When students work with a non-Euclidean distance formula, geometric objects such as circles and segment bisectors can look very different from their Euclidean counterparts. Students and even teachers can experience the thrill of creative discovery when investigating these differences among geometric worlds. In this article, the author describes a…

In this paper, we trace the development of our theorizing about students' mathematical understanding, showing how the adoption of an enactivist perspective has transformed our gaze in terms of the objects of our studies and occasioned for us new methods of data analysis. Drawing on elements of Pirie-Kieren (P-K) Theory for the Dynamical Growth of…

Acquaintance with various ways of inculcating concepts in any studied area of knowledge is one of teachers' duties, particularly mathematics teachers. Studies indicate errors and difficulties when inculcating concepts in mathematics and learning them. Many concepts have different meanings in different contexts. Hence, teachers should deal with the…

This discussion paper put forwards variation as a theme to structure mathematical experience and mathematics pedagogy. Patterns of variation from Marton's Theory of Variation are understood and developed as types of variation interaction that enhance mathematical understanding. An idea of a discernment unit comprising mutually supporting variation…

Surface area is consistently identified as a curriculum standard for K-12 students and it regularly appears on national and international assessments. Recently, many schools began acquiring digital fabrication and advanced manufacturing equipment. The growing use of digital fabrication in classrooms raises the question of whether or not this…

Within recent years, history of mathematics (HoM) has become an increasingly popular topic. Studies have shown that student reactions to it depend on the ways they use history of mathematics. The present action research study aimed to make students deduce volume rules of frustum pyramids using the dissection method. Participants were 24 grade…

An engaging activity which prompts students to listen, talk, reason and write about geometrical properties. The "Bag of Tricks" encourages students to clarify their thoughts and communicate precisely using accurate mathematical language.

In line with the Common Core and Standards for Mathematical Practice that portray a classroom where students are engaged in problem-solving experiences, and where various tools and arguments are employed to grow their strategic thinking, this article is the story of such a student-initiated problem. A seemingly simple question was posed by…

Geometry is the branch of mathematics that addresses spatial sense and geometric reasoning. Students begin to understand geometry through direct interaction with their physical world. Because it is the study of the physical attributes of the environment, geometry has relevance for every student; the world becomes a big classroom. As students see,…

Counting complete subgraphs of three vertices in complete graphs, yields combinatorial arguments for identities for sums of squares of integers, odd integers, even integers and sums of the triangular numbers.

The complementary use of the theories of instrumental genesis and shifts of attention enabled me to obtain insights into students' understandings of the concepts of eigenvector and eigenvalue. My analysis suggests that the use of different dragging modalities can provide evidence of shifts in structure of a student's attention and consequently can…

This study aimed to reveal the effects of teaching with concrete learning objects taken from the history of mathematics on student achievement. Being a quasi-experimental study, it was conducted with two grade 8 classes in a secondary school located in Trabzon. The experimental group consisted of 27 students and the control group consisted of 25.…

In Part 1 of their article, Erin Host, Emily Baynham and Heather McMaster use a combination of digital technology and concrete materials to explore the concept of "corners". They provide a practical, easy to follow sequence of activities that builds on students' understandings. [For "Using Digital Technology to See Angles from…

How can geometry teachers design great tasks that allow students to make connections among interrelated concepts and expand their geometric reasoning skills? Many curricular materials provide problems for students to apply a single geometric concept. However, these problems do not always promote reasoning opportunities for students, because…

Designing an optimal Norman window is a standard calculus exercise. How much more difficult (or interesting) is its generalization to deploying multiple semicircles along the head (or along head and sill, or head and jambs)? What if we use shapes beside semi-circles? As the number of copies of the shape increases and the optimal Norman windows…