Representations are used throughout school mathematics for students to both think through a problem and communicate their ideas. Students also often come to mathematics class with a repertoire of representations. This article presents the ideas of different and multiple representations with conceptual connections as their underlying distinguishing…

Understanding the concept of area requires an understanding of the relationship between geometry and multiplication. The multiplicative reasoning required to find the areas of regular figures is used in many courses in elementary mathematical education. This paper explores various methods in which multiplicative reasoning is incorporated into the…

Developing and supporting understanding of the meaning of multiplication and multiplicative relationships in students with mathematics learning disabilities requires carefully designed instruction that emphasizes strategic representation use. This article discusses three ways in which teachers can incorporate multiple representations within…

This article introduces unit transformation graphs (UTGs) as a tool for diagramming the ways students use sequences of mental actions to solve mathematical tasks. We report findings from a study in which we identified patterns in the ways preservice elementary school teachers relied on working memory to coordinate mental actions when operating in…

Math facts are foundational for later math skill development, but many students demonstrate difficulties with math facts in early schooling. Drill-based interventions have been shown to effectively target foundational academic skills; however, there is limited research examining drill-based interventions for teaching math facts compared to basic…

This study presents the results of a series of design experiments that aimed to engage twelve fourth-grade students in mathematical activity exploring the volume of right prisms and cylinders as a dynamic sweep of a surface through a height, an approach that is referred to as Dynamic Measurement for Volume (DYME-V). This article describes this…

Progressing from additive to multiplicative thinking is a key outcome of school mathematics, making ratios an essential topic of study in junior secondary. In this study, 15 Australian Year 8 students were administered a ratio test followed by semi-structured interviews to explore their conceptions of ratio prior to formal instruction. In this…

In November 2022, ChatGPT, an Artificial Intelligence (AI) large language model (LLM) capable of generating human-like responses, was launched. ChatGPT has a variety of promising applications in education, such as using it as thought-partner in generating curricular resources. However, scholars also recognize that the use of ChatGPT raises…

The authors have found that having students learn accessible standard algorithms by explaining them using mathematics drawings increases students' sense of place--value numbers and enables students to articulate their understanding of what is actually happening with the numbers and why. In this article, they will discuss three standard algorithms…

Responsive teaching requires teachers to have knowledge of children's thinking. Prospective teachers need support in developing such knowledge because they often enter teacher preparation programs with fragile mathematical knowledge and anxiety about teaching mathematics to children. We designed a learning trajectory-based intervention to support…

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…

A one-on-one dialogue-based mathematics intelligent tutoring system (ITS) for learning multiplication and division of fractions was developed and evaluated in this study. This system could identify students' error types and misconceptions in real-time by using a block-based matching method. The adaptive dialogue-based instruction was supported by…

This study aims to develop a conceptual framework regarding practitioner knowledge and describe how one mathematics teacher generates her practitioner knowledge based on the framework. The characteristic of this framework is a combination of static and dynamic aspects of knowledge based on the teacher's perception and interpretation. The research…

The purpose of the present article is to investigate the definition of matrix multiplication as a central issue in linear algebra courses. Applying both historical and pedagogical approaches, it focuses on the philosophy of generating the usual matrix multiplication, as a special binary operation, with its partly unexpected form compared with the…

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…