The blogger at College@Home sent me an interesting infographic illustrating just how unprepared the great majority of American high school students are for college level instruction.
While the readers of our blog (focusing on real analysis) are not in this
category, they may likely end up at some time in their career teaching freshman level mathematics. The phenomenon is particularly acute for mathematics instruction: many students arrive with their stellar math grades from high school and end up completely crushed by a simple calculus course. There is a huge gap between the understanding expected at the high school level and the demands and ideas at the level just above.
Much can be written about this (I won't). Here I want to mention that the same thing can happen with an elementary real analysis course. I have never taught this subject without a few "weepers." These are students who have always been convinced that they are "good at mathematics." They aced the calculus courses, and yet are overwhelmed at the elementary real analysis level. They can understand everything in class (they claim) but simply cannot write a correct proof for even the most trivial limit theorems of the course. They have miserably failed the first midterm, something that they couldn't conceive would happen to them ever.
Why is analysis so difficult? They aren't much amused when I tell them that this material was routinely taught in the former Soviet Union to 15 year olds.
Early study of any subject may not prepare you for the next level. Real analysis courses are very subtle (not difficult really) and demand greater insight and thought than the computational courses that precede them.
Learning a new subject does not always create challenges, but you should be aware that at some time in your life you will certainly encounter a new subject that overwhelms you at first. Perhaps the most important skill you can learn is how to dig yourself out of trouble when a subject bites you hard. You will have to do it eventually.
For high school students, if this happens in the first weeks of college---well enjoy the experience and learn the necessary study skills. You will need them again. For math students--don't get too smug. At some point in your life you will have to work very hard to get to the next level.
We will always find ourselves at some point in our life "unprepared" for the next level of study. Consider it a question of character as to whether you can salvage yourself. As the College@Home blogger points out, probably 75% of college students arrive "unprepared." If we can't improve that percentage, at least we can warn them.