Interesting conservation properties that exist in parallelograms are well known, such as: diagonal intersection, pairs of interior triangles that overlap each other, and other properties. This article presents a dynamic study of choosing any point inside and outside the parallelogram and connecting it with two of the vertices of the parallelogram.…

In this paper we deal with a geometric machine that has gone somewhat unnoticed in mathematics education: the Perfect Compass. It allows conics to be drawn using a special two aperture compass, based on Apollonius' conic sections. Proposed in the 10th century, it seems that it was never built in medieval times--at least no archaeological remains…

Throughout the years of the history of the development of mathematics, starting with the ancient mathematicians, new things were discovered from time to time, which made it a central component of the computerized world, which serves us in many fields, with the young "swimming" in it, and the adults asking: how could we live without it!…

In high school geometry, students are expected to deepen their understanding of geometric shapes and their properties, as well as construct formal mathematical proofs of theorems and geometric relationships. The process of helping students learn to construct a geometric proof can be challenging given the multiple competencies involved (Cirillo…

We introduce the free web-app PyGgb -- the combination of the well-known programming language Python and the dynamic mathematics system GeoGebra. Motivated by the desire to provide an introduction to Python coding in the familiar context of mathematics and geometry, PyGgb allows GeoGebra constructions to be created using Python code. We outline…

The ability to solve geometric construction problems is justly regarded as an essential component of mathematical culture. The dynamic, general nature of objects provided by dynamic geometry systems allows the development of intuitive methods for solving construction problems. The core mathematical concept underlying this approach is the loci…

The complexity of the art of teaching and learning, especially in the field of mathematics, calls for a broader approach than the traditional status quote. Proof without words in geometry coupled with the dynamic and interactive GeoGebra applet have the capacity to be used in all levels of mathematics education from grade schools to higher…

This paper seeks to answer the question: How does the handling of the contradictions that emerge in activities of initial mathematics teacher education contribute to the learning of geometric knowledge when pre-service teachers and the teacher educator solve geometric construction problems with GeoGebra? A dialectical-materialist approach is used…

This article is a discovery-based instructional resource for faculty to use as a capstone course or exploratory project for undergraduates who are familiar with (but not necessarily fluent in) calculus, linear algebra, complex variables, and geometry. The explorations flow from a simple change in sign: in the complex numbers a + bi, where…

A primary goal of ethnomodeling is to value, sustain, and in some instances introduce cultural practices and traditions that have often been lost or overshadowed because of an emphasis on modern methods (Orey & Rosa, 2010). Through an ethnomodeling task focusing on geometric transformations, the authors share a Model It! activity to engage…

How might a 4-year-old pupil come to conceive a line, created by himself with a continuous gesture, as constituted by a series of points? In this paper, we theoretically introduce and discuss a design study of the educational path in which this possibility emerged. The path aimed at leading pupils to "experience," in the sense of the…

The study of mathematics is often described as the science of patterns (Resnik 1981), and the cognitive tasks involving the search, recognition, and assessment of patterns are fundamental to learning mathematics. Patterning activities lead students to a better understanding of dependent relations among representations, create a transparent way for…

For a century and a half, the scene of mathematics competitions underwent a remarkable transformation from isolated and geographically scattered events to a full-scale and a full-featured vibrant global ecosystem comprising an impressive variety of competitions, school students, university students, teachers, mentors, scientists, schools,…

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…

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…