We present an investigation of the infinite sequences of numbers formed by calculating the pairwise averages of three given numbers. The problem has an interesting geometric interpretation related to the sequence of triangles with equal perimeters which tend to an equilateral triangle. Investigative activities of the problem are carried out in…

In this study, we explore automated reasoning tools (ART) in geometry education and we argue that these tools are part of a wider, nascent ecosystem for computer-supported geometric reasoning. To provide some context, we set out to summarize the capabilities of ART in GeoGebra (GGb), and we discuss the first research proposals of its use in the…

An allegorical thought experiment occurring in a pseudo Huxleyean world in the future is conducted, in which "Euclidean" geometry has been forgotten and can only be retrieved by traversing backward in time, physically and virtually by two protagonists, Alpha and Beta, inspired by the challenge issued by their teacher Aleph 1. In the year…

Proof, as a central and integral part of mathematics, is an essential component of mathematical education and is considered as the basic procedure for revealing the truth of mathematical propositions and for teaching productive reasoning as part of human civilization. Is there therefore room for conjectures in mathematics? In this paper after…

The present paper describes beautiful conservation relations between areas formed by different geometrical shapes and area relations formed by algebraic functions. The conservation properties were investigated by students at an academic college of education using a computerized technological tool and were subsequently accompanied by justified…

Unprecedented investments are being made in mathematics tutoring interventions for K-12 students, but results from these interventions are not always promising. Traditional online or distance math tutoring can treat learning as disembodied, and not give learners access to embodied resources like gestures, movements, and actions. Virtual Reality…

I explore students' discourses in small groups working on mathematical problems using GeoGebra, focusing on the Cartesian connection between algebra and geometry. Specifically, the interest lies in what is internally persuasive for students in upper-secondary school (11th grade) with histories of low attainment. Three problem-solving episodes are…

We present an investigative activity that was set as part of a course of pre-service teachers of mathematics. The emphasis in the course was set on the importance of using the computerized technological tool for teaching the subject. The activity focused on investigating interesting geometrical conservation properties which are not known to the…

Access to body-based resources has been shown to augment cognitive processes, but not all movements equally aid reasoning. Interactive technologies, like dynamic geometry systems (DGS), potentially amplify the link between movement and geometric representation, thereby deepening students' understanding of geometric properties. This study…

This paper concerns the design of a specific computer-based educational environment fostering students' shift from argumentation to proof in geometry. In particular, we focus on the language difficulties that such a shift might present and on the need for suitable interventions to overcome them. In this respect, we designed a specific device,…

Pre-service mathematics teacher (PST) education often addresses within Geometry Classes how to utilize Dynamic Geometric Software (DGS). Other classes may also incorporate teaching pre-service teachers about the history of mathematics. Although research has documented the use of Dynamic Geometric Software (DGS) in teaching the history of…

This document introduces, describes and exemplifies the technical features of some recently implemented automated reasoning tools in the dynamic mathematics software GeoGebra. The new tools are based on symbolic computation algorithms, allowing the automatic and rigorous proving and discovery of theorems on constructed geometric figures. Some…

Dynamic geometry environment allows us to investigate geometric relationships and problems more efficiently than the pencil-and-paper method. Continuous variation of geometric configurations shows the relationship from different points of view. This paper aims to provide some indications of how a dynamic geometry environment, especially GeoGebra…

We explored transformations that teachers made to modify geometry proof problems into investigation problems and analyzed how these transformations differ in teachers who use a dynamic geometry environment (DGE) in their classes and those who do not. We devised a framework for the analysis of problem transformations and types of teacher-generated…

Proving in school geometry is not just about validating the truth of a claim. In the school setting, the main function of the proof is to convince someone that a claim is true by providing an explanation. Students consider proving to be difficult; in fact, they find the very concept of proof demanding. Proving a claim in planar geometry involves…