Proof in the curriculum
Maria Alessandra Mariotti
Università di Pisa
Last fall (4 - 6 October 2001) the Qualifications and Curriculum Authority
(QCA) in England hosted an international seminar on the theme:
Reasoning, explanation and proof in school mathematics
and their place in
the intended curriculum.
The purpose of the seminar was to share experiences with colleagues
from different countries and to explore the dynamic between the intended
curriculum in algebra and geometry -particularly with respect to reasoning,
explanation and proof - and pupils' real classroom experiences of these
areas of mathematics.
The seminar focussed on how successful different countries have been
in implementing their intended curricula, on what construction principles
underpinned their curricula, and how central the skills of reasoning,
explanation and proof are to development of pupils' mathematical understanding
and learning.
You can find a brief account of that meeting (QCA-Proof-Conference
) by the organizer, Jack Abramsky and a personal feedback (QCA-Proof-Overview.doc)
by one of the participants, John Mason.
The theme is of great interest and in my opinion our readers should
be stimulated by some of the questions which were discussed in the meeting.
For this reason we now repropose them to the audience of our newsletter,
in order to open a larger debate (forum).
Although each of us has his/her own answer, may be no definite answer
is possible. But a debate will be very productive in order to construct
a shared base of ideas.
There are at least two levels of questioning:
Epistemological level: discussion about proof and its place in mathematics,
and consequently its relevance in mathematics education
Pedagogical level: discussion about proof in the curriculum: when
and how introduce a theoretical perspective in mathematics classes.
Differences, related to many cultural aspects, are reflected into the
curricula of the different countries, and deeply affect teachers' attitude
to proof and its place in school mathematics. Sometimes, people are
not aware of most of these differences. As Balacheff clearly pointed
out, (http://www-didactique.imag.fr/preuve/Newsletter/990910.html),
far from being shared, different cultural perspectives are present so
that mathematics epistemology presents a high variety of complexity
which make difficult to understand each other. Even professional mathematicians
do not always share a common point of view on mathematics, why should
we, math educators do? Becoming aware of this complexity may be a first
fundamental achievement.
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 (Knuth, 2000)
(http://www-didactique.imag.fr/preuve/Newsletter/000708).
To give an example, consider the following excerpts drawn form the last
version of the Principles and standards for school mathematics published
in year 2000.
"Reasoning and Proof as fundamental aspects of mathematics .
Reasoning and Proof are not special activities reserved for special
times or special topics in the curriculum but should be a natural,
ongoing part of classroom discussions, no matter what topic is being
studied."
(Principles and standards for school mathematics, NCTM, p. 342)
By the end of the secondary school, student should be able to understand
and produce mathematical proofs - arguments consisting of logically
rigorous deductions of conclusions from hypotheses - and should appreciate
the value of such arguments. [
] Reasoning mathematically is
a habit of mind, and like all habits, it must be developed through
consistent use of many contexts"
(Principles and standards for school mathematics, NCTM, p. 56)
Other countries, for instance Italy and France, have a long standing
tradition of including proof in the curriculum. Never the less, the
idea of "proof for all" is not a view most teachers hold;
on the contrary, a strong belief in the intrinsic difficulty of proof
is widely shared by teachers at any school level. Futhermore, the main
difficulties encountered in introducing pupils to proof have lead many
teachers to abandon this practice. Either proofs are ignored, or they
are presented by the teacher, but they are never requested from the
pupils.
This reality seems not to take into account most of the results coming
form a number of research studies developed in the last years and reported
in the literature.
In other terms, the active dabate devolped within the community of math
educators has not yet penetrated in the school practice, and seems to
be very far from it .
The issue is to find the proper bridge between what we learn from research
(which is situated in the reality of schools, but still related to research
aim) and what practitioners can grasp from it. A strategy should be
found to make understandable and useful the results obtained in the
last years.
The distance separating research studies and school practice should
find a way to be overcome, and this can be done by re-elaborating research
results through a debate in the community of mathematics education researchers.
I think it is time to start, rethinking our research work in terms of
what to say to school practice both in terms of curricula and teachers
training.
Reactions?
Remarks?
The reactions to the contribution of Maria Alessandra
Mariotti will be
published in the Summer 2002 Proof Newsletter
©
Maria Alessandra Mariotti
|