Difference between high school and college calculus courses

Question

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:

1. They learn to use the traditional "plug and chug" formulas for differentiation and integration.
2. They learn to "translate" word problems into mathematical problems they can solve with the tools of calculus (esp problems involving physics and engineering applications).
3. They learn how to take limits (but we do not do delta-epsilon proofs).
4. 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.
5. 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.

Answer 1

Assuming we're talking about mostly US students, most American high schools teach calculus in a way that's very focused on the AP test. The pressure to get students through that with an adequate success rate seems to have lead to some streamlining and cutting corners in ways that mean that what gets taught doesn't really stick, and also doesn't set students up for success when they retake calculus in college.

I've noticed a number of ways that my students' high school courses seem to cause them trouble in college:

1. Reliance on formulas: As you noticed, a lot of high schools teach calculus in way that's overly reliant on formulas. That has some direct effects - students struggle with figuring out how to apply formulas which are more abstract (like the definition of the derivative), and aren't prepared for problems which call for a non-algorithmic decision about which formula to use. I think it also has an indirect effect: students have learned that formulas are what math is, and some of them have learned to tune out anything other than formulas. I've seen students who rely on their notes leave out whole sections of class because we spent half an hour on an idea rather than a formula, and they didn't see anything in it worth writing down.
2. Pre-calc preparation: High schools are under a lot of pressure to push students through calculus, so they often skimp on algebra and trigonometry. Calculus courses, on the other hand, often work with problems that assume students can reliably do algebra. Even when students can eventually do the algebra, it's slow and difficult for them; for instance, I notice that some of my students have trouble learning calculus because they get distracted by the algebra - it takes them forever and they have to spend a lot of time thinking about it, so they can't focus as much on the new material.
3. Streamlined problems: To accommodate the previous item, high school courses seem to primarily give problems where the algebra is simple. I notice that my students are sometimes uncomfortable with answers that aren't very "clean". For example, my students can actually factor quadratics on sight very well (better than I can), but aren't very comfortable with the quadratic formula (to the point that some of them don't remember having seen it). That presumably reflects that in high school, problems were always rigged so that the quadratic would factor.
4. Straightforward problems: My students expect problems to always be short, with only one idea in play. I see a lot of students completely flummoxed by problems which need them to apply two or three steps in a row - even when they've mastered each step. Again, this is a way that the high school and college versions can say they cover the same material, and yet mean something very different by it. (A student once told me that she wasn't prepared to start problems unless she already knew what all the steps were going to be to finish it. I think it reveals a lot about what sort of problems she was being asked to solve in high school that she'd made it to college with that approach.)
5. Overconfidence: Finally, as already noted in the comments, because students don't realize that a college course has different expectations, some of them are overconfident and don't put in the necessary work to relearning things they think they know.

Answer 2

This is not really an independent answer, but more of a supplement to Henry Towsner's answer, proposing some more possible reasons.

7. AP classes in the US have become such an entitlement that there is often little pretense of actually preparing students for the AP exam. Particularly in school districts that draw from low levels of socioeconomic status, it's not uncommon to see that zero students pass the AP exam in a particular course. In my area (near Los Angeles), this has prompted some high schools to get the community college to teach classes at the high school campus, for college credit. That way the students don't have to pass the AP exam.

8. Students' skills decay, and this decay is especially rapid among students who memorized rules and algorithms without understanding why they worked. The less you understand, the more you have to rely on memorization.

9. You don't say whether you're at a state university, but if so, then that may explain a lot. There has been a huge, long-term trend of grade inflation, but the one place where this has been least pronounced is in STEM courses at state universities, especially state universities with low admissions standards. While these schools have continued to apply relatively consistent academic standards over the decades, public high schools, private universities, and the humanities and social sciences have dropped their standards gradually but, in the end, dramatically. Therefore the students who show up in your class may be experiencing real grading standards for the first time.

Since many old-timers like to groan about grade inflation, and there may be some skepticism about my claims above, here is some info:--

Stuart Rojstaczer is retired Duke professor who has made a project out of researching this sort of thing. His web site is gradeinflation.com. Here is a newspaper op-ed piece summarizing some of his research. An interesting book-length treatment, with original research and a prescription for reform, is Valen Johnson, Grade Inflation: A Crisis in College Education, 2003. A big theme of Johnson's is that grade inflation is driven by teaching evaluations, and he offers clever experimental evidence demonstrating that there really is a cause-and-effect relationship between grades and evaluations. He also gives data demonstrating differences between standards in STEM and other fields, including pretty convincing comparisons of grades students earn within their major and outside it.

Answer 3

This is a follow on to Ben Crowell's follow on answer. First, I would like to agree with much of what he says, and say that my personal experience agrees with it. I hold my students to a high standard (and sometimes get poor reviews for it) but am in a fortunate position to have the support of my department (at a public engineering university) in doing so. By way of illustration, my grade distribution in the current semester is:

2/63 - A

13/63 - B

17/63 - C

19/63 - D

12/63 - F

along with 18 drops since the start of the semester. I place a heavy emphasis on basic number sense and word problems. I heavily penalize errors in basic algebra (as a graduate student I teach primarily college algebra, trigonometry, and pre-calculus) and do not spend class time on things they should already know, though I offer to help in office hours. I also categorically refuse to give students a "I showed up" C. I have sometimes looked at a students Grade in my course vs their overall GPA and can say with confidence that there is grade inflation in other disciplines.

Anecdotally, after trying out multiple majors, the harsh grading is part of what drew me to mathematics. It was the first subject I found both interesting and challenging enough to hold my interest, without being told to memorize reams of information.

Having taught high school before, I would say that there is how I wanted to teach my courses and how I was told to teach my courses. That said, there are school districts with excellent calculus classes. The issue is it is very hit or miss. It requires both a good teacher and a district willing to care about more than just the AP exam pass rate. I was fortunate to have such a course in my own high school career.