ENRICHING HIGH-SCHOOL ALGEBRA THROUGH TEE USE OF HISTORICAL MATERIALS A Thesis Presented for the Degree of Master of Arts by ,,, ,/ Paul BJI Boeckerman,.. B.s • . .:: : :.. ... .... .... TBE OHIO STA'Th: UNIVERSITY l 9 4 7 Approved by: ~J:J'I- LIST OF ILLIDSTFJ1.'rIONS NlAPS Page Map 1. Spread of Ancient Babylonian, Egyptian, and Greek Influence. 24 Map 2. The ~xtent of the Known World at the Time of Ptolemy The Lands Conquered by the Barbarian Invaders in Vie stern Europe. The Extent of the East Roman Empire. 25 Map 3. Extent of Moslem Conquest Rise of Universities CrusaO.es 43 Map 4. Caravan and Trade Routes of the Micidle Ages Teaching Centers of Hindu Mathematicians 44 PIC·ruHES Numerals of the 28th Century B.C., A Sumerian Tablet. 77 E6yptian Hieroglyphic Notation. 81 A Problem in Hieratic from the Ahmes Papyrus and its Equivalent liie roglyphic Form. 82 ·rhe Ancient Maya Calendar-Script. 91 Two Sixteenth Century Modes of Reckoning. 94 Counter Reckoning in 1514. 94 Early Forms o:f the Hindu-Arabic Numerals. 97 Egyptian Word Proble~. 130 Symbols for Roots of Algebraic Polynomials. 131 Symbols for the Representation of Unknown Quantities. 132 The Sign of Equality 132 Variants of the Radical Sign. 133 Methods of Indicating Plus and Minua 133 Map of the Worlu by Hecataeus, 517 B.C. 139 Map of the World according to Eratosthenes. 139 Ptolemy's Map of the World. 140 Page Chapter I Int r·oO.uct ion • 1 Statement of the Problem 7 hel&tea Stuc:i.ies 7 Hesolution of the Problem 8 Foot-Notes 9 Chapter II Chronolo6ical Outline Introduction 11 First Period c. 50,000 B.C. to c. 500 A.D. 16 Second Period c. 500 A.D. to c. 1600 A.D. 33 Third Period. c. 1600 A.D. to 1700 A.D. 51 Foot-Notes 63 Chapter III Extension of the Number System Introduction 68 The Number Concept in Pre-History 70 The Natural Numbers Babylonian Number System 75 Egyptian Number System 79 Greek Number System 83 Roman Number System 86 Mayan Number Systerr: 90 Hindu-Arabic Number System 92 Extensions to the Natural Numbers 101 Fractions 102 Irrationals 110 Negative Numbers 113 Complex Numbers 117 Foot-Notes 121 Pae;e Chapter IV Sug'-:;estions for tbe Use of Historical Materials in the Classroom 124 Foot-Motes 150 Chapter V Surrnnary 153 Sug";estions for :b,urther Study 156 Bibliography 158 CHAPTER I INTRODUCTION There is little doubt that mathematics can be taught, and taught well, without any reference being made to its historical development, or to its in£luence in the develop­ ment of civilization. Such a method may, perhaps, have certain definite advantages. Mathematica is an abstract science, more abstract today in its t·oundations than ever before, and, as such, may be studied independently of any physical reality. But to teach it without occasional refe:P.. ence, at least, to its historical development is to with­ hold from the student an enrichment of understanding that only the history of the subject can give. Of what value is the history of mathematics to the teacher of mathematics? In addition to information a well- informed cultured person should know, the history of mathe­ matics provides the teacher with a perspective that enables him to discriminate between the essentials and non-essen- 1 tials in his teaching field. History shows how the mathe- matics of the Greeks, though sufficient for the builders of their civilization, is wholly inadequate for the builders of present-day civilization. The arithmetic of the Greeks has long ago been discarded for better methods. Advances and refinements are characteristic of living mathematics, but there has always been a time lag between an advance and its incorporation into the educational program of the age. Statistics has been an important and relatively recent mathe- - 2 - matical development, It is widely used today in many differ­ ent fields, ranging from Gallup polls to nuclear physics. But how many secondary schools, at present, give students a working knowledge of statistics? Some authorities, even, are of the opinion that if all obsolete material were dis­ carded from high-school mathematics there would be sufficient time to give the f'undamentals of calculus, already in the secondary schools. Even a reading knowledge of what modern scientists imagine the world to be requires a working know- 2 ledge of the elements of the calculus. As mathematics and the teaching of mathematics advance it is the well-informed and alert teacher who compensates for the time lag in text- book material. There is yet another important reason why teachers of mathematics should be familiar with the history of their subjects. The inherent difficulties of many topics of mathe- 3 matics may be seen in their historical backgrounds. "History," says 9ajori, "emphasizes the importance of giving graphical representation to negative numbers in teaching algebra. Omlt all illustrations by lines, or by the thermometer, and nega- tive numbers will be as absurd to modern students as they 4 were to the early algebraist a. 11 Negative numbers were 1 ab- surd' or 1 fictitious 1 to early writers until Fibonacci, in the early thirteenth century, suggested a meaning for them by interpreting a negative number in a money problem as a 5 loss instead of a gain, and until Descartes, in the seven- teenth century, gave them a geometrical interpretation in - 3 - his analytic geometry. It may be important to point out that negative numbers today are defined in a manner independent of any geometrical interpretation. Of what value is the history of mathematics to the students? It is quite easy to see that, for a student with a natural inclination for the study of civilization and its developnent, the story of mathematics may be a fruitful ap­ proach to a much wider study of mankind. All students, on the other hand, may deepen their appreciation of the value of such attitudes as tolerance and cooperation by underatand- 7 ing the nature of mathematical growth. Mathematica is rarely conscious of nationalistic, racial, or religious prejudice. The Arabs eagerly sought after Greek and Hindu mathematics. Thia learning the Moslema later passed to Christian Europe. The personalities of mathematicians in the eighteenth cen- tury took on an international character, as these mathema- ticians lectured and carried on research at the various royal courts and universities of ~urope. The development of mathematics, both pure and applied, moreover, has largely been cooperative in nature, the product of the activity of many men, united by a common interest. Eighteen hundred years after Apollonius wrote his treatise on conics Kepler read the work and applied it. Kepler discovered that his calculations, made from the data so painstakingly collected by Brahe, led to the conclusion that the bodies in the soler system revolved about the sun in elliptical orbits. - 4 - The history of mathematics as an aid to teaching is a point we cannot af'f'ord to overlook. In all newspaper re- porting, pictorial or written, it is the human interest aspect of an event that is stressed. Why not give the hum.an interest side of mathematics, those numerous episodes and interesting achievements in the lives of mathematicians? Fourier was a great mathematician and a great teacher, and it is said of him that he "enlivened his lectures on mathe- matics by out-of'-the-way historical allusions (many of which 8 he was the fir·st to trace to their source)". J. V. Collins has developed an interesting variation of the idea of intro- ducing human-interest stories into courses. There are in- tervals of relaxation in all labor, and as a relief during the progress of the teaching period he suggests relating anecdotes, which are mathematical in nature, from the lives of such prominent men as the students meet in their study 9 of literature, history, and science. The human-interest side of mathematics may stimulate student interest. The histories of special topics, on the other hand, when skill- fully woven into teaching procedures, may make the learning of these topics easier, through an understanding of how they developed. The Committee on the function of mathematics in general education in their report, Mathematics in General Education, recognizes the values in an understanding of the history of 10 mathematics. The Committee points out that the history of mathematics, in aGaition to giving new light and life to - 5 - the subject, leads to an appreciation of mathematics as one of the main instruments by means of which man has construe- ted for him.self the world in which he now lives. The Committee, likewise, points out common misconcep- tions as to the nature of mathematics and its place in the history of civilization: 1) Mathematics always had its present form and few additions will ever be made. 2) Mathematics is the product of steady inexorable progress from primitive prehistoric fonns of counting and space perception to present com­ plex techniques. 3) Mathematics is a system of proposition and re­ lations without relation to personalities and human affairs. 4) Political and universal histories ignore the in1'luence of mat;hematics in historical trends. To correct such misconceptions and to attain the posi- tive values, the Committee proposed that the teacher help students gain understandings of the following sort: 1) Understandings related to the development of mathematics in response to the needs of the times. a) Mathematics as a direct response to the spirit and needs of the times. b) The insuf'f'iciency of the needs and the spirit of the times as an explanation of all mathematical development.