Math Musings: When Should We Work on Memorization?

Having knowledge in long-term memory can be very helpful in solving problems.

But master problem-solver Sherlock Holmes was concerned that if he had too much knowledge in his mind, new facts would crowd out the old and cause him to forget something important:

“I consider that a man’s brain originally is like a little empty attic, and you have to stock it with such furniture as you choose. A fool takes in all the lumber of every sort that he comes across, so that the knowledge which might be useful to him gets crowded out, or at best is jumbled up with a lot of other things so that he has a difficulty in laying his hands upon it.

“Now the skillful workman is very careful indeed as to what he takes into his brain-attic. He will have nothing but the tools which may help him in doing his work, but of these he has a large assortment, and all in the most perfect order.

“It is a mistake to think that that little room has elastic walls and can distend to any extent.

“Depend upon it: there comes a time when for every addition of knowledge you forget something that you knew before. It is of the highest importance, therefore, not to have useless facts elbowing out the useful ones.”

—Sherlock Holmes, A Study In Scarlet

Even though most of us don’t try to file away nearly as many bits of precisely detailed information as Holmes, we do want to fill our brain attic with useful stuff. If we memorize a math fact or formula, we want to be able to pull it out of storage when we need it.

So what should we work on memorizing, and when?

How Memory Works

Memory works by association. Seeing something we’ve seen before, smelling a familiar scent, or recognizing something new as related to something from our past draws a memory to the fore. We were not thinking of it, then we noticed the related thing, and suddenly the memory came into our minds.

You can’t force memory; it is automatic. Seeing and paying attention to something triggers the related memory without conscious thought.

Therefore, if we want to be able to recall what we store in memory, we need to have connections that will prompt the recall. There is no point in memorizing unconnected facts because we will have no way to recall them to mind when needed. But the more connections we have to a fact, the more things that can trigger it, and the stronger those connections, the more easily the item will be drawn to mind.

So we want to memorize items to which we have plenty of strong connections — such as items that we find ourselves using over and over in the course of solving problems. We do not want to memorize isolated facts or theorems unconnected to our life experiences.

So that is the first test:

• Is this fact, equation, or theorem something you are using often and having to look up each time?

Memory Happens Naturally

When we use a fact or equation often, we may discover that we have memorized it without trying to, a natural consequence of frequent use.

For example, I found this note in an old journal from back when my youngest daughter was in middle school. We had been doing buddy math, going through a prealgebra chapter on exponents.

“I already knew a lot about powers, having gone through algebra myself many years ago and also having taught Kitten’s four older siblings. I knew the squares up to 12² = 144, but the cubes only up to 3³ = 27. But today, I had a problem that involved adding perfect cubes together, and I surprised myself by knowing the values up to 6³, twice as many as I expected. I haven’t tried to memorize them, but repeated use has done the job.”

If I had decided at that point that it would be valuable to learn the first ten or twelve perfect cubes, it wouldn’t be hard for me to memorize the rest of them. They would naturally form connections to all the knowledge I already had in my mind.

So that is our second test:

• Is this fact, equations, or theorem closely related to something (or better, to several things) that you’ve already memorized naturally, through repeated use?

What This Means for Our Students

Wise teachers will not ask students to memorize lists or charts of disconnected facts. Whether we are talking about math, vocabulary, history, or science, disconnected facts may go into memory, but they cannot easily be pulled back out. (Can you tell what the 17th letter of the alphabet is without reciting the ABC song?)

Wise teachers will also not say, “You are going to need this someday, so you should memorize it now.”

Instead, we will pay attention to our students as they work. When we notice them having to check the same fact or look up the same equation many times (I’m looking at you, quadratic formula!), we may suggest that perhaps some memory drill would help.

But There’s a Caveat

If we’re dealing with basic items like the multiplication math facts, we’re probably better off helping children develop a variety of thinking strategies instead.

The problem with math facts is that they seem almost too close to random numbers. 6 × 9 = 54 and 7 × 8 = 56, and there is nothing inherent in the numbers to keep them from getting cross-wired in a child’s brain.

With math facts, our students may be better off not trusting their memories, but stopping to think, “Does this make sense?”

For example, they can learn to reason the answer out by thinking of related information. They might know that 6 × 9 is almost 6 × 10, so the answer must be one six less than ten sixes: 60 – 6 = 54. Or consider that 7 × 8 is a next-door neighbor to 7² and to 8², so it must be 49 + 7 or 64 – 8, and either of those goes to 56.

So that gives us our third test:

• If you memorize this and then get it partially jumbled in your mind, will the memory do you more harm than good?

A prime example of memory gone wrong is the fractions mnemonic “Keep, Change, Flip.” By the time they get to high school, many students have forgotten what this phrase refers to, and they proceed to change signs or flip fractions almost at random, whenever they don’t know what else to try.

Conclusion

In the end, there isn’t really all that much our students need to memorize in math.

Young children need to memorize the counting sequence, since that’s an arbitrary series of sounds. We have picture books and counting songs to help, but it still takes plenty of practice and repetition.

Older children need to memorize the number symbols and the symbols for math operations. One of the reasons I like the algebra before arithmetic courses by Sonya Post (see my article Math Manipulatives Part 3) is that they give meaning to symbols like “+” and “×,” so children can focus on mathematical reasoning.

Vocabulary must be learned by memory, which is best done a little at a time, as they need the word to communicate their own ideas about numbers, shapes, and patterns.

But the cool thing about math is that pretty much everything else follows by logic. We can think our way through the ideas and concepts, using what we know to make sense of new concepts.

As podcaster Pam Harris says, math is figure-out-able.