Reflection on the theme of the paper makes one to remember Sir Snow's 1959 lecture on The Two Cultures in which Snow had advanced the thesis that the sciences and the humanities had become split into two cultures and had argued that this division had become a major handicap to solving the world's problems. Mathematics education itself has always…

This study aimed to examine whether a computational thinking (CT) intervention related to (a) number knowledge and arithmetic (b) algebra, and (c) geometry impacts students' learning performance in primary schools. To this end, a quasi-experimental, nonequivalent group design was employed, with 61 students assigned to the experimental group and 47…

This is an article about abstraction, generalization, and the beauty of mathematics. We claim that abstraction and generalization in of itself may very well be a beauty of the human mind. The fact that we humans continue to explore and expand mathematics is truly beautiful and remarkable. Many years ago, our ancestors understood that seven stones,…

The ability to solve mathematical problems has been an interesting research topic for several decades. Intuition is considered a part of higher-level thinking that can help improve mathematical problem-solving abilities. Although many studies have been conducted on mathematical problem-solving, research on intuition as a bridge in mathematical…

Analyzing and interpreting student thinking through written work is a key, often challenging, practice of teaching. It entails noticing students' mathematical thinking and drawing on mathematical knowledge for teaching. This study investigates how pre-service secondary teachers at the beginning of their preparation reason about students' written…

The purpose of this article is to present and discuss two recommended sequences of learning the areas of polygons, starting from the area of a rectangle. Exploring the algebraic derivations of the two sequences reveals that both are valid teaching progressions for introducing the area formula for various polygons. Further, it is suggested that…

In line with the latest positions of Gottlob Frege, this article puts forward the hypothesis that the cognitive bases of mathematics are geometric in nature. Starting from the geometry axioms of the "Elements" of Euclid, we introduce a geometric theory of proportions along the lines of the one introduced by Grassmann in…

Given two vectors u and v, their cross product u × v is a vector perpendicular to u and v. The motivation for this property, however, is never addressed. Here we show that the existence of the cross and dot products and the perpendicularity property follow from the concept of linear combination, which does not involve products of vectors. For our…

Maryam Mirzakhani is the first and the only female winner of the Fields Medal since its establishment in 1936. She is arguably one of the greatest mathematicians of our generation. This biographical paper outlines her life and work. Her mathematical theorems and noteworthy accomplishments are just as impressive as her determination, imagination,…

Students are faced with many transitions in their middle school mathematics classes. To build knowledge, skills, and confidence in the key areas of algebra and geometry, students often need to practice using numbers and polygons in a variety of contexts. Teachers also want students to explore ideas from probability and statistics. Teachers know…

Given three planes in space, a complete characterization of their intersection is provided. Special attention is paid to the case when the intersection set does not exist of one point only. Besides the vector cross product, the tool of generalized inverse of a matrix is used extensively.

The cubic polynomial with real coefficients has a rich and interesting history primarily associated with the endeavours of great mathematicians like del Ferro, Tartaglia, Cardano or Vieta who sought a solution for the roots (Katz, 1998; see Chapter 12.3: The Solution of the Cubic Equation). Suffice it to say that since the times of renaissance…

This paper utilizes concept mapping as a tool for conscious and deliberate knowledge building in mathematics and its extension to algorithms. Currently, alleged defects in mathematics education are obvious: instead of conceptual elaboration, everyday praxis relies on routine computations that are likely to lead into alienated concepts with weak…

The purpose of this paper is to present an analysis of the difficulties faced by students when working with different representations of vectors, planes and their intersections in R[superscript 3]. Duval's theoretical framework on semiotic representations is used to design a set of evaluating activities, and later to analyze student work. The…

Complex functions, generally feature some interesting peculiarities, seen as extensions of real functions. The visualization of complex functions properties usually requires the simultaneous visualization of two-dimensional spaces. The multiple Windows of GeoGebra, combined with its ability of algebraic computation with complex numbers, allow the…