Misunderstandings about Understanding

The issue of balance between procedural fluency and conceptual understanding in mathematics has served as a dividing line in education. Some believe that understanding of a procedure or algorithm must precede the procedure/algorithm itself—and if it doesn’t precede it, it should come about quickly. Failure to do this results in students who some call “math zombies”. Others believe that procedural fluency and conceptual understanding is an iterative process where one feeds the other.

The learning process encompasses a spectrum that begins at the novice level and extends to expert levels. Students benefit…