This article discusses a set of tasks that introduce dilations using nonstandard figures, incorporating snippets of classroom conversations as students try to make sense of the specified tasks, to show how they progress in their understanding. The authors provide a theoretical foundation for the tasks and discuss the pedagogical implications for…

This paper discusses an interesting, classic problem that provides a nice classroom investigation for dynamic geometry, and which can easily be explained (proved) with transformation geometry. The deductive explanation (proof) provides insight into why it is true, leading to an immediate generalization, thus illustrating the discovery function of…

In this paper, we highlight examples from school mathematics in which invariance did not receive the attention it deserves. We describe how problems related to invariance stimulated the interest of both teachers and students. In school mathematics, invariance is of particular relevance in teaching and learning geometry. When permitted change…

In this "Delving Deeper" article, the author introduces the slip-slide method for solving Algebra 1 mathematics problems. This article compares the traditional method approach of trial and error to the slip-slide method of factoring. Tools that used to be taken for granted now make it possible to investigate relationships visually,…

This article describes a geometrical solution to a problem that is usually solved geometrically as an example of how alternative solutions may enrich the teaching and learning of mathematics. (Contains 11 figures.)

The authors present 3 experiments demonstrating ways to reduce illegal moves in problem-solving tasks. They propose a 3-stage framework for the rejection of illegal moves. An illegal move must come to mind and be selected, checked for legality, and correctly rejected. Illegal move reduction can occur at any stage. Control group participants…