There is much to be learned and pondered by reading "Proofs and Refutations," by Imre Lakatos. It highlights the importance of mathematical definitions, and how definitions evolve to capture the essence of the object they are defining. It also provides an exhilarating encounter with the ups and downs of the mathematical reasoning process, where…

This article was written as a result of the author reading "MT177," a special issue dedicated to the teaching of "proof" in mathematics. He used the ideas in this special issue for planning his session "mathematical reasoning and proof," which was part of a weekend course for primary trainees. It consisted of three activities: (1) How many…

This article "opens up" the study of the relationship between quadrilaterals and circles to two sets of quadrilaterals--those that can be drawn either inside (cyclic) or outside (tangential) a circle (or both). Of course, "all" triangles are both cyclic and tangential. Or, to use the traditional Euclidean language for triangles, a unique…