This study examines the behavior of individuals when two intuitive rules in their minds lead to two different outcomes concerning a problem. In the absence of the formal knowledge, intuitive rules can affect the mathematical thinking. Hence, studies have usually compared the correct answer of the related formal knowledge with the incorrect one…

We implemented GeoGebra software in Rwandan secondary schools to check its effectiveness during teaching and learning geometry concepts. The quasi-experimental design was used, and four schools were purposefully selected. Two schools were from Northern Province, while the other two were selected from Kigali city. A geometry-based test composed of…

"Representational transformation diagrams" are used to compare and contrast standard textbook presentations of vector line integrals in undergraduate courses in both mathematics and physics. These presentations are taken as the lower anchor in a learning trajectory. Two principal approaches in the lower division are identified, roughly…

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…

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…

In this paper, we introduce a technology-enhanced pedagogical sequence for supporting lower secondary school students' sense-making of the concept of volume in a non-procedural and non-formula-driven way. Specifically, we illustrate a novel approach of using dynamic geometric environment (DGE) to introduce the meaning of volume and then deriving…

Geometrography, "alias the art of geometric constructions" was proposed by Emile Lemoine between the late 1800s and the early 1900s. It consisted originally of a system to measure the complexity of ruler-and-compass geometric constructions, capable of: designate every geometric construction by a pair of values that manifests its…

Much like one's experiences with the wonders of the world, concrete and multiperspective representations of mathematical concepts are key to understanding. Ideally, "learning environments should give students the opportunity to get to know different representations of a mathematical object in order for them to develop an appropriate…

Language can alter our mental conceptions of space, time, and categories. While there is compelling evidence that thought can be shaped by syntactic, morphological, and lexical features of a language, less is known about the impact of phonology on thought. This dissertation uses novel objects (alien cartoon figures) and pseudoword names in three…

The regular pentagon had a symbolic meaning in the Pythagorean and Platonic philosophies and a subsequent important role in Western thought, appearing also in arts and architecture. A property of regular pentagons, which was probably discovered by the Pythagoreans, is that the ratio between the diagonal and the side of these pentagons is equal to…

This paper concerns "planes-coordination" and "long-term-prediction" difficulties. These are specifically the case when students attempt to visualize solution curves of autonomous differential equations for predicting the long-term behavior of various initial conditions. To address these issues, a study was conducted in which…

Although geometry is important and strongly related to our surroundings, students still have few chances to learn it through practicing in authentic contexts. Therefore, ubiquitous geometric (UG) APP in mobile devices was developed in this study and allowed students to apply and practice geometry in authentic contexts. This study focused on the…

The study investigated the effects of a technology-integrated guided inquiry-based approach on pre-service mathematics teachers' conceptual understanding of geometry when compared to a guided inquiry approach and a traditional teacher-centered approach. A non-equivalent pretest-posttest control group quasi-experimental design was used. A…

Aiming to enhance understanding of visual obstacles inherent in two-dimensional (2-D) sketches used in high school spatial geometry instruction, we propose a measure of visual difficulty based on two attributes of the sketches: potentially misleading geometrical information (PMI) and potentially helpful geometrical information (PHI). The…

Manipulation of physical models such as tangrams and tiles is a popular approach to teaching early mathematics concepts. This pedagogical approach is extended by new computational media, where mathematical entities such as equations and vectors can be virtually manipulated. The cognitive and neural mechanisms supporting such manipulation-based…