MEMORY AND MATHEMATICS

My stepmother told me that memory was not necessary to learning math. It would be enough if I just understood it.

I was too young at the time to realize the contradiction in what she was saying. After all, she was the one who encouraged me—urged me—to learn my times tables. Wasn’t that necessary memorization?

Math teachers have required me to memorize various facts since then. For Algebra 1 we had to memorize the squares of the first twenty numbers. I did so, and never regretted it.  Being able to recognize and remember those squares has made factoring considerably easier than stumbling around in the dark. It was also necessary to memorize various formulae—for instance the Pythagorean theorem, the quadratic formula, the trigonometric ratios, and more. In trigonometry I memorized still more formulae and the trigonometric functions of certain basic angles.

During all this time, it never occurred to me that I was memorizing, and going against the advice of my stepmother. In other subjects I railed against memorization. That was why I truly disliked history. I was willing to memorize for art history, where the memorization was a necessary evil; but I felt it was wrong.

As a new teacher, I was introduced to Bloom’s taxonomy, a six-layer list of cognitive verbs, from boring old recitation and memorization, to the exalted evaluation. Again here was an indictment of memorization.

But then, during my graduate studies, a professor advocated that we memorize the proof of a difficult theorem. His thesis was that though we might not understand the proof at first, memorizing it would lead to understanding. This was totally unexpected at this level of math. Memorization for Advanced Topology was a startling idea, and I had to think about it. It occurred to me that the professor was right, and I began to re-examine my stand on memorization all the way through school.

Memory is necessary for everything. Whether we mean to or not, we memorize telephone numbers and street names, the times of our favorite television shows and websites, birthdays, clothing sizes, and on and on. Not being able to remember is recognized as a very serious malady, not just an inconvenience.

The only reason I was able to get through high school and college math without memorizing was that I had worked enough practice problems that I had memorized facts in spite of myself and my misguided convictions. I had actually been memorizing all along!

I began to look at my high school and college students who were struggling. I found that, in fact, they were not memorizing material; and, since they were struggling and doing as little as possible, they were not getting the amount of practice they needed in order to memorize automatically, either. I became an advocate of memorization.

Let’s go back to Bloom’s taxonomy. As far as I can tell now, this is a non-judgmental list. There are six categories, but readers have superimposed the hierarchy. Instead, let’s imagine that we must educate (and learn) at every level.

What would be more worthy of evaluation or appraisal than a synthesis of ideas, something we originated or compiled or theorized?

What would be more worthy of syntheses or extension than material we had analyzed, something we had ascertained or examined or inferred?

What would be more worthy of analysis or examination than an application of ideas, some discovery or solution or demonstration?

What would be more worthy of application or classification than ideas we truly comprehended, concepts we could alter or discover or summarize?

And what would be more worthy of comprehension or summary than things we knew, that we could cite, name, identify, or reproduce? Items that we had memorized?

Memorization is essential to learning. It is not bad to memorize; it is bad to stop at memorization. We must aspire to all the cognitive levels, all the way to evaluation as critical thinking. But, as it turns out, we can never get there if we leave out the first steps: the acquisition of knowledge and the attainment of comprehension.