This article explores a conceptual relationship between learner choice and mathematical sense-making. It argues that when learners can exercise choice in their mathematical activities, mathematical sense-making can be enhanced. The literature around mathematical modelling suggests a link between sense-making and learner choice. A three-tiered…

Research indicates that learning about teaching methods is not the same as learning to enact teaching methods. In our mathematics and science teacher education programs, we employ a practice-based pedagogy approach requiring novice teachers to learn, practice, enact, and reflect on their teaching. This approach draws on work highlighting core…

Given two vectors u and v, their cross product u × v is a vector perpendicular to u and v. The motivation for this property, however, is never addressed. Here we show that the existence of the cross and dot products and the perpendicularity property follow from the concept of linear combination, which does not involve products of vectors. For our…

We aim to open up discussion about the intertwined roles of teachers and tasks that involve students communicating about mathematics when working in groups. Over many years we have observed, researched and ourselves have taught students working on mathematics in groups and find that it is often easier to pay attention to the forms of communication…

The emerging field of mathematics educational neuroscience provides researchers with new approaches to understanding mathematical development, as well mathematics itself. This paper focuses on the role of the hand in constructing mathematics through activity. We rely on Piaget's distinction of three kinds of activity: sensorimotor activity,…

Persistent disconnects within and among education research, practice, and policy are limiting the reach of professional mathematics teacher communities, one of the most promising levers for humanizing mathematics teaching and learning in schools. An overarching goal of this commentary is to convince the field of mathematics education to broaden…

While many semiotic and cognitive studies on learning mathematics have focused primarily on students, this study focuses mainly on teachers, by seeking to bring to light their awareness of the semiotic and cognitive aspects of learning mathematics. The aim is to highlight the degree of awareness that teachers show about: (1) the distinction…

Whether using yoga as a physical activity break in a math class or measuring postural alignments and shapes in the gym, yoga and mathematics can complement each other. This article focuses on elementary school standards and outcomes of physical education and math, to help teachers use yoga as a math literacy tool.

Action-Process-Object-Schema (APOS) theory and tools resulting from dialogue with the Anthropological Theory of the Didactic (ATD) were used to analyse data from semi-structured interviews and teaching materials to study students' understanding of the relationship between tangent planes and the total differential. Results of the study show…

The Mathematics Scan (M-Scan), a content-specific observational measure, was utilized to examine the extent to which "standards-based mathematics teaching practices" were present in three focal lessons. While previous studies have provided evidence of validity of the inferences drawn from M-Scan data, no prior work has investigated the…

This article describes two heuristic strategies for problem solving: the "use of false assumption" strategy and the "use of double false assumption" strategy. Both of these strategies have their roots deep in history. In this article, we define these two strategies and illustrate their use in a problem developed by Frances…

This article investigates the potential of utilizing technology to apply Lesh's representations for teaching mathematical concepts. Lesh's model includes written symbols, verbal symbols, real life situations, pictorial representations, and concrete representations. The author suggests that, with the wealth of technological applications, teachers…

Although algebraic reasoning has been considered as an important factor influencing students' mathematical performance, many students struggle to build concrete algebraic reasoning. Metacognitive training has been regarded as one effective method to develop students' algebraic reasoning; however, there are no published meta-analyses that include…

This study investigated the impact of analyzing student work on preservice teachers' knowledge and beliefs about mathematics teaching. Forty-two prospective teachers participated. Data were collected using established instruments from the Integrating Mathematics and Pedagogy (IMAP) project and from the Learning Mathematics for Teaching (LMT)…

Finding intersections, unions, and complements of sets is an essential issue in elementary mathematics. It builds the foundation for set theory, probability, logic, and other topics. It is commonly recognized that drawing a Venn diagram, which was first introduced by the British philosopher and mathematician John Venn in 1881, is a classic and…