[Math Lair] Memorization

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Many students prepare for a mathematics test by memorizing large numbers of formulas, procedures, and the like. This isn't the best way of preparing. It's often better to gain an understanding of the underlying concepts instead. In my book How to Ace the Multiple-Choice Mathematics Test, I used the following analogy:

Say that you've moved to a new town. You're unfamiliar with the town, but you need to figure out how to get to various places. Let's also say that you don't have a GPS or something similar. There are two approaches that you could take. The first would be to memorize the directions to every place that you need to go ("go north two blocks, turn left at the stop sign, proceed three blocks, the store is on the right-hand side"). The second approach would be to gain a general understanding of how the city is laid out, where the main streets are and what they are called, how the street numbers run and so forth. The first approach works well if there are only a few places you need to go, but if there are dozens or hundreds of places, it will become impossible to remember everything. If there are a lot of places that you need to go, the second approach works much better because the streets are all interconnected. Once you have an understanding of the main streets, you can get to all the main places in town and just need a small amount of extra information to find places off the beaten path, and even if there's someplace you can't find, you'll probably be able to get close enough that you might be able to find it by just driving around.

Similarly, as you learn more and more about mathematics, the more difficult it will be to memorize every formula, procedure, algorithm, etc. However, because all mathematical concepts are interconnected with each other, just like all of the streets in your city are connected with each other, having a strong understanding of the underlying concepts makes it a lot easier to understand it all and means that you have that much less to memorize. If you can understand the basic concepts that underlie and link the formulas and procedures, memorization is often unnecessary. Most math problems can be solved through an understanding of mathematical concepts combined with problem solving techniques.

To take an elementary example, say that you needed to know how to calculate the area of various shapes (say you had to write a test on this information). One way of going about this is to memorize the formulas for the area of triangles, squares, rectangles, rhombuses, circles, ellipses, and so on, but if you need to work with a large number of shapes, this will get very complicated. It would be a better idea to understand basic concepts of area and also understand how shapes can be divided up into other shapes. For example, you might divide a parallelogram or a rhombus into two triangles, or into a rectangle plus two triangles.

One danger of excessive reliance on memorization is that you might begin to think of mathematics as not requiring any thought. If you don't spend time thinking about and understanding the basic concepts, when you run into a question that doesn't exactly fit the formulas that you've memorized, you'll often find yourself lost, with no way of knowing how to begin tackling it. On the other hand, if you've spent time thinking about and understanding mathematical concepts, you can often apply those, along with problem solving skills and number sense to make progress on the question.

Keep in mind that this doesn't mean that you should stubbornly refuse to memorize anything at all. If you become successful in mathematics, you will almost certainly be able to recite a large number of mathematical facts; however, most successful mathematicians don't come to memorize these facts by sitting around attempting to fill their brains with fact after fact, but but by taking a keen interest in their work, through which this sort of learning often occurs automatically. At the same time, a mathematician will freely consult books, papers, tables, web sites, and the like, for formulas and other information that he or she may not know, or may not need to know, offhand.