Abstract

There has always been a tremendous and varied amount of work on problem-solving in mathematics education research. However, despite its variety, most if not all work in problem-solving shares similar epistemological assumptions about the fact that there is a problem to be solved and that solvers make an explicit selection of a strategy and apply it to solve the problem. Problem: Varela’s ideas about problem-posing provide a means of going beyond these assumptions about problem-solving processes. We propose to explain and illustrate the way we found inspiration from these ideas in our work through a discussion grounded in data excerpts collected in our research studies on mental mathematics. Method: Concrete data and observations are referred to for discussing issues related to problem-solving processes and activity in mathematics. Results: Engaging with Varela’s work led us to revisit and reformulate many common notions in relation to mathematical problem-solving, namely concerning the meaning of a problem and of a strategy, as well as the relationship between the posing and solving of a problem. Through this, these notions are conceived as dynamic in nature and co-constitutive of one another. This leads us to engage in what we call the dialectical relationship between posing and solving. Implications: We illustrate the sort of educational insights that might be drawn from such conceptualizations, mostly in terms of affecting the way we look at students’ productions and engagements in mathematics not as pre-fixed or pre-definable entities, but as activities that emerge in the midst of doing mathematics.