The main aim of this paper is to propose a critical view of the application of Toulmin's model of argument to mathematics education, focusing on aspects that have strong teaching implications. It is claimed that Toulmin's description of arguments in mathematics, in particular as regards the relations between form and meaning, is misleading and out…

Problem solving is widely regarded as a fundamental feature within the school mathematics curriculum. However, there is considerable disagreement over what exactly problem solving is, and if and how it can be taught. In this article, I define problems as non-routine tasks and propose the explicit teaching of domain-specific problem-solving tactics…

The tradition of studies involving the combinatorial approach to recurring numerical sequences has accumulated a few decades of tradition, and several problems continue to attract the interest of mathematicians in several countries. This work specifically discusses the Fibonacci, Pell, and Jacobsthal sequences, focusing on Mersenne sequences. The…

We explore the various Mathematica functions for performing Gaussian elimination, specifically when the linear system contains input parameters. On the one hand, we want the solution, perhaps heuristically found, to be in terms of the input parameters, and as general as possible. On the other hand, we must adhere to all necessary conditions and…

This essay extends beyond the characteristics and discourse of word problems to, more generally, school mathematical problem-solving and the implications entailed by a predominant paper-and-pencil mode of learning and instruction since the modern era of education. Contrasting what I call "one-handed" (with paper-and-pencil) with…

Years of failed attempts to engage my mathematics teacher students in exploration and co-creation led me to radically change my teaching approach. Inspired by literature that uses the jazz improv metaphor for teaching, I derived three tenets based on principles from jazz improv. First, I would bring no more than a single problem-solving task to…

In research on mathematical problem posing, a broad spectrum of different situations is used to induce the activity of posing problems. This review aims at characterizing these so-called problem-posing situations by conducting three consecutive analyses: (1) By analyzing the openness of potential problem-posing situations, the concept of…

Even though there is a great temptation as teachers to share what is known, many are aware of an idea called "rules that expire" (RTE) and have realized the importance of avoiding them. There is evidence that students need to understand mathematical concepts and that merely presenting rules to carry out in a procedural and disconnected…

The research on mathematical argumentation has mainly adopted a dialectic lens which focuses on understanding the abstract and logical development of reasoning in argumentation. However, this approach may have overlooked other key aspects of mathematical argumentation, including the unfolding of the meaning-making experience and process during…

Evidence in South Africa of poor performance on word problems points to particular challenges among learners relating to difficulties with, on the one hand, making sense of mathematical word problems and, on the other, moving beyond inefficient counting based calculation methods. While many studies in mathematics education in South Africa have…

This article draws from Hiroko Kawaguchi Warshauer's conceptualisation of productive struggle, which capitalises on experiences that elicit struggle, that is, those that require students to 'expend intellectual effort' and that are productive, in the sense that they seek to 'advance thinking and deepen' understanding of mathematical concepts and…

A Prolog program consists of a set of facts and rules rather than imperative statements, commonly used in most other programming languages. Therefore, the Prolog language is used to encode logic, from which the inference engine deduces logical conclusions. In this article, we argue that the use of the Prolog language can be useful to help students…

The results of this study revealed students' solution strategies progressed from inefficient strategies e.g., counting on fingers to more efficient strategies with mathematical connections. These findings should enable educators to gain more insight to use Number Talks to help student develop computational fluency with numbers.

Tentativeness is often framed as a deficit, synonymous with timidity or a lack of confidence. In this article, I situate the notion of tentativeness in an enactivist framework and describe its role as both a strategy and affordance in a spatial visualization exercise. Drawing on insights from mathematics education and ecological psychology, I…

Lockwood has argued that taking a set-oriented perspective is critical for successful combinatorial enumeration. To date, however, the research literature has not yet captured the cognitive processes involved in taking such a perspective. In this theoretical paper, we elaborate the constructs of spatial structuring and spatial-numerical linked…