StudentsÕ understanding of proofs: a historical analysis and implications for the teaching of geometry and linear algebra

The process of observing and analyzing studentsÕ behaviors is interesting and complex but also unstable. It is unstable because it involves countless variables, many of which are uncontrollable. Despite this, what we learn from this process is useful, even essential, in designing and implementing mathematics curricula for both students and teachers. This presentation is about studentsÕ behaviors in relation to justi®cation and proof. Some of these behaviors are assumed to be due to faulty instruction in school; others seem to be unavoidable, in the sense that they are of human developmental nature. Analyzed from a historical perspective of mathematical development, these studentsÕ understandings of proof can be classi®ed into three categories: · Category 1: In this category, studentsÕ understandings of proof (viewed in relation to those of their instructors) seem to parallel the Greek conception of mathematics (viewed in relation to that of modern days). · Category 2: In this category, studentsÕ understandings of proof are reminiscent of the 16±17th century conception of mathematics. · Category 3: In this category, studentsÕ understandings of proof seem, to a large extent, to be a result of faulty instruction in the elementary and secondary schools. www.elsevier.com/locate/laa