Investigating high-school students’ reasoning strategies when they solve linear equations

A cross-curricular structured-probe task-based clinical interview study with 44 pairs of third-year high-school mathematics students, most of whom were high achieving, was conducted to investigate their approaches to a variety of algebra problems. This paper presents results from one problem that involved solving a set of three linear equations of the form ax ± b = cx ± d. One of the equations had a unique solution, another was an identity, and the third resulted in a contradiction. Most pairs of students were able to solve the equation resulting in a unique solution using symbol-manipulation algorithms or by a graphical approach. On the equations resulting in an identity or a contradiction, most student pairs did not know how to interpret the results of their symbol manipulation, and few turned to another representation when symbol manipulation failed them. The constructs of versatility and adaptability [Sfard, A., & Linchevski, L. (1994). The gains and the pitfalls of reification—The case of algebra. Educational Studies in Mathematics, 26, 191–228] guided our interpretation of the findings.