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,…

Existing studies have quantitatively evidenced the relatedness between problem posing and problem solving, as well as the magnitude of this relationship. However, the nature and features of this relationship need further qualitative exploration. This paper focuses on exploring the interactions, i.e., mutual effects and supports, between problem…

"Attribute" and "measurement" are two fundamental concepts in mathematics and physics. Teaching these concepts is essential even in elementary school, but numerous studies have pointed out pupils' difficulties with them. These studies emphasized that pupils must learn about attributes before being taught how to measure these…

Inequalities are one of the foundational subjects in high school math curricula, but there is a lack of academic research into how students learn certain types of inequalities. This article fills part of the research gap by presenting the findings of a study that examined high school students' methods of approaching absolute value inequalities,…

In this paper we describe two episodes of instructional interaction, in which examples are used in order to help students face their misconceptions. We introduce the notions of pivotal example and bridging example and highlight their role in creating and resolving a cognitive conflict. We suggest that the convincing power of counterexamples…

With our conceptualization of Harre and van Langenhove's (1999) positioning theory, we draw attention to immanent experience and read transcendent discursive practices through the moment of interaction. We use a series of spatial images as metaphors to analyze the way positioning is conceptualized in current mathematics education literature and…

When we consider the gap between mathematics at elementary and secondary levels, and given the logical nature of mathematics at the latter level, it can be seen as important that the aspects of children's logical development in the upper grades in elementary school be clarified. In this study we focus on the teaching and learning of "division with…

Three teaching experiments are reported which study aspects of a diagnostic teaching methodology. An experiment in the field of directional quantities showed a positive relationship between the intensity of discussion and amount of learning; one on fractions and another on geometric reflections showed good two-month retention under the…

Sketches a theory for designing teaching based on mathematical activity, situations, tasks, and interventions, exposing and resolving cognitive conflicts, changes of structure and context, feedback, reflection and review. Reviews the main psychological principles underlying this theory, then considers some examples of teaching designs in light of…