This paper frames children's mathematics as mathematics. Specifically, it draws upon our knowledge of children's mathematics and applies it to understanding the prime number theorem. Elementary school arithmetic emphasizes two principal operations: addition and multiplication. Through their units coordination activity, children construct two…

This study enquires into the embodied processes of children in solving multiplication tasks, considering how such processes can expand access to spatial reasoning skills and simultaneously develop students' understanding of multiplication. The analysis focused on four Year 2 students as they completed two embodied tasks. The aim was to understand…

In this study I investigate Saldanha and Thompson's (1998) claim that conceptualizing a coordinate pair in the Cartesian coordinate system as a multiplicative object, a way to unite two quantities' values, supports students in conceptualizing graphs as emergent representations of how two quantities' values change together. I presented three…

An important decision that professional development (PD) facilitators must make when preparing for activities with teachers is to select an appropriate tool for the intended learning goals of the PD (Sztajn, Borko, & Smith, 2017). One important and prevalent tool is artifacts of student thinking (e.g. Jacobs & Philipp, 2004). In this paper…

When viewed through a lens of embedded cognition, algorithms may enable aspects of the cognitive work of multi-digit multiplication to be "offloaded" to the environmental structure created by an algorithm. This study analyses four multiplication algorithms by viewing different algorithms as enabling cognitive work to be distributed…

Counting problems have applications in probability and computer science, and they provide rich contexts for problem solving. Such problems are accessible to students, but subtleties can arise that make them surprisingly difficult to solve. In this paper, students' work on the Groups of Students problem is presented, and an important issue related…

We examined how a student with learning disabilities (SLD) in mathematics constructed a scheme for differentiating, selecting, and properly operating on/with units that constitute a multiplicative situation, namely, singletons ('1s') and composite units (abbreviated UDS). Conducted as part of a larger teaching experiment in a learning environment…

The purpose of this study was to document and trace how students' understanding of situations that involve finding a fractional part of a fraction developed during a course of instruction in which students explored and represented various kinds of situations using each of the five representational systems. A class of 5th grade students (n=22)…

This research focused on preservice elementary teachers' understanding and reconstruction of understanding about multiplication in the domain of fractions. At the start of the study, 44% of the 50 preservice teachers studied reported that they had a method for reasoning about multiplication with fractions, and 28% were able to describe a situation…

This paper studies the difficulties children have solving word problems that correspond to one of the situations modeled by multiplication; that of a ratio (e.g. "candies per box"), multiplied by an extensive quantity (e.g. "boxes") (I x E). Sixty children were individually observed solving and repeating I x E problems. Various…

A computer model of memory (based on input, retrieval, and reaction time) was applied to the rapid processing of simple arithmetic facts (addition, subtraction, and multiplication) by 18 educable retarded adolescents when compared with 18 nonretarded fourth graders (of equal mental age) and 18 normal adolescents. Results for addition indicated…