-- From The New Mathematics and an Old Culture: A Study of Learning Among the Kpelle of Liberia by John Gay and Michael Cole

Educators, like politicians, seem keen on approaching issues in binary terms. Student driven curriculum or direct instruction, but never both. Phonics or whole language, but never both.

And oddly, memorization of knowledge or the ability to retrieve knowledge from references, but never both.

I say oddly because it is clear in any education there is some memorization going on. If nothing else to be able to read one needs to know the letters of the alphabet; this can be extended further to “basic” knowledge like locating one’s own country on a map*. Conversely not every fact can be memorized, and an ability to retrieve has always been essential (age of Google or not).

The real question then is more particular: where are the borders? Where is memorization meaningless and where is it helpful? These questions do not lend themselves to sweeping generalizations (as per the binary approach) and require painstaking trawling through content.

The issue can result in long digressions (that geography has always been on the chopping block due to atlases, whether all types of memorization can really be considered “rote”, or the right moment to relearn material**) but I’ll stick here with just mathematics.

Times Tables

Let me make very clear: having to use a calculator to multiply single digit numbers (say, 6 times 7) is impairment.

Problems that would normally be straightforward for a student:

 Simplify: 3(4x+5y+6z)

take triple the time to solve. Problems such as:

 Factor: x² + 9x + 20

become nearly impossible, because the student needs to come up with all the ways of multiplying two numbers to equal 20.

More importantly, a level of fluency is gone. It is analogous to a reader who needs to check a dictionary for every fifth word; it is difficult to think about a text holistically when there are so many roadblocks.

Formulas

Tackling whether or not formulas need to be memorized is trickier, but the is certainly some level where they don’t – who needs to calculate the volume of a sphere off the cuff?

For example, students should be able to recognize the formula:

 sin²x + cos²x = 1

as the Pythagorean Theorem quite naturally. In geometry there are often right triangles with missing sides, and without having a2+b2=c2 memorized, it is unlikely the student would remember to apply it.

Also from trigonometry, the relationships like sine = opposite / adjacent has be recognized in every possible context – off an equation, off a diagram, forwards, backwards – to enough of an extent that the relationships should be memorized with the same level of recall as the multiplication tables.

In General

A rule that seems to apply well to memorization in mathematics is:

Will the knowledge need to be recognized or used in reverse?

Multiplication needs to be done in reverse for factoring (which shows up as early as simplifying of fractions). Trigonometric ratios need to be recognized in any fashion that they appear.

The only famous equation left out of this rule is the quadratic formula.

I argue it is possible for a student to function without the formula memorized. I realize this goes against 100+ years of mathematics teaching, but I have three reasons:

1. It isn’t memorized in the classic sense: even students who have the formula memorized need to write it in its algebraic form before applying it. It’s like having the times tables “memorized” by having to recreate them each time before using them.
2. The “plus or minus” is misleading; the equation is really two formulas. This brushes intuition of the Fundamental Theorem of Algebra (that every polynomial has as many solutions as its degree, so x⁴+x³-2x²+3x-1 has 4 solutions) under the rug.
3. The situations where the formula is needed is well-broadcast. There is no subtlety later in mathematics that will go unrecognized due to a lack of memorization.

Caveat

I stated earlier that choices as to what or what not to memorize do not lend themselves to sweeping generalization; hence my own general rule can be considered just a light suggestion.

*Reports that a fifth of US students can’t find the US on a world map are exaggerated.

**“Practice too soon and you waste your time. Practice too late and you've forgotten the material and have to relearn it. The right time to practice is just at the moment you're about to forget.”

Jason Dyer holds degrees in Fine Arts Studies and Math and teaches at Pueblo High School in Arizona. His school mascot is the Warriors and his other blog of residence is The Number Warrior.

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