A 12-episode constructivist teaching experiment with two pairs of elementary preservice teachers was conducted to examine how they reason distributively and proportionally. Specifically, I studied how prospective teachers reason with ratios as multiplicative comparisons. Prospective teachers solved different problems in one specific context (Step…

We examine a hypothesis implied by Steffe's constructivist model of children's numerical reasoning: a child's "spontaneous" additive strategy may relate to a foundational form of multiplicative reasoning, termed multiplicative double counting (mDC). To this end, we mix quantitative and qualitative analyses of 31 fourth graders' responses…

Promoting an understanding of multiplication of fractions has proved difficult for mathematics educators. I discuss a research study aimed at developing a concept of multiplication that supports both multiplication of whole numbers and multiplication of fractions. The study demonstrates how domain-specific theories grounded in two major…

In this chapter, I propose a stance on learning fractions as multiplicative relations through reorganizing knowledge of whole numbers as a viable alternative to the Natural Number Bias (NNB) stance. Such an alternative, rooted in the constructivist theory of knowing and learning, provides a way forward in thinking about and carrying out…

Informed by a constructivist-based, student-adaptive pedagogical approach, this study explores the benefits of teacher-learner discourse moves for the mathematics learning of students with learning disabilities (LD). During a constructivist teaching experiment for nurturing the multiplicative reasoning and problem solving of five third-grade…

This study investigated the developmental influence of collaborative games, in the form of game-based worksheets, on the performance and attitudes of sixth-graders in the mathematics classroom. It is posited that games have the potential to enhance learning through positive emotional experiences. Non-digital game-based learning methodologies are…

As today's classrooms become more and more diverse, there is a growing need to explore the intersection between English Learners (ELs) and students with learning disabilities (LD) in the content-specific instruction of mathematics problem solving. The aim of this study was to determine which types of instructional scaffolds may be used by math…