The concept of function is critical in mathematics in general and abstract algebra in particular. We observe, however, that much of the research on functions in abstract algebra (1) reports widespread student difficulties, and (2) focuses on specific types of functions, including binary operation, homomorphism, and isomorphism. Direct, detailed…

This paper initiates a teaching sequence that focuses on building up equivalent definitions to the standard ones for the limit concept in Real Analysis. It comprises two parts: The first provides a classroom assignment where students, guided by Analysis lecturers, are led to develop an alternative definition to the [epsilon] - [delta] one for…

This paper reports on five secondary school mathematics prospective teachers' conceptions of "extreme point." The analysis of the data addressed students' definitions, examples, and evaluation of given examples, with special attention to the related domain. Written assignments and individual interviews uncover salient, erroneous concept…

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…

Definitions play an integral role in mathematics and mathematics classes. Yet, expectations for definitions and how they are intended to operate, i.e., mathematical norms for definitions, can remain hidden from students and conflict with other discursive norms, explaining differences in mathematicians' and students' understandings of the nature of…

Instructors manage several tensions as they engage students in defining, conjecturing, and proving, including building on students' contributions while maintaining the integrity of certain mathematical norms. This paper presents a case study of a teacher-researcher who was particularly skilled in balancing these tensions in a laboratory setting.…

The notion of definition is a central building block of mathematics. In addition to specific definitions, it is desirable that students learn about meta-mathematical aspects of definitions such as their role of classification, and about characteristics of definitions such as arbitrariness. We investigated how such meta-mathematical aspects emerge…

Problem solving is widely regarded as a fundamental feature within the school mathematics curriculum. However, there is considerable disagreement over what exactly problem solving is, and if and how it can be taught. In this article, I define problems as non-routine tasks and propose the explicit teaching of domain-specific problem-solving tactics…

Knowing functions and functional thinking have recently moved from just knowledge for older students to incorporating younger students, and functional thinking has been identified as one of the core competencies for algebra. Although it is significant for mathematical understanding, there is no unified view of functional thinking and how different…

The aim of this paper is to contribute to the ongoing discussion about the role of intuition and ambiguity in doing, teaching and learning mathematics. We start by discussing the ways in which limits and continuity are presented in Calculus textbooks, to illustrate some of the ambiguities, and to contrast them with precise and rigorous definitions…

This study investigates the potential of rehearsal interjections to provide opportunities for novice teachers and teacher educators to discuss topics related to teaching the practice of mathematical defining. Through analysis of video recordings of seven elementary novice teachers' rehearsals about geometric definitions, we identified the problems…

This study aims to examine pre-service elementary mathematics teachers' knowledge of quadrilaterals and evaluate their understanding within the framework of Van Hiele's Geometric Thinking Levels. In particular, the study investigates how pre-service teachers define quadrilaterals and establish relationships between different types of…

Typically, vocabulary instruction is associated as part of reading instruction. However, vocabulary instruction is also a necessary component of mathematics instruction. In fact, state and national standards (e.g., Common Core State Standards) require elementary students to communicate about mathematics to provide clear explanations, construct…

In current teaching materials, when using Dedekind cuts to construct real numbers, the definition of a Dedekind cut is always involved in defining addition and multiplication. In this paper, as it is done in many current textbooks, Dedekind cuts are used to construct the set of real numbers. Then the order in it is defined, and the…

This study explores opportunities to learn definite integrals in three widely used textbooks in the U.S. Definitions, worked examples, and exercise problems were coded using research-based cognitive resources in definite integrals. The results show that limited opportunities for students to explore multiplicative relationship between two…