Generalizing, conjecturing, representing, justifying, and refuting are integral parts of algebraic thinking and mathematical thinking in general (Lannin et al., 2011). The activity described in this article makes a case for generalizing as an overall mindset for any introductory algebra or geometry class by illustrating how generalization problems…

This lecture reports on aspects of a larger research programme focused on studying mental mathematics in elementary and secondary mathematics classrooms. It specifically addresses an unplanned aspect that became salient through the work conducted in these classrooms. In this research programme, mental mathematics sessions are designed on a variety…

The surface area of a sphere as a function of radius, "r," is a derivative of the volume of the sphere, and a few other families of solids with this property are known. Sometimes, the derivative relationship can be reached with a tricky choice of size parameter "r." This paper presents an activity for students that discovers…

The effect of using Geoboards in teaching Euclidean geometry in Grade 11 mathematics learners is described. A qualitative research design was adopted in the study. This design was preferred as it provides the conceptual understanding of the teaching methods of the teachers. The participants were taught Euclidean Geometry with the use of Geoboard…

These notes discuss several related propositions in geometry that can be explored in a dynamic geometry environment. The propositions involve an unexpected property of quadrilaterals.

A classic TV commercial once asked, "How many licks does it take to get to the center of a Tootsie Roll Tootsie Pop?" The narrator claims, "The world may never know" (Tootsie Roll 2012), but an Internet search returns a multitude of answers, some of which include rigorous systematic approaches by academics to address the…

This "Activity for Students" article presents a taxicab geometry problem that engages students in plotting points and observing surprising shapes and underlining reasons for the appearance of figures when working with street grids. With this activity, teachers can provide an extra challenge by writing additional problems introducing a…

The study addresses the question of what makes a mathematical task interesting to the 9th year students. Semi-structured interviews were carried out with 15 students of purposive selection of the 9th year. The students were asked to recall a task they found interesting and engaging during the past three years. An analysis of the tasks was made…

This study surveyed and analysed four secondary school students' writing about a square. Sfard's discursive approach to understanding mathematical discourse was used to analyse the responses collected from 214 Australian secondary school students. The results showed that geometric knowledge was developed experientially and not developmentally.…

The advent of dynamic geometry software has changed the way students draw, construct, and measure by using virtual tools instead of or along with physical tools. Use of technology in general and of dynamic geometry in particular has gained traction in mathematics education, as evidenced in the Common Core State Standards for Mathematics (CCSSI…

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…

This paper refers to the concept of semiotic and theoretic control describing resources to conduct decisions in epistemic processes. We consider an argumentation process from a complex problem-solving activity involving different conceptual frames related to parabolas. Using a micro-analytical interpretative lens, we will show that, in order to…

In this paper, we explore an approach to understanding how multimodality works in a community of practice. Using a social learning framework, we show how a community of practice, involving a pair of high school students, engaged in perceptual, bodily, and imaginary experiences while discussing about calculus concepts in a dynamic geometry…

We live our lives in three-dimensional space and encounter geometrical problems (equipment instructions, maps, etc.) every day. Yet there are not sufficient opportunities for high school students to learn geometry. New teaching methods can help remedy this. Specifically our experience indicates that there is great promise for use of geometry…

This study examined teachers' use of student tasks involving dynamic geometry software, in which a figure is constructed then altered while maintaining its constructed properties. Although researchers, professional organizations, and policy makers generally have been proponents of dynamic geometry for instruction, there is little research about…