Ana C. Stephens

Middle School Students’ Understanding of Core Algebraic Concepts: Equivalence & Variable

Algebra is a focal point of reform efforts in mathematics education, with many mathematics educators advocating that algebraic reasoning should be integrated at all grade levels K-12. Recent research has begun to investigate algebra reform in the context of elementary school (grades K-5) mathematics, focusing in particular on the development of algebraic reasoning. Yet, to date, little research has focused on the development of algebraic reasoning in middle school (grades 6–8). This article focuses on middle school students’ understanding of two core algebraic ideas—equivalence and variable—and the relationship of their understanding to performance on problems that require use of these two ideas. The data suggest that students’ understanding of these core ideas influences their success in solving problems, the strategies they use in their solution processes, and the justifications they provide for their solutions. Implications for instruction and curricular design are discussed.

Advancing the Math Skills of Middle School Students in Technology Education Classrooms

While curriculum specialists and committees often decide how mathematics is taught, it is ultimately principals who influence the extent to which these initiatives are carried out. The overall goal of this article is to provide school leaders with classroom-based research that describes one way of improving the math skills of middle school students. The study employed a randomized pretest-posttest comparison group design to examine the effects of two versions of Enhanced Anchored Instruction (EAI) and a Business as Usual (BAU) condition on the math skills of middle school students in technology education classrooms. Results showed that both EAI conditions were effective at improving the math skills of students over those of students in the BAU classes. The findings suggest that technology education teachers can make important contributions in helping students develop their computation and problem-solving skills.

A Learning Progression for Elementary Students’ Functional Thinking

ABSTRACT In this article we advance characterizations of and supports for elementary students’ progress in generalizing and representing functional relationships as part of a comprehensive approach to early algebra. Our learning progressions approach to early algebra research involves the coordination of a curricular framework and progression, an instructional sequence, written assessments, and levels of sophistication describing students’ algebraic thinking. After detailing this approach, we focus on what we have learned about the development of students’ abilities to generalize and represent functional relationships in a grades 3–5 early algebra intervention by sharing the levels of responses we observed in students’ written work over time. We found that the sophistication of students’ responses increased over the course of the intervention from recursive patterning to correspondence and in some cases covariation relationships between variables. Students’ responses at times differed by the particular tasks that were posed. We discuss implications for research and practice.

Developing a Learning Progression for Curriculum, Instruction, and Student Learning: An Example from Mathematics Education

ABSTRACT Learning progressions have been demarcated by some for science education, or only concerned with levels of sophistication in student thinking as determined by logical analyses of the discipline. We take the stance that learning progressions can be leveraged in mathematics education as a form of curriculum research that advances a linked understanding of students learning over time through careful articulation of a curricular framework and progression, instructional sequence, assessments, and levels of sophistication in student learning. Under this broadened conceptualization, we advance a methodology for developing and validating learning progressions, and advance several design considerations that can guide research concerned with engendering forms of mathematics learning, and curricular and instructional support for that learning. We advance a two-phase methodology of (a) research and development, and (b) testing and revision. Each phase involves iterative cycles of design and experimentation with the aim of developing a validated learning progression. In particular, we gathered empirical data to revise our hypothesized curricular framework and progression and to measure change in students. thinking over time as a means to validate both the effectiveness of our instructional sequence and of the assessments designed to capture learning. We use the context of early algebra to exemplify our approach to learning progressions in mathematics education with a focus on the concept of mathematical equivalence across Grades 3-5. The domain of work on research on learning over time is evolving; our work contributes a broadened role for learning progressions work in mathematics education research and practice.

Build an early foundation for algebra success

Research tells us that success in algebra is a factor in many other important student outcomes. Emerging research also suggests that students who are started on an algebra curriculum in the earlier grades may have greater success in the subject in secondary school. What’s needed is a consistent, algebra-infused mathematics curriculum all throughout the earlier grades. The authors outline three foundational principles for students to learn in the early grades, which can help them with the algebra they encounter in secondary school.