I am curious why students who take calculus in high school often do so poorly in college calculus. I am an instructor at an engineering college and I've noticed a decent number of students who have calculus 1 on their high school transcript and yet fail abysmally at the college level. Is there such variance among what constitutes "calculus" in high school curriculum or that big a difference between high school and college calculus?
To give you a bit more context about my situation, my institution is a moderately competitive engineering school that offers a 3 sequence calculus class that is mandatory for our engineers. With the one qualifier that we do not allow our students to use advanced calculators that can do graphing or symbolic manipulation on our exams, I would characterize our calculus 1 class as a pretty standard typical one where students focus on 5 main areas:
- They learn to use the traditional "plug and chug" formulas for differentiation and integration.
- They learn to "translate" word problems into mathematical problems they can solve with the tools of calculus (esp problems involving physics and engineering applications).
- They learn how to take limits (but we do not do delta-epsilon proofs).
- They develop an understanding of what an integral and derivative mean mathematically by applying the limiting processes along within a geometric setting to a few simple power functions.
- They learn and apply some of the standard theorems and definitions (continuity, IVT, Extreme Value Theorem, MVT, FTC).
One observation I have made is that the students who take calculus in high school are very reliant on formulas: I think those that struggle at the college level find it, for example, hard to apply the definition of the limit to actually derive a derivative. Which leads me to believe that at least some high school curricula teach very little of the underlying math and teach calculus as a bunch of formulas. But at the same time, I doubt this can account for the huge disparity in performance between I would expect from a student who took high school calculus and what I see.
Any insight would be appreciated.