Binary operations are one of the fundamental structures underlying our number and algebraic systems. Yet, researchers have often left their role implicit as they model student understanding of abstract structures. In this paper, we directly analyze students' perceptions of the general binary operation via a two-phase study consisting of task-based…

The algebraic structure is one of the axiomatic mathematical materials that consists of definitions and theorems. Learning algebraic structure will facilitate the development of logical reasoning, hence facilitating the study of other aspects of axiomatic mathematics. Even with this, several researchers say a lack of algebraic structure sense is a…

Isomorphism and homomorphism appear throughout abstract algebra, yet how algebraists characterize these concepts, especially homomorphism, remains understudied. Based on interviews with nine research-active mathematicians, we highlight new sameness-based conceptual metaphors and three new clusters of metaphors: sameness/formal definition, changing…

Cognitive development of eighth-grade students, as identified by Jean Piaget, occurs during a time when many of them are transitioning between concrete operations and formal operations where the ability to think in abstract concepts becomes possible. Because of this period of transition, many eighth-grade students find difficulty in demonstrating…

Analogical reasoning has played a significant role in the development of modern mathematical concepts. Although some perspectives in mathematics education have argued against the use of analogies and analogical reasoning in instructional contexts, some attempts have been made to leverage the pedagogical power of analogies. I assert that with a…

A study was conducted to explore the beliefs and practices of 49 New South Wales (NSW) primary school teachers regarding their beliefs and practices concerning the use of concrete materials in the learning and teaching of Number and Algebra. This paper reports on elements of the study regarding why and how teachers use concrete materials. Not only…

To contribute to the sparse educational research on student understanding of eigenspace, we investigated how students reason about linear combinations of eigenvectors. We present results from student reasoning on two written multiple-choice questions with open-ended justifications involving linear combinations of eigenvectors in which the…

This study examined student actions, interpretations, and language in respect to questions raised regarding tabular, graphical, and algebraic representations in the context of functions. The purpose was to investigate students' interpretations and specific ways of working within table, graph, and the algebraic on notions fundamental to a…

Adult developmental mathematics students often work under great pressure to complete the mathematics sequences designed to help them achieve success (Bryk & Treisman, 2010). Results of a teaching experiment demonstrate how the ability to reason can be impeded by flaws in students' mental representations of mathematics. The earnestness of the…

Trouche's [Third Computer Algebra in Mathematics Education Symposiums, Reims, France, June 2003] presentation at the Third Computer Algebra in Mathematics Education Symposium focused on the notions of instrumental genesis and of orchestration: the former concerning the mutual transformation of learner and artefact in the course of constructing…

A major goal of mathematics teaching is the involvement of students in the personal process of discovering mathematical ideas and formulating problems. The process of an inductive leap followed by a deductive argument is used in mathematics courses at Phillips Exeter Academy (New Hampshire). Mathematical problems are presented in which the givens…

This article examines students' ability to use the balance model to solve for unknowns. A teaching experiment was conducted in four Year 3 classrooms. This experiment focused on exploring the application of the balance model as an analogue for representing equations and solving for unknowns. The teaching experiment promoted a shift by students…

Evidence for suggested basic differences in the abstract reasoning capacity of Negro and Caucasian children includes consistent findings of significantly poorer performance by Negroes on Raven's Progressive Matrices (PM). This study investigated the PM performance of Negro children taught algebra via a discovery method of instruction. It was…