What is Homework Good For, Anyway?

If you wanted to learn how to do something --- fly an airplane, swim, play a violin --- you would probably spend a lot of time practicing. If you buy a whole bunch of Karate books and spend lots of time just reading them carefully, you will not become very good at Karate. You actually have to practice Karate ... practice a lot of Karate ... for many years ... before you become any good at it.

Let's look at some examples.

First of all, many of us (myself included) would like to learn by reading. In Those Magnificent Men and their Flying Machines, a movie about an airplane race from London to Paris in 1910, a German officer learns how to fly by reading the book of instructions. This doesn't work very well, and the plane winds up in the English channel. In real life, aspiring pilots fly, with an instructor, learning to take off, to land, to take off, to land, to take off, over and over and then they learn further exercises. And further exercises. And further exercises. And only after forty hours in the air are you permitted to take the test for flying itty-bitty planes in daytime in good weather.

There are many books on swimming, including publications by the American Red Cross. But you can't get certification for swimming, much less lifeguarding, by reading some pamphlets. You have to get wet, and try different kinds of strokes, while some helpful teacher says, ``turn your head sideways, and keep those legs straight.''

Some people would prefer to learn by watching. An old Peanuts cartoon portrayed Linus saying that his grandfather went to a concert to learn how to play the violin. Even though the old man watched the violinist very carefully, when he went back home, he couldn't duplicate the performance on his own violin. So Linus's grandfather is going to another concert and will try to get a better view.

Reading about doing something, or watching someone do something, is not the same as doing it oneself. Would you want to be operated on by a surgeon who had read about and observed but never done the operation? The brain is economical, and will only develop the ability to do things that it is regularly called on to do. So if a student reads and watches, he will develop the ability to ... read and watch. But not to do.

Let's turn to mathematics.

Mathematics (and other academic fields) can be deceptive, because it seems that what a student is supposed to do is study the subject until they know the subject. Assuming that it is knowledge that we are after, how do you know you know the subject? This can be tricky because if you are exposed to a subject, you become familiar with it. Practically all of us are familiar with Leonardo da Vinci; that does not mean that we know, for example, whether he ever lived in France. But if we are familiar with something, we tend to think that we know things about it: studies on familiarity suggest that people tend to overestimate their knowledge of subjects on which they have only a passing acquaintance. But whatever we are after, it is more than mere familiarity.

In the real world, knowledge means "you can use it." If you know some mathematics, that means that you can apply it to solve problems. Familiarity is not sufficient; in fact, we all have had experiences when we were familiar with a subject, and then took a test on it and ... WHAM! Thus sayeth the resulting grade: familiarity is not knowledge. In fact, knowledge is more like a skill. And as with airplanes, swimming, and violins, the only way to develop the skills for solving mathematics problems is by doing lots and lots of mathematics problems. For more on the difference between repeated exposure (e.g., skimming 53 times) versus careful study (reading carefully, doing the examples, working out anomalies, etc.), see Daniel T. Willingham's article on how Students Remember...What They Think About, the point being that what you remember will be, for better or worse, what you actually are thinking about during the study period.

Typically, I find that students who do most of the homework assignments do a lot better in exams than those who do just a few, and I don't mean by a little bit. Quite often in a calculus test, all the students who did most of the homework do better than all the students who did little of the homework. The graph of test scores sometimes shows two big mounds: one mound (mostly grade B-ish) of scores of students who worked hard on the homework before the test, and one mound (mostly grade D-ish) of scores of students who did not. The reality is this: while homework may officially count for little (if any) of the grade, unofficially homework drives the grades. This should not surprise anyone who looks at exams as performances, and homework as rehearsal: as any musician, actor or athlete will tell you, if you do not practice, you bomb on stage.

And there is another detail: unless you are training to become a musician or athlete, your long-term goals involve few test-like situations. If you are trying to do a project for your job, for your church or temple, for your political party, or for your family, that project may require a lot of thought, some experimentation, perhaps some checking of references, etc. In other words, a lot of what you will be doing will be more like homework than like tests. If you learn how to do homework, you will be better off later on. That is another reason why homework -- even homework problems different from anything on the exams -- is assigned and graded: what you will need to learn is how to do extended projects.

Let's get technical: what does it mean to ``learn'' something? This means that:

• You have a new ability.
• You can look at things in a new way.