Learning algebra is difficult for many students in part because of an emphasis on the memorization of abstract rules. Algebraic reasoners across expertise levels often rely on perceptual-motor strategies to make these rules meaningful and memorable. However, in many cases, rules are provided as patterns to be memorized verbally, with little overt…

In this article, we describe the design principles undergirding "productive failure" (PF; M. Kapur, 2008). We then report findings from an ongoing program of research on PF in mathematical problem solving in 3 Singapore public schools with significantly different mathematical ability profiles, ranging from average to lower ability. In…