In this paper, a simple proof of the Morley's Trisector Theorem is presented which involves basic plane geometry only. The use of backward geometric approach, trigonometry or advanced mathematical techniques is not required. It is suitable for introducing to secondary or undergraduate students, as well as teachers or instructors for learning or…

This paper provides a case study account of a preservice secondary mathematics teacher's thinking while engaging in slope tasks using dynagraphs. The data included audio recordings and screen captures of a small group of preservice teachers engaging with these tasks, with our analysis focusing on the case of Robin. Despite familiarity using slope…

In this pedagogical action research study, we, as post-secondary mathematics teacher educators, built on an existing effort to improve pre-service teachers' mathematical vocabulary understandings by intentionally addressing their struggles related to polygonal area formulas. Utilizing cognitive load theory and Bruner's levels of developmental…

The paper focuses on students' understanding of the concepts of axial and central symmetries in a plane. Attention is paid to whether students of various ages identify a non-model of an axially symmetrical figure, know that a line segment has two axes of symmetry and a circle has an infinite number of symmetry axes, and are able to construct an…

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…

This research aimed to develop a definition maps-based plane geometry module to improve the student teachers' mathematical reasoning skills based on the level of Van Hiele. This development research was to produce a product in the form of a module at the college level. This research referred to the ADDIE model involving five stages, i.e. analyze,…

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…

A student, Stuart, related perimeter to pixels and the professor, Beth, moved back and forth between "reserved believing" and "reserved doubting" and "doubting" teacher actions (Elbow, 1986; Harkness & Noblitt, 2017) while assessing the merit of his conjecture in the moment. Video allowed the researchers to…

In our study, we looked for an answer to the research question "How prospective teachers comprehend the concept of a plane angle and what kind of variation there is between the conceptions? The study shows that the prospective teachers interpret the concept "plane angle" which in principle is familiar to everybody in many different…

To interpret in detail the meaning-making in the classroom and the corresponding teacher semiotic mediation, we have resorted to Peirce's triadic ign theory, interpreted by Sáenz-Ludlow and Zellweger. We present an example of the use of a few elements of that theory in the analysis of a classroom episode in which meaning is constructed with the…

How many rods does it take to brace a square in the plane? Once Martin Gardner's network of readers got their hands on it, it turned out to be fewer than Raphael Robinson, who originally posed the problem, thought. And who could have predicted the stunning solutions found subsequently for various generalizations of the problem?

This report summarises a study examining preservice elementary teachers' geometry content knowledge with a focus on parallelogram. It was a descriptive study examining 45 preservice teachers' reasoning in the process of determining which quadrilateral can be classified as a parallelogram. Findings have revealed that these elementary teachers do…

Research on naive physics and naive optics have shown that people hold surprising beliefs about everyday phenomena that are in contrast with what they see. In this article, we investigated what adults expect to be the field of view of a mirror from various viewpoints. The studies presented here confirm that humans have difficulty dealing with the…

The Department of Mathematics and Statistics at Utah State University offers two courses for preservice secondary school mathematics teachers that complement each other in working toward a vision for school mathematics in which "all students have access to high-quality, engaging mathematics instruction." Through these courses, students…

This note presents geometric and physical interpretations of the sufficient condition for a critical point to be a strict relative extremum: f[subscript xx]f[subscript yy] - f[superscript 2][subscript xy] greater than 0. The role of the double derivative f[subscript xy] in this inequality will be highlighted in these interpretations. (Contains 14…