Symposium on the Occasion
of the 100th Anniversary of ICMI
(Rome, 58 March 2008)
WORKING GROUP #1
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
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?
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?
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?
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
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)
- 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
- 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
- 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 20th century
Disciplinary mathematics and school mathematics: an overview, by Bill Barton and Frédéric Gourdeau