Keith Devlin thinks math is hard but possible:
Any mathematician who says she or he finds math easy isn't tackling sufficiently challenging problems. The fact is, what most of our students don't realize is that mathematicians are not people who find math easy. We don't. We find it hard. The key factor is that we recognize that, given enough effort, and enough time, it is nevertheless possible.
But for some reason, American students don't seem to get that. Rather than sticking with a problem over a period of days (or weeks, or months, etc.), they tend to give up when faced with a difficulty or false start. They lack the "spark" that Devlin claims to have seen in the past:
In contrast to most of the students I dealt with at home [the U.K.], many of my American students were highly motivated, hard working, fiercely competitive, and determined to show they were the best in the world. They would go to heroic lengths to avoid being defeated by a problem. Two decades later, living in the US now, I still encounter such students from time to time. But they no longer seem to be in the majority. For most of the young people I meet, the spark I used to see in their predecessors seems to be absent.
Devlin asks how this came about, and what can be done about it (if anything).
I think there are several factors that help explain why students have lost the ability (or interest) to persist in mathematics.
- More options. There are so many more things for students to do with their time that they don't have time for thinking about mathematics. It would be easier to say that thinking is hard and it's easier to watch t.v. or surf the internet, but I don't think that gives students enough credit. The kinds of activities that they engage in, from sports to cultural, simply provide them with more interesting and stimulating experiences.
Gone are the days that someone discovers a predilection for mathematics because they were cooped up inside on a rainy afternoon and came across a collection of math problems in the family library. Students have more things to do than hard problems in mathematics.
- Denigration of hand calculations. Many educators argue that the calculator should be incorporated early on in mathematics education. In doing so, students are supposedly freed from having to perform tedious, "mindless" calculations and instead can focus on "the big picture".
But there is something to be said for carefully carrying out a long sequence of calculations, especially in calculus. Students learn to think carefully, pay attention to detail, and develop an attention span that allows them to follow more than three algebraic steps without taking a t.v. break. They also start to see patterns and forms that can encourage mathematical thinking. And it only takes a few instances of reaching the end of a calculation and finding out you've made a mistake to make you more careful, and thus better, in the future.
That's not to say that students can't be careful thinkers without doing lots of calculations by hand. It's just that it may be tougher to develop good habits/skills without them. Students who rely primarily on calculators tend to limit themselves — they can't imagine doing mathematics without a calculator and so are never quite able to go beyond the numeric and experience mathematics at a higher level.
- Lack of instant gratification and external acclaim. As Devlin points out, "large numbers of [American youth] bring a feverish intensity to sporting endeavors, putting in endless hours of dedicated training to become the best in their school, their district, their country, or even the world, yet only a few will put in the same kind of effort to mastering mathematics."
But in mathematics, there are no cheering fans ready to lift you on your shoulders when you finish off that proof. There is very little external recognition of mathematical accomplishments — or really any academic pursuit. Why work hard at an assignment when no one, except possibly the instructor, is going to pat you on the back and say "good job"? Unfortunately, many students don't receive positive comments from their peers for doing well or having solved a difficult problem.
- No prior experience with challenges. It is not in vogue, pedagogically, to challenge students with problems that they cannot solve (at least not immediately). That might damage their fragile psyches and discourage them from pursuing mathematics. There is some wisdom in that, but overall I think students are intrigued by puzzles and problems — they want to solve things. The challenge for the curriculum is to find the right mixture of accesibility and difficulty. It's not easy to do, I'll grant, but I think more students should come to college with the experience of having tackled open-ended problems on a regular basis.
More colleges/universities should be working with local public schools to develop "Math circles". These would expose students at all levels to the world of mathematics as a method of problem-solving, not as simply a set of rules to be exercised.
So what can be done? Of course I wouldn't want to limit options of students — except in the classroom where I would put strong limitations on calculator use. In the past, I have gone so far as to ban them on tests altogether; some students were okay with that, others struggled because they never really learned how to solve an equation (just type it in and press 'Solve', right?), what the graphs of fundamental functions look like (so they can't look at the definition of a function and know how that function will behave), or how to approximate quickly the value of a given expression. They simply don't have an intuitive feel for mathematical concepts.
I would also encourage teachers in high school to give students a wide variety of problems, some easy, some challenging and some hard. Encourage the students to work on them over a period of days or weeks by not asking for the solution right off. Instead, just ask for "progress reports" on the problems — which ones were they able to solve? How did they solve them? Which ones are still open, and what have they tried? Not every assignment should be like this, of course, but it would be a good start to have a few of these types of problems going at any one time.
And professors could do a better job of exposing students to math/problem solving outside of the classroom. In the past I have written puzzle columns for the school newspaper; I plan to do that again in the future. The response was not overwhelming (in fact, it was rather disappointing), but I think some students enjoyed the chance to flex their analytic muscles. And perhaps a few did some extended thinking on a non-course related mental exercise that they would not have otherwise.
But another, bigger, change will have to come from academia itself. The more that the university is run like a business, where the "product" is the student, or the students (or their parents) are the "customers", the more difficult it will be to get students away from their instant gratification desires. After all, they aren't paying X dollars a year just to be challenged — they expect a degree that leads to a good job. That, however, is another issue.