Why Johnny struggles when familiar concepts are taken to a new mathematical domain: towards a polysemous approach

This article is concerned with cognitive aspects of students’ struggles in situations in which familiar concepts are reconsidered in a new mathematical domain. Examples of such cross-curricular concepts are divisibility in the domain of integers and in the domain of polynomials, multiplication in the domain of numbers and in the domain of vectors, and roots in the domain of reals and in the domain of complex numbers. The article introduces a polysemous approach for structuring students’ concept images in these situations. Post-exchanges from an online forum and excerpts from an interview were analyzed for illustrating the potential of the approach for indicating possible sources of students’ misconceptions and meta-ways of thinking that might make students aware of their mistakes.     

Minding mathematicians’ discourses in investigations of their feedback on students’ proofs: a case study

Educational Studies in Mathematics - This article presents a research apparatus for investigating and making sense of stories that emerge from feedback that mathematicians provide on...     

Dissecting success stories on mathematical problem posing: a case of the Billiard Task

“Success stories,” i.e., cases in which mathematical problems posed in a controlled setting are perceived by the problem posers or other individuals as interesting, cognitively demanding, or surprising, are essential for understanding the nature of problem posing. This paper analyzes two success stories that occurred with individuals of different mathematical backgrounds and experience in the context of a problem-posing task known from past research as the Billiard Task. The analysis focuses on understanding the ways the participants develop their initial ideas into problems they evaluate as interesting ones. Three common traits were inferred from the participants' problem-posing actions, despite individual differences. First, the participants relied on particular sets of prototypical problems, but strived to make new problems not too similar to the prototypes. Second, exploration and problem solving were involved in posing the most interesting problems. Third, the participants' problem posing involved similar stages: warming-up, searching for an interesting mathematical phenomenon, hiding the problem-posing process in the problem's formulation, and reviewing. The paper concludes with remarks about possible implications of the findings for research and practice.