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Xamples 53 Suggested Problems/Topics for Student Research 63 References for Student Research 64 Bibliography 68 IV California State University, Northridge lVIATHEIV;ATICS SElVIINAR COURSE A paper submitted in partial satisfaction of the requirements for the degree of Master of Science in Mathematics by Harry James Pappas May 1972 The paper of Harry James Pappas is approved: California State University, Northridge May 1972 TABLE OF CONTENTS Page I, INTRODUCTION 1 Bibliography 15 II. THE MATHEMATICS SEMINAR 17 Bibliography 25 III. HISTORY OF MATHEMATICS 26 Behavioral Objectives 26 Preface 26 Introduction to the Topic 27 Examples 53 Suggested Problems/Topics for Student Research 63 References for Student Research 64 Bibliography 68 IV. NON-EUCLIDEAN GEOMETRY 69 Behavioral Objectives 69 Preface 69 Introduction to the Topic 81 Examples 96 Suggested Problems/Topics for Student Research 103 References for Student Research 103 Bibliography 106 v. LINEAR PROGRAMMING 108 Behavioral Objectives 108 Preface 108 Introduction to the Topic 109 Examples 114 Suggested Problems/Topics for Student Research 134 References for Student Research 135 Bibliography 137 TABLE OF CONTENTS Page VI, GAME THEORY 138 Behavioral Objectives 138 Preface 138 Introduction to the Topic 140 Examples 144 Suggested Problems/Topics for Student Research 153 References for Student Research 153 Bibliography 155 VII. THEORY OF NUMBERS 157 Behavioral Objectives 157 Preface 157 Introduction to the Topic 159 Examples 165 Suggested Problems/Topics for Student Research 176 References for Student Research 177 Bibliography 179 CHAPTER I INTRODUCTION We should provide courses that reveal mathematics as a way to understanding "almost every corner of human thought."(3, P• 46) The role of mathematics in modern society is a very significant one. Mathematical methods have penetrated many other fields of knowledge, changing them in substance and power, and have actually brought new disciplines into being. Mathematics has become an indispensable tool in economics, sociology, biology, linguistics, etc. New mathematics- based fields have been created, such as computer science, information theory, and cybernetics. Today any domain that desires rational thinking is more and more in debt to mathematics. Within mathematics itself has come a great revolution since the turn of the century; its foundations have been reappraised and unifying mathematical disciplines have been created, making possible a better view of classical mathe­ matics and opening new areas and methods of research and application. Until 1930, mathematics applications were restricted to small areas accessible only to a very limited number of people. Now mathematics has become an important "social factor," making the planning of a good mathematics 2 curriculum an educational challenge and responsibility of the highest order.(l6 ) Although there has been much difference of opinion among secondary mathematics teachers as to just how curri­ culum should be changed, there seems to be general agreement that changes were necessary--thus giving rise to so-called accelerated programs, such as a full year of calculus; or enrichment, within existing courses, by introducing at a simplified or intuitive level topics that will be studied in depth at a later time.(S) A current problem in high school mathematics curricula is what mathematical topics ought to be taught to seniors and advanced juniors. Woodby (1965) in a survey of emerging twelfth•grade programs, concluded that no particular program seems to be the most appropriate one at the present; however, he noted that both acceleration and enrichment were included in what he labeled "strong mathematical programs." Many writers feel that acceleration is often overemphasized; for example, Grossman (1962) argues that too often enrichment is slighted in favor of acceleration. The twelfth-grade mathematics program offers an excel­ lent opportunity for enrichment by means of seminars, mathematics laboratories, or independent study. (5) Although at present some degree of agreement has been evolved regarding the mathematics curriculum through grade eleven, various proposals are still under discussion 3 involving the twelfth-grade program: some probability theory, matrix algebra and elementary functions, calculus, computer mathematics, and number theory. The program remains undecided throughout the country, Essentially, the updating of mathematics programs involves a change in content and approach. The change in content presents a desire to impart knowledge of topics now of greater importance than formerly, such as inequalities, number fields and probability, necessitating a drop from the curriculum of such items as extensive logarithmic calculations and solid-geometry proofs. However, no longer can a student learn in school all the mathematics he might be expected to use in later life. By concentrating on any topic, certain others are omitted, and students are unprepared for possible new uses of mathematics. This, then, ties to a change in approach. By emphasiz­ ing the structure of the subject, it is desired that students become sufficiently skillful in handling mathematical systems that they will be able to learn new mathematics more easily in the future. (7) As Wagner< 20 • p. 454) so aptly stated: Since no one can predict with certainty his future profession, much less foretell which math skills will be required in the future by a given profession, it is important that math be taught that students will be able in later life to learn new mathematical skills which the future will surely demand of them. 4 Basically, in the evolving mathematics curriculum (as only one part of our changing knowledge and information explosion), the teacher functions differently to promote a different type of learning, Emphasis is shifting away from the mastery of old knowledge to techniques for developing access to new knowledge (in this process existing knowledge remains important and cannot be discarded; it is simply reorganized). Thus, the teacher functions less autocrati­ cally and instead directs student inquiry, creating an environment where learning is maximized, The teacher functions as a resource and a coordinator, retaining the decision-making role in curriculum. Much of the classroom activity is open-ended, with the teacher the operational curriculum writer or developer, initiating changes and modifying existing format where necessary,(ll) In class, the mathematics teacher is aware of teaching only a small fragment of one of the great mathematics systems, The teacher should convey this to the students. Almost every course has little mathematics significance except as it relates to the mathematics that follows and to that which has preceded, To equalize this emphasis, teachers should feel more of a responsibility for giving new meaning, new interpraation to previous work and to developing better programs for the training of mathematics scholars. 5 In selecting subject matter for a course, a good teacher of mathematics should think through these three points:(l2) 1. Selecting material so that the resulting structure gives the impression of unity and completeness and provides an adequate founda­ tion for future extensions. 2. The utilitarian aspects of mathematics that have made possible the great technological and scientific achievements should be stressed-­ this can coexist with the pure aspects of mathematics, all too often stressed to the exclusion of all else. No teacher has a better opportunity to achieve a balance between the practical and the impractical. 3. Mathematics, along with all other subjects, should accept the responsibility for providing experiences to enhance cultural breadth and appreciation. A teacher of mathematics must believe and emphasize that there is a litera­ ture of mathematics as important to man's development as the great classics. It has been clearly shown as far back as 1950 that high IQ often accompanies low creativity because of adult and peer lack of attention or sympathy toward creativity. However, this originality, lack of conformity, and a tendency to see something different and new even in widely accepted usages and procedures can and should be fostered and directed. The gifted are reluctant to accept the obvious and reach the final solution too soon; they can combine ideas that are usually considered unrelated. Repetitions, reviews and 6 routine methods of presentation bore the gifted. Their pace of learning is high; they keep seeing new things and raising new questions. In mathematics, it becomes not just a ques- tion of solving harder and harder problems for these students, but imaginative thinking--as mathematics has areas of "simplicity, ingenuity, beauty and utility, like linear programming, finite mathematics, nonmetrical geometry, and theory of games."(lB, P• 396) Teachers must cope with the phenomenon of highly creative individuality mixed with a good deal of ambiguity and absence of rigor in speculation. The content of mathematics courses taught continues to be determined largely by the basic textbook. Although most schools have reorganized their mathematics curricula to some degree since 1960,<14) this has been effected mainly by the adoption of recently copyrighted textbooks and sub­ sequent changes in course content. Student boredom with textbooks among the gifted should lead in addition to such work as, for example, the more stimulating records of the triumphs, breakthroughs, and pitfalls of the masters. "The genius of Archimedes in working with a very inconvenient numeration system, the simple but profound contributions of Euler and Gauss in the theory of
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