The Article
In this article, Professor Jack Smith (College of Education) and Senior Specialist Betty Phillips (Department of Mathematics) examine the work of an eighth grade algebra class to determine what new algebra skills and understandings students develop in a Standards-based middle school curriculum (the Connected Mathematics Project), and what evidence suggests students possess these skills.

The Discussion
With content and assessment issues at the core of the algebra debate, Smith and Phillips decided to focus their research on what algebraic thinking is possible for students by the end of middle school. They observed a class of 24 students, concentrating on how students understood linear and nonlinear relationships and the mathematical terms they used to express this understanding. By "attending carefully" to students' language, the researchers came to understand both the positive insights and limitations in the students' thinking. The researchers had the students work on a problem for 10 minutes and then asked them to explain their work. Smith and Phillips were surprised to find the extent of variation in how students reasoned about linear relationships. They outlined a "starter set" of competencies that they believe can serve as a solid foundation for developing more advanced algebraic and mathematical abilities. These important skills include the student's ability to (a) identify the quantities that vary in problem situations; (b) describe the rates of change and y-intercepts of those relationships, represented in tables, graphs, and symbolic expressions; (c) think "across" and make connections among those representations; (d) understand the equivalence of algebraic expressions in multiple ways; and (e) most of all, make sense of algebraic expressions and equations in relation to the contexts in which they appear. Through their interviews with students, Phillips and Smith learned that powerful ideas in algebra are accessible to middle school students and that their initial ideas are neither flawless nor useless. Helping students explore the world of algebra means appreciating and supporting their early insights. Learning algebra is a complex, multiyear process that involves many intellectual challenges.

What It Means to You
How can your middle school mathematics program incorporate better ways to hear and appreciate how students express their thinking, especially in the crucial area of algebra? Do your teachers help students explore how symbolic expressions have meaning in algebra? In what ways can math educators in your district support the early insights of introductory algebra students?

For More Information
Smith, J. P. & Phillips, E. (November 2000). Listening to middle school students' algebra thinking. Mathematics Teaching in the Middle School, 6(3), 156-61.

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