Re-conceptualizing Procedural Knowledge: the Emergence of "intelligent" Performances among Equation Solvers

This paper explores the development of students' knowledge of mathematical procedures. Students' tendency to develop rote knowledge of procedures has been widely commented on. I postulate an alternative, more "intelligent" endpoint for the development of procedural knowledge, where students choose to deviate from established solving patterns on particular problems for greater efficiency. The purpose of this study was to explore the instructional conditions that facilitate the emergence of this outcome. Students with no prior knowledge of formal linear equation-solving techniques were taught the basic transformations of this domain. After instruction, students engaged in problem-solving sessions in two conditions. In the treatment group, students completed "alternative ordering tasks," where they were asked to re-solve previously completed problems but using a different ordering of steps. Completing alternative ordering tasks was found to lead to more intelligent solving.