Multiple Approaches to Problem Posing: Theoretical Considerations Regarding its Definition, Conceptualisation, and Implementation

TL;DR

This theoretical paper organizes the diverse literature on mathematical problem posing across three levels: definitions, relations to other constructs, and implementation in research and teaching. It argues that the field lacks consensus on a definition and that researchers should make their own conceptualizations transparent rather than necessarily converging on one.

Key findings

• Analysis of existing definitions yielded five categories of problem posing: (1) only generating new problems, (2) only re-formulating given problems, (3) both generating and/or re-formulating, (4) raising questions, and (5) an act of modelling; these categories are not necessarily disjunctive.
• Problem posing is conceived by the research community either as an autonomous construct (the goal of an activity) or as interwoven with problem solving, creative mathematical thinking (CMT), and modelling.
• Three main ways of implementing problem posing were identified: as a tool for teacher training (enhancing subject didactic competence), as a pedagogical/educational tool, and as a diagnostic/assessment tool.
• The authors note that empirical research on connections to problem solving or CMT focuses mainly on the products (the problems posed), with a lack of research evaluating the process of problem posing.

Methods

• Theoretical/narrative synthesis (not a full systematic literature review) drawing on representative literature compiled by two research groups working on problem posing.
• Organized around three research questions: how existing definitions can be categorized, how problem posing is conceived relative to other constructs, and possible ways of implementing it in research and teaching.
• The theoretical considerations within each subsection are the product of an inductive category development in a qualitative content analysis (Mayring, 2014), analyzing definitions with regard to their content-related key aspects.

Limitations

• The authors state they do not provide a full systematic literature review but rather present and discuss a broad spectrum of literature, aiming to stimulate reflection rather than propose irrefutable answers; they note a future systematic literature review may provide more valid insights.
• The authors acknowledge the five definitional categories are not necessarily disjunctive and that it is not their goal to provide distinct categories.
• Evident from the text: the paper is a conceptual synthesis without empirical data of its own, so its claims rest on the selection and interpretation of cited literature.