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Definition in High-School Geometry THE NATURE 0? DEFINITION IN HIGH-SCHOOL GEOMETRY A CRITIQUE OF CURRENT PRACTICES DISSERTATION Presented In Partial Fulfillment of the Requirements For the Degree Doctor of Philosophy in the Graduate School of The Ohio State university By SHELDON STEPHEN lyîYERS, B.8., B.E.» M, Ed, ****** The Ohio State University 1955 Approved by: Adviser Department of Education AOKNOWLEDCaÆHNTS The writer would like to express his sinoere eppreoiation to Professor Nathan Lazar for making unstint- Ingly available his Intellectual resources and wise counsel during the completion of this study. The writer would also like to express his appreciation to Professor Harold P. Fawcett who first stimulated his Interest In the teach­ ing of geometry. 11 t able o f C0NTMT3 CHAPTER page I. INTRODUCTION............................... 1 Statement of the problem.................. 1 Importance of the p r o b l e m ................ 3 Scope and limitations of the study........ 4 Definitions of terms ..................... 5 Preview of the study................... , 6 II. ARISTOTELIAN ORIGINS OF SOME DEFINING PRACTICES IN PLANE GE0L5ETRY TETTBOOKS . 8 Definitions should be in the form of "genus at differentia,"....................... 10 Definitions should not use obscure language and should use terms which are prior and more intelligible ................... 18 Prior and more intelligible terms .... 18 Obscure language ..... .............. S7 Definitions through opposites, circular definitions, and definition by coordi­ nate species .................. SO Circular defiiiitions................ 31 Sequences of ùufinition .............. 33 undefined teriLs............... 36 S u m m a r y ............................. 51 iii CHAPTER PACE Definitions should not oontain redundant or superfluous information •••••••* 55 Definitions must be reversible........ 58 There is only one definition of a thing • • 74 Definitions state what a thing is but not the foot that the thing exists...... 85 Purposes of definitions given by textbooks . 99 Chapter s u m m a r y ..........* ............. 101 III. TREATMENT OF DEFINITION IN I.iATHEMATIOS METHODS BOOKS ............................... 103 Per genus et differentiam method of defining 103 The question of whether there is only one, or more than one, definition of a term . 106 Use of simpler or previously defined terms in definition..................... # 110 Undefined t e r m s ................... 112 The relation of meaning to definition • , * 118 The problem of existanoe and definition. • . 121 The question of reversibility and definition......................... 124 Redundancy or overloading in definition • • 124 Other pedagogical sources on definition in geometry ....................... 128 iv CHAPTER PAGE Chapter Summary •••.«•*•«••••• 13£ 17. TREATMENT BY LOGIC AND MATHIWATICS BOOKS OF CERTAIN DIFFICULTIES CONCERNING DEFINITIONS FOUND IN CONTEMPORARY GEOMETRY BOOKS .... 136 Dlffloultlea arising out of the Aristotelian distinotion between essenoe and property . 136 Dlffioulties arising out of the Aristotelian notion that Glasses and rank of intelli­ gibility are absolute ..................... 145 We do not know the meaning of a term until we define it......................... 151 Definitions can be true or f a l s e ............. 153 Definitions establish exlstenoe ............. 155 Dlffioulties relating to the reversibility of definitions............................. 158 Other theoretical points about definition. 161 Real and nominal d e f i n i t i o n ............... 16£ Stipulative and lexical definition .... 164 Analytic and synthetic definition .... 166 Causal or genetic definition .......... 167 Operational definition ................... 167 Explicit definition and definition in use. 169 ▼ CHAPTER PAGE SxtenslTe or donotatlv# definition and oonnotative definition • • .......... 170 Oateneive definition .................. 171 Implioatire definition ........ ••••• 171 Coordinating definition ................ 173 Descriptive definition......... • . • • 173 Definitions in mathematical systems • • • 174 Chapter s u m m a r y ...................... 176 V. smiMARY, CONCLUSIONS, AND RECOLîMENûATIONS . 178 Recommendations........................ .. 181 Recommendations for other s t u d i e s .... 185 BIBLIOGRAPHY... ...................................... 188 Part I. High School Geometry Textbooks Examined. 189 Part II. Methods manuals in mathematics and related w o r k s .................. 193 Part III. Logic and philosophy-of-mathematics b o o k s .......................... 199 Part IV. Periodical l i t e r a t u r e............ 200 ▼i LIST OF TABLES TABLE PAGE I. Number of Plane Geometry Textbooks^ by PubXloatlon Date, yfhloh Do or Do Not Mention Undefined Terms ........ 47 vii CHAPTER I INTRODUCTION Statement of the Problem Almost universally students in plane geometry are told to define a figure by placing it in its next larger class and by giving characteristics which will distinguish that figure from all others in the class* Thus, they will see that the textbook definition, "A triangle is a polygon with three sides,*' comfortably meets the requirements which they have been taught, Since no suggestion is made that there can be another way to define, they do not question the method. Later, in the same textbook, the follcwing definition may be encountered, "'./hen one straight line meets another straight line so as to form equal adjacent angles, the lines are said to be perpendicular." Usually there is no hint that this definition does not follow the rule prescribed earlier. Later on, when a student attempts to define a parallel­ ogram as a quadrilateral with the opposite sides equal, he is rebuked and informed that "equality of sides" is a deducible property and that the essential, or defining property, of a parallelogram must be "opposite sides are parallel," Yet, were he to glance furtively into several £ other textbooksr he might find in one the definition, *▲ rectangle is an equiangular qualdrilateral," and in another, "A rectangle is a parallelogram with one right angle." But not having other books available, usually, which would enable him to mcounter alternate definitions, he is thereby given the impression that every figure has an essential property that must be used for the definition. As a result of this, he is given the additional impression that there is only one correct definition for each figure. Early in the course, the student is often informed that if he wants to understand a term, he must memorize its definition carefully. After memorizing the definition of adjacent angles, he later finds that he is unable to iden­ tify them and has a very confused idea of what they are. Contrary to the widely-held opinion that the function of definitions is to convey the meaning of terms, they did not do so in the case just cited. In the light of the three, foregoing illustrations, the question might well be raised as to what is the basis of these difficulties and what can be done to correct them. One suspects further that the above difficulties are symptomatic of underlying causes which may have produced other problems with regard to definition in geometry. The purpose of this study will therefore be to investi­ gate the treatment accorded definitions in contemporary, hlgh-BOhool, geometry teztbooke» to determine the natare and extent of tmeound defining jo^aetioee, to determine the origin of theme praotioea, and to reoomaend improvement in aoeordanoe with m o d e m pedagogical, logical, and mathe­ matical theory* Probl*» An examination of contemporary geometry textbooks reveals that there is firmly imbedded in them a tradition about definition which gives rise to a number of unsound practices* This can have serious conséquences for the future intellectual growth of the student, for definitions occupy a key role in all organized knowledge systems, such as physics, economics, philosophy and law* Geometry has long been accepted as an important training ground for the development of critical and logical thinking* Therefore, it would seem extremely important in plane geometry, where the student first studies a logical system, that he utilize the soundest defining practices* This study should serve to point out the source and nature of defining difficulties in plane geometry. It should also focus the attention of teachers, textbook writers, and mathematics educators, on the most approved theories and methods of defining today with particular reference to geometry* This should make it possible for 4 teaohers to work out eroatlvaly in tha olaasrooa thalr own mathodologr of definition# within the framawork of sound theory* 8eOP* SB&. T.imltmtion. Of $6* Sj-SOX In particular, this study will undertake to show the extent to which defining practices in contemporary geometry textbooks and in books and other souroes related to the teaching of mathematics originate In the logical works of Aristotle* Certain of these practices will then be criti­ cized by means of documented comments from the current mathematical and logical literature* This literature will then be used to present the various acceptable methods of defining* The relation of these methods to geometry is pointed out* The various types of information in this study will be gathered from the following sources: (1) Contemporary practices in geometry— high-school geometry textbooks, contemporary mathematics methods books, and articles from Journals (2) Origin of contemporary practices— the logical works of Aristotle and commentaries on these works (3) Mathematical and logical
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