RE-"CONCEPTUALIZING " PROCEDURAL KNOWLEDGE IN MATHEMATICS

Many mathematics educators have lost sight of the critical importance of the mathematical "understanding" which underlies procedural competence, in part because we do not have a language to refer to this kind of understanding. The modal way of categorizing mathematical knowledge-- conceptual and procedural knowledge--is limited in that: (a) it is almost exclusively focused on the mathematics of elementary school, and (b) in studies using this framework, knowledge of procedures and concepts are assessed in very different ways. Three students' solutions to a linear equation are presented and compared. I suggest that the differences between what these three students know about linear equation solving can be framed in terms of students ' planning knowledge of the procedures-- in other words, conceptual knowledge about procedures. By many accounts, the U.S. is beginning to experience a backlash against current mathematics education standards-based reform efforts, with many advocating a renewed emphasis on "basic skills". Many mathematics educators recognize the similarities between this current push toward basic skills and prior backlashes against mathematics reforms in years past. Why does this oft-commented-on pendulum between “skill-based ” and “reform-based ” curricula appear to be swinging yet again? One reason for the emergence of the current back-to-basic