# History of mathematics teaching in the national curriculum

Santos, editorsp. Dedicated to Maria Fernanda Estrada, with friendship and respect Abstract: Some comments are made on the use of History in the teaching of Mathematics. 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 mathematics teaching in the national curriculum 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: 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. 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.

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 history of mathematics teaching in the national curriculum 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.

### Jessica Furr (Waggener)

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.

- A most common misconception arises in the form of lack of clarity about the distinction between number, base and notation since children are not formally introduced to any other numeral system or made to think about the significance of representations.
- In cities with business interests, certain mechanical mathematics skills were still needed and taught in a few schools Willoughby, 1967, p. ED 253 969 Grouws, D.
- Very rarely did they understand an operation. Some topics in algebra and geometry were recommended to be introduced in junior high and it was suggested that all students complete the program through eighth grade, with only those who mastered continuing on.

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. In late secondary school, and especially in the university, the situation changes.

### Jessica Furr (Waggener)

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. 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 onebut for a Mathematics teacher it is an obligation, because a teacher must be a person of culture. Katz editorUsing history to teach mathematics. Portuguese translation published by Porto Editora, 2001.