As algebra teachers, the authors explore the following question in this article: "How can algebra 1, algebra 2, and precalculus teachers support students to develop algebraic reasoning and understanding of structure that can serve them in day-to-day algebraic computation?" The article shows how the algebraic identity "(a +…

This article presents a checklist of 10 evidence-based practices for educators to apply in mathematics instruction for students with learning disabilities. The checklist is "actionable," meaning the items on the checklist can be put into action immediately. It provides practical strategies teachers can adopt to fit their lessons…

Concern about children's mathematics performance in Ireland and elsewhere has prompted a range of responses from researchers, policymakers, educators and the media. While policy-level responses in Ireland include revising curricula and implementing a numeracy strategy that calls for increased tuition hours, teachers have also drawn on a wide array…

Tasks that ask students to label rational number points on a number line are common not only in curricula in the upper elementary school grades but also on state assessments. Such tasks target foundational rational number concepts: A fraction (or a decimal) is more than a shaded part of an area, a part of a pizza, or a representation using…

A steady stream of research has shown that many elementary school teachers have weak mathematical knowledge in some areas, including place value and fractions. Since a teacher's mathematical knowledge affects their students' performance, improving elementary school teachers' knowledge is critical. A better understanding of the mathematical…

In this article, pre-service teachers' mathematics content knowledge is explored through the analysis of two items about ratio from a Mathematical Competency, Skills and Knowledge Test. Pre-service teachers' thinking strategies, common errors and misconceptions in their responses are presented and discussed. Of particular interest was the range…

The basic rules of elementary algebra can often appear beyond the grasp of many students. Even though most subjects, including calculus, prove to be more difficult, it is the simple rules of algebra that continue to be the "thorn in the side" of many mathematics students. In this paper we present a result intended to help students achieve a…

The aim of the present study is to provide further evidence that the errors that arise from improper application of the linear model are not random and not easy to overcome. Using three different types of test, we attempt to show that the errors referred to in the literature as "pseudo-analogous" are the result of an epistemological…

This study investigated middle school students' conceptual understanding of algebraic equations. Participants in the study--257 sixth- and seventh-grade students--were asked to solve one set of algebraic equations and to generate story problems corresponding with another set of equations. Structural aspects of the equations, including the number…

Mathematics departments rarely require students to study very much logic before working with proofs. Normally, the most they will offer is contained in a small portion of a "bridge" course designed to help students move from more procedurally-based lower-division courses (e.g., abstract algebra and real analysis). What accounts for this seeming…

Certain misunderstandings in science, mathematics, and computer programing reflect analogous underlying difficulties. These misunderstandings are examined through four knowledge levels: (1) content; (2) problem-solving; (3) epistemic; and (4) inquiry. Analysis of several examples shows that misunderstandings have causes at multiple levels, and…

Previous research has shown that--due to the extensive attention spent to proportional reasoning in mathematics education--many students have a strong tendency to apply linear or proportional models anywhere, even in situations where they are not applicable. This phenomenon is sometimes referred to as the "illusion of linearity". For example, in…

In the study of mathematics instruction the focus has changed from the outcomes of learning to the cognitive processes involved in mathematics learning. This new emphasis presents a serious challenge for curriculum design, but it may also lead to more success for students in learning mathematics. (Author/VM)

The transformation of a solid to its net is based on something quite different from simple perceptual impression. It is a mental operation performed by manipulating mental images. The aim of this study was to observe pre-service and in-service teachers' ability to visualize the transformation of a curved solid to its net and vice versa, and to try…

We report a study of 10-14 year old children's use of additive strategies while solving ratio and proportion tasks. Rasch methodology was used to develop a diagnostic instrument that reveals children's misconceptions. Two versions of this instrument, one with "models" thought to facilitate proportional reasoning and one without were…