This paper presents a novel figure for teaching multiple geometric proofs of the Pythagorean theorem. Because it consists only of congruent given right triangles, the figure can be constructed using a template of the given right triangle or, if available, a computer program. Within the figure, called a Pythagorean multi-proof square, there are…

Vectors have important applications both within and outside mathematics, but the concept of vectors is often taught to students in a less-than-engaging way, leading to students feeling inadequate and frustrated. This article describes the use of a mathematical microworld, "Driving with Vectors," to explore vectors using equitable…

This practitioner article describes a lesson carried out in a high school classroom at the conclusion of a unit on exponential growth. Two teachers use a series of music-related activities to engage students in using and connecting multiple representations of exponential growth while exploring musical frequencies on a piano.

Three-dimensional symmetry plays a crucial role in crystallography education. The educational process can be facilitated by introducing the analysis of friezes and plane patterns such as parquets before presenting more sophisticated crystal structures. Analysis of the symmetry of parquets can serve as an opportunity to follow the full routine of…

Sequences are an important topic--not only in mathematics but also in the mathematics curriculum. Exploring sequences related to medial triangles offers learners opportunities to solve problems involving sequences within geometric contexts. Sequences related to medial triangles are also a rich source of beautiful and interesting geometric patterns…

By having students practice constructing diagrams for geometric theorems, teachers can develop students' understanding of mathematical claims, vocabulary, and notation methods. This practice can also strengthen students' ability to interpret mathematical diagrams and recognize their limitations.

A key motivational tactic in undergraduate mathematics teaching is to launch topics with fundamental questions that originate from surprising or remarkable phenomena. Nonetheless, constructing a sequence of tasks that promotes students' own routes to resolving such questions is challenging. This note aims to address this challenge in two ways.…

Models are an important tool for geography teaching, as they can be used to represent features of human and/ or physical systems. In addition, they enable the student to identify with the demonstrated reality, since they work with iconic images and symbols specific to each culture, used to represent the elements contained in the models (da Silva…

Fractals are geometric objects used to describe irregular shapes that have a fractional dimension and commonly appear in nature. Although several proposals for the study of perfect fractals at the basic level are present in the literature, only few proposals for the study of real fractals exist, which does not seem reasonable considering the…

Perceptually rich concrete manipulatives benefit mathematics learning. Because students should be challenged to consider connections between two-dimensional (2D) and three-dimensional (3D) perspectives of conics (NCTM 2000), this article describes hands-on teaching techniques that promote this type of student reasoning. This article aims to (1)…

Minecraft is an open world survival computer game that was originally created in Sweden in 2011 and later purchased by Microsoft® in 2014. It is among the most popular computer games with students of all ages because it gives players a sense of ownership and opportunities for creative exploration. The game has three unique features: (1) no clear…

Students learn mathematics by solving problems. Mathematics textbooks are full of problems, and mathematics teachers use these problems to test students' understanding of mathematical concepts. This paper discusses how problem-solving skills can be fostered with a geometric tiling problem.

Using the correct materials and manipulatives appropriately allows the richness of geometrical concepts to be fully investigated to enable students to explore and reason about shapes. This article explores some of the key ideas involved in learning about two dimensional shapes and discusses how materials can be used to support learning.

When teaching how "tension" and "compression" relate to geometrical structures such as bridges, arches, and domes, The author shows a picture of the Notre Dame Cathedral in Paris, completed in the 14th century, and presents a lesson on using the curve called a catenary to explain how he teaches about tension and compression…

Research suggests that embodying mathematical concepts through movement supports improvements in students' spatial fluency, communication, collaboration, critical thinking, and problem-solving skills (Alibali and Nathan 2012; Cone and Cone 2011; Sarama and Clements 2009). This article describes a dance-based strategy that middle school mathematics…