Learning Mathematics in College

Pointing at Homework

These pages are composed and maintained by Greg McColm, Department of Mathematics, University of South Florida.

Sometimes, doing homework can be like pointing at the Moon.

There is a Zen ``koan,'' or enlightening story, of the Sixth Patriarch HuiNeng. HuiNeng pointed at the Moon, and said that the word ``moon'' figuratively points at the Moon, just as his finger was literally pointing at the Moon. The word ``moon'' is not the Moon, any more than his finger was the Moon.

So what does homework have to do with pointing at the Moon?

When you are working on a homework problem, you are thinking. Now comes the Zen part: are you thinking about the homework problem or are you thinking about thinking about the homework problem?

• The Moon. If you are thinking about a homework problem on, say, the shortest path from a point A to a point B that bounces off a mirror M, you might be thinking about straight lines, points, and maybe the Pythagorean distance formula.

• The Finger. If you are thinking about thinking about a homework problem on, say, the shortest path, you might be thinking about how hard the problem is, how little progress you are making, what will happen if you can't solve it, and maybe what your inability to solve this problem indicates about your mathematical ability.

There are several kinds of thinking about thinking about something (when you should be thinking directly about that something).

• Reverie, or just daydreaming. We all do this, but if we spend more than a few minutes on it, it can consume a lot of time. Victor Hugo called it a poison of the mind (sounds like the great French novellist had it bad). In several Sherlock Holmes stories, Dr. Watson falls into a short reverie (Holmes author Arthur Conan Doyle probably had better control of himself). Reverie can be useful in visualization techniques, but just lying back and fantasizing is a recreational (in)activity, not work.

• Listening to the monster. We've all heard that we shouldn't get discouraged when things get tough, but there's more to it than that. We indulge --- there is no better word for it --- in gratuitous self-criticism. We imagine that we will never get over the current toughness, that we were never any good at mathematics anyway, and that we should give up. I will discuss this monster in more detail in my page on attitude, but for the moment, let's just observe that this sort of thinking does not help at all.

There are two aspects to thinking about thinking about thinking ... .

• Attitude. If you have a positive attitude, you can accomplish more. This is not just a sentiment. Psychologists do measure optimism and pessimism and compare what optimists accomplish to what pessimists accomplish. Optimists live longer, have fewer accidents, publish more papers, and are more frequently elected to public office. Moreover, optimism is not something out of your control, or controllable only by pharmaceuticals. By getting in the habit of writing down positive things (so that you are aware of them) and your worries (so that they are more clearly identified), you can actually become more positive. For more on this, see the page on attitude. And your attitude towards homework influences how much time and effort you spend thinking about homework, as opposed to thinking about thinking about ... .

• States of Consciousness. If you are sitting at a desk, gazing at a piece of paper, you may be thinking about the shortest path from point A to point B bouncing off a mirror M, or you may be thinking about thinking about the shortest path, or you may be worrying about your inability to concentrate on the shortest path, or you may be doodling short paths, etc. The point is that the only constructive thought process is the first one: only by actually thinking about the shortest path itself will you figure out what the shortest path is. The other thought processes are merely distracting or even demoralizing. For more on thinking about the shortest path, rather than on scenic routes, see the page about states of consciousness.

• Or ears. Or fingers. We are primates, so about 90 % of us have vision as our dominant sense. We are good with pictures, less so with formulas. But nearly 10 % of us are primarily auditory, and such people are not as good with pictures, but often better with formulas. And there are a few of us whose dominant sense is tactile/kinesthetic: such people rely on motions to work things out. It is important to learn what one's own orientation is, for that will indicate how one would best learn things. (Note that most people are probably mixtures.) It should also be noted that teachers tend to teach to people like themselves, so that visual teachers will tend to rely on pictures, auditory teachers on formulas, and tactile/kinesthetic teachers on gestures. In a class, one sometimes has to orient onesself to a teacher who has different dominances than onesself.

• Finally, when one is working on a hard problem, one is not doing all the work consciously. In fact, a lot of the effort goes into prodding the Unconscious to do the work. But there are tricks to doing this, and it helps if one understands the Unconscious.