Mathematics educators and researchers have advocated for the use of manipulatives to teach mathematics for decades. The purpose of this article is to provide illustrative uses of a readily available manipulative rather than a complete list. From an Australian perspective, Pop-it fidget toys can be used across the mathematics curriculum. This paper…

We address the question: How can a student's conceptual transition, from attending only to singleton units (1s) given in multiplicative situations to distinguishing composite units made of such 1s, be explained? We analyze a case study of one fourth grader (Adam, a pseudonym) during the course of a video recorded cognitive interview. Adam's case…

While previous studies mainly focused on children's additive and multiplicative reasoning abilities, we studied third to sixth graders' "preference" for additive or multiplicative relations. This was investigated by means of schematic problems that were "open" to both types of relations, namely arrow schemes containing three…

This study explored the impact of numbers used in instructional tasks on the construction and generalization of multiplicative reasoning by fourth grade students designated as having learning difficulties or disabilities in mathematics (SLDs). In particular, this study addressed the following research questions: (1) In what ways do SLDs'…

Theoretical analysis of whole number-based calculation strategies and digit-based algorithms for multi-digit multiplication and division reveals that strategy use includes two kinds of reasoning: reasoning about the relations between numbers and reasoning about the relations between operations. In contrast, algorithms aim to reduce the necessary…

In contrast to previous studies on Spontaneous Focusing on Quantitative Relations (SFOR), the present study investigated not only the "extent" to which children focus on (multiplicative) quantitative relations, but also the "nature" of children's quantitative focus (i.e., the types of quantitative relations that children focus…

This article reports on a study in which researchers asked children to "make up as story and a picture about marbles for this number sentence: 3 x 5 = 15." Students in this study came from pre - dominantly low- to average-income families living in three distinct geographical areas within the United States. A similar division task was…

Algorithms and representations have been an important aspect of the work of mathematics, especially for understanding concepts and communicating ideas about concepts and mathematical relationships. They have played a key role in various mathematics standards documents, including the Common Core State Standards for Mathematics. However, there have…

In this case study with Devin (pseudonym), which was part of a larger, constructivist teaching experiment with students identified as having learning difficulties in mathematics, we examine how a fourth grader constructed a dual anticipation involved in monitoring when to start and when to stop the simultaneous count of composite units (numbers…

The Common Core State Standards (2010) involve the demonstration of conceptual knowledge of numbers and operations. For students who struggle with mathematics and have not responded to instruction, it is important that interventions emphasize this understanding. In order to address conceptual understanding of numbers and operations in meeting the…

The maturation of multiplicative thinking is key to student progress in middle school as rational number, ratio, and proportion concepts are encountered. But many students arrive from the intermediate grades and falter in developing this essential disposition. Elementary students have historically learned multiplication and division as operation…