La lettre de la Preuve

ISSN 1292-8763

Some thoughts after ICME 9

The Topic Study Group on Proof (TSG-12) met in Tokio last summer, within the 9th International Conference of Mathematics Education (ICME). We have decided to ask some questions to Paolo Boero, co-organising and co-chairing the TSG-12 with G. Harel and C. Maher. Our aim was to clarify some of the aspects that emerge from the presented reports.
  The interview highlights the representativeness and diversity of the contributions and refers to different research trends s in the field of "Proof and proving in Mathematics Education".

What trends can be described in the studies about mathematical proof?

A basic distinction may be drawn between trends concerning methodologies of research and trends concerning the issues being studied.
  As far as methodologies of research are concerned, purely quantitative studies on the behaviours of pupils--which were rather common in the field some time ago--are becoming less and less frequent, in this as in other fields.
  Qualitative studies about pupils' behaviour in individual proving tasks have decreased in comparison with the past, while increasing relevance is acquired by qualitative analysis of the evolution of individual behaviours in the long term. At the same time, analysis of individual behaviours in relation to individual tasks are carried out alongside the development of analysis of interactions (either among peers or between teacher and pupils) within tasks proposed either to small groups or to the class-group.
  As far as the issues dealt with are concerned, the prevailing orientation is toward studies on the teaching and learning of proof in relation to other aspects of the teaching and learning of mathematics. Thus researchers are interested, for instance, in the relationships between argumentation and proof in mathematics; in those themes (either mathematical or related to the applications of mathematics) which might support a more complete and a faster development of skills and abilities concerning proof: in the role that Information and Communication Technologies might have in fostering the learning of proof.
  As for the latter aspect, it seems remarkable that the interest of part of the research community has shifted from the identification of the potentialities of computer based learning environments to the study and modelisation of the specific teaching and learning processes that may be promoted in such environments.
  A number of research projects deal with the issue of teachers and students' conceptions about mathematical proof as well as with the influence that such conceptions may have on teachers and students' behaviours in the teaching and learning of mathematical proof.
  Finally it is remarkable that the interest for historical and epistemological aspects of proof constitutes a research trend that has been taking up relevance (since the 80's) as a need for many researchers in the field.

The criteria, followed in the selection of the contributions to the Topic Study Group, suggest some questions. For instance, I quote:
       diversity of countries and research paradigms (in order to cover a wide spectrum of orientations);
In what sense do cultural differences influence research in the field of mathematical proof? Can you give us an example?

I would like to start off with an example: the permanent importance of mathematical proof in the French curricula and the oscillations in the indications provided by NCTM (1989 and 2000) for the USA are influenced by different cultural positions regarding priorities in the teaching of mathematics in the two countries. The diversity of such positions may also explain--at least partially--the diverse characteristics of the research on proof carried out in both countries during the last twenty years.
  Other differences come from the specific educational research paradigms adopted by researchers (at times within the same country): for instance, the choice of a "constructivist" paradigm rather than a "socio-cultural" one has usually important consequences on the choice of the object of study, on the way in which experimental investigations are carried out and, finally, on the instruments used to analyse students' behaviour.

The theme of mathematical proof is nowadays dealt with in a wide literature: is it reasonable to wonder whether there are results of preceding pieces of research which may be considered as acquired, in the sense that they are generally accepted by the community and do not constitute an object for debate? If so, is it possible to identify transformations in the design of curricula that may be related to those results?
  I think that nowadays (differently from ten years ago) there is a general consensus on the fact that research on proof concerns an important objective of the mathematical formation: such objective is strictly intertwined with other objectives (for instance the development of logico-linguistic abilities and competence within mathematics), which require strategies of intervention in the long term and within an encompassing curricular perspective.

I also think that there is a wide consensus on the fact that it is not possible to separate out the analysis of issues related to learning and the analysis of issues related to teaching and this holds for the study of mathematical proof as well as for any other object of study in mathematics education.
  As for the transformations in the design of curricula, we can highlight that the change in the NCTM-2000 standards with respect to the 1989 standards has been determined not only by pressures from the academic world but also by a re-consideration--in the educational research field--of the importance of proof within the mathematical activity and within the mathematical formation.

Is it possible to outline a general framework in which different researchers may find themselves? Or rather, are the divergences so deep as to determine opposite and contrasting points of view?

Deep divergences do exist and they concern the role of the cognitive study, of the cultural and epistemological study and of the sociological study in the educational research concerning mathematical proof, as well as the theoretical frameworks chosen in order to carry out such studies.

Historic-epistemological analysis has played a crucial role in the research studies you have been carrying out. Which is the contribution of this kind of analysis?

It is important, under distinct viewpoints: for framing the existing didactical practices concerning mathematical proof; for identifying the main aspects of the "culture of proof" to be investigated both experimentally and theoretically; for orienting an innovative didactical planning.

Which do you think can be considered the innovative trends on which studies will focus in the next years?

- Proof and constitution of mathematical objects (in relation with the study of the discursive constitution of both mathematical concepts and procedures, which constitutes an important trend in the present educational research).
- A comparative analysis of the "cultures of proof" which are proposed in the schools of different countries, with relation to the specific cultural features of curricula and, more generally, to the cultural values characteristic of each country.
- Analysis of students and teachers' conceptions about proof.
- Modelisation of teaching and learning processes related to proof, carried out according to a number of criteria (in relation with the diverse paradigms chosen by researchers).

Should you indicate an emerging research theme, which one would you choose?

The choice of a single theme is not easy and it would be influenced by my personal preferences. It seems to me that the study of the diverse components of the "culture of proof" and of the teaching and learning strategies that may allow students to appropriate such culture, identifies a research issue which is wide enough to include a number of different themes which have been emerging in the last few years.

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