How should our students think about external forcing in differential equations setting, and how can we help them gain intuition? To address this question, we share a variety of problems and projects that explore the dynamics of the undamped forced spring-mass system. We provide a sequence of discovery-based exercises that foster physical and…

Attention to mathematical reasoning in curriculum standards is part of an international trend, but identifying and understanding reasoning continues to challenge teachers. We report on one component of an Australia-wide initiative supporting teachers to implement innovative pedagogies. This paper contains insights from design research that focused…

Perseverance is vital for the process of learning mathematics with understanding. To do so, students must wrestle with obstacles and amend their problem-solving plan when necessary. I collected qualitative data to investigate the nature of perseverance for 11 ninth-grade algebra students working on a challenging task for which they affirmed…

The formal acceptance of a mathematical proof is based on its logical correctness but, from a cognitive point of view, this form of acceptance is not always naturally associated with the feeling that the proof has necessarily proved the statement. This is the case, in particular, for proof by contradiction in geometry, which can be linked to a…

The mathematical domain of fraction division continues to be an area of great difficulty for many teachers and preservice teachers (Lo and Luo, Journal for Mathematics Teacher Education 15:481-500, 2012; Newton, American Educational Research Journal 45:1080-1110, 2008; Rizvi and Lawson, International Education Journal 8(2):377-392, 2007; Young and…

In this article, we question the prevalent assumption that teaching and learning mathematics should always entail movement from the concrete to the abstract. Such a view leads to reported difficulties in students moving from manipulatives and models to more symbolic work, moves that many students never make, with all the implications this has for…

The Three-Square Puzzle shows a remarkable relationship between three angles. What happens when the number of squares increases? This article explores that question and brings in Fibonacci and Lucas sequences.

Explaining the complexity of mathematics teaching to others can be challenging. The Knowledge Quartet, a pedagogical content knowledge (PCK) framework from the UK, provides a fresh lens to reflect on mathematics classroom practice. Developed from videotaped mathematics lessons taught by final year pre-service teachers, the Knowledge Quartet…

Adding Digital Technologies curriculum content into primary classrooms may feel, to many primary teachers, like a new burden and more work--but many aspects of computational thinking can be explored in the context of our existing mathematics programmes. This article highlights the connection between computational thinking and mathematics, and…

The empirical law of large numbers is an important content in secondary school mathematics. Tasks used to analyze students' understanding of this law are often based on the hospital problem, but vary in various features, leading to mixed and conflicting empirical results. To identify task features that support students when approaching this type…

All students need to discuss mathematics to develop and deepen understanding. However, for English Learners (ELs), peer dialogue is imperative and indispensable to conceptual understanding as they participate in mathematical practices and engage in increasingly sophisticated uses of language (Heritage, Walqui, and Linquanti 2015). As ELs share…

We recognise that though teachers may participate in various forms of professional development (PD) programmes, learning that they may have gained in the PD may not always lead to corresponding perceivable changes in their classroom teaching. We offer a theoretical re-orientation towards this issue by introducing a construct we term…

A core aim of contemporary science, technology, engineering, and mathematics (STEM) education is the development of robust problem-solving skills. This can be achieved by fostering both discipline knowledge expertise and general cognitive abilities associated with problem solving. One of the most important cognitive abilities in STEM education is…

The purpose of this mixed-methods study was to explore how undergraduate students in an applied calculus course used the written curriculum (i.e., textbook, online homework system, and previous exams). The study design included three phases: (a) an online survey, (b) individual observations, and (c) student interviews. Students' survey responses…

In this paper, we analyse how N = 31 in-service teachers study the question that could generate a Study and Research Path (SRP): How does a parabolic antenna work? We also consider how they design lessons based on this question, by adopting the pedagogy of questioning the world according to the Anthropological Theory of the Didactic (ATD). The…