Entanglements of Mathematics Education Research and Large-Scale Assessment: Rethinking Formulas as Relational

In this article, we examine eighth graders’ incorrect responses to a specific task on a national standardized assessment of mathematics. The task asked students to write the formula for the perimeter of a given figure as a function of a variable. We focus on incorrect responses to better understand students’ difficulties with the algebraic thinking demanded by the task, especially with the formula and the variable. We show how these responses identify a variety of approaches to the solution of the task, which we name routes. We use these data to conclude that, at the end of middle school in our country, there still appears to be a lack of a relational view of formula, pointing to a need to reconceptualize formulas as relations rather than procedures. The strength of large-scale data in fueling mathematics education research is also discussed.