This study examines preservice elementary teachers' (PTs) knowledge for teaching the associative property (AP) of multiplication. Results reveal that PTs hold a common misconception between the AP and commutative property (CP). Most PTs in our sample were unable to use concrete contexts (e.g., pictorial representations and word problems) to…

This paper explores how models can support productive thinking. For us a model is a "thing", a tool to help make sense of something. We restrict attention to specific models for whole-number multiplication, hence the wording of the title. They support evolving thinking in large measure through the ways their users redesign them. They assume new…

Using qualitative data collection and analyses techniques, we examined mathematical representations used by sixteen (N=16) teachers while teaching the concepts of converting among fractions, decimals, and percents. We also studied representational choices by their students (N=581). In addition to using geometric figures and manipulatives, teachers…

This paper looks at 21 fifth grade students as they discuss a linear graph in the Cartesian plane. The problem presented to students depicted a graph showing distance as a function of elapsed time for a person walking at a constant rate of 5 miles/h. The question asked students to consider how many more hours, after having already walked 4 h,…

In school, at least in the US, we were taught to multiply by hand according to a standard algorithm. Most people find that algorithm difficult to use, and many children fail to learn it. We propose a new way to make sense of this difficulty: to treat explicit computation as perceptually supported physical and mental action. Based on recent work in…

This article reports on the activity of two pairs of sixth grade students who participated in an 8-month teaching experiment that investigated the students' construction of fraction composition schemes. A fraction composition scheme consists of the operations and concepts used to determine, for example, the size of 1/3 of 1/5 of a whole in…

In line with current efforts to understand the piece-by-piece structure and articulation of children's mathematical concepts, this case study compares the reversibility schemes of two eighth-grade students. The aim of the study was to identify the mechanism through which students reverse their thought processes in a multiplicative situation. Data…

Although children partition by repeatedly halving easily and spontaneously as early as the age of 4, multiplicative thinking is difficult and develops over a long period in school. Given the apparently multiplicative character of repeated halving and doubling, it is natural to ask what role they might play in the development of multiplicative…