (Hint: Assume AD is greater than Be and construct AE * BC, ABCE is a Saccheri quadrilateral*) 29* Prove that the base and summit of a Saccheri quadri lateral are non-intersecting lines* 30. Prove that in a Lambert quadrilateral the sides adjacent to the acute angle are greater than their respective opposite sides. 31. Which is greater, the base or the summit of a Saccheri quadrilateral? Why? 32. Prove that if perpendiculars are drawn PROM the extremities of one side of a triangle TO a line passing through the mid-pointa of the other two sides, a Saccheri quadrilateral is formed. 33* Prove that the segments Joining the mid-points of two sides of a triangle is less than one half the third side* We have now developed enough power to Investigate the Interior angle suns of polygons. Some people describe the difference between Euclidean and Hyperbolic in terms of the sum of the Interior angles of a triangle. At this point you probably vron't be surprised to hear that. In hyperbolic, the sum of the interior angles of a triangle is not 180. THEOREM 44. The sum of the interior angles of any right triangle is LESS THAN 180. A GIVEN* Right triangle ABC with right angle at C. C PROVE* ZCAB + /.B + £C <180 E 1. Construct 7. ZAFM Is a right angle 7. 8. 11 = L 2 8. 9. E, H, and F are collinear 9. 10. AFEC is a Lambert 10. quadrilateral 11. ZCAF < 90 11. 12. /CAF “ Z3 + ZB 12. 13. ZC « 90 13. 14. ZCAB + IB + ZC z 180 14. THEOREM 45* Ttae sum of the Interior angles of any triangle is LESS THAN 100. PROBLEMS 34. Prove theorem 45. 35. Using the diagram below, prove that ^A + ZABC + ^ACB <180. Given: & triangle ABC Kith angles B D and C acute, M is the mid-point of AC B Prove: ZA + ^ ABC + ZACB< 180 Hint: Use the results of problem J2 and the role that A? plays in the proof of that problem. 36, The sum of the Interior angles of a quadrilateral Is less than 3^0, 37. Prove that the sum of the Interior angles of a polygon of n sides is leas than (n-2) 180. Although this unit will conclude at this point. It should be strongly emphasized that there is much more to be studied in the area of hyperbolic non-Euclldean geometry, Much of what would naturally follow is of a more rigorous nature and, in moot cases, would require a degree of mathematical maturity beyond what should be expected of a tenth grade student. The author will close the unit with what would be the next theorem. A rather unusual result occurs when we look at what would be similar triangles In Euclidean geometry and examine the analogous situation in hyperbolic. It turns out that there are no similar triangles In hyperbolic geometry. Hopefully, some of you will retain enough interest in the fascinating world of geometry. Euclidean and non-Euclldean, to continue your investigations in these areas. THEOREM 45. If the three angles of one triangle are equal respectively to the three angles of another triangle. then the two triangles are CONGRUENT# BIBLIOGRAPHY 1. Eves, Howard: Newsom, Caroll V. An Introduction to the Foundations and Fundamental Concepts of .-lathematlos# New lorlt: Rinehart and Company, Inc., 195^* 2. Kattsoff, Louis 0. "The Saccheri Quadrilateral." THE MATHEMATICS TEACHER, LV (Dec, 1962), pp. 630-636. 3. fcalres, Wesley W. "Introduction to Non-Euclidean Geometry," THE hATHEKATICS TEACHER, LVII (Nov. 196*0 PP. 457-461. 4. Wolfe, Harold E. Non-Euclldean Geometry. New York: Holt, Rinehart, and Winston, 1945. APPENDIX D NON-EUCLIDEAN GEOMETRY TEST State below either Euclid's Fifth Postulate or the form of It known as Playfair's Axiom. State below the Characteristic Postulate of Hyperbolic Geometry. The word "parallel" means something different in hyperbolic non-Euclldean geometry than It does in Euclidean geometry. Explain briefly below Just what that difference is. B Given: line 1 with BA intersecting 1 and R is a point within ifABC (a) If BC is parallel to line 1 then how is 6R related to 1? (Intersects, non-intersecting, or parallel) (b) If and 1 are non-intersecting then how is BR related to 1? (c) If Be intersects 1 then how is BR related to 1? Gperating in hyperbolic geometry the figure at the right has DA_LAP, CD _l AB, and AD « BC. State below all the accurate conclusions you can make about 1 and 2 and how they are related. 138 6. Given: /CAB = /.DBA - 90 T A Prove: 1^ DOES NOT Intersect lg * :r 19 ■— 2 B D 7. Given: OARl, M the midpoint B of AB, /A = Z B M P r o v e : M il J, AB A 8. Suppose a quadrilateral In hyperbolio geometry has 3 right angles. .‘/hat conclusions would you he willing to make about the fourth Interior angle? Try to Justify your conclusions and use the back of this paper for your answer. APPENDIX E TEST INSTRUCTIONS 1* Please print your name In Pencil on the answer sheet and darken the matching grids directly under your name. Continue to use pencil on the exam. 2. On this test a 11 mathematical system" contains unde fined terms, definitions, postulates, theorems, and .laws of logic. 3. The term "postulate" will mean the same thing as the term "axiom". if. Some sections of the test that do not have 5 multiple choice answers have special instructions preceding the questions. 2s sure to read these special Instructions carefully so that you are able to mark the answer sheet correctly. 5* You will have to work quickly and carefully. When you finish the exam be sure your name Is on the answer sheet. At the bottom of the answer sheet In the place for your Instructors name, print his name and the name of your school. 6 . Please hand in BCTH the test and the answer sheet. 139 140 1. In proving: If x2 is an odd number, then x la an odd number, a valid alternative would be to (1) Prove: If x2 la not an odd number, then x la not an odd number. (2) Prove: If x la not an odd number, then x2 la Kot an odd number. (3) Prove: If x is an odd number, then x2 la an odd number. (4) Pick 100 odd perfect squares and show their square roots are odd. (5) None of the above are valid alternatives. 2. In a mathematical system definitions are (1) Statements which are agreements as to how a symbol or term Is to be used, and they do require p^oof. (2) Statements which are agreements as to how a symbol or term la to be used, and they do not require proof. (3) Statements that should be flexible so they can be used in a. variety of situations. (4) Self evident truths. (5) None of the above. 1 3. Theorem: If a triangle la equilateral, then it Is isosceles. An equivalent form of this theorem Is (1) Some triangles are equilateral and some are isosceles. {2) If a triangle Is Isosceles, then It Is equilat eral. (3) It will never happen that a triangle will be equilateral and not Isosceles. (4) If a triangle Is not equilateral, then it might be isosceles. (5) Both (1) and (4). 4. Given a mathematical system In which the undefined terms are "mugs" and "chairs" and In which the undefined relations "mug on a chair" and "chair on a mug" mean the same thing. The axioms are: Axiom 1. There exists exactly three distinct nugs. Axiom 2. Not all mugs are on the sane chair. Axiom 3. bn any two distinct mugs there is exactly one chair. Axiom 4. Cn any two distinct chairs there Is at least one oommon mug. 141 The least number of chairs that satisfies this system is? (1 ) 1 (2 ) 2 (3) 3 (5 ) more than ^ 5* A ticket read: f,If It rains, there will be no picnic on Friday." On Friday it did not rain. A valid conclusion from this information is (1) There was a picnic, (2) There was not a picnic, (3) Not enough information is given to determine If there was or was not a picnic. (*M Friday turned out to be a good day for the picnic, (5) Even if it did rain there might be a picnic. 6 . In trying to prove i/S is not rational using the indirect method of proof, one should (1) Assume 7. Given a set of undefined terms, definitions, postulates, and laws of logic for proving theorems, then the theorems deduced and proved in this system (1) Would be different on Mars than on Earth. (2) Would be different on the moon than on Earth. (3) Would be the same on all planets and moons. (4 ) May or may not be the same on other planets depending on the individual theorems. (5) Both (1) and (2) arc correct. 8 . Given a mathematical system in which the undefined terms are "point” and "line" and in which the unde fined relations "point on a line” and "line on a point” have the same meaning. The axioms are: Axiom 1. Not all points are on the sane line. Axiom 2. There exists exactly three distinct 142 points. Axiom 3* On any two distinct points there Is exactly one line. Axiom On any two distinct lines there Is at least one common point. The least number of lines that satisfies the system is? ' (1) 1 (2 ) 2 (3) 3 <<0 * (5 ) more than ** The following questions are to be answered "True" or "False1', It is understood that those Items that are true only some of the time aro to be considered false. Remember that on the answer sheet that 1. la True and 2. is False. 9. An axiom could be a statement that Is the opposite to ones basic Intuition. 10. Regardless of the axioms used, the sum of the interior angles of a quadrilateral is 360 degrees. 11. Undefined terms are not really necessary in geometry. 12, When a mathematician remarked: "One must be able to say at all times - Instead of points, straight lines and planes - tables chairs and beer mugs." he was referring to undefined terms. 13, Cnee a theorem is proved in one mathematical system, it is true in any other mathematical system. 1^. Definitions never have to be proved. 15. A theorem la a statement that must be proved. 16. Some axioms must be proved. 17* It is impossible to define all words in a manner that axoids any confusion about their meanings. 18. It Is desirable in a mathematical system to be able to prove some of the axioms from the other axioms. 19* A statement and its negation' can not both be true in a mathematical system. 143 Classify the following as 1. axiom (used Interchangeably with "postulate" on this teat), 2, definition, or'3* theorem. Determine the correct choice using the same classification that your textbook dons, 20. An Isosceles triangle is a triangle In which at least two sides are equal, 21. If two angles of a triangle are equal, then the sides opposite those angles are equal. 22. Through a point outside a line there is exactly one parallel to that line, 23. Point B Is said to H e between points A and C if and only If all three points are distinct points on a line and AB + BC « AC. THIS PARAGRAPH APPLIES CNLY TG ITEfriS 24 THROUGH 30. A postulate you have used about parallel lines states: Given a point. A, not lying on line, 1, there exists one and only one line through A which does not Intersect 1. In the questions 24-30 assume that wo have replaced this postulate with: ANY TWO STRAIGHT LINES INTERSECT. Everything else, except, of course, those Ideas that are a consequence of, or follow from the postulate we have now changed, remains the same. It will be helpful to know that the only congruence theorem that you may use is the SAS (Side-Angle-Side) theorem. 24. If angle 1 = angle 2, then i^_ (1) 1. intersects I2 (2) 1- is parallel to lg (3) it is not known if intersects I2 or if 3-2 is parallel to 13 (4) angle 1 can't equal angle 2 (5) both (1) and (4) 144 REI'.EMBER THE INSTRUCTIONS ON THE’ PAGE BEFORE THIS PERTAINING TO ITEKS 24-30 APPLY TO QUESTIONS 25-29 ON THIS PAGE* 25. Given a line 1 ’an d a point P not on 1 , then the number of straight lines that can be drawn through P that do not intersect 1 is (1) 0 (2) 1 (3) 2 (4) 3 (5) more than 3 1 26. If OA AC and CB AC and BC = A3, then 0 (1) DC = OA (2) 00 AC (3) CA and OB can't both be perpendicular to AC (4) both Cl) and (2) A B C (5) neither (1) or (2) 27. Given AC AB and DB AB, then (1) AC RD (2) AC and BD (3) AC and DB can11 both be perpendicular to AB B (4) impossible to determine what AC and BD do when extended (5) both (2) and (3) 28. Given OA AB and AO * AO* and CB AB, C ,A,Cf colllnear with points C* and B connected, then (1) triangle AOB = triangle AO'B (2) angle 0 1 BA is a right angle (3) O, B, and O' are on the same line (6 ) (1 ). (2 ), and (3 ) are correct but not (3 ). 29. The least number of lines that can inclose an a r e a Is (1 ) 0 (2 ) 1 (3) 2 (4) 3 (5) 4 145 REMEMBER THE INSTRUCTIONS PERTAINING TO ITEMS 24-30 APPLIES TO QUESTION 30 ON THIS PAGE. 30. If angle 1 * angle 3, then (1) lj Intersects I2 (2) li Is parallel to I2 (3) it is not known If 1^ is parallel to or If li intersects Ip (4) angle 1 can't - angle 3 (5) both (2) and (4) 31. It is known that "if a function Is differentiable, then it is continuous," Which of the following follows from this fact? (1) If a function is not differentiable, then it la not continuous. (2) If a function is continuous, then It is differen tiable, (3) If a function Is not continuous, then it is not differentiable. (4) It is desirable for a function to be differen tiable, (5) The question can't be answered unless the terms are understood, 32. Accept as being true the statement: "If each of two sets, A and B are compact, then the union of these two sets is compact." Given two sets A and B, and the knowledge that the union of A and B Is compact, we can conclude (1) A is compact and B is compact. (2) Either A or B is compact. (3) Neither of the two sets is oompact. (4) Nothing definite about A or R can be concluded. (5) The question can't be answered unless the terras are understood. 33. Accept as being true, "If a number is composite, then it can be expressed as a product of primes.1' Whioh of the following statements below implies that 12,243 can be expressed as a product of primes? (1) 12,243 is a prime number. (2) 12,243 is a composite number, (3) 12,243 is a large nurabefr. (4) 12,243 is an odd number. (5) In order to answer this question it is necessary 146 to understand what prime and composite numbers are. 3 it-. Accept as being true the statement: "If a nurd is a dingle, then the nurd is a nood.M Which of the following Implies that Ned, a nurd, is also a nood? (1) Ned is a nurd. (2) Ned is a dingle. (3 ) Ned is not a dingle. (4) Ned is not a nurd. (5) Impossible to tell without understanding the terms. 35. Mathematicians include undefined terms In a mathe matical system (1) In order to keep the system flexible, (2 ) Because they don*t want to be bothered defining the obvious. (3) To add difficulty to the system. Because some terms are hard to define. (5) To avoid circularity of definitions. 36.' A statement that is a self evident truth, accurately describes (1) An undefined term (2) A definition (3) An axiom (if) A theorem (5) None of the above BIBLIOGRAPHY BIBLIOGRAPHY Adler, Irving, "What Shall We Teach in High School Geometry?" The Mathematics Teacher, LXI (March, 1968), 22 6-38. Allendoerfer, Carl B. "The Dilema in Geometry." The Mathematics Teacher, LXII (March, 1969), 165-69, Bidwell, James K., and Clason, Robert G. Readings in the History of Mathematics Education. Mational Council of Teachers of Matheraati.cs, 1970. Brumfiel, Charles. "Conventional Approaches Via Synthetic Euclidean Geometry." Unpublished paper, University of Michigan, Ann Arbor, undated. (Mimeographed.) Buchalter, Barbara Diane Elpern. "The Validity of Mathematics Textbook Series in Grades 7-14 VJith Structure as an Objective." Dissc-rtat 1 on Abstracts , XXX, 198-A. Buck, Charles. 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