Validating proofs and counterexamples across content domains is considered vital practices for undergraduate students to advance their mathematical reasoning and knowledge. To date, not enough is known about the ways mathematics majors determine the validity of arguments in the domains of algebra, analysis, geometry, and number theory--the domains…

This paper is a case study of the teaching of an undergraduate abstract algebra course with a particular focus on the manner in which the students presented proofs and the class engaged in a subsequent discussion of those proofs that included validating the work. This study describes norms for classroom work that include a set of norms that the…

The purpose of this study was to investigate elementary children's conceptions that might serve as foundations for integer reasoning. Working from an abstract algebraic perspective and using an opposite-magnitudes context that is relevant to children, we analyzed the reasoning of 33 children in grades K-5. We focus our report on three prominent…

One of the basic components of algebraic thinking is functional thinking. Functional thinking involves focusing on the relationship between two (or more) varying quantities and such thinking facilitates the studies on both algebra and the notion of function. The development of functional thinking of students should start in the early grades and it…

Frequently, in the US students' work with proofs is largely concentrated to the domain of high school geometry, thus providing students with a distorted image of what proof entails, which is at odds with the central role that proof plays in mathematics. Despite the centrality of proof in mathematics, there is a lack of studies addressing how to…

The Isis problem, which has a link with the Isis cult of ancient Egypt, asks: "Find which rectangles with sides of integral length (in some unit) have area and perimeter (numerically) equal, and prove the result." Since the solution requires minimal technical mathematics, the problem is accessible to a wide range of students. Further, it is…

We used a teaching experiment to evaluate the preparation of preservice teachers to teach early algebra concepts in the elementary school with the goal of improving their ability to generalize and justify algebraic rules when using pattern-finding tasks. Nearly all of the elementary preservice teachers generalized explicit rules using symbolic…

This study is part of a large research and development project aimed at observing, describing and analyzing the learning processes of two seventh grade classes during a yearlong beginning algebra course in a computer intensive environment (CIE). The environment includes carefully designed algebra learning materials with a functional approach, and…

This study examined conceptions of algebra held by 30 preservice elementary teachers. In addition to exploring participants' general ''definitions'' of algebra, this study examined, in particular, their analyses of tasks designed to engage students in relational thinking or a deep understanding of the equal sign as well as student work on these…

A cross-curricular structured-probe task-based clinical interview study with 44 pairs of third-year high-school mathematics students, most of whom were high achieving, was conducted to investigate their approaches to a variety of algebra problems. This paper presents results from one problem that involved solving a set of three linear equations of…

Differing perspectives have been offered about student use of recursive and explicit rules. These include: (a) promoting the use of explicit rules over the use of recursive rules, and (b) encouraging student use of both recursive and explicit rules. This study sought to explore students' use of recursive and explicit rules by examining the…

This paper examines the development of student functional thinking during a teaching experiment that was conducted in two classrooms with a total of 45 children whose average age was nine years and six months. The teaching comprised four lessons taught by a researcher, with a second researcher and classroom teacher acting as participant observers.…