How do teachers’ approaches to geometric work relate to geometry students’ learning difficulties 教师的几何作业方法与几何学生学习困难有何关系？.pdf

Educ Stud Math (2011) 77:129– 147 DOI 10.1007/s10649-011-9304-7 How do teachers’ approaches to geometric work relate to geometry students’ learning difficulties? Alain Kuzniak Jean-Claude Rauscher Published online: 1 March 2011 # Springer Science+Business Media B.V. 2011 Abstract Various studies suggest that French students (grades 7 to 10) may solve geometric problems within a paradigmatic framework that differs from that assumed by teachers, a situation prone to misunderstandings. In this paper, we study the extent to which secondary school teachers recognise the conflicting paradigms and how they handle the geometric work conducted, in sometimes unintended ways, by their students. This is done by analysing teachers ’ reactions to specific answers students offered to the Charlotte and Marie problem, an “ambiguous” problem with various solutions depending on the paradigm adopted. As a result of the study, we found that, beyond similarities due to a shared mathematical background, the way secondary schoolteachers handle students ’ answers varies with their conceptions of geometric work. Implications are drawn regarding the teaching of geometry and the training of teachers. Keywords Geometry . Geometrical paradigms . Geometric work . Epistemic conception 1 The general purpose 1.1 On knowledge required for teaching mathematics Since the early 1970s, numerous studies carried out in several countries have evaluated teachers ’ mathematics subject knowledge based upon the common belief that the greater the subject knowledge, the better the teaching. In the case of geometry, mathematical subject knowledge among teachers appears uneven with a lot of gaps especially among elementary school teachers. Yet, the link between limitations in subject knowledge and the quality of the teaching is not easy to study. Identifying the influence of thes