Examples play a variety of roles in proving and disproving. Buchbinder and Zaslavsky (2019) have produced an a priori mathematical framework for assessing students' understanding of the role of examples when proving and disproving universal and existential statements. In this paper, I highlight three important aspects that suggest an extension of…

This article shares activities that engage students in quality peer dialogue to build up and break down networks, revealing underlying structure and equivalence. These investigations offer students three kinds of mathematical arguments: (1) proofs of impossibility; (2) proofs by induction; and (3) proofs by exhaustion. The authors describe a…

Mathematical representation is one of the skills that students need to master because it can help them in various problem-solving tasks. This research is a literature review of articles related to mathematical representation in mathematics education. The articles are those published in journals indexed by Scopus from 2020 to 2024. The results of…

In sports, youth are coached to see persistence and hard work as important paths to personal improvement and success. They come to understand through practice that mistakes are tools to help them improve and that collaboration and teamwork are keys to success in team sports. These ideas about the importance of effort, persistence, mistakes, and…

Initial algebra is a critical stage in the teaching of algebra and occurs when students are transitioning from arithmetic to algebra. Irish mathematics education at post-primary level has undergone a period of major reform beginning in 2010, which encompassed a radical change in the methods for teaching algebra. Despite this reform evidence…

Several studies on student's understanding of the concept of the tangent line agree that the difficulties students have when dealing with lines tangent to curves at inflection points or tangent lines that have more than one point in common with the curve are due to a misconception of tangency that is limited to the geometric relationship between…

Our field is brimming with insights into how children learn mathematics. Yet, much of this treasure fails to reach a group that holds significant influence over children's learning: parents. While most parents want their child to succeed in the subject, many do not know how to help, or are so wounded from their own mathematics education that they…

Using the sawtooth map as the basis of a coupled map lattice enables simple analytic results to be obtained for the global Lyapunov spectra of a number of standard lattice networks. The results presented can be used to enrich a course on chaos or dynamical systems by providing tractable examples of higher dimensional maps and links to a number of…

This article presents a theoretical-didactic perspective on the formation and development of concepts concerning mathematical objects as a result of research on didactic difficulties in dealing with concepts in school. The theoretical foundation that supports the research is based on the contributions of the materialist approach to the theory of…

Industrial-scale models require considerable setup time; hence, once built, they are used in myriad ways to consider closely related cases. In practice, the code for these models frequently evolves without appropriate notational choices, largely as a result of the lengthy development time of, and the number of individuals contributing to, their…

As students have struggled to use the "chip model" (i.e., red and yellow chips representing positive and negative numbers) to model integer addition and subtraction and have found it confusing, the authors developed a series of activities based on adding and removing opposite objects to and from a boat to better help students in this…

Mathematical flexibility is thought to be a critical component of mathematical proficiency, and the term "mathematical flexibility" has been used by teachers, researchers, and policy makers for more than 2 decades. Although there seems to be consensus on the importance of mathematical flexibility as a construct, the way it is defined and…

We introduce learning modules in cryptography that can be crafted to motivate many abstract mathematical ideas, and we illustrate with a sample module. These modules can be used in a variety of ways, such as the core for a cryptography course or as motivating topics in other courses such as abstract and linear algebra or number theory.

In the early 1980s, the "theory of didactic situations" put the analysis of mathematical activity at the core of the didactic analysis, thus initiating a new research programme in mathematics education: the "epistemological programme." This paper describes and interprets some of the contributions of the "anthropological…

Functions are an essential concept in mathematics. The studies that have examined functions in advanced contexts have primarily focused on students' reasoning about specific types of functions (such as binary operations and isomorphisms) but not on the core characteristics of well-definedness and everywhere-definedness. Here, we report on a study…