The primary goal of advanced mathematics courses is to equip students with proof skills. That is why the competence of pre-service mathematics teachers in proof is considered an assessment of their performance. Despite the emphasis on mathematical proof in undergraduate education, students studying higher mathematics at university often face…

In this paper, we identify various forms of geometric work carried out by student teachers who were asked to perform a geometric task for the estimation of a land area. The theory of Mathematical Working Spaces is used to analyze and characterize the work produced. This study provides evidence that students developed forms of geometric work that…

This study was conducted to examine the effectiveness of proof tasks in the transition from conjecture to proof in Euclidean geometry on freshmen's proof schemes. In line with this aim, the proof schemes of the freshmen who performed conjecture-proof and theorem-proof tasks were compared. The freshmen were composed of 109 pre-service middle school…

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…

This study investigates the associations of spontaneous "dynamic gesture" and "transformational speech" with the production of "deductive proofs" in participants' reasoning about geometric conjectures (N=77). Although statistical analysis showed no significant association, the result suggests that purposefully…

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…

Increasing students' exposure to mathematical reasoning (Lannin, Ellis, & Elliot, 2011) has been a constant call from mathematics education reform efforts. However, studies have raised concerns about whether teachers are prepared to teach reasoning. This paper explores one aspect of reasoning, refutations, by examining the geometric…

The purpose of this study was to examine prospective teachers ' uses of diagrams and approaches to congruence while solving proof tasks. Eight prospective high school mathematics teachers were given three proof tasks to solve at the beginning and end of a mathematics education course. The course used the history of mathematics as the context for…

We perform dynamic investigation of two surprising geometrical properties, each of which involves additional properties. In the first task the property belongs to two regular polygons with the same number of sides and with one common vertex. It is found that all the straight lines that connect corresponding vertices of the two polygons intersect…

It is of critical importance, in particular, for mathematics teachers who will teach future generations to understand and do mathematical proofs. It is important to determine future teachers' beliefs about and difficulties with proofs because their knowledge of this issue affects their teaching. This study aims to determine and compare the proof…

The present paper describes a dynamic investigation of polygons obtained by reflecting an arbitrary point located inside or outside a given polygon through the midpoints of its sides. The activity was based on hypothesizing on the shape of the reflection polygon that would be obtained, testing the hypotheses using dynamic software, and finding a…

Grounded and embodied cognition (GEC) serves as a framework to investigate mathematical reasoning for proof (reasoning that is logical, operative, and general), insight (gist), and intuition (snap judgment). Geometry is the branch of mathematics concerned with generalizable properties of shape and space. Mathematics experts (N = 46) and nonexperts…

Representational competence is essential for the acquisition of conceptual understanding in physics. It enables the interpretation of diagrams, graphs, and mathematical equations, and relating these to one another as well as to observations and experimental outcomes. In this study, we present the initial validation of a newly developed…

Grounded and embodied cognition (GEC) serves as a framework to investigate mathematical reasoning for proof (reasoning that is logical, operative, and general), insight (gist), and intuition (snap judgment). Geometry is the branch of mathematics concerned with generalizable properties of shape and space. Mathematics experts (N = 46) and nonexperts…

Students use GeoGebra to explore the mathematical relations among different radii of circles in a triangle (circumcircle, incircle, excircles) and the sides and other segments in the triangle. The more formal mathematical development of the relations that follows the explorations is based on known geometrical properties, different formulas…