Reasoning, proof and proving
in mathematics education
ICME 10
Those of you who have had a look at the scientific programme of the next ICME-10
may have noticed that it contains two components on reasoning, proof and proving,
namely TSG 19 and ST 2.
This testify the interest that the theme of proof has in the community of mathematics
educators.
According to the direction of the programme commette, the division of labour
between these programme elements is the following.
The task of ST 2, is to "excavate", survey and review the state-of-the-art
in its theme, with particular regard to identifying and characterising important
new knowledge, recent developments, new perspectives, and emergent issues, the
outcomes of which will be presented in a regular lecture .
The task of TSG 19 will be to present a carefully selected set of papers
to the TSG participants, thus addressing the "expert" or the "member
of the congregation" who wants to deal with reasoning, proof and proving
for several
timeslots.
In the following more information can be found concerning the organization of the two programme elements.
Survey Theme 2:
Reasoning, proof and proving in mathematics education
"What do we know and where are we today as far as reasoning, proof and
proving in mathematics education is concerned."
M. A. Mariotti
Since two decades, mathematical proof has been at the core of
an active debate in the community of mathematics educators: often blamed as
responsible of pupils' difficulties, but also recognised as a crucial aspect
of mathematics activity.
In the recent past the role and the place that proof takes in the mathematical
curriculum have often changed. For instance, in the United States, after a period
of 'banishment' proof has got a central position in the new Standards 2000.
In math education that on proof is a very active field of research, as witnessed,
for instance, by the number of contributions to the last PME conferences.
This lecture aims at providing a panorama of the state of art in this field,
it is organized around the following three key questions.
1. What is the status of proof in the Curriculum, across different countries.
What is the real situation in schools? Both the perspective of students and
that of teachers will be considered.
2. Is proof so crucial in the mathematics culture to be worth to be included
in school curriculum?
3. How has research in math education approached the problem of proof. In particular,
is it possible to overcome the difficulties so often described by teachers in
introducing pupils to proof?
In answering these question we aim to provide a living panorama of different
curricula, in different countries, introducing the educational problem of proof
in the reality of school. A historic and epistemological discussion will attempt
to clarify the role and status of proof within the community of mathematicians,
taking into account the main role played in evolution and systematisation of
mathematics knowledge throughout the centuries.
The complex and sometimes controversial relationship between argumentation and
proof will be discussed, especially in the perspective of its relevance in the
educational field. Different epistemological assumptions differently direct
research studies, and lead to different approaches to proof as an the educational
issue: a short account of the main streams will be given
A number of studies have been devoted to describe and analyse students' difficulties,
in different school context. The rich collection of data and the living discussion
accompanying them have provided a solid base for a number of research projects
aiming to experiment new approaches to proof.
Different experiences have been carries out, at different age levels in different
countries. Especially interesting, appears the proposal of introducing a "proof
culture" at the primary school level. Actually, an early exposition to
a practice of argumentation seems fundamental for establishing a correct relationship
between different components of mathematics activity.
A final remark will be devoted to the special issue raised by the introduction
of new technologies and in particular of microworlds. For instance, Geometry,
traditionally related to the first approach to proof, has seen a dramatic change
in ideas and practice, after the introduction of dynamic Geometry environments.
What, if any, is the contribution of technologies to the solution of the educational
problem of proof?
The team is composed by
Professor Maria Alessandra Mariotti
Dipartimento di Matematica
Università di Pisa
Via Buonarroti 2
ITALY
e-mail: mariotti@dm.unipi.it
Professor Celia Hoyles
Mathematical Sciences Group
Institute of Education
20 Bedford Way
London WC1H 0AL
UK
e-mail: c.hoyles@ioe.ac.uk
Professor Hans Niels Jahnke
Fachbereich 6 Mathematik
Universität Duisburg Essen
DE 45117 Essen
GERMANY
e-mail: njahnke@uni-essen.de
Professor Alexander S. Mishchenko
Department of Mathematics
Moscow State University
Moscow 119992, GSP-2
RUSSIA
e-mail asmish@higeom.math.msu.su
Dr. Ren Zichao,
National Education Examinations Authority
# 39 Zhongguancun,
Beijing, 100080
CHINA
e-mail: renzz@mail.neea.edu.cn
Topic Study Group 19:
Reasoning, proof and proving in mathematics
G. HAREL
The aim of TSG 19 is to provide opportunity for the participants to share their
research in reasoning and proving in mathematics education. The focus is on
transition from informal argumentation to formal proof in
mathematics classrooms, including classrooms where technology is used.
There will be eight paper presentations; the are listed below in the order they
will be given at the conference:
Mathematicians perspectives on the transition to formal proof - Presenter:
Lara Alcock
A pilot study on five mathematicianspedagogical views on proof - Presenter: Kirsti Nordström
Proofs as a tool to develop intuition- Presenter: Alexander Khait
Key ideas in the context of a proof from collegiate calculus - Presenter: Manya Raman
The role of logic in teaching proof - Presenter: Susanna Epp
Improving reasoning abilities of 5th-6th grade pupils using a specially designed teaching unit in pre-formal logic Presenters: Raisa Guberman, Marita Barabash
Polyminos: A way to teach the mathematical concept of implication- Presenter: Virginie Deloustal-Jorrand
The nature of studentsrule of inference in proving: The case of reflective symmetry - Presenter: Takeshi Miyakawa
Team Chairs are:
Professor Guershon Harel
Department of Mathematics 0112, University of California, San Diego
La Jolla, CA 92093 0112, USA
e-mail: harel@math.ucsd.edu
Professor Sri Wahyuni,
Department of Mathematics, Faculty of Mathematics and Natural Science, Gadjah
Mada University
Sekip Utara, Yogyakarta, 555281, Indonesia
e-mail: swahyuni@indosat.net.id
Team Members is composed by
Gudmundur Birgisson
Iceland University of Education, Iceland
e-mail: gudmundur@birgisson.com
Christine Knipping
University of Hamburg, Germany
e-mail: knipping@uni-hamburg.de
David A. Reid
School of Education, Acadia University, Canada
e-mail: david.reid@acadiau.ca