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.…

With math scores stagnating in the United States and achievement gaps widening between demographic groups, improving student outcomes is imperative. Previous studies have shown that using visual representations to teach math concepts has been shown to be an effective component of equitable elementary math teaching. This action research study…

We report findings from a longitudinal study of students' beliefs about empirical arguments and mathematical proof. We consider the influence of an 'Introduction to Proof (ITP)' course and the consequences of the observed changes in behaviour. Consistent with recent literature, our findings suggest that a majority of the thirty-eight undergraduate…

Contribution: This article identifies possible ruptures between the ways fundamental notions of exact differential and exact differential equations (EDEs) are employed in mathematics courses and professional engineering disciplines. Background: Engineering students often experience difficulties with mathematics courses which may even lead to…

In response to recent trends in the field of mathematics education, real-world contexts are frequently found in curricula and classrooms across the world. Numerous scholars contend that, in addition to making instruction more engaging and relevant, rich, imaginable contexts can provide a semantic grounding that helps students deeply understand…

In this article we address the historical and epistemological study of infinity as a mathematical concept, focusing on identifying difficulties, counter-intuitive ideas and paradoxes that constituted implicit, unconscious models faced by mathematicians at different times in history, representing obstacles in the rigorous formalization process of…

The study of loops and spaces in mathematics has been the subject of much interest among researchers. In Part 1 of "The Theory on Loops and Spaces," published in the "Mathematics Teaching Research Journal," introduced the concept and the basic underlying idea of this theory. This article continues the exploration of this topic…

One approach to understanding how the human cognitive system stores and operates with quantifiers such as "some," "many," and "all" is to investigate their interaction with the cognitive mechanisms for estimating and comparing quantities from perceptual input (i.e., nonsymbolic quantities). While a potential link…

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…

In a study of student understanding of negative definite integrals at two institutions, we administered a written survey and follow-up clinical interviews at one institution and found that "backward integrals", where the integral was taken from right to left on the x-axis, were the most difficult for students to interpret. We then…

This paper introduces the basic knowledge of integral factors of first-order ordinary differential equations and Lie symmetry analysis. It then discusses the principle of constructing an integral factor of the first-order ordinary differential equation by the Lie symmetric method. Finally, it presents some examples to show the process of…

In the literature on numerical cognition, the presence of the capacity to distinguish between numerosities by attending to the number of items, rather than continuous properties of stimuli that correlate with it, is commonly taken as sufficient indication of numerical abilities in cognitive agents. However, this literature does not take into…

The authors discuss the use of "real life" activities that are interesting to children in the teaching of measurement and number concepts. In this article, they present a series of lessons targeting volume and capacity that seamlessly incorporate genuine "real-life" experiences with practical mathematical applications. In…

In this paper, we explore the following research questions: How do first-grade students define even and odd numbers? What types of justifications do they use to support their generalizations using these definitions? We report the ways in which first-grade students define even and odd numbers and how they justify generalizations that use their…

In this article, the author shares two investigations that relate the variables of height and volume using different contexts. Each of the investigations surfaced unfinished learning for students, regardless of their starting place, and facilitated growth in understanding of measurement and algebra. The Functions and Volume of Vases task…