NOTE ON THE USE OF HISTORY IN THE TEACHING OF MATHEMATICS
João Filipe Queiró
Departamento de Matemática - Universidade de Coimbra
Research Seminar on History and Epistemology
of Mathematics - Proceedings (A. Azevedo, M. E. Ralha, L.
Santos, editors), p. 91-95
Centro de Matemática da Universidade do Minho, 2010.
Dedicated to Maria Fernanda Estrada, with friendship and respect
Abstract: Some comments
are made on the use of History in the teaching of Mathematics.
1. Mathematics and its History
The French philosopher of Mathematics Jean Cavaillès, in the Introduction to his 1938 book Remarques sur la formation de la théorie abstraite des ensembles [1], writes the following:
“L’histoire mathématique semble, de toutes les histoires, la moins liée à ce dont elle est véhicule; s’il y a lien, c’est a parte post, servant uniquement pour la curiosité, non pour l’intelligence du résultat: l’après explique l’avant. Le mathématicien n’a pas besoin de connaître le passé, parce que c’est sa vocation de le refuser: dans la mesure où il ne se plie pas à ce qui semble aller de soi par le fait qu’il est, dans la mesure où il rejette autorité de tradition, méconnaît un climat intellectuel, dans cette mesure seule il est mathématicien, c’est-à-dire révélateur de nécessités.”
It’s true that he immediately adds:
“Cependant avec quels moyens opère-t-il? L’oeuvre négatrice d’histoire s’accomplit dans l’histoire.”
And he uses the remainder of the Introduction to analyse the relations between mathematical creation and historical conditions.
But the insight contained in those first sentences, which I read while still a student, left me with an impression which, after many years, with more information and maturity, has never gone away. The creation and the study of Mathematics in the present can be carried out ignoring History. It is possible to conceive of a Fields Medal mathematician who knows absolutely nothing about the origins and historical evolution of his field of expertise. Of course, we can also consider him an uncultivated mathematician — or human being — but that is a kind of moral judgment, external to Mathematics itself.
It is a fact that intelligibility of Mathematics is independent of the knowledge of its past. One who studies Mathematics, be it at research level be it while learning the subject, does not need to know the History of what he is studying, apart from, possibly, in the case of the researcher, the recent contributions on the problem under investigation.
The History of Mathematics — understood as the history of mathematical ideas — is today an established academic field. Gone are the days when the history of a scientific subject was considered a part of the subject itself, and its knowledge a precondition to any attempt towards progress in the field. This classical justification for studying the history of any scientific discipline no longer exists (see [3] for a discussion).
As to Mathematics itself, its legitimacy is twofold: it is the area, among human discourses, which most radically and rigorously questions itself, its correctness and internal coherence; on the other hand, it is an essential component of all fields of knowledge with any aspiration to be called scientific.
Why Mathematics possesses the latter characteristic is a classical and fascinating question, analysed, for example, in the famous article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, by the Nobel Prize winner Eugene Wigner. As noted by another Nobel Prize, Paul Dirac, the mathematician plays a game for which he invents the rules, while the physicist plays a game in which the rules are given by Nature; but it turns out, over time, that the rules are the same.
What is of interest to us here is that History plays no part in the legitimization of Mathematics, except in the broad sense that the subject has been around for a long time (plus, of course, the historical component of the twofold legitimization mentioned in the previous paragraphs).
History of Mathematics, on the other hand, cannot claim total independence from Mathematics, at least for the last few centuries, for obvious technical reasons. This is what makes it such a difficult field, beyond linguistic, cultural and methodological requirements.
2. History and teaching
If Mathematics, as a current scientific field, is independent of its history, as discussed, a fortiori the teaching of Mathematics does not formally necessitate the use of the History of Mathematics. Such use, therefore, can only be justified by pragmatic considerations, for example, bringing about a stronger student motivation and thus improving the quality of their learning.
My pragmatic claim is that the History of Mathematics should not be used in the teaching of Mathematics too early, surely not before the 6th grade. By “used” I mean really used, not giving the name of Pythagoras to a theorem and the like.
Until the 6th grade, students are in a very immature phase of their intellectual development. Teaching and learning should concentrate on the acquisition of basic knowledge and techniques. Any reference to the origins and historical evolution of introductory algorithms and results — even assuming that such origins and evolution are easy to unravel — would have, at these levels, a serious effect of distraction and confusion.
It is unproductive, for example, to try to teach children two or more algorithms for the arithmetical operations, mentioning different historical or cultural contexts. In the very imperfect environment of elementary schools, the question is not whether children will learn more than one algorithm, the alternative is between learning one or none at all.
The idea behind learning more than one algorithm seems to be that it enhances understanding of the underlying concept. The emphasis on “understanding” basic material in these age groups is misguided. In Mathematics, there are many examples of skills that have to be acquired before the corresponding concepts are fully understood.
Between the 6th and the 9th grade, there will occasionnaly be an opportunity for interesting and relevant use of historical material. A few times, with motivated groups of students, a reference to short biographies of mathematicians and to historical contexts for the appearance of certain results or techniques — including the re-enactment of famous calculations — may be justified and useful. But this will be the exception, not the rule, and it most certainly should not be mandated by the national curriculum.
Historical references should never crowd out the real purpose of mathematical study, which is the acquisition of important knowledge and techniques — either as ends in themselves or as prerequisites for further study — and the progressive development of a logical and rigorous mind. Time constraints, and student mental overload, are legitimate considerations here.
On the other hand, Mathematics is a human activity, which progresses and changes like any other. But truth criteria in Mathematics have been the same for centuries, even millennia, only progressively more refined. So one should not overuse historicism, nor make vague “historical” references as illustrating an alleged temporal contingency of mathematical activity. If this is to be the role of History of Mathematics in teaching, it’s better to leave it out altogether.
In late secondary school, and especially in the university, the situation changes. The book [2] contains several examples of the use of history in teaching at those levels, and there are many other references on the subject.
This makes sense. A cultivated relation with Mathematics, which supposes an historical vision, can only be attained at university level, where contents already possess some sophistication. So it is clearly an option to use history to enhance and enrich teaching of Mathematics at that level.
For students enrolled in teacher training, a course dealing exclusively with the History of Mathematics should be mandatory. A Mathematics teacher should know some History of Mathematics, not because he is going to teach it, but because, unlike researchers — and even more so people who apply Mathematics to other fields such as Engineering — a teacher should have the above mentioned cultivated relation with Mathematics. To have a reasonable knowledge of the History of Mathematics makes for a more cultivated researcher (possibly even a better one), but for a Mathematics teacher it is an obligation, because a teacher must be a person of culture.
References