Isomorphism and homomorphism appear throughout abstract algebra, yet how algebraists characterize these concepts, especially homomorphism, remains understudied. Based on interviews with nine research-active mathematicians, we highlight new sameness-based conceptual metaphors and three new clusters of metaphors: sameness/formal definition, changing…

Instruction in many STEM domains heavily relies on visual representations, such as graphs, figures, and diagrams. However, students who lack representational competencies do not benefit from these visual representations. Therefore, students must learn not only content knowledge but also representational competencies. Further, as learning…

Over half a century has passed since Bruner suggested his three-stage enactive-iconic-symbolic model of instruction. In more recent research, predominantly in educational psychology, Bruner's model has been reformulated into the theory of instruction known as concreteness fading (CF). In a recent constructivist teaching experiment investigating…

Despite the critical role of teachers in the educational process, few advanced learning technologies have been developed to support teacher-instruction or professional development. This lack of support is particularly acute for middle school math teachers, where only 37% felt well prepared to scaffold instruction to address the needs of diverse…

In this report, we present an analysis of two prospective secondary mathematics teachers' generalizing actions in quantitative contexts. Specifically, we draw from a teaching experiment to report how Lydia and Emma engaged in different generalizing processes for the same task. Based on these differences, we found Lydia's generalizing actions…

Learning from expository science texts is challenging. These studies explore whether difficulties can be attributed to poor memory or poor reasoning. To eliminate the need for memory during testing, some students took the tests with the texts available. To test for the effects of reasoning on performance, some students were prompted to engage in…

As a multifaceted concept, the learning of angle concepts takes years to achieve and is beset with challenges. This paper explores how the processes of constructing and validating a learning progression in geometric reasoning can be used to generate targeted teaching advice to support the learning of angle concept. Data from 1090 Year 4 to Year 10…

Diagrams are powerful opportunities for grappling with and learning abstract relationships, for example learning the relations between elements in an ecosystem rather than simply memorizing the objects within the system. Further, what is crucial from any diagrammatic learning opportunity is the ability to use this relational knowledge in a new…

Physical phenomena and relations can be described mathematically by functional dependencies. These are represented in the form of tables, graphs, algebraic expressions or verbal descriptions. Only the combination of different representations enables students to experience a physical phenomenon or concept in a holistic way. Therefore, students have…

The capacity to recognise, represent, and reason about relationships between different quantities, that is, to think multiplicatively, has long been recognised as critical to success in school mathematics in the middle years and beyond. Building on recent research that found a strong link between multiplicative thinking and algebraic, geometrical,…

One of the challenges that commencing university students in computing degree programs face is the difficulty in engaging with the abstract and complicated theories in the computing discipline. In particular, it is hard for beginner computer architecture students to visualise the link between the theory of digital logic and the behaviour of the…

Conventional coordinate systems are often considered representational tools for reasoning about mathematical concepts. However, researchers have shown that students experience persistent difficulties as they engage in graphing activity. Using examples from research and textbooks, we present a framework based on a conceptual analysis of the use of…

The development of computational thinking is as important as writing, reading and arithmetic, and, it should start as early as kindergarten (Wing, 2008). However, little has been done in terms of investigating the factors influencing the development of computational thinking in preschool education (Bers et al., 2014; Ching et al., 2018; Kazakoff…

We present study findings that depict the natural and fractional number knowledge of one third grade student with learning disabilities (LDs) in seven experimental sessions. We utilize qualitative analysis methods to illustrate how this student evidenced her knowledge of natural number and fractions through her interactions with varied learning…

The positive correlation between spatial ability and mathematical ability has been well-documented, but not well-understood. Examining student work in spatial situations that require numerical operations provides us with insight into this elusive connection. Drawing on student work with angle, length, volume, and area, we examine the ways in which…