This article considers why (-1)(-1) = +1 and how and why a teacher might go about explaining this concept to high school students without using pseudoreasoning. Furthermore, it provides a precise explanation, through the use of the distributive property, as to why (-1)(-1) = +1. (Contains 1 figure.)

Relatively low participation in the hard sciences (mathematics, science, engineering and technology) has become a concern with respect to the capacity of Australia to meet critical infrastructure projects. This problem has its roots in poor student attitudes towards and perceptions about the study of prerequisite subjects including mathematics and…

Jean Piaget (1970) defined "disequilibrium" as a conflict between new ideas and current conceptions. In this article, the author describes how introducing this concept to her first graders improved formative assessment results and the overall classroom climate during mathematics class. Students feel success in mathematics even without mastering a…

This article describes how children use an expressive microworld to articulate ideas about how to make a game seem fair with the use of randomness. Our aim in this study is to disentangle different flavours of fairness and to find out how children used each flavour to make sense of potentially complex behaviour. In order to achieve this, a spatial…

The study explores the nature of students' conceptual understanding of calculus. Twenty students of engineering were asked to reflect in writing on the meaning of the concepts of limit and integral. A sub-sample of four students was selected for subsequent interviews, which explored in detail the students' understandings of the two concepts.…

The present study examined children's implicit and explicit knowledge of linear and non-linear processes. Five- and nine-year-olds (N = 60) were asked to forecast linear and exponential growth by providing the corresponding number of beads. Implicit knowledge was assessed via the magnitudes of the forecasts; explicit knowledge was investigated…

In order to get undergraduates interested in mathematics, it is necessary to motivate them, give them good reasons to spend time on a subject that requires hard work, and, if possible, involve them in undergraduate research. This article discusses how cryptography can be used for all these purposes. In particular, a special topics course on…

This note presents geometric and physical interpretations of the sufficient condition for a critical point to be a strict relative extremum: f[subscript xx]f[subscript yy] - f[superscript 2][subscript xy] greater than 0. The role of the double derivative f[subscript xy] in this inequality will be highlighted in these interpretations. (Contains 14…

This qualitative study investigated young children's mathematical engagement in play with wooden unit blocks. Two boys, ages 6 and 7, were independently observed completing the task of filling outlined regions with the various sets of blocks. Three major mathematical actions were observed: categorizing geometric shapes, composing a larger shape…

The article describes the performance of several individual students in a college algebra/precalculus course that focuses on the development of conceptual understanding and the use of mathematical modeling and discusses the likely differences in outcome if the students took a traditional algebra-skills focused course.

The new EYFS (The Early Years Foundation Stage, DfES, 2007) reinforces that mathematical language and understanding should be developed through stories, songs, games and imaginative play; that practical activities should provide opportunities for children to practise and talk about their developing mathematical understanding. Despite a huge number…

Concept cartoons are cognitive drawings or "visual disagreements" that use a cartoon-style design to present mathematical conversations inside speech bubbles. The viewpoints portrayed are all different and it is this difference that acts as a catalyst for further conversations, as learners talk together to discuss their thinking. They make…

Fractal geometry offers teachers great flexibility: It can be adapted to the level of the audience or to time constraints. Although easily explained, fractal geometry leads to rich and interesting mathematical complexities. In this article, the authors describe fractal geometry, explain the process of iteration, and provide a sample exercise.…

Brian Griffiths (1927-2008) was a British mathematician and educator who served as a member of the founding editorial board of "Educational Studies in Mathematics". As a mathematician, Griffiths is remembered through his work on what continue to be known as "Griffiths-type" topological spaces. As a mathematics educator, his…

We describe the design of a simple homemade apparatus for the measurement of the refractive indices of liquids and demonstration of refraction. A circular transparent plastic tank and a lazy Susan are held concentrically. A laser pointer is mounted on the lazy Susan with its laser beam pointing radially through the centre of the plastic tank.…