William Thurston on mathematical education.
Along with the emphasis on tests has come an emphasis on precocity and acceleration in mathematics. It is relatively easy for a bright student to work through the mathematics curriculum far more quickly than the usual pace.
There are several problems associated with precocity. People who skip ahead in the curriculum often have gaps in their background which only show up later. At that point, the person may be too embarrassed to admit the gap and tries to fake understanding. This regularly leads to disastrous results.
Another problem is that precocious students get the idea that the reward is in being 'ahead' of others in the same age group, rather than in the quality of learning and thinking. With a lifetime to learn, this is a shortsighted attitude. By the time they are 25 or 30, they are judged not by precociousness but on the quality of work. It is often a big letdown to precocious students when others who are talented but not so precocious catch up, and they become one among many. The problem is compounded by parents in affluent school districts who often push their children to advance as quickly as possible through the curriculum, before they are really ready.
A third problem associated with precociousness is the social problem. Younger students are often well able to handle mathematics classes intellectually without being able to fit in socially with the group of students taking them. Related to precociousness is the popular tendency to think of mathematics as a race or as an athletic competition. There are widespread high school math leagues: teams from regional high schools meet periodically and are given several problems, with an hour or so to solve them.
There are also state, national and international competitions. These competitions are fun, interesting, and educationally effective for the people who are successful in them. But they also have a downside. The competitions reinforce the notion that either you ‘have good math genes’, or you do not. They put an emphasis on being quick, at the expense of being deep and thoughtful. They emphasize questions which are puzzles with some hidden trick, rather than more realistic problems where a systematic and persistent approach is important.
This discourages many people who are not as quick or as practiced, but might be good at working through problems when they have the time to think through them. Some of the best performers on the contests do become good mathematicians, but there are also many top mathematicians who were not so good on contest math. Quickness is helpful in mathematics, but it is only one of the qualities which is helpful. For people who do not become mathematicians, the skills of contest math are probably even less relevant.
These contests are a bit like spelling bees. There is some connection between good spelling and good writing, but the winner of the state spelling bee does not necessarily have the talent to become a good writer, and some fine writers are not good spellers. If there was a popular confusion between good spelling and good writing, many potential writers would be unnecessarily discouraged.
I think the answer to these problems is to build a system which exploits the breadth of mathematics, by allowing quicker students to work through the material in greater depth and to take excursions into related topics, before racing ahead of their age group.
2 comments:
An example of an educational policy based on a complete misunderstanding of the nature of mathematical ability is provided by the current British policy of encouraging the "acceleration" of able students.
In British beaurocratic newspeak, "acceleration" refers to the strategy of moving stronger students ahead to take tests, or to study material, from subsequent school years "when they are ready". As a policy "acceleration" is cheap, easy to administer, and does not require any additional professional development on the part of teachers---and so requires no additional effort from those responsible for administering the educational system! But in most hands, such "acceleration" offers simply "more of the same". "Acceleration" generally fails to ensure both that earlier techniques become sufficiently robust and that they are linked together to form a sufficiently strong foundation for subsequent work. And---most important of all---it deprives students of those experiences, which are known to be most valuable in the long run: namely the daily reminder that, while school mathematics may be "elementary" (in the sense that the beginner can insist on understanding, rather than being obliged to trust the teacher's, or the textbook's authority), it is by no means always ``easy".
"Acceleration", combined with overexamination and "teaching to
test", is soul-destructive. In modern Britain, school students are overexamined; they are working under stress and exhibit sensible defensive reflexes, refusing to study anything which is not part of the future examination.
[I cannibalised here text from my essay Mathematical Abilities and Mathematical Skills.]
I have found that some degree of acceleration for precocious students is fine. A challenged young student is better than a bored one.
I know that I was terrifically bored for years in school and wish I had some opportunity to be moved forward.
But there is a social limit to such acceleration. I know some very socially maladjusted teenagers who began university at a very early age and it took them years to recover, so to speak.
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