La lettre de la Preuve

       

ISSN 1292-8763

Spring 2002

 

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

 
 

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