By Dan Willingham
Commenters on the teaching of mathematics sometimes express impatience with the idea that attention ought to be paid to conceptual understanding in math education. I get it: it sounds fuzzy and potentially wrong-headed, as though we’re ready to overlook inaccurate calculation so long as the student seems to understand the concepts—and student understanding sounds likely to be ascertained via our mere guess.
Impatience with the idea that conceptual aspects of math ought to be explicitly taught is often coupled with an assurance that, if you teach students to calculate accurately, the conceptual understanding will come. A new experiment provides evidence that this belief is not justified. People can be adept with calculation, yet have poor conceptual understanding.
Bob Siegler and Hugues Lortie-Forgues asked preservice teachers (Experiment 1) and middle school students (Experiment 2) to make quick judgments (true or false) of inequalities in this form:
N1/M1 + N2/M2> N1/M1
N and M were two digit numbers, making it hard to calculate the problem quickly in one’s head. Instead, you needed to evaluate what must be true. In this case, subjects easily recognized that the sum on the left side of the inequality had to be larger than the value on the right side. Likewise, they made few mistakes with inequalities of this form
N1/M1 – N2/M2> N1/M1
But when multiplication or division was called for, subjects made errors. Specifically, when N2/M2 amounted to less than one and was multiplied by N1/M1, subjects incorrectly thought the result would be larger than N1/M1. And division was required, subjects thought the result would be smaller than N1/M1. In fact, they answered correctly less often than chance.
Yet these same subjects were quite accurate when asked to calculate answers to problems entailed multiplication or division of fractions; middle-school students got about 80% correct. AND they showed quite good understanding of the magnitude of fractions between 0 and 1 (as shown by placing marks on a number line to represent fraction quantities).
This is a small sample, and the absolute level of performance should not be taken as representative of preservice teachers or of middle-school students. But Siegler and Lortie-Forgues suggest that the disconnect between computation and understanding is typical. That conclusion is in line with the evaluation of the National Math Panel.
So what’s to be done? Teach concepts. Among other ideas, Siegler and Lortie-Forgues suggest that, once they have some competence in calculation, students might compare the results of
8/7 * 1/2
6/7 * 1/2
There are sure to be many methods of helping with conceptual understanding, some best introduced before calculation, some concurrent with it, and some after. This latest finding points to the necessity of greater attention to understanding in instruction.
Siegler, R. S. & Lortie-Forgues, H. (in press) Conceptual knowledge of fraction arithmetic. Journal of Educational Psychology.
Reprinted with permission of the author.