David A Reid, Acadia University

- Aims
- Rationale
- How the symposium will be conducted
- References
- Call for contributions
- Short descriptions of practices in different contexts

- coming to a better understanding of cultural differences in the teaching of proof, especially as they relate to access to learning about proof in different cultural contexts, and
- discussion of the need for universal access to instruction in proof and proving.

Some factors that might relate to these differences on a national level include differing priorities and opportunities in rich and poor countries, different philosophical histories in Eastern and Western culture, and the influence of linguistic structures on reasoning patterns and the role of reasoning. Within countries variations occur regionally, in public and private school systems, in urban, suburban and rural areas, in different types of public schools (e.g., in Germany and China), and in areas with students from mono-cultural or multi-cultural backgrounds. Even within a single school of classroom differences can occur, related to the backgrounds of students (e.g., poor or rich, academic or non-academic families, or by class) and the way they have been classified by the school system (e.g., "gifted" or "non-gifted").

Work has been done on describing differences in mathematical achievement and curricula across cultures, but there are issues in both the way such comparisons have been done and used, and in the attention (or lack of it) paid to proof and proving, that suggest that more work is needed on cultural differences in the teaching of proof.

Studies such as TIMSS focus on differences in mathematical achievement and the production of "league tables" comparing national scores. This means that their results can be used to support efforts to reduce education policy to a service for business and market forces and to view education as a tradable good with an exchangeable value. They can also nurture the myth of the possibility of global education standardisation and hence to push convergence of curricula across cultures. This can lead to the undervaluing of diversity in teaching, with negative implications for efforts to use culturally appropriate approaches to support the learning of students from varied social and cultural backgrounds (Keitel & Kilpatrick, 1999).

There is also a gap between research on social and cultural issues, and research on proof and proving. Proof and proving are rarely mentioned in discussions of social and cultural influences on teaching and learning, except occasionally in a stereotypical way presenting proof as a uniquely European/Western practice that supports an image of mathematics as authoritarian and culturally blind. It is equally the case that research into proof and proving rarely addresses social or cultural differences. Results from work with students from specific background in specific cultural contexts is often treated as if they apply generally.

This symposium will bring together educators from a range of cultural contexts with insights into practices in a wide range of milieus in order to promote exchange and communication. We seek to create a context for the acknowledgement of cultural diversity and different cultural, social and political contexts, beliefs and attitudes; the valuing of the teaching traditions of countries and cultural/social groups that have not, as yet, had any voice in education, and the promotion of the goal of "education for all".

Two central questions arise from consideration of the above issues:

- To what extent are practices advocated in one context are suited for other contexts?
- Should all students have access to instruction in mathematical reasoning, proof and proving?

That all students should have the opportunity to learn to prove is by no means universally acknowledged, and even when it is, the implications of such a goal are not fully explored. For example, access to instruction in proof and proving might look different for lower v higher achieving students, students from mono-cultural v multi-cultural backgrounds, students from academic v non-academic families and students classified as "gifted" or "non gifted". Given such differences, what does the claim for a need for universal access to instruction in proof and proving imply? What kind of goals and practices do we envisage? How might this conflict with other goals of mathematics education (e.g., emphasis on "real-world" mathematics problems and applications)? (see, e.g., Chenk Kai-Ming)

The information gathered and the discussions of the symposium
will be summarised after MES on a Web page, both to make the findings of
the symposium available to a larger audience, and as a focus for a network
of collaboration on this and related topics.

- Balacheff N. (1991) 'Benefits and limits of social interaction: The case of teaching mathematical proof', in Bishop A., Mellin-Olsen S., and Van Dormolen J. (Eds.) Mathematical knowledge : Its growth through teaching, Dordrecht : Kluwer Academic Publisher, pp. 175-192.
- Chenk Kai-Ming. 'Education and Development: the neglected mission of cross-cultural studies' in Alexander, R.; Osborn, M. and Phillips, D. (Eds.). Learning from Comparing, Vol. 2: Policy, Professionals and Development. Oxford, Symposium Books, 81-92.
- Keitel, C. & Kilpatrick, J. (1999) 'The Rationality and Irrationality of International Comparative Studies' in Kaiser, G.; Luna, E. ; Huntley, I. (Ed.). International Comparisons in Mathematics Education, Vol. 11. London, Falmer Press, pp. 241-256.
- Krummheuer G. (1995) 'The ethnography of argumentation', in Cobb P. and Bauersfeld H. (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures Hillsdale, NJ: Lawrence Erlbaum Associates, pp. 229-269.
- Sekiguchi, Y. and Miyazaki, M (2000) Argumentation and Mathematical Proof in Japan, International Newsletter on the Teaching and Learning of Proof, Jan/Fev. 2000. Online at http://www-leibniz.imag.fr/DIDACTIQUE/preuve/Newsletter/000102Theme/ 000102ThemeUK.html

Based on my observations of teachers in Canada, the purpose of proving most often given by teachers is to verify or justify a statement. In many cases however no purpose is given beyond stating that proving is part of mathematics. Most teachers I have spoken to are of the opinion that proving does not have a large role in mathematics classes, and that their students find proving very difficult.

Curricula in Canada are set on a provincial level, so approaches to proving vary across the country. Perhaps the largest differences are between the two largest provinces: Québec and Ontario. Most of my experiences with proving in Canada have been in the Atlantic provinces and Québec.

Curricula across Canada have been reformed recently, under the influence of the National Council of Mathematics Teachers' Standards documents of 1989-1995. Some effort has been made to extend proving to other areas of mathematics. For example in the Atlantic provinces 16 year olds still study inscribed angles in circles but are encouraged to prove theorems using co-ordinate geometry and transformations as well as traditional Euclidean methods.

Social and cultural influences in Canada are difficult to identify as there is more regional variation than there is national cohesion. Canadian culture is often defined in opposition to that of the US, but US influences are very strong in Canada. The English and French speaking communities are also influenced by the UK and France respectively. One feature of Canadian culture that may influence proof teaching is an emphasis on self-reliance. This would suggest that being able to independently verify mathematical statements without reference to authorities such as book or teachers would be valued, as it is. If this is a societal influence on proof teaching, then perhaps it is related to Canada's history as a colony far from Europe and sparsely populated. One would then expect this influence to be present also in the US, Australia and possibly South America.

Contributed by David Reid

The youngest students I observed in China were in grade 7 (aged about 13 years). They were proving statements about measures of segments and angles in Euclidean geometry. The teachers I spoke to indicated that their students enter grade 7 able to reason deductively, but unable to formulate their thinking and to express it mathematically. The only non-geometry class I observed (having asked to see classes where proving was involved) concerned factoring polynomials, where the proving amounted to algebraic manipulations to prove that specific polynomials had particular factors.

While the teacher gave verification as the reason for proving in mathematics, in class proving was usually used to explain given statements and occasionally to discover new facts about a geometric situation.

Social and cultural influences in China might include the authoritarian nature of the present and past governments of China, and a cultural emphasis on respect for elders and other authorities. Independent verification of statements would not be valued in such a context, but explanation might be. There is much room for discussion on this issue.

Contributed by David Reid

My observations of proving in French classes have mostly been in the Paris area, expanded by some further impressions in northern and south- eastern districts of France. Regionally I could not locate any substantial differences in proof teaching. Teachers explained to me that they respected and followed the national curriculum even if they personally disagreed with some of its goals or approaches. Nevertheless a variety of school books is used in the country, chosen collectively by the stuff of each school and reflecting regional preferences. The latest change of curricula have been from 1996 on for secondary level, laying down a cautious initiation to mathematical reasoning as preparatory to proving from the first year of secondary level, when students are about 11 years of age.

The national curriculum is compulsory not only for all French districts but for all schools in lower secondary as the "collège" is the exclusive school form for all students up to 14 years of age. In spite of all that differences in aspirations and teaching proving exist. For example, classes with students who learn Latin or German, or students of "bi-national" classes, studying some subjects in a language other than French, are regarded as classes of high achieving students. Due to the capability and high motivation of students in these classes differences in mathematics teaching also become obvious. Proving practices were on a high level compared to ordinary classes I have observed. As argumentation and justification is highly valued in upper secondary mathematics, demanding practices in this domain turn out to be an important preparation for later mathematical and academic success.

Contributed by Christine Knipping

Most of my experiences with proving in schools have been in the north- western states of Germany. As a consequence of the federal tradition of education in Germany approaches to proving vary across the country. No national curriculum defines a common standard, but centralised exams at the end of year 10 and 13 in some states are likely to homogenise teaching practices in these parts of Germany more than in other districts. Major educational changes in the last decade concerned in particular the states in the eastern part of Germany after the unification. Proving had been highly valued in former (national) curricula in East Germany, but this may have changed in light of the other changes to education in eastern Germany. Due to my lack of experiences with teaching practices in these states I am not qualified to remark on changes concerning proving in class, but I hope that someone with experiences in this field might be encouraged to add her/his perspective.

Important differences in curricula and proving practices are also to be found due to different types of schools in lower secondary. With regard to different professional careers students were traditionally decreasing focus on geometry in all school types has more and more devalued proving.

Contributed by Christine Knipping

In this framework, students in Greece experience proving for the first time in the 3rd grade of the lower secondary school when they are about 14 years of age. Initially, they are asked to verify simple statements in the context of algebra, where proofs cover statements concerning properties and relations of rational numbers and require primarily arguments involving quantitative transformations and algebraic manipulations.

Later on, and usually during the second semester of the same school-year, proofs are more formally introduced in the context of geometry. Proofs in geometry cover simple statements concerning congruent triangles, properties of transversals of parallel lines, angles and special lines in a triangle and other configurations, proportions of lines and congruencies of triangles, as well as simple trigonometric identities involving sine and cosine. Proving practices in this context rely exclusively on Euclidean methods and the purpose of proving is to verify or justify a given statement.

In Greece there exists a unique form of lower secondary school, which is compulsory for all students up to 14 years of age, a national curriculum compulsory for all schools in the country and the use of a single mathematics text-book for each school grade is obligatory. So substantial differences between schools in proof teaching may not be expected. However, according to my experience, clear differences in proof teaching and thus in proving practices may be traced in those school classes that are composed of highly motivated and high achieving students, usually of upper socio-economic backgrounds.

The teaching of proof in Greek secondary schools is, in my view, rather highly valued and a number of cultural and societal factors have in the past contributed to this fact. The prevailing image of the nature of mathematical knowledge and the mainstream conception of mathematics education as a medium for developing, and a field for refining, logical reasoning are two such factors, however closely related to a particular view of mathematics and its history.

Contributed by Dimitris Chassapis

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