of the 100th Anniversary of ICMI

WORKING GROUP #1

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?

**MAIN ISSUES:**

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.

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?

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)**

**PAPERS:**

- V.B. Alekseev Algorithms and proofs in school
- Arlete de Jesus Brito Case study about how bourbakism became implemented via international agencies in a key region Of Brazil
- Brent Davis What’s wrong with this title? “Disciplinary mathematics and school mathematics”
- Viviane Durand-Guerrier Some epistemological question addressed by school mathematics to mathematicians
- Johann Engelbrecht The moves of the game of mathematics
- Peter Galbraith Some thoughts around disciplinary mathematics for teaching
- Gila Hanna Beyond verification: Proof can teach new methods
- David W. Henderson What do we wish students to know with respect to mathematics when they come to us from school to the university?
- Magali Hersant “Problèmes pour chercher” : Experience, possible and necessity in pupils' reasonings
- Derek Holton What VIEW of mathematics should we learn at school?
- Kwon, Oh Nam An inquiry-oriented approach to undergraduate mathematics: Contributions to instructional design of differential equations
- Lins, Romulo Where would Leonhard Euler stand today as a school teacher?
- Laura Martignon & Elke Kurz-Milcke Educating children in stochastic modeling: Games with stochastic urns and colored tinker-cubes
- Maria Lucia Faria Moro & Maria Tereza Carneiro Soares About sequences and developmental stages in the teaching of mathematics – Some contributions
- Jarmila Novotná Non-standard mathematical structures in mathematics teacher training
- Siu, Man Keung Disciplinary mathematics and school mathematics : New/old wine in new/old bottle?
- Klaus Volkert The problem of solid geometry
- Anne Watson School mathematics as a special kind of mathematics
- Terry Wood, Megan Staples, Sean Larsen & Karen Marrongelle Why are disciplinary practices in mathematics important as learning practices in school mathematics?
- Rina Zazkis “Working like a mathematician” with school mathematics
- Luciana Zuccheri & Verena Zudini
An example of the influence of cognitive theories in mathematics instruction,
concerning the introduction of the elements of differential and integral calculus in
some Austrian secondary schools at the beginning of the 20
^{th}century

**OVERVIEW PAPER:**

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