"It is perhaps pertinent to make a comment or two here about the problems of the text. There is a distinction between what may be called a PROBLEM and what may be considered an EXERCISE. The latter serves to drill a student in some technique or procedure, and requires little, if any, original thought. Thus, after a student beginning algebra has encountered the quadratic formula, he should undoubtedly be given a set of exercises in the form of specific quadratic equations to be solved by the newly acquired tool. The working of these exercises will help clinch his grasp of the formula and will assure his ability to use the formula. An exercise, then, can always be done with reasonable dispatch and with a minimum of creative thinking. In contrast to an exercise, a problem, if it is a good one for its level, should require thought on the part of the student. The student must devise strategic attacks, some of which may fail, others of which may partially or completely carry him through. He may need to look up some procedure or some associated material in texts, so that
he can push his plan through. Having successfully solved a problem, the student should consider it to see if he can devise a different and perhaps better solution. He should look for further deductions, generalizations, applications, and allied results. In short, he should live with the thing for a time, and examine it carefully in all lights. To be suitable, a problem must be such that the student cannot solve it immediately. One does not complain about a problem being too difficult, but rather too easy."
"It is impossible to overstate the importance of problems in
mathematics. It is by means of of problems that mathematics develops and actually lifts itself by its own bootstraps. Every research article, every doctoral thesis, every new discovery in mathematics, results from an attempt to solve some problem. The posing of appropriate problems, then, appears to be a very suitable way to introduce the student to mathematical research. And it is worth noting, the more problems one plays with, the more problems one may be able to pose on one's own. The ability to propose significant problems is one requirement to be a creative mathematician."
Eves, Howard (1963). A Survey of Geometry volume 1. Boston: Allyn and Bacon, page ix.
>Is math class too easy for you? Looking for a greater challenge? You've come to the right place.
My friend laments something like it was not around when he was in grade school (in the Bitnet/Compuserve days, ha!)
P.S. AoPS is where I first took on my screen name.
You write with ease, to show your breeding;
But easy writing's vile hard reading.
This, by the way, is why I still prefer to read The Economist and Skeptic and other magazines to reading blogs on the same subjects. More learning for less of my reading time, because the writing has been subjected to editing.
I find it a bit miserable though - how can you head into something knowing that it will take a decade of focused practise to be good at it? It's a bit intimidating, and limiting - how many decades do you have left? Wouldn't want to waste them on something not worth it, eh?
Another quotation that I've tried to dig up and verify, but have never been able to find a primary source for, is "Because thinking is the easiest of all things to do, it is the hardest thing to do well." Great thought. I would be very much in the debt of any reader who can help me attribute it to its original author.
As for the danger of misplacing the decade of deliberate practice on the wrong discipline, I advise my children to get to know a lot of different adults, the better to choose a passion to pursue that they really like well enough to devote practice to and become good at. And I think if people learn to truly reach an expert level in something, they can find personal satisfaction and reasonable income doing it. There are a lot of journeymen, but few genuine experts.
"Thinking is the hardest work there is, which is the probable reason why so few engage in it."
It's reminiscent of your quote, though probably not close enough.
As an aside, I'm skeptical that Henry Ford actually said this.