Christine Knipping, Universität Hamburg
David A Reid, Acadia University


Two aims will guide the work of the symposium:


While mathematical reasoning is given a central role in mathematics education worldwide there exist substantial differences in curricula and educational practices across and within countries. For example, in Germany there are different curricula for different types of schools and not all curricula include proof and the teaching of proof. In parts of Canada the same curriculum applies to everyone, but lower achieving students are not expected to prove. In France there is just one curriculum for all students; but differences exist in French schools concerning the role, the form and importance of mathematical reasoning and proof. Such differences are apparent whenever education systems in different cultural contexts are compared.

 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:

Differences between cultures raise the question to the extent to which practices advocated in one context are suited for other contexts. For example, teaching of proof in France and the US has emphasised argumentation in a "debate" format (Balacheff, 1991;Krummheuer, 1995), but this format seems to be unsuitable in contexts like Japan were open disagreement is discouraged (Sekiguchi & Mayazaki). This symposium will provide an opportunity for discussion of the relation of the role of proving in mathematics to larger cultural contexts.

 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)

How the symposium will be conducted

The symposium is planned to occur over two sessions. Prior to MES information will be shared by the organisers and prospective participants about the teaching of proving in contexts with which they are familiar. This will be done via the Web. Early in 2002 the organisers will call for contributions and provide examples of short written descriptions of practices with which they are acquainted. From this information a summary of differences in practices will be generated and the sessions at MES will open with exploration of the reasoning behind these differences. This in turn will provide a basis for discussion of the extent to which expectations for the teaching of proving can be extended across cultures, and the implications this has for comparisons of achievement and movement in people between cultures.

 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.


Call for contributions

To prepare for the symposium, the organisers invite prospective participants to share short descriptions about the teaching of proving in contexts with which they are familiar. Examples appear below. Contributions may describe practices in contexts not yet included, or offer other perspectives on the contexts described by previous contributions. From this information a summary of differences in practices will be generated and the sessions at MES will open with exploration of the reasoning behind these differences. This in turn will provide a basis for discussion of the extent to which expectations for the teaching of proving can be extended across cultures, and the implications this has for comparisons of achievement and movement in people between cultures. Contributions can be sent to David Reid at

Short descriptions of practices in different contexts

Proving in Canada

Most students in Canada encounter proving in the context of Euclidean geometry, when they are about 15 years of age. The study of geometry is one of several topics in each year of schooling. In their first encounter with proving students are asked to prove simple consequences concerning conditions of congruency of triangles, and properties of transversals of parallel lines. They are likely to see a proof of the Pythagorean theorem but unlikely to be asked to prove it themselves. The following year they prove statements concerning inscribed angles in circles and derive trigonometric identities, a task which is described as proving, but which involves little more than algebraic manipulations.

 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

Proving in China

In 1997 I observed teachers in three schools in Beijing. This hardly qualifies me to comment on proving in China, but I hope the following comments will inspire someone with wider experiences to add their perspective.

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

Proving in France

Students in France, at least those in my comparative study, experience proving in the context of geometry when they are about 13 years of age. In my classroom observations I found students being encouraged to prove statements in form of justifying their solutions of geometric problems. Proofs presented by the teacher function as a model showing how to come to a mathematical conclusion. Mathematical reasoning is not performed primarily to validate a statement, but rather to connect it via sound references to already studied mathematical concepts and theorems. Proofs are usually written in a variety of formats that emphasise this function of proofs. Proofs in lower secondary cover statements concerning properties of transversals of parallel lines, angles in triangles and other configurations, trigonometric identities, triangles and quadrilaterals inscribed in a circle, and special lines in a triangle. Students are likely to see a proof of the triangle midsegment theorem (theorème de milieux) and the Pythagorean Theorem but unlikely to be asked to prove these theorems themselves. Further, transformations are already introduced in lower secondary and play an important role within proving besides Euclidean methods.

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

Proving in Germany

Most students in Germany come into contact with proving in the context of geometry, when they are about 14 years of age. In my experience students are unlikely to be asked to prove statements themselves. Instead, they encounter proofs presented by the teacher or produced through Socratic questioning. In general the teacher's purpose is not only to demonstrate a theorem, but to motivate it and to help the students to understand the statement. In class proving processes, comprising the development and proving of a statement, can go on for more than one lesson, whereas written proofs are rarely carried out. Proofs in lower secondary cover carefully chosen theorems, of which the Pythagorean Theorem is probably considered the most important. Students might also see proofs in the context of particular lines in a triangle, congruencies of triangles, theorems about similarities or trigonometric identities.

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

Proving in Greece

One of the main objectives of teaching mathematics in lower secondary schools in Greece is “to initiate students in proving processes and to help them realize that proving is a useful tool for the verification of general principles”.

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

[top] [MES] [Comments to David Reid at]