In current teaching materials, when using Dedekind cuts to construct real numbers, the definition of a Dedekind cut is always involved in defining addition and multiplication. In this paper, as it is done in many current textbooks, Dedekind cuts are used to construct the set of real numbers. Then the order in it is defined, and the…

The algorithm for fraction multiplication is relatively easy to memorize and implement with accuracy. The simplicity of the algorithm masks the conceptual complexity involved in making sense of what fraction multiplication means in quantitative situations. One important interpretation of fraction multiplication involves fraction composition,…

In this article, I argue that the common practice across many school mathematics curricula of using a variety of different representations of number may diminish the coherence of mathematics for students. Instead, I advocate prioritising a single representation of number (the number line) and applying this repeatedly across diverse content areas.…

This study sought to use indigenous ("Adinkra") artefacts to present "Concreteness Fading" in multiplication of one-digit and one-digit numbers. The researcher used simple random sampling technique to select 51 participants from 300 student teachers. Two sets of tests were used to collect the data. The results were equally…

How can students achieve an understanding of multiplication that allows them to go beyond recall to explain their thinking? Author and mathematics education professor Juli Dixon introduces a program that teachers can seamlessly integrate into existing mathematics instruction. Learn six tactics to help you shift from an anxiety-producing,…

This article describes "Build It!--The Rectangle Game" task that uses the context of a game to develop mathematical generalizations based on strategy. The underlying mathematics in this game-based task is for students to discover factors and prime and composite numbers through 100. The playful use of "The Rectangle Game"…

When teaching the multiplicative domain in New Zealand primary schools, teachers tend to spend a greater proportion of time on the meaning and processes of multiplication, to the detriment of a specific focus on understanding the concept of division. When division is taught, it tends to be by reversing the context and turning the division…

This study examined a cohort of middle school pre-service mathematics teachers' understanding of the rate of change as they engaged in a model development sequence. By adopting a design-based research perspective, a model development sequence on the concept of rate of change has been designed and implemented as part of a mathematical modeling…

Structural reasoning is a combined ability to "look for structures, recognize structures, probe into structures, act upon structures, reason in terms of general structures, and see how a piece of knowledge acquired resolves a perturbation experienced" (Harel and Soto, 2017). The purpose of this study was to explore the cognitive…

The purpose of this mixed-methods study was to measure the impact of a sequential use of manipulatives and visual models on third-grade students' multiplication fluency, math attitudes, and fact retention. According to results of this study, there was significant evidence that the students who gained a conceptual understanding of multiplication…

This article explores the role that gestures play in the development of mathematical understanding. Using Pirie Kieren's (1994) notion of image making and Lunney Borden's (2011) idea of verbing mathematics, we share two examples of how students respond to teacher requests to demonstrate what they know about arrays.

Within a study of student reasoning in abstract algebra, we encountered the claim "division and multiplication are the same operation." What might prompt a student to make this claim? What kind of influence might believing it have on their mathematical development? We explored the philosophical roots of "sameness" claims to…

State standards include fluent use of standard computational algorithms. However, learning and using them require conceptual understanding of numbers and operations. Previous research using the concrete--representational--abstract (CRA) sequence has been effective in teaching computational algorithms to students at risk of and students with…

Multiplicative thinking underpins much of the mathematics learned beyond the middle primary years. As such, it needs to be understood conceptually to highlight the connections between its many aspects. This paper focuses on one such connection; that is how the array, place value partitioning and the distributive property of multiplication are…

In this study, based on the analysis of a teaching experiment with middle school students, we propose a framework for describing meanings of a point represented on a plane in terms of multiplicative objects in the context of graphing. We classify those meanings as representing: (1) non-multiplicative objects; (2) quantitative multiplicative…