We present action research of a problem posed as part of a multi-participant national (Israeli) test checking the mathematical knowledge of high school students at the ages of 16-17, where some of those who solved this problem made an error by using the converse to a well-known theorem, where the converse is not true. In order to examine the…

In this paper, we consider mathematical competitions for pre-university students, such as the "International Mathematical Olympiad" (IMO) and many national and regional Olympiads following a similar model. The problems proposed in these contests must be solvable by 'elementary' methods (i.e., without using calculus) and belong…

In many mechanical engineering undergraduate curriculums, there are topics that are vital to the students' future careers in the manufacturing and design workforce that are not taught in-depth. As one of those topics, Geometric Dimensioning and Tolerancing, or GD&T, is vital to companies who develop and manufacture products because it allows…

Geometry, calculus and in particular integrals, are too often seen by young students as technical tools with no link to the reality. This fact generates a loss of interest in students with a consequent removal of motivation in the study of such topics and more widely in pursuing scientific curricula. With this note we put to the fore a simple…

Fermat's principle states that a light ray refracted across different media will traverse the fastest path as the physics for Snell's law. A geometric proof of Fermat's principle will be demonstrated as an intuitive approach to learn high school geometry and physical optics. It will be proved explicitly by showing that all alternative paths need…

Augmented Reality (AR) and Virtual Reality (VR) platforms now support shared, immersive experiences that enable people to directly physically and perceptually engage with mathematical objects, including shared objects. These new forms of AR/VR technology -- which we call "shared holographic AR/VR" (shAR) -- enable multiple learners to…

The recent increase in the number of mathematics museums has given rise to a need for tools with which to design and assess mathematics educational proposals in this non-formal context. This study proposes the use of the van Hiele model, a benchmark in mathematics instruction, for the design of interactive museum models focusing on mathematics…

The authors describe their approach to teaching a course on finite fields and combinatorial applications, including block designs and error-correcting codes, using a hybrid of lectures and active learning. Under the discussed classroom model, there are two lecture days and two discovery-based discussion days each week. Discussions center around…

Derek Hurrell shares a number of geometry tasks designed to tease out students' reasoning. Concrete materials are used to initiate discussions and develop a culture of reasoning in the mathematics classroom.

Many mathematics teachers and students are familiar with the typical "box problem." In this type of problem, one takes a rectangular (or a square) sheet of paper and cuts out four squares from the four corners of the sheet and then folds the four strips up to form a box. Math problems like this are seen in middle school, high school,…

The Coronavirus pandemic has thrown public schooling into crisis, trying to juggle shifting instructional modes: classrooms, online, home-schooling, student pods, hybrid and blends of these. This poses an urgent need to redesign curriculum using available technology to implement approaches that incorporate the findings of the learning sciences,…

In this paper, I discuss the potential transformations to teach K-12 geometry topics with the technology of 3D Pens, which enabled 3D models to be created instantly via one's moving hands. I draw on an example of using constructionist learning activities with 3D Pens to teach young children, 'how tall is the triangle?'

As children progress through the elementary grades, they are expected to begin to use attributes of shapes to name and classify them. When listening to children talk about shapes, it becomes clear that the process of learning to reason geometrically is complex. Recognizing the complex nature of students' geometric reasoning, the authors present in…

A solid ball placed on a rotating turntable is known to roll slowly around a circular path, at a speed 3.5 times slower than the turnable itself. If the ball is located in a straight track across a diameter of the turntable, then it accelerates rapidly to the edge. Both effects were filmed in slow motion using a video camera and a cake decoration…

This article offers a blueprint for how middle-school geometry and physical education units naturally intersect for authentic integration of both subjects. This disciplinary intersection is called the "Geometry Racket" and illustrates how mathematics knowledge provides and almost unfair advantage (i.e. racketeering) in net/wall games.