Backward transfer refers to the influence on prior knowledge of the acquisition and generalisation of new knowledge. Studies of backward transfer of mathematical knowledge have focused on content that is closely related in time and in curricular sequencing. Employing the notion of thickening understanding, we describe instances of transfer that…

Cognitive load theory's incorporation of evolutionary perspectives has generated several instructional designs based on movement, including the tracing effect, occurring when learners benefit from explicit instructions to trace out specific elements of lesson materials with the index finger. Historical descriptions of children's tracing behaviours…

Every student has mathematical strengths beyond knowing basic facts, solving problems quickly, or showing work clearly. In this article, the author presents Smiles as an "on-ramp" task that supports students working together by unveiling and leveraging mathematical strengths. Nielsen describes on-ramp mathematics tasks as scaffolds that…

The COVID-19 pandemic has affected student learning; according to the National Assessment of Educational Progress (2022), scores for students in the United States have shown a decline in mathematics. . One way educators can strengthen students' current mathematical knowledge and bridge to new content is through Number Talks. Number Talks are brief…

In this article, the author describes strategies to support students in conjecturing in ways that promote their agency. Conjecturing involves justifying (i.e., a practice that promotes epistemic agency and engages students in mathematical reasoning to defend their claims; Bieda & Staples, 2020) by which a conjecture becomes proven, modified,…

This paper provides an overview of the research landscape for the kind of professional learning needed for project-based learning to succeed. The evidence is clear that such professional learning must ensure teachers are active, self-directed learners who receive support that is meaningful and relevant to their experiences and context. The authors…

Problem-based learning (PBL) is an innovative educational approach that dates back to the 1960s. However, the twenty-first century goal of sustainable education poses a challenge to PBL, especially as it relates to isolation. Here we discuss the underlying issue of isolation in three respects. First, the information-processing model of PBL depends…

Students enter classrooms equipped with a rich foundation of skills and knowledge from their everyday experiences. High-quality science education builds on students' wealth of experience in making sense of the world. In science classrooms, "sensemaking" is a collaborative practice that includes students sharing initial ideas with peers…

The transition to tertiary mathematics requires students to use definitions of mathematical objects instead of intuitions. However, routines of defining and of proving with definitions are difficult to engage in, as they are not familiar to students who come from secondary school mathematics. Defining is highly complex because of its underlying…

This article begins by describing a class scenario in which a teacher gives her students the L problem (Watanabe 2008), which involves finding the area of a composite figure. Her students inappropriately extend the area formula for rectangles (A = L x W) to composite figures by multiplying all given dimensions. After class, the teacher concluded…

A detailed characteristic of teaching and learning approaches used within the new concept of Learning by Being (LBB) is given. The evolution of educational paradigms from Learning by Doing (LBD) and Learning by Understanding (LBU) toward LBB is analyzed. The basic idea of LBB is students' ownership on cognitive goals, or the assumption of learning…

One of the foundational topics in first-year algebra concerns the concept of factoring. This article discusses an alternative strategy for factoring quadratics of the form ax[superscript 2] + bx + c, known as "factoring for roots." This strategy enables students to extend the knowledge they used when the leading coefficient was 1 and…

When presented with a problem in mathematics class, students often function as problem performers rather than problem solvers (Rigelman 2007). That is, rather than understanding the problem, students focus on using an operation to complete it. Students' tendencies to act as problem performers can prevent them from suggesting problem-solving…

This chapter focuses on the interaction of two first graders as they attempt to make sense of a particular instructional context for learning negative numbers. The context is one where they move an elevator to a building's floors above and below ground in order to model integer addition and subtraction problems. In particular, the focus of the…

Why are math modeling problems the source of such frustration for students and teachers? The conceptual understanding that students have when engaging with a math modeling problem varies greatly. They need opportunities to make their own assumptions and design the mathematics to fit these assumptions (CCSSI 2010). Making these assumptions is part…