La lettre de la Preuve

ISSN 1292-8763

 

Proof in the curriculum: an international seminar

The Qualifications and Curriculum Authority (QCA) - the body with responsibility for the National Curriculum in England - hosted an international seminar on 'Reasoning, explanation and proof in school mathematics and their place in the intended curriculum'. The invitation seminar, held in Cambridge during October 2001, had delegates from 16 countries. The delegates numbered policy makers, researchers in mathematics education and pedagogy, teachers and academic mathematicians.

QCA convened the seminar to help clarify and justify the central positions accorded to algebra and geometry in the mathematics curriculum and to ask key questions about the significance of developing reasoning and proof in school mathematics.

The seminar included contributions about the role of reasoning and proof in the intended curricula of a range of different countries. The similarities and differences between the countries were explored, as were decisions underpinning the construction of the mathematics curricula in these counties.

Some highlights from the seminar are discussed below.

John Mason (UK) opened the conference with a challenging presentation which posed many questions and extracted significant themes from some of the seminar papers:

· What is the purpose of proof? Does the level of acceptable proof vary with this purpose?
· Where lies the warrant for the authority of proof?
· How do pupils move from 'because it just is' to convincing themselves and others (present or at a distance) that a particular mathematical statement is necessarily true?
· How often do pupils experience their own teachers using chains of deductive reasoning?
· How can mental imagery be developed to support the development of mathematical reasoning?
· Should proof be assessed? How can assessment be designed so that it does not become routine?
· Is algebraic proof of the same nature as geometric proof?
· Should proof be core or optional for all students?
· Does learning about mathematical proof help enhance one's ability to make informed decisions and so become a better citizen?
· Does the use of ICT undermine or enhance the role of reasoning and proof in mathematics? The technology increases the 'arithmetisation' of mathematics, yet algebra is needed to generalize. How can these tensions be reconciled?

Tibor Szalontai (Hungary) argued strongly that pupils should reason almost every time that they have a mathematics lesson. He emphasized the significance of extension and generalization. The importance of 'early algebra' in primary school helped to develop mathematical intuition, reasoning and notions of true or false.

Abraham Arcavi (Israel) discussed the importance of developing symbol sense; the significance of reading and manipulating algebraic expressions; the engineering of suitable examples and contexts to develop learning and to distinguish the different roles played by symbols. He stressed the necessity of intervention by the teachers and the creation of a classroom culture which promotes symbol sense through active participation and discussion and a shared sense of purpose towards the acquisition of mathematical knowledge. It was essential to challenge students to the limits of their capability (but not beyond) and to develop their patience so they do not give up if they cannot see an immediate way through a problem.

Kaye Stacey (Australia) argued the case for the use of new technology to both support and challenge the current curriculum, and to force a reassessment of the mathematical content and the value of that content. For the less able, in particular, it was important to develop realistic workplace skills and to remove the wall that abstract algebra tended to erect for such students. Technology, despite the fact that it highlights numerical mathematics, can (through the use of computer algebra systems, or CAS) give meaning to the more algebraic aspects of mathematics and allow more students to tackle complex and more realistic problems.

The theme of developing algebraic reasoning through the use of spreadsheets was further developed by Teresa Rojano (Mexico). But a key question remained: Did spreedsheets encourage integer arithmetic only?

James Kaput (USA) argued that CAS and other specialist software would change the nature of the mathematics that is taught in schools. The basic mathematical object of the 21st century would no longer be a simple entity like a linear equation, but a parametrised family of mathematical objects. The range of representational infrastructures is vastly enhanced by computer technologies and this will radically change what we think and how we notate mathematics and derive mathematical results.

Michele Artigue (France) described the current breakdown of the strong tradition of mathematical rationality in France and how the insistence of formal proof in geometry is now proving an obstacle to progress. The French are moving away from transformation geometry and are now looking at invariant properties associated with angle, or area or congruence. To develop an appropriate pedagogy, it was important to construct situations which optimize knowledge gained and help devolve responsibility down to student level. Proof must always begin where ther is some uncertainty. There has to be a progression from the student's 'private knowledge' to that accepted as 'institutional knowledge'; situations have to be engineered to ensure that such a transition takes place. Students need to progress from computations to strategies, then to conjecture, and then finally to proof. Dynamical geometry packages can play a central role in developing the necessary geometrical reasoning to enable students to make this progression.

The discussions were far ranging. Some themes that emerged included:

· why teach proof?
· the nature of proof itself;
· is the idea of proof more important, say, than the ability to handle data meaningfully?
· teacher expertise and continuing professional development;
· the role of discussion and explanation in classroom pedagogy;
· equity issues - should proof be taught to all students? Should there be differentiated curricula for different student need?
· the importance of valid and reliable assessment of reasoning and proof;
· the role of textbooks, in which 'proof' is often merely explanation;
· the role of computer technology;
· the importance of engineering suitable problems and contexts with the appropriate degree of challenge;
· the extent to which mathematics (and reasoning and proof in particular) contributes towards good citizenship;
· the importance of perseverance in mathematics;
· the social contact between student, teacher and learning institution.

© Jack Abramsky

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