**Project Directors: Celia
Hoyles, Lulu Healy**

This project ,funded by the ESRC, conducted a nationwide survey into the conceptions of proof amongst high-attaining year 10 students (aged 14-15 years) who have followed a statutory National Curriculum. It identified the characteristics of proof as recognized by the students, how they constructed proofs and the extent to which these productions fulfilled their own criteria.

Analysis of the survey showed that students were most aware of the verification and explanation functions of proof, with its role in systemization and discovery largely absent from student views. A major finding was that the arguments students considered would receive the best mark from their teacher were not the same as the arguments they would create themselves. For the former, algebraic arguments were popular in the number domain, and formal proofs in geometry but to a lesser extent. For the latter, students preferred arguments that they were better able to evaluate and which they found to be convincing, both criteria which tended to exclude algebra and deductive proofs. Pragmatic arguments predominated in students' proof constructions, although the majority were aware of their shortcomings. The students who were the most successful in constructing proofs presented their reasoning in everyday language, sometimes including examples and pictures.

Students' responses were also influenced by their individual competence in mathematics and their sex. Some curricular influences were identified: performance was better in algebra then geometry, displayed wider variation in response in geometry which is less-specified in the National Curriculum, and was clearly influenced by the data-driven investigations encouraged in the curriculum.

Following the survey, the project designed computer-integrated teaching experiments in these two areas of the mathematics curriculum using a Logo microworld in algebra and a dynamic geometry system in geometry. These were evaluated according their influence on students' conceptions of proof and the proving process.

Selected publications

The Curricular Shaping of
Students' Approaches to Proof, *For the Learning of
Mathematics*, 17, 1. pp.7-15. (February 1997)

Steering Between Skills and
Creativity: A Role for the Computer?*Proceedings of the First
ICMI-East Asia Regional conference on Mathematics Education*,
(Park H.S, Young M. Choe, Shin, H., Kim, S.H. (eds)) Korea, Aug.
1998, (Also translated into Korean), pp. 211-226,
pp227-242

Students Performance in
Proving: Competence or Curriculum? *Proceedings of First Conference
of the European Society for Research in Mathematics Education August
1998, Osnabruck, Germany*.

Students' Views of Proof,
*Mathematics in School, * May, pp. 19-21, 1999

Can they Prove it?,*
Mathematics in School, *1999 (in press)

Linking Informal Argumentation
with Formal Proof Through Computer-Integrated Teaching Experiments,
*Proceedings of the 23rd Conference of the International Group for
the Psychology of Mathematices Education,* Haifa, 1srael
1999

A Study of Proof conceptions in
Algebra,to be published in *Journal for Research in Mathematics
Education*, 1999