Symposium on the Occasion
of the 100th Anniversary of ICMI

(Rome, 5–8 March 2008)


Disciplinary mathematics and school mathematics


The mathematics taught and learned in school is a selection from and a transformation of the subject that mathematicians have taken as their discipline over the course of time. This working group will consider the ways in which ICMI has addressed school mathematics. For example, how do (should) the goals of school mathematics reflect the nature of disciplinary thinking and practice? How can (should) high-level mathematics be made accessible to all students?


We wish to indicate some of the issues which could be addressed as part of our work. This is not to be taken as limiting the ways in which this theme should be addressed, but rather as a tentative description of what it may encompass. A starting line could be the general question: What sort of mathematics do we learn at school?

Focusing on content leads to the question of how we could describe the content which is to be mastered by a student? A first attempt would be simply to describe school mathematics as a subset of disciplinary mathematics. We believe that this does not provide a satisfactory framework, and wish to engage in a broader discussion than this. In particular, mathematics does not reduce itself to a list of topics, but also comprises processes such as abstraction, symbolization and axiomatisation. More fundamentally, school mathematics may differ in nature from disciplinary mathematics.

So, how can we describe the mathematics which is taught and learned in school? If we are trying to describe it in terms of disciplinary mathematics, then what, exactly, are we comparing it with? The nature of mathematical knowledge and mathematical certainty are not static. As we look at mathematics since 1908, the year ICMI was founded, we can easily identify major events which have impacted deeply on the perception of what is mathematics. One can think of the crisis in the foundations of mathematics, of Gödel and Russell, of constructivism and Brouwer, of Bourbaki, of the advent of computer-assisted proofs, of probabilistic proofs. The great development of applied mathematics, whether it is seen as challenging the notion of proof with its reliance on experimentation to test validity, or as changing the nature of the tasks we can face by allowing us to model situations of great complexity without needing to resort to simplification in the same way as before, is also to be considered.

The constant evolution of disciplinary mathematics means that one cannot simply contrast mathematics taught in school with a fixed identity. Nevertheless, it is important to reflect on school mathematics and somehow to contrast it with mathematics as the subject that mathematicians have taken as their discipline over time. Let us look at some aspects which could be used to approach this question.

Proofs and proving

What is the role of proofs in disciplinary mathematics and in school mathematics? How are they different? If the nature of proof in mathematics is changing, then does this change the nature of proofs in school mathematics? Are the standards for accepting a proof different? What is the role and nature of so-called visual proofs? Can these questions be asked for computer assisted school mathematics as well? In what way is proving as done in school mathematics a preparation for proving in research mathematics?

One can also think of truth. In mathematics, truth is not only a logical and autonomous construct: it is also a social construct and the validity of statements can be accepted by a mathematical community even if not all if its members have read (yet understood) the proofs of those statements. Recent examples are Wiles’ proof of Fermat’s Last Theorem or the recent solution to Poincaré’s conjecture by Grigori Perelman, but this is also true of much simpler results. Is it, or should it, be the same in school mathematics?

The Development of Mathematical Ideas

The way fundamental mathematical ideas are presented to learners of mathematics will, of necessity, be shadow versions of the concept as held by mathematicians. It is hoped that these shadows develop more substance as the learning becomes more sophisticated. But this raises a new set of questions. Is there a best developmental sequence for any particular mathematical idea? Are there stages of students’ development where particular concepts are best taught? What are the consequences if a student stops learning mathematics before an idea is fully developed? Are there any mathematical concepts that cannot be taught as a sequence but must be presented in full from the beginning?

Social aspect

The way in which mathematicians view their subject area is deeply rooted in the way they do mathematics and learn mathematics themselves. Without going into caricatures, it is clear that a school classroom and a research seminar in mathematics are not identical. However, we may wish to look at the differences between disciplinary mathematics as experienced by undergraduates in mathematics, and school students. Is the nature of the interactions part of this picture? How is how we learn linked to how we see mathematics?

Applied Mathematics

Although we recognize that there is not a clear distinction between Pure and Applied mathematics (indeed, the dichotomy may not even be a productive one), nevertheless there are some features embedded in this distinction that cause us to reflect on the teaching of mathematics in schools. Should the extremes of mathematical investigation for its own sake and modelling real world processes both be represented in school mathematics? How should the different philosophies inherent in each of these be presented? Even within modelling there are two different approaches: the classical mode of trying to find the fundamental mathematical laws that govern physical processes (as exemplified in physics); and using mathematical methods to add layer upon layer of complexity to our understanding of systems such as the human body or climate change. Should (and how) can both these be represented?

High-level mathematics

Educational systems have different means to ensure that some students master high-level mathematics, and the purpose of education is construed differently for such students. We do not wish to discuss the appropriate goal for education, but we could reflect on the ways in which high-level mathematics has been made accessible to students. How is this different from school mathematics? Is it a good idea to present mathematics differently to students of different levels or abilities?

There are many other aspects of school and disciplinary mathematics that might be discussed: language; assessment; formalism; role of technology; creativity and discovery; and so on.

Co-chairs: Bill Barton (New Zealand), Frédéric Gourdeau (Canada)



Disciplinary mathematics and school mathematics: an overview, by Bill Barton and Frédéric Gourdeau