What Math Students Need Beyond the “Why”

A comment to page 165 of Jo Boaler's new book Math-ish

It’s great when students learn the “why” behind a mathematical procedure! But…

1. The explanation has to be crystal-clear. 3/4 x 2 = 6/8??? That notational disaster is a recipe for confusion.

2. The student still needs plenty of retrieval practice (eventually, in a timed setting) with carrying out the procedure, i.e., the “what.”

3. During the timed retrieval practice, the student needs to use the most efficient strategy – which, in the case of fraction division, is “multiply by the reciprocal,” NOT “convert both fractions to have the same denominator and then divide the numerators.”

(And by the way, the most efficient strategy is sometimes memorization – for instance, while a student should of course be able to compute 6 x 5 = 5 + 5 + 5 + 5 + 5 + 5 = 30, they absolutely need to memorize 6 x 5 = 30.)

Why? To free up mental resources for higher-level thinking.

In hierarchical skill domains like mathematics, students need to get the point that they can perform low-level actions instantaneously and effortlessly.

Otherwise, low-level actions will constantly be interrupting their flow of thought and they won’t be able to see the forest for the trees.

It’s just like how, if a basketball player had to consciously think about the mechanics of running and dribbling, they would not be able to do both at the same time, and they would not have enough brainspace to think about strategy.

The scientific name for this idea is “automaticity,” which I’ve previously posted about here: Fast, Correct Answers Do Matter in Mathematics