When Is a Problem? Contribution in Honour of Jeremy Kilpatrick

Abstract

Some of Jeremy Kilpatrick's contributions to mathematics education are used to contextualise the question of 'when is [something] a problem?' leading to different approaches to or value systems for engaging students in mathematical thinking. My own preferred approach is experiential and hence phenomenological, in which the reader is invited to compare their experience of working on some task exercises with my commentary. The product of such enquiry is possibly revised and honed task exercises which can be used to sensitise others to the same issues.

Peer assessment processes in a problem-solving activity with future teachers

“…Since the publication of Polya's (1945) book, How to solve it, problem-solving has become one of the main lines of research in mathematics education (e.g., Felmer et al, 2019;Liljedahl et al, 2016). The large amount and diversity of research has generated a range of meanings associated with the terms "problem" and "problemsolving" that sometimes detract from the clarity of the studies conducted (Mason, 2015). At the present time, there is a broad consensus that "problem" refers to a mathematical task that generates in someone a sense of problematicity, of not knowing a direct way to solve it and desire to find it (Mason, 2015;Schoenfeld, 1992).…”

“…The large amount and diversity of research has generated a range of meanings associated with the terms "problem" and "problemsolving" that sometimes detract from the clarity of the studies conducted (Mason, 2015). At the present time, there is a broad consensus that "problem" refers to a mathematical task that generates in someone a sense of problematicity, of not knowing a direct way to solve it and desire to find it (Mason, 2015;Schoenfeld, 1992). Thus, for a mathematical task to be a problem, it must have the potential to generate an intellectual challenge to the person trying to solve it, in a way that enhances its mathematical development, promotes its conceptual understanding and mathematical reasoning, and the competence to communicate mathematical ideas (Cai & Lester, 2010).…”