Blumenau, Brasil
2527 July 1994
History of Mathematics in the classroom:
hopes, uncertainties and dangers
Jaime Carvalho e Silva
Departamento de Matemática
Universidade de Coimbra, Portugal
A few years ago a portuguese secondary school teacher asked me how should the history of mathematics be used in the classroom. I had to answer I had no idea. After that I was involved in a lot of different things: I wrote a few papers on the now wellknown XVIII century portuguese mathematician José Anastácio da Cunha (Silva and Duarte 1990b),(Silva and Duarte 1990a), I was one of the cofounders of the portuguese SNHMNational Seminar on the History of Mathematics (now chaired by Luís Saraiva), I gave an 18hour course to highschool teachers on the possible uses of the history of mathematics in the classroom, I wrote a Calculus book where the history of mathematics is one of the components, I gave a course on the history of the derivative using original texts, and a few other things related to the history of mathematics.
If that secondary school teacher made the same question now, what would I answer?
Well, now I have a few ideas on that subject but I still cannot offer a very clear picture. And I doubt that a completely clear picture can be offered. That's why in the title of my talk I included uncertainties and dangers alongside with the hopes.
In fact I think the discussion on this subject is not enough. Mathematics Education is discussed, History of Mathematics is discussed, but, except for an overwhelming optimism, very few concrete discussions are made on the "hows and whys" of the History of Mathematics in the classroom.
Just before coming to Brasil, I received the British Society for the History of Mathematics Newsletter 25/26 and I saw the summary of a talk by Bob Burn in the HIMED 94, "History for the antihistory teacher" where some practical difficulties are discussed.
Of course I see a great danger here. It is obvious that the interest for the History of Mathematics is quite recent and growing, that fashion in Mathematics Education includes history. My question is: will it vanish in some years and be forgotten like all fashions?
In Portugal there is a very sad example of this. The portuguese mathematician José Sebastião e Silva (1914  1972) wrote several books for the highschool in the 50's and 60's. All of them included very interesting references to the history of mathematics.
The first volume of his "Compêndio de Álgebra"(Silva and Paulo 1970) includes a lot of very detailed and readable historical notes about the evolution of the concept of number (with a photo of a "modern" computer), the notions of function and limit and an history of the calculus in a total of 20 pages in a book of 314 pages. The second volume includes 16 pages out of a total of 283 about the history of algebra and of logarithms (with a reference to the portuguese mathematician Pedro Nunes (15021578) and to calculations with computers). He justifies the history of mathematics as a mean of placing the subjects in a general cultural setting "that will temper and humanize the abstractedness inherent to mathematics, trying to explain it as an historical process"(Silva and Paulo 1970) (the first edition dates from 1956). In another book (Silva 1964) the author introduces the study of the complex numbers with a digression through the equations of the third degree and their history. In a book of Analytical Geometry (Silva 1970) he includes 9 of the 146 pages with historical notes about Descartes and Fermat.
I must add that these books were the only ones available (the official textbooks) for almost 20 years. The books that replaced these ones (form 1975 to 1992) included absolutely no reference to the history of mathematics and we can say history was out of the mathematics classroom. Now the history of mathematics is returning to the textbooks but only as a minor illustration (and some books still do not include any reference to it  one of the authors said to me that the absence was due to the fact that there was no material available).
The main reason for the return of history
of mathematics to the textbooks is that the official syllabus for secondary
schools now include explicit references to the use of history of mathematics
in the classroom. Two main goals for the use of the history of mathematics
are:
"Recognize the contribution of
Mathematics for the understanding and solution of mankind problems along
the times" (MinistériodaEducação 1991), p. 10.
"Relate steps of the history of mathematics with the evolution of mankind" (MinistériodaEducação 1991), p. 11. 
For the grades 79 there are only four
explicit references to the use of history of mathematics, like the following
one:
"If opportune, the pupils can make
a small work in group about equation solving in the History of Mathematics
(solutions of particular equations in antiquity, Pedro Nunes and the solution
of equations, the symbolic writing and its contribution to the advance
of equation solving and its uses, etc.)" (MinistériodaEducação
1991),p.59

The main criticism I make to these
official texts is:
i) the references are too vague,ii) there are almost no texts available in portuguese,
iii) the curriculum is already surcharged.
Is "history of mathematics in the
classroom" a topic used by government officials only as a proof of modernization,
as a cosmetic and not structural change? I fear so!
When I began planing the calculus book I wrote(Silva 1994), one of the problems I faced was: how much history of mathematics? One of my preoccupations was to give the corrected historical names to the theorems. I had of course to rename the BolzanoCauchy condition for the convergence of series as CunhaBolzanoCauchy condition. But I was puzzled with other denominations. Different sources give different names and I don't know which one to choose. For the Gregory series I added a warning that it was discovered independently by Leibniz, Gregory and the 15th century indian mathematician Nilakhanta (Roy 1990). But in the same BSHM newsletter the Gregory series is called the MadhavaGregory series (pg. 49)... Different textbooks have different namings for different theorems, and sometimes they even say the usual naming should not be adopted. For example, in the excellent Calculus book by Richard Courant (Courant 1963) we can see in a note about the MacLaurin formula: "A special case of this theorem is often quoted, without historical justification, as the theorem of MacLaurin. We will not adopt such designation."
I think it would be helpful to have a reference work with the correct naming (as far as the current research can tell us, of course). The incorrect naming can suggest the idea that only this particular "genius" could think about it, when in fact the same thing is discovered by different people in different times and different contexts.
In my calculus book, I decided to include the history of mathematics in two ways: in the beginning I wrote a short history of calculus stressing only the problems that were solved by calculus (sometimes with very few details). Inside the book I included references when I thought their inclusion would enhance the text. For example: the continuousnondifferentiable example of Weierstrass and the recent work of Risch and Bronstein about the automatic calculation of antiderivatives, or the Malthus model of population growth.
In the exercise section (to be published as a separate volume) I am going to include some historical problems or some problems with historical background but I am not sure they will be successful. Of course I will choose problems I like, but will the students like them? Is this a good choice for a calculus book?
I would definitely like to see some research on problem solving with an historical background!
Another uncertainty is the inclusion of biographies in textbooks. Are they useful? Is it useful or interesting to know when Cauchy was born and died, when he began teaching, etc.? I remember my highschool physics book had some biographies and I never found any interest on them (although I must confess I liked to see the faces, how these genius looked like).
I would also like to see some convincing research on the use of biographies in the mathematics classroom and textbooks.
So far, it may seem I am only discussing the history of mathematics in the classroom, because it is the fashion now. Not at all. I think the history of mathematics is very important for mathematics education for a number of reasons I will discuss later. My fear is that these reasons will be lost when fashions will change, and all the uncertainties today about the history of mathematics in the classroom may destroy our hopes.
If the groups interested in the "history of mathematics in education" do not intensify their action producing written materials and carrying research that sheds light on the correct practices, I fear all will be forgotten in some years.
There a few papers on the "whys and hows" of using the history of mathematics in the classroom. Two of the ones I prefer are
In the very interesting text of André Weil we can find the following concluding remark: "(...) the essential business of both [historians and mathematicians] is to deal with mathematical ideas, those of the past, those of the present, and, when they can, those of the future. Both can find invaluable training and enlightenment in each otherís work. Thus my original question ëWhy mathematical history?í finally reduces itself to the question ëWhy mathematics?í, which fortunately I do not feel called upon to answer."
Struik: Why study the history of mathematics? (Struik 1980)A. Weil: Hstory of mathematics: why and how? (Weil 1978)
Of all the arguments I agree with,
I can see two groups:
a) The point of view of the teacherb) The point of view of the student
I think the study of the history
of mathematics is very important for teachers of all levels (including
the university) because:
i) it gives a better idea of the evolution of ideas, offering him an opportunity to connect different chapters of the syllabus;ii) the teacher will be more tolerant with the students, knowing that certain things like functions, limits, derivatives, etc., were a big obstacle that took sometimes hundreds of years to overcome; If the teachers studied papers like:
I. Kleiner: Rigor and proof in mathematics: an historical perspective (Kleiner 1991)J. Grabiner: The changing concept of Change: The derivative from Fermat to Weierstrass (Grabiner 1983)
then they would have a better perspective of what mathematics was, is and will possibly be. In this last paper the author calls our attention to the fact that: "The derivative was first used; it was then discovered; it was then explored and developed; and it was finally defined." And afterwards: "This point is important for the teacher of mathematics: the historical order of development of the derivative is the reverse of the usual order of textbook exposition." It is a point that should be considered carefully by every teacher.iii) The teacher will have a greater choice of concrete examples to choose form (and use them if he feels it will increase the interest or motivation of the students. That's the approach used in the courses I gave to secondary school teachers and a paper about the subject (Silva 1993). Books like the ones produced by the "Commission InterI.R.E.M. Épistémologie et Histoire des Mathématiques" are excellent sources of "historical mathematics"(for example the books "Mathématiques au fil des âges"(I.R.E.M. 1987) and "Histoires de ProblèmesHistoire des Mathématiques"(I.R.E.M. 1993))
From the point of view of the student
I think that when he finds the history of mathematics (mixed with another
subject or as a subject itself) he will have a broader and better idea
of mathematics. He will see that mathematics is not a rigid subject but
something that is built along the centuries in a nonlinear way. He will
learn that all civilizations made important contributions to the history
of mathematics (and so he will become a better citizen, respecting cultures
different from his own).
I think it is not only important to include references to a broad range of civilizations (like the often forgotten islamic, indian, or chinese, or the even more forgotten african, maya, aztec, ...) but also from the student's own country. It will teach him also something about his own culture (science or technology is culture) but also contribute to his selfesteem as citizen of his country. Like the portuguese historian Luís de Albuquerque (19171992) said when he was claiming History of Science in Portugal should be more fully studied: "However modest these manifestations may be, it does not mean they concern less the history of our culture, because they represent also a tradition that, even if it is less vivid, does count to our common patrimony. And maybe the perfect knowledge of the vicissitudes that it went through were still today for us a beneficial lesson."(Albuquerque 1961).
So many positive aspects in the use
of the history of mathematics in the classroom must convince us we must
work hard to support the research and dissemination of the history of mathematics
and its uses in the classroom.
References
Courant, Richard. Cálculo Diferencial e Integral. 3ª imp. da 1ª ed., Rio de Janeiro: Editora Globo, 1963.
Grabiner, Judith. "The changing concept of change: The derivative from Fermat to Weierstrass." Mathematics Magazine 56 (4 1983): 195206.
I.R.E.M. Mathématiques au fil des âges. Paris: GauthierVillars, 1987.
I.R.E.M. Histoires de ProblèmesHistoire des Mathématiques. Paris: EllipsesEd. Marketing, 1993.
Kleiner, Israel. "Rigor and proof in mathematics: an historical perspective." Mathematics Magazine 64 (5 1991): 291314.
MinistériodaEducação. Programa de Matemática  3º ciclo. Lisboa: INCM, 1991.
Roy, Ranjan. "The discovery of the series formula for 1 by Leibniz, Gregory and Nilakhanta." Mathematics Magazine 63 (5 1990): 291306.
Silva, Jaime Carvalho. Princípios de Análise Matemática Aplicada. Lisboa: McGrawHill, 1994.
Silva, Jaime Carvalho. "A reforma curricular e a História da Matemática." Educação e Matemática 27 (1993): 2731.
Silva, Jaime Carvalho and António Leal Duarte. "Os "Principios Mathematicos" de José Anastácio da Cunha." In Anastácio da Cunha, o matemático e o poeta, 8195. Lisboa: INCM, 1990a.
Silva, Jaime Carvalho and António Leal Duarte. "Sobre a influência da obra matemática de José Anastácio da Cunha." In Anastácio da Cunha, o matemático e o poeta, 133145. Lisboa: INCM, 1990b.
Silva, José Sebastião. Compêndio de Matemática, 1º vol  2º tomo. Lisboa: Min.Educação/OCDE, 1964.
Silva, José Sebastião. Geometria Analítica Plana. 2 ed., Lisboa: Emp. Lit. Fluminense, 1970.
Silva, José Sebastião and J. D. Silva Paulo. Compêndio de Álgebra, 1º vol. 2 ed., Vol. 1. Braga: Livraria Cruz, 1970.
Struik, D. J. "Why study the history of mathematics?" UMAP journal 1 (1 1980): 328.
Voolich, Erica Dakin. "Using biographies to ëhumanizeí the mathematics class." Arithmetic Teacher 41 (1 1993):
Weil, André. "History of mathematics: Why
and How?" In International Congres of Mathematicians in Helsinki,
227236, 1978.