In agreement with the main tenets of the anthropological theory of the didactic (ATD), this study uncovers dependencies between what students can learn, the established curriculum and the current state of mathematicians' mathematics ('scholarly mathematics'). One main result is that the mathematics taught, too often taken for granted by curriculum…

In this article, the author will demonstrate how gestures--often used in combination with offloading information on scratch paper--can be used to help students with difficulties in mathematics succeed with secondary level algebra. The authors will provide key examples from three studies of the algebra learning of students with mathematics…

The mode of assessment is one of the most important factors influencing learning. E-assessment usually includes only checking the final answer, thus limiting teacher's ability to check the complete solution, and it does not allow inclusion of math proofs problems that constitute an important part of math content. The e-assessment module of Halomda…

This article revisits and expands upon a previous phenomenographic study characterising the qualitatively different ways in which South African undergraduate physics students may experience the use of +/- signs in one-dimensional kinematics (1DK). We find the original categorisation as applicable for interpreting Swedish university-level students'…

This article aims to analyze the process of the progressive mathematization of the concept of the function by secondary school students through the resolution of a free fall situation. A free-fall mathematical situation was designed and experimented using the Tracker software to obtain movement data. Worksheets and conversations were collected…

Prior research highlights a positive effect of incorrect worked examples on mathematics learning. Yet the mechanisms underlying these benefits are unclear. To investigate potential mechanisms of the benefits of various worked example types, we examined process data from a previously published classroom-based experiment. More specifically, we…

The mathematical representation ability is essential in solving mathematical problems, especially in algebraic expressions problems. Therefore, it is crucial to have a valid design of learning activities to support students' mathematical representation ability. Several previous studies claimed that realistic mathematics education is one of the…

This article provides insights into Grade 11 learner understanding of the roots of quadratic equations, how learners used solution strategies to find equations with given roots, how they determine the unknown values and the other root when one root is given and how they determine the unknown values when two roots of the equation are given. The…

Productive failure has shown positive effects on conceptual and transfer measures, but no clear effects on procedural measures. It is therefore an open question whether, and to what extent, productive failure methods may be used to enhance the learning of procedural skills. A typical productive failure study focuses on a single, complex concept;…

Realistic mathematics education (RME) has proven to be an effective model for mathematics elsewhere. However, students and teachers still grapple to confront the circle equation to learn and teachers to teach. This study explored students' didactical phenomenological discourses in the circle equation as an alternative. The mixed methods research…

In this paper I analyze a problem solving event in a secondary mathematics classroom. As the event unfolds, the teachers, including me, understand the event as involving interactions that were not related to learning. By adopting an expansive view of learning, I advance a different interpretation, that the interactions tell a story of student…

This article uses the concurrent mixed methods design to explore the errors made by 171 Grade Seven learners in algebraic problem-solving within the Assin Central Municipality in Ghana. The participants were categorized into low-achieving and high-achieving groups based on their performance in a pretest, to help provide a detailed examination of…

Generalisation is a skill that enables learners to acquire knowledge in general, and mathematical knowledge in particular. It is a core aspect of algebraic thinking and, in particular, of functional thinking, as a type of algebraic thinking. Introducing primary school children to functional thinking fosters their ability to generalise, explain and…

Conjectures are a key component of mathematical inquiry, a process in which the students raise conjectures, refute or dismiss some of them, and formulate additional ones. Taking a design-based research approach, we formulated a design principle for personal feedback in supporting the iterative process of conjecturing. We empirically explored the…

The present study examines the effectiveness of incorporating worked examples with prompts for self-explanation into a middle school math textbook. Algebra 1 students (N = 75) completed an equation-solving unit with textbooks either containing the original practice problems or in which a portion of those problems were converted into a combination…