The paper focuses exclusively on the famous geometry lesson given by Socrates to the slave-boy in Plato's Meno, providing an in-depth analysis that emphasizes the pedagogical aspects of the lesson. This approach allows for an examination of the lesson that teachers, educators and students alike--regardless of their interest in the philosophical…

Transitioning from material to theoretical geometry is often considered a significant challenge in compulsory education. Over the last twenty years, research rooted in the Theory of Didactical Situations (TDS) has explored how material geometry can support the teaching and learning of theoretical geometry, promoting continuity in teaching. These…

It is well known that there is a unique circle passing through three specified points if and only if they are non-collinear. We show that there is also a unique circle passing through the endpoints of diameters of three specified circles if and only if they have non-collinear centres.

We introduce the concept of the "sequence of the ratios of convex quadrilaterals," identify some properties of these sequences and use them to provide new characterizations for some classic quadrilateral families. The research involves aspects of geometry, arithmetic and mathematical analysis, which converge to produce the results.

This paper focuses on the importance of identifying students' Prior Knowledge (PK) for the teaching and learning of mathematics. Specifically, we present a framework that exemplifies the core elements teachers need to consider in relation to the purpose, design and implementation of learning experiences that enable PK to be activated and assessed.…

It's often useful extending students beyond the limiting geometry of triangles and quadrilaterals to regularly consider generalizations of results for triangles and quadrilaterals to higher order polygons. A brief heuristic description is given here of the author applying this strategy, and which led to an interesting result related to the…

Mulla Sadra's philosophy has an ontological approach. However, his explicit texts or philosophical foundations in social issues such as education and especially the curriculum and its relationship with learning methods can be extracted. Mulla Sadra's different approach to learning has distinguished his educational system from other educational…

Today math teachers can best reach their students and show them how math surrounds us by using manipulatives, children's literature, and GeoGebra while teaching mathematics. These are some of the best pedagogical practices for teaching mathematics today. In our high-tech world, students need to be proficient in science, technology, engineering,…

Grade 8 students were given a task that had pairs of shapes that had been transformed (i.e., preimages and images). Students were to identify which of the three rigid transformations--reflections, rotations, or translations--were used to map the preimage to the image. Students were also asked to give a justification for their choice of…

Eye-tracking (ET) method provides a promising channel for educational researchers to connect learning outcomes to cognitive processes. The main principle of ET is that our gaze and our focus of attention are connected. Due to the advent of digital technologies, eye tracking studies are increasingly growing in different fields and in mathematics…

This paper highlights the pedagogical importance of gaps in mathematical proofs to foster students' learning of proofs. We use the notion of 'gap-filling' (Perry & Sternberg, 1986) from literary theory to analyze a task based on a Proof Without Words, which epitomizes the notion of gaps. We demonstrate how students fill in gaps in this…

One goal of inquiry-oriented instruction is student engagement with others' mathematical ideas. This paper analyzes a relatively short episode in which students engaged with others' ideas; the instructor facilitated engagement in order to support students in making mathematical progress. Students expressed some bafflement pertaining to the…

The need to develop consistent theoretical frameworks for the teaching and learning of discrete mathematics, specifically of graph theory, has attracted the attention of the researchers in mathematics education. Responding to this demand, the scope of the Van Hiele model has been extended to the field of graphs through a proposal of four levels of…

Continuing our critique of the classical derivation of the formula for the area of a disk, we focus on the limiting processes in geometry. Evidence suggests that intuitive approaches in arguing about infinity, when geometric configurations are involved, are inadequate, and could easily lead to erroneous conclusions. We expose weaknesses and…

The purpose of this work is to explore alternative geometric pedagogical perspectives concerning justifications to 'fast' multiplication algorithms in a way that fosters opportunities for skill and understanding within younger, or less algebraically inclined, learners. Drawing on a visual strategy to justify these algorithms creates pedagogical…