The concept of determinant plays a central role in many linear algebra concepts and is also applied to other branches of mathematics and science. In this study, we focus on the application of determinant and inverse matrix concepts, in solving systems of equations by a group of 116 in-service mathematics teachers who were studying the topic at a…

In many mathematics curricula, the notion of limit is introduced three times: the limit of a sequence, the limit of a function at a point and the limit of a function at infinity. Despite the use of very similar symbols, few connections between these notions are made explicitly and few papers in the large literature on student understanding of…

A model solution to a proof question on an examination is explored and subjected to a detailed analysis in terms of Toulmin's scheme of argumentation. In doing so, the ways in which the scheme has been variously used in the mathematics education and philosophical literature are contrasted. The analysis raises a number of issues concerning the…

The Mathematics education community has long recognized the importance of diagrams in the solution of mathematical problems. Particularly, it is stated that diagrams facilitate the solution of mathematical problems because they represent problems' structure and information (Novick & Hurley, 2001; Diezmann, 2005). Novick and Hurley were the first…

This paper introduces an applied problem-solving task, set in the context of cryptography and embedded in a network of computer-based tools. This designed learning environment engaged students in a series of collaborative problem-solving activities intended to introduce the topic of functions through a set of linked representations. In a…

The goal of the study reported here is to gain a better understanding of the role of belief systems in the approach phase to mathematical problem solving. Two students of high academic performance were selected based on a previous exploratory study of 61 students 12-13 years old. In this study we identified different types of approaches to…

We examine whether the array representation can support children's understanding and reasoning in multiplication. To begin, we define what we mean by understanding and reasoning. We adopt a "representational-reasoning" model of understanding, where understanding is seen as connections being made between mental representations of concepts, with…

This study examined how selected school mathematics textbooks in China, Singapore, and USA at the lower secondary grade level represent problem-solving procedures. The analysis of problem-solving procedures was carried out in two layers--general strategies, which adopted Polya's four-stage problem-solving model, and specific strategies, which…

The functions of a mathematician in industry are analyzed and suggestions are given concerning the mathematical training that would be appropriate. (DT)

Problem solving is presented as a resource for creating curriculum through reflections on intuition such as: exploring available intuitions, making explicit the intuitions used, and using intuition to generalize. (MP)

Data from interviews with pupils who attempted to solve problems involving the appropriate weighing and combining of individual means into an overall mean is reported. For many, dealing with the mean seems to be a computational rather than a conceptual act, with many subjects unable to correctly carry out the task. (MP)

Elementary combinatorial problems may be classified into three different combinatorial models: (1) selection; (2) partition; and (3) distribution. The main goal of this research was to determine the effect of the implicit combinatorial model on pupils' combinatorial reasoning before and after instruction. Gives an analysis of variance of the…

Reports on a study that focused on three teachers and their ways of teaching problem solving. Findings indicate that the participants unconsciously constructed personal metaphors that became the basis of their conceptualization of problems. "Community", "adventure" and "game" were the key metaphors. The outcome…

Analyzes the notion of structuring resources and their role in shaping mathematical activity. Examines ways in which nurses conceptualize ratio and proportion problems and provides a case study of two nurses discussing conflicting strategies when monitoring the fluid balance of a patient. Draws out some implications for the design of viable…

Reports on an intensive study of problem-solving activity of female students at the senior secondary level. Focuses on both the mathematical processing and the underlying cognitive and metacognitive activities that led to that processing. Concludes that all successful groups displayed a high number of key points where metacognitive decisions could…