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Maybe a Mathematics Teacher Can Profit from the Study of the History of Mathematics ABRAHAM ARCAVI, MAXIM BRUCKHEIMER, RUTH BEN-ZVI

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Maybe a Mathematics Teacher Can Profit from the Study of the History of Mathematics ABRAHAM ARCAVI, MAXIM BRUCKHEIMER, RUTH BEN-ZVI Maybe a Mathematics Teacher can Profit from the Study of the History of Mathematics ABRAHAM ARCAVI, MAXIM BRUCKHEIMER, RUTH BEN-ZVI Introduction them labelled and opened when the time-table an­ "Never in the history of mathematics has so much been nounces the history of the subject matter: owed by so few to so many" (with apologies to Churchill) 4. The argument clo.sest to hand - and the most often So many advance the case for the history of mathematics heard- is that knowledge of the history ofa subject area and its value in mathematical education for students and helps in understanding the subject matter itself I doubt especially for mathematics teachers Through the 20th it- at last (sic) as far as mathematics is concerned. century, reports [MAA, 1935; Ministry of Education, We took the first two of Freudenthal's statements as axi­ UK, 1958; among others] and single authors [for example, omatic in our particular situation; i e , as two consciously­ Barwell, 1913; Jones, 1969; Shevchenko, 1975; Grattan­ chosen variables ("Seminar" may have different connota­ Guinness, 1978; Rogers, 1980; Struik, 1980] suggest so tions: in our context the seminar took the form of a much possible positive influence on attitudes, understand­ workshop in which the participants were active and ing, etc "worked" most of the time, as will be further described In general discussions of the sort mentioned, as in gen­ later ) eral discussions of almost every educational "case", the in­ The third statement corresponds in our context to the fluence of whatever one wants to talk about (in our case, following. The approach was not chronological, nor bio­ history of mathematics), depends not only upon the sub­ graphical (about mathematicians) as such; our purpose ject matter itself but upon many other variables rarely ex­ was to create a picture of the development of a topic of plicitly taken into account.* relevance to the teacher, and by the way, into that picture, The case for the history of mathematics has to be care­ enter mathematicians, dates, etc., that are subsidiary to fully stated within a particular context in terms of varia­ the study of the mathematical story. This was our conclu­ bles such as: target audience, their mathematical back­ sion on considering a further variable: the objective ofthe ground, the sources, the approach, the topics etc. If, in whole exercise. We do not, at this stage, or perhaps ever, another context, the variables are not identical, and they expect the teachers to go away and teach the history of never are, the case has to be restated with possible modifi­ mathematics, but what we do expect will be part of the fol­ cations. lowing description Freudenthal [1981] explicitly mentions some of these This article is intended to take issue with the fourth of variables. Our purpose is to refer to some aspects of his ar­ Freudenthal's statements There is no doubt that this par­ gument, in the process of describing an experience of "do­ ticular statement is seriously dependent upon the varia­ ing" history of mathematics with in-service teachers. The bles, and we will describe a situation in which the "doubt" teachers (as the target audience) are our first variable: they was shown to be unfounded. This cannot be done by argu-­ were, in the main, junior high school teachers whose ing in generalities, but only by describing a course, its mathematical background barely extends beyond the cur­ materials and the outcomes The value of such a descrip­ riculum which they are expected to teach tion lies in that others who may be interested can then de­ Freudenthal makes the following statements: velop and plan their own courses suited to their chosen 1 History is worth being studied at the murce rather than and enforced variables. by reading and copying what others have read and co­ pied before. 2 The most appropriate way to learn and teach the history The course and its material of mathematics is through seminars rather than courses The material for a two-day teacher workshop on the con­ 3 I stress the history of science as integrated knowledge ceptual development of negative numbers was created as a rather than items stored in well-stocked drawers, each of series of worksheets whose general form was as follows * We exclude controlled studies such as McBride & Rollins [1977] and Delaney [1980] that confirm some limited aspects of the general claims 'These studies, and other unreported courses, which certainly A brief biographical-chronological introduction in or­ exist, represent ''the few". How few, can perhaps be judged from John­ der to set the historical scene son and Byars [1977}, who report that, on average, of a total of more An historical source; as far as possible a primary source than 30 semester hours devoted to mathematics in pre-service teacher training courses, a mere 0 23 hours was devoted to the history of math­ Leading questions on the source material and on math­ ematics ematical consequences thereof For the Learning of Mathematics 3, 1 (July 1982) 30 FLM Publishing Association. Montreal. Quebec, Canada I o each worksheet an extensive discussion/solution leaflet The second contradiction, attributed to Wallis was prepared, containing detailed written solutions to the (1616-1703) among others, is as follows: questions and further source material and background, as If we extend the sequence of inequalities appropriate to complete the historical-conceptual infor­ mation < 1/3 < 1/2 < 1/1 The series of worksheets consists of an introductory to the left, we have sheet, seven worksheets and a summary sheet The rela­ tively short time available for the workshop, and the rela­ < 1/3 < 1/2 < 1/1 < 1/0 < 1/-1 tively time-consuming fOrm of activity (two further varia­ from which we "conclude" that the negatives, which are bles), implied a very careful selection of material less than zero are also greater than infinity In particular, since the early histmy of the negative The teachers are asked to explain these paradoxes numbers and the original sources are fairly sparse, we Io answer the Wallis paradox, the discussion/solution chose to introduce the subject using a secondary source leaflet includes a section from Euler's Vol/standige An­ [Bell, 1940], which compactly summarizes the history of leitung zur Differenzial = Rechnung (a German translation negative numbers The specific purpose of the introductory of the Latin original published in 1790), in which Euler sheet is twofold: (1707-1783) refutes the argument by considering these­ - to provide the teachers, who knew almost nothing about quence the topic (see pretest results later) with some orienta­ tion; that is, a framewmk in which to work, and 1/9 ' 1/4' 1/1 ' 1/0 ' 1/1 ' 1!4 ' 1!9 - to fill in some of the stages of the development of nega- whose general term is l/n2 and from this "surely no one tive numbers not treated in the wmksheets would maintain that they (terms to the right of 1!0) are I ime was given to read the sheet and then it was dis­ greater than infinity" * cussed. The seven worksheets which then followed are de­ Is is interesting to note that Euler did no mme than re­ scribed briefly below The text of one of them (the sixth) pulse the suggestion that the negative numbers were and its accompanying discussion/solution leaflet is given, greater than infinity by showing that the same argument as an example, in the appendix led to an absurd conclusion .. The teachers in the workshop experienced considerable difficulty in arguing against Ar­ I) F Viete (1540-1603), for an example of nauld and Wallis, and in discussion we began to sense the i) subtraction and the law of signs, but only in those main objective source of this difficulty, i.e. the absence of a cases in which all the expressions involved are definition of negative numbers. This turned out to be a positive, and continuing theme until, by the time we reached sheet 7, ii) the complete non-recognition of negative quanti­ the teachers felt that the definition was begging to be intro­ ties duced The primary source on which this wmksheet was based, 3 N Saunderson (1682-1739), for an example of didactics was taken from In Artem Analytzcem Isagoge (1591) as in the 18th century quoted (in translation) by Struik (1969] Because of the Saunderson was essentially a university teacher We chose difficulty of reading source material in general, and the un­ a paragraph dealing with multiplication of negative num­ familiar language of the older primary sources in particu­ bers from his posthumously published book The Elements lar, the first questions on this and other worksheets were of Algebra (1741] His didactical strategy is what we usually devoted to "compelling" the teacher to read In would regard today as the extension of patterns observed this case he was asked to translate the mathematics of fOr positive numbers to the negative numbers Viete's text into modern symbolism Here, as a byproduct, Interesting issues occur, and one of the questions we met the teacher is encouraged to practice reading mathematics was the distinction between definitions and assumptions, in a situation which is partly in his favour: the language is which are not distinguished in the text. This led to a dis­ unfamiliar but the content, when "translated", is familiar cussion of the problem of proof in the absence of the basic Thus a possible bonus, which we have not yet seriously definition evaluated, maybe the enhancing of the teacher's math­ 4 L Eulet, for another didactical example We chose a ematical literacy section from the English version of his Elements of Alge­ bra (1828, Fourth Edition).
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