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

Abstract

The importance of mathematical problem posing has been acknowledged by many researchers. In this theoretical paper, we want to capture different meanings and aspects of problem posing by approaching it from three different levels: (1) by comparing definitions, (2) by relating it to other constructs, and (3) by referring to research and teaching settings. The first level is an attempt to organise existing definitions of problem posing. The result of this analysis are five categories, which shows that there is no consensus regarding the conceptualisations of problem posing. In the second level, we examine how problem posing is conceived by the research community compared to other mathematical constructs, such as problem solving, mathematical creativity, or modelling. Finally, in the third level, we summarise possible ways of implementing problem posing in research and teaching settings as they are depicted in the relevant literature. Given this broad variance regarding the conceptualisations of problem posing, we attempt to provide some arguments as to whether there is a need for consensus on a commonly accepted concept of problem posing.

Problem Posing and Problem Solving in Primary School: Opportunities for the Development of Different Literacies

“…Problem solving has had a more pronounced emphasis in curricular guidelines; however, its relationship with problem posing is close and has been highlighted for a long time by several educators and researchers. Einstein and Infeld emphasized the importance of formulating problems when they stated that "the formulation of problems is often more essential than their resolution" [20] (p. 15). Despite not using the expression "problem posing", Polya highlights the relevance of encouraging students to analyze problems they have solved and to create new problems based on them, arguing that this practice helps them to consolidate their knowledge, develop their skills, improve the solving process, and understand the problems posed to them [21,22].…”

“…There are different categories for problem posing based on their intentionality or context: only with the intention of creating new problems (students' experiences, in or outside of school, are contexts that favor the formulation of new problems, both to respond to everyday needs and to interpret real-life phenomena); only with the intention of reformulating existing or proposed problems (it may occur on the initiative of students, when faced with a problem they find interesting, or be proposed to them by someone, such as the teacher); with both intentions, creating new problems and reformulating problems; with the intention of proposing questions and viewing already known issues from another angle (although it may seem similar to reformulating problems, in this case, the focus is on the questions posed in a problem and not on the problem as a whole, that is, creating new problems from questions of a problem); as an act of modeling (when approached as an opportunity to interpret and analyze reality through its mathematization) [20]. The authors emphasize that the previously mentioned categories are not necessarily disjoint, and the formulation of problems can be approached in contexts in which more than one of them prevails.…”

“…The presented study focuses on posing and solving problems in contexts close to the students' reality involving Financial Education and Consumer Education. When formulated problems, students were asked, based on a familiar scenario [20] and through a semi-structured situation [18] involving a hypothetical hypermarket leaflet, to formulate their own questions. In general, students carried out what had been asked of them without many difficulties.…”

The tradition of problem-posing in Hungarian mathematics teaching

“…According to Cai and Hwang (2020), problem-posing in mathematics education involves teachers and students conceiving (or reformulating) and conveying a problem or assignment in a particular pedagogical context. According to Papadopoulos et al (2021), there are five types of problem-posing methods: generating new problems only; reformulating existing or given problems only; generating and reformulating problems simultaneously; posing questions; and modeling. Despite the diversity, one of two concepts of problem-posing appears in most research papers (Baumanns & Rott, 2021).…”

“…According to Papadopoulos et al (2021), there are five types of problem-posing methods: generating new problems only; reformulating existing or given problems only; generating and reformulating problems simultaneously; posing questions; and modeling. Despite the diversity, one of two concepts of problem-posing appears in most research papers (Baumanns & Rott, 2021). According to the first concept, problem-posing includes developing a new problem or reformulating an existing problem before, during, or after problem-solving.…”