Students who know high-level math before going to college

Question

There is a high school in the city I live in which has some high-level math courses in their curriculum. It's a special math class mentored by some university lecturers, and the children basically do the first year of university math courses (single-dimensional analysis, linear algebra, basics of set theory and some miscellaneous things).

Students who come from this high school are generally very bright people, frequently participate in national math contests and achieve successes in various areas.

It would be all great, but there is a problem: there is a subgroup that never really had to learn (i.e. put in a serious effort), being smart was enough. When they come to the university, the first year is a breeze (while all the others learn hard work), but the second gets tough (because there is many new things and they are unable to manage their time) and in the third year they drop out (which is a shame because they were very promising). Also, there are issues with ego (they were always good, and now what), neglect (skipping homework, because "I can always do it") and attitude (like "of course I'm the best", or at the beginning not paying attention because "it's obvious" and later because "I don't know what is going on, why bother catching up, I can always do it later").

I have heard about similar observations from people from different places, always with regard to math or computer science. What's wrong?

The university offers a wide range of courses of varying difficulty, from basic to insanely hard, so it's not a problem of "not enough challenge". Also, the university doesn't have resources to mentor them, so it's not an option. It seems a kind of problem of maturity (they frequently have some childish behaviors), like doing all this stuff in high school (their schedules had to be packed) would leave too little time for growing up, but that's just a hypothesis.

Is there any related research? Are there any solutions or suggestions? Is it a problem of the university, or perhaps the high-school?

Answer 1

The question you are asking has little to do with the particular subject in which the student excels and everything to do with student motivation. The students have not developed the skills needed to study and persevere through difficult classes because everything to this point has come naturally for them. They haven't needed to set time aside for a study group or tutor because they've never needed it before and they likely don't even think about it as a problem until it's too late.

Unfortunately, I can't find any reputable studies on this in the myriad of articles from other sources. It seems that every newspaper columnist has written about student motivation at some time. But it's my firm opinion that some students are not mature enough to attend college after they graduate high school. I know that I was not ready for college at 18 years of age and dropped out after a semester. It was the best decision of my life. If I had continued on that path, I would never have ended up on the road that I am on now to become a teacher. Now, I spend hours per night on homework and studying, then read about interesting math topics and write about math in my free time.

Anecdotally, I have seen four students this year drop out of the Math Ed program because they can't handle the workload. And they are all intelligent students who simply would not apply themselves to study and do the homework. I think that a teacher can help motivate students through a number of different methods and the success of each depends entirely on the individual student. I've seen teachers bribe students (turn in this homework by X day and get extra points on the test). I've seen teachers strike the fear of God into their students (A quiz with 3 of the most difficult problems from the homework once a week on random days). I've seen teachers plead, joke, or be apathetic. Regardless of the method, the student is ultimately the one who makes the decision to try. Imparting the wisdom of experience to a college student is frustratingly fruitless. But every now and then, they get it.

Answer 2

Apart from quite general issues about smart kids' study skills or lack thereof, my observations over many years gives me the impression that mathematics and computer science offer special hazards/advantages. I'll address mathematics, since most of my experience lies there, but I suspect similar observations apply to computer science (which I've also paid some attention to since late 1960s).

First, beyond the obvious connection to immediate tactile reality, mathematics allows "precocity" and forms of "problem-solving" that do not necessarily depend heavily on wider experience, in contrast to, for example, history or philosophy. And timed low-level problem-solving contests, in which speed plays a key role? Nothing like that in much else. Even music performance, which can be very much "of the moment" and allowing precocity, is not just about playing as many notes as possible. The "problem" is that subsequent development of mathematics in school or in real life does depend more and more on wider experience (even if only somewhat interior to mathematics), and considerably less on raw quickness or cleverness. The sense of pace changes enormously. Kids may reasonably see this as a bait-and-switch.

The typical school curriculum (what I've seen in the U.S.) and ambient cultural attitudes (manifest in the curriculum) reinforce a peculiarly shallow picture of what mathematics is, and what it means to excell at it. Contests are understandable, but more substantive things aren't. Use of low-level math ("(pre) calculus"?) to "filter" for other life paths, is rather strange, but accounts for the size of the math departments in U.S. public universities.

In particular, kids can be declared "successes" in mathematics already at a strangely young age, based on criteria that mislead them, thus selecting for traits that are not ... in fact ... best longer-term.

But there'd not really be anything good accomplished by eliminating contests, for example. The fact that this aspect is easily and popularly misinterpreted seems to me inescapable. There should be a strong counterpoint to this attitude, but our school system itself traditionally rewards similar traits, so...?

There is a further, complementary problem, in my observation, that (in the U.S., anyway) "the system" often selects against mathematical traits that would in fact be beneficial long-term, in cases where the quick-clever-contest-math attributes are not strongly present. It's so easy to over-simplify, and identify quickness with talent, that even in math grad school lack of facility in timed exams is misconstrued as lack of mathematical talent. I confess that it took me a long time to recover from this misunderstanding, myself, and I have no good recommendation about over-coming either side of this false identification, except to keep it in mind and remind others.