While agreeing with van der Linden (this issue) that test equating needs better theoretical underpinnings, my comments criticize several aspects of his article. His examples are, for the most part, worthless; he does not use well-established terminology correctly; his view of 100 years of attempts to give a theoretical basis for equating is…

In this article, the authors talk about transformation geometry being treated as little more than a set of tricks rather than as a mathematically rigorous topic. This appears to lead to pupils seeing little point in studying "reflections, rotations and translations" as other than examinable items in some future test. Following the argument…

Students learn more when they attempt to make sense of a mathematical situation they face. For example, a question like why square root of 25 is not + or - 5. Providing the intermediate steps and the reasoning of a technique with graphs can help make better sense of mathematics.

Analysis of students' work on tasks relating to reflections, translations and rotations is discussed. Findings from research with regard to students' understanding of a subject would help teachers prepare for the class.

In this paper it is argued that the characterization of mathematical justifications as inductive or deductive is incomplete. Promoting in classrooms the development of and discourse about a third type of reasoning, transformational reasoning, which considers the results of operations on a set of objects, may contribute to mathematics learning in…

This essay concentrates on those aspects of music that can be classified as purely mathematical or musical form, and is restricted to secondary-level concepts. (MP)

Presents a qualitative and global method of graphing functions that involves transformations of the graph of a known function in the cartesian coordinate system referred to as graphic operators. Explains how the method has been taught to students and some comments about the results obtained. (MDH)

The LOGO subroutine turtle geometry and Euclidean geometry are compared with respect to their treatment of similarity and difference of plane figures. The problem and its proposed solution of introducing a FLIP operation are viewed briefly from the perspectives of mathematics, computer science, and education. (MDH)

Describes one teacher's reflection concerning the quest to develop an understanding of school mathematics that promotes and sustains students' opportunities for exploration and conjecture. Recounts that a particular student's exploration of the features of parabolas eventually led to an understanding of the quadratic formula precisely because of…

Argues that Turtle Geometry has a great deal to offer mathematics teaching and learning. Suggests that it helps develop general problem-solving strategies and encourages new approaches to teaching and learning. (PK)

Presents 13 examples in which the intuitive approach to solve the problem is often misleading. Presents analysis of these problems for five different sources of misleading intuitive generators: lack of analysis, unbalanced perception, improper analogy, improper generalization, and misuse of symmetry. (MDH)

This paper considers how the algebra curriculum in secondary mathematics might be reformulated by rethinking both the content of algebra and the approaches to teaching and learning algebraic concepts. It considers how the concept of function can be made a central theme of the algebra curriculum and further suggests that computer software, when…