The beauty of abstract simplicity

Gary Davis over at Republic of Mathematics has an interesting hypothesis about how less able mathematics students are making problem solving unnecessarily complicated. He uses some good illustrations to show how an astute mathematics student will take the time to formulate the simplest method of approach. He hypothesizes that a less able student will even choose more complicated examples in studying a particular idea (see his example of multivariable functions).

Excelling in mathematics, or in any theoretical field, requires an ability to make things simpler. It seems to me that what holds people back is that they erroneously equate "abstract" with "complicated," when in fact it's just the other way around. We create abstractions precisely to make simple the things that are complicated.

Consider this example. Suppose I care about the product of 132479 and 42398. That's a tough product to compute without a calculator, but now suppose all I care about is getting a good estimate on the product. Well, the first number is greater than 130000 and the second is greater than 40000, so the product is greater than 5200000000, or 5.2 billion. On the other hand, the first number is less than 150000 and the second is less than 43000, so the product is less than 6450000000, or 6.45 billion. So the product is between 5.2 and 6.45 billion. This may be a good enough estimate for many problems.

How did I gain this information? I gained it precisely by giving up some of the concrete knowledge I had and instead using more abstract knowledge. Lots of numbers (infinitely many) are greater than 130000 or greater than 40000, but this was all I needed to know about the two numbers in question to show that their product was greater than 5.2 billion. I reduced the amount of knowledge I needed to use, and in so doing I made the problem easier.

Davis's example of guessing a number between 1 and 1000 is another good example of using abstract information. It's a logical first step to decide whether the number is greater than or less than 500. This is an abstract piece of information that reduces the complexity of the problem. A further level of abstraction allows a good problem solver to realize that any collection of numbers can be evenly divided in this way, so that the simple idea of dividing the remaining collection in half can be repeated over and over again to achieve a quick solution to the problem. Seeing how the same approach can be used in many different settings is at the heart of good problem solving. By favoring the abstract over the concrete knowledge you have of the situation, you can make the problem much simpler than it at first appears. To the poor problem solver, the situation seems hopeless because there are 1000 numbers to deal with. To the advanced problem solver, the situation is quite simple because there is really just a collection of numbers which can be repeatedly divided in two. Thus the advance problem solver uses less concrete information to gain more problem solving power.

Many people falsely believe that the abstract is necessarily less familiar, and therefore harder to grasp. In many cases precisely the opposite is true. If I were to tell you that someone looked frightened, this would immediately communicate to you an idea that would only get muddled if I tried instead to describe the way in which a person's face and body looked. Many abstractions, such as emotions like fear, are so familiar to us that we think of them as very concrete; but in fact it is usually the concrete details associated with these abstractions which are harder to identify. Just try to identify all the concrete details associated with "happiness" or "love," and you will say how very much more difficult the concrete can be than the abstract. All language is, in fact, an abstraction which is used to represent what we experience.

The frustration people have with math is that it's not equally obvious to all what is the relevant abstract information for a given problem. Whereas from the features of a person's face most will recognize a certain emotion (like fear or happiness), a problem involving numbers will not necessarily reveal the same essential features to all. Yet it would appear that the only difference is experience. Just like children learn language by imitating their parents as they experience the world around them, so I think students must learn mathematical problem solving by imitation. I don't think learning a particular problem solving method is the answer. The methods we actually use to solve problems are probably too complicated to ever be stated in words. Trying to explain how we solved a problem is probably a lot like trying to explain why we thought a person's face exhibited fear. It's always nice to gain helpful tips in the hopes of applying them to related problems, but a general theory of problem solving is probably impossible.

Instead, repetition of the whole problem solving process is key. Students need to watch instructors solve problems, and then they need to be challenged with problems of their own, so that they begin to see patterns through personal experience. Mathematics seems to be one of the subjects most infected by naive rationalism, which tends to diminish the value of experience in favor of formal logic. But at the heart of problem solving is an element of personal experience that no amount of formalism can ever fully express. Once students can learn to appreciate the power of abstracting key information for themselves, then they will start to enjoy mathematics.