Based on a study with six elementary school teachers and 115 students in grades 3, 4, and 5, this article discusses ways that teachers might support an understanding of variable as students explore pattern-finding activities. The practices emerged from interviews with teachers and a yearlong observation of teaching algebraic thinking during an…

This case study describes the ways in which problems involving additive differences with unknown starting quantities, constrain the problem solver in articulating the inherent quantitative relationship. It gives empirical evidence to show how numerical reasoning takes over as a Grade 6 student instantiates the quantitative relation by resorting to…

This article reports on a study that investigated the extent to which 57 Grade 6 students at a particular school have misconceptions regarding equality, with the equal sign as focus. It also investigated this school's three Grade 5 and 6 teachers' mathematical knowledge for teaching regarding equality, again focusing on the equal sign. The…

The aim of this study is to investigate student engagement from the point of view of the teacher, by focusing on teacher's didactical strategies used to engage students during algebra introduction. Eight teachers in grade 6 and 7 participated in a focus-group interview study. The findings are based on episodes of student engagement in algebra and…

This two-part investigation (a) assessed the impact of the Jaime Escalante Math Program (JEMP), a structured summer mathematics intervention program, on the math achievement of urban middle school students, (b) identified the characteristics of the program that the administrators and teachers perceived to contribute to student achievement, and (c)…

Linear algebra is one of the most challenging topics to learn and teach in many countries. To facilitate the teaching and learning of linear algebra, priority should be given to epistemologically analyze the concepts that the undergraduate students have difficulty in conceptualizing and to define their ways of reasoning in linear algebra. After…

This study of fourth-grade students and teachers explores mathematics teaching and learning that focuses on discovering and modeling algebraic relationships. The study has two parts: an investigation of how students learn to construct algebraic statements and models for comparisons and measurement situations in the multiplicative domain, and an…

This dissertation explores fifth and eighth grade students' interpretations of three kinds of mathematical representations: number lines, the Cartesian plane, and graphs of linear functions. Two studies were conducted. In Study 1, I administered the paper-and-pencil Linear Representations Assessment (LRA) to examine students'…

In this article, the authors seek to re-conceptualize the perspective regarding students' difficulties with algebra. While acknowledging that students "do" have difficulties when learning algebra, they also argue that the generally espoused criteria for algebra as the ability to work with the syntactical rules for solving equations is…