In this study, we take a sociocultural perspective on teaching and learning to examine how teachers in an urban Algebra 1 classroom constructed opportunities to learn. Drawing on analyses of discourse practices, including videotaped classroom lessons as well as other classroom artifacts and telephone interviews, we describe ways that two teachers…

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…

A sophisticated and flexible understanding of the equals sign (=) is important for arithmetic competence and for learning further mathematics, particularly algebra. Research has identified two common conceptions held by children: the equals sign as an operator and the equals sign as signaling the same value on both sides of the equation. We argue…

Taking as the point of departure the vision of school algebra as a formalized meta-discourse of arithmetic, we have been following five pairs of 7th grade students as they progress in algebraic discourse during 24 months, from their informal algebraic talk to the formal algebraic discourse, as taught in school. Our analysis follows changes that…

This article investigates the numbers [image omitted], originally studied by Catalan. We re-confirm that they are indeed integers. Using the close relationship between them and the Catalan numbers C[subscript n], we develop some divisibility properties for C[subscript n]. In particular, we establish that [image omitted], where f[subscript k]…

In a previous study, the onto-semiotic approach was employed to analyse the mathematical notion of different coordinate systems, as well as some situations and university students' actions related to these coordinate systems in the context of multivariate calculus. This study approaches different coordinate systems through the process of change of…

Equivalence is mostly associated with the strand of Number and Algebra and is fundamental in both of the substrands. Equivalence is considered a key and foundational concept of algebra, and whereas an assumption might be made that equivalence is a relatively "easy" concept; studies have shown that when asked to balance an equation such…

This study looks at a mixed ability group of 21 Year 5 primary students (aged 9-10 years old) who had previously never had formal instruction using letters to stand for unknowns or variables in a mathematics context; nor had they been introduced to formal algebraic notation. Three lessons were taught using the computer software "Grid Algebra"…

The Sliding along a Side task was presented at a National Council of Teachers of Mathematics (NCTM) conference session for high school teachers. According to the authors, while exploring this task with this group of experienced mathematics educators and classroom teachers, they shared a "gasp!" moment. With the aid of the dynamic mathematics…

We illustrate a method, which is modeled on "breaching experiments," for studying tacit norms that govern classroom interaction around particular mathematical content. Specifically, this study explores norms that govern teachers' expectations for the doing of word problems in school algebra. Teacher study groups discussed representations of…

In this paper, we propose an approach to analysing teacher arguments that takes into account field dependence--namely, in Toulmin's sense, the dependence of warrants deployed in an argument on the field of activity to which the argument relates. Freeman, to circumvent issues that emerge when we attempt to determine the field(s) that an argument…

The middle grades are a critical transition period in students' mathematics trajectories, as students move from arithmetic to the more complex and abstract concepts of algebra. Teachers' and parents' judgments of students' math abilities in these years are important to instructional planning and decision making for teachers, and can advise parents…

Compared to boys, many girls are more aware of a social context in the learning process and perform better when the environment supports frequent interactions and social relationships. For these girls, embodied agents (animated on-screen characters acting as tutors) could afford simulated social interactions in computer-based learning and thereby…

This study compared student engagement and achievement levels between students enrolled in a traditional college algebra lecture course and students enrolled in a "flipped" course. Results showed that students in the flipped class had consistently higher levels of achievement throughout the course than did students in the traditional…