Divisibility: A Problem Solving Approach Through Generalizing and Specializing

This paper describes a divisibility rule for any prime number as an engaging problem-solving activity for preservice secondary school mathematics teachers

Encounters With “Love and Math” A Belated Review of Edward Frenkel’s Love and Math: The Heart of Hidden Reality

In “Love and Math” the author intertwines his personal experiences as a student of mathematics and as a research mathematician with an exposition of modern mathematics, focusing on its elegance and beauty. Much is said in previous reviews (e.g., Grosholz, 2015) of this New York Times bestseller and winner of the Euler Book Prize from the Mathematical Association of America. All praise the author for his fascinating narratives in which he reveals the joy of intellectual discovery and provides a passionate account of exciting ideas of modern mathematics. There is no need in repetition, so this review is rather personal. I start with my personal encounters with “Love and Math” and then share what I learned from the book as a teacher and mathematics educator, focusing on issues that are often overlooked in acclaimed accounts of Frenkel’s mathematical expositions

On Convincing Power of Counterexamples

Despite plethora of research that attends to the convincing power of different types of proofs, research related to the convincing power of counterexamples is rather slim. In this paper we examine how prospective and practicing secondary school mathematics teachers respond to different types of counterexamples. The counterexamples were presented as products of students’ arguments, and the participants were asked to evaluate their correctness and comment on them. The counterexamples varied according to mathematical topic: algebra or geometry, and their explicitness. However, as we analyzed the data, we discovered that these distinctions were insufficient to explain why teachers accepted some counterexamples, but rejected others, with seemingly similar features. As we analyze the participants’ perceived transparency of different counterexamples, we employ various theoretical approaches that can advance our understanding of teachers’ conceptions of conviction with respect to counterexamples

Fooled by Rounding

For any convex quadrilateral, joining each vertex to the mid-point of the next-but-one edge in a clockwise direction produces an inner quadrilateral (as does doing so in a counter-clockwise direction). In many cases, a dynamic geometry measurement of the ratio of the area of the outer quadrilateral to the area of the inner one appears to be 5:1. It turns out, however, that this is due to rounding. We generalise the construction by replacing mid-points by more general ratios, finding the maximum and minimum values of the area ratio and determining the conditions on the original quadrilateral that achieve those two extremes

Do we need to insist on REAL numbers?

Do we need to insist on real numbers? We present our discussion of this question in the style of duoethnography

Definitional ambiguity: a case of continuous function

Definitions are an integral aspect of mathematics. In particular, they form the backbone of deductive reasoning and facilitate precision in mathematical communication. However, when an agreed-upon definition is not established, its ability to serve these purposes can be called into question. While ambiguity can be productive, the existence of multiple non-equivalent definitions for the same term can make the truth value of certain mathematical statements unclear. In this study, we asked mathematics educators to determine the truth of a definitionally ambiguous mathematical claim. Based on their responses, we identified several factors that influenced the teachers’ choice of definitions. Finally, we consider the pedagogical implications of employing such a task in teacher preparation programs