The notion of "understanding", and its place in the learning process, is often placed in sharper focus when considered in the context of learning and teaching algebra. There are tensions between "doing" algebra, following algorithms, using "rules", or "tricks", and understanding "what you are doing". How does this impact the "way" that algebra is…

As with previous reports of activity at the annual Institute of Mathematics Pedagogy, this does not disappoint. The tasks used to support teaching and learning in mathematics can vary in so many ways, but this variation is compounded when learners begin to access the tasks. A task that really motivates and challenges one learner is likely to…

Algebraic notation has been bedazzling learners for generations, yet despite research, learned papers, comparative studies, and a great deal of hard work on the part of classroom practitioners the situation remains a stubborn constant. In this article some small-scale research has highlighted some issues that might find a resonance in many…

Unlike old perceptions of algebra as a subject that needs to be taught in the upper grade levels of schooling, much of the recent research reports that young students are capable of reasoning algebraically. It is important to note that the recommendation to include algebraic experiences in the early grades is not made simply to introduce typical…

This article reports on work undertaken by schools as part of Qualifications and Curriculum Authority's (QCA's) "Engaging mathematics for all learners" project. The goal was to use in the classroom, materials and approaches from a Royal Institution (Ri) Year 10 master-class, "Number Sense", which was inspired by examples from…

A method is given for generalizing Pascal's triangle to three dimensions. (DT)

A paper staircase drawn on a grid forms the basis for a discovery lesson in algebra. (DT)

The relationships between arithmetic, geometric, and harmonic means and the root mean square of two numbers are examined. The problem of generalizing these relationships to three variables is discussed. (SD)

Notes on divisibility using manipulative materials, cyclic groups of functions, Horner's method of synthetic division, arc lengths of cycloids and cardioids, number squares, and students' concepts of mathematics. (MM)

A series of puzzle-type activities combining arithmetic reasoning and geometric figures is described and illustrated. (SD)

These three brief articles concern problems students encounter in solving simple linear equations, children's conceptions of number, and the distinction between adding objects and adding numbers as illustrated by students' problems with Euler's formula. (SD)

Students examined patterns emerging when numbers placed at vertices in a braided pattern were summed. (SD)