How is teaching calculus in high school different from teaching calculus in college?

I've taught calculus in college for five years, and it's always interesting to see students coming in who already had calculus in high school. Many of them do very well, and don't even seem like they needed the course. Others struggle.

I took Calculus in high school, and I don't remember doing Newton's method or his law of heating and cooling. In fact, even though my high school class was a 'double class' (counting for two credits), I don't remember covering as much material as we did in college; but I didn't feel behind in my next classes.

In the United States, what are the main differences between the way high school Calculus and college Calculus are taught?

I'm interested in amount of time spent in class, average homework loads, topics covered, and any other differences you feel are important.

Edit: I'm specifically talking about AP Calculus BC in highschool compared to a one year sequence of Calculus at a four-year university. Both as currently taught.

• I doubt that it is possible to give an accurate characterization of a typical US AP calculus course. There's an excellent science magnet school in my town where I'm sure AP calc is far more rigorous than at the community college. But within the last decade or so, there has also been a vast proliferation of AP courses and vastly increased numbers of students taking the courses. It's not uncommon for a certain school to offer an AP course in a certain subject, and have 10% of their students or less get a passing grade on the AP exam. – Ben Crowell May 8 '14 at 14:55
• There are wildly different expectations at different high schools and colleges, and even major differences between different teachers at the same schools. For instance, my high school Calculus experience was 1 hour a day of lecture, about 4 hours of homework, and we were expected to know (in full detail) the proofs of all of the major theorems, derive the formula for arclength, epislon delta proofs, etc. I have never had a math course since which has matched the level of that Calculus course. – Steven Gubkin Oct 2 '16 at 19:43
• It is also not clear whether you mean to ask about "rigor" per se, or "substance" (which might have a broader, open-ended sense). E.g., I have some interest in rigor, but far more in substance (which is what would, as a side effect, generate some interest in rigor). – paul garrett Jun 29 '17 at 22:52
• One observation: in high school, kids taking calculus are among the relatively elite. In college, everyone takes calculus. This obvious affects the ambience, the self-images, and so on. – paul garrett Jun 29 '17 at 23:02
• @paulgarrett I'm not sure taking calculus is so common in universities. It should be, but, is it? – James S. Cook Jun 30 '17 at 1:56

I took AP calculus BC in high school, and since TA's and taught calculus at two universities since (I'm a grad student, but I've lectured as well as TA'd). These experiences have amalgamated together into this answer.

I had 170 days of calculus instruction in high school, each one hour long. We used a standard college text of Larson and Edwards (whatever edition it was at the time). There were weekly homeworks, a test roughly each month, and a final each semester (and an AP test afterwards, I suppose). We did not prove things, and we did not need to use the $\epsilon- \delta$ formulation of a limit. In practice, most functions were continuous and differentiable. In fact, most functions were elementary and smooth. We did learn Newton's law of cooling, some numerical integration bits like the the trapezoid rule and Simpson's rule, Taylor series with remainder, u-substitution, integration by parts, trig substitution (to mention topics of various difficulty that I feel might be omitted in some high school classes). We did not talk about many differential equations (we learned initial value problems, and Euler's method, but I simply cannot remember if we did much else).

A very important component that was missing from my high school education was understanding why anything was true. We might heuristic our way through things and prove a (small) subset of the things we came across, but in general there were few proofs. Math was a gift, set down whole before us like a swaddled babe from a stork.

With respect to the homework, I never did any calculus homework at home - many of us were able to finish it all during school hours. In this regard, we couldn't have done too much homework timewise. But the homework was graded completely and returned with notes, and we did an incredibly large amount number-of-problems-wise.

Now, university calculus. At my current university, calculus meets for almost 4 hours per week: roughly 2.5 hours for lecture and roughly 1.5 hour for recitation. There are roughly 15 weeks weeks, so that there are approximately 60 hours of instruction per semester. My high school calculus BC course is the equivalent of two semesters of calculus at my university (roughly corresponding to differentiation and then integration), so we should really be thinking of 170 hours vs. 120 hours. (At my undergraduate university, calculus courses met for 6 hours per week and BC calculus was roughly the equivalent of the first semester and the first fourth of the second semester of calculus - the other three-fourths were actually linear algebra).

We teach Newton's law of cooling, but we do almost no numerical integration. In fact, we barely cover Riemann sums at all. The lectures will talk about Riemann sums insofar as to define integration, and they aren't really brought up again. (At my undergrad, we similarly skimmed over Riemann sums, though there was more variability: some professors care and others don't). Thus we don't talk about any numerical integration techniques. We don't talk about "differentials," by which I mean approximating $f(x+\delta)$ by using some sort of $f(x) + f'(x)\delta$ style argument, whereas in high school we did, including some with multiple steps (like calculating $\arctan{1.5}$ using three steps of length $.5$ from $0$, or something). We similarly skipped Euler's method for approximating solutions to differential equations. This is a recurring theme: we skimp out on numericals.

I partially attribute this to calculator usage. In high school, we used calculators as part of the AP test uses calculators. At my university, we use no calculators ever - so no numerics and usually the problems themselves are easier. I don't know whether this is a good thing or a bad thing.

At my university, we teach $\epsilon-\delta$ but do not test it (a conceptual hurdle that the powers that be have decided isn't worth the struggle, but which is not skipped at my undergrad). We teach Taylor remainder estimation but do not test it (another numerical problem skipped). This was a large and challenging part of high school calculus entirely missed. (This was not skipped at my undergrad).

On the flipside, we do teach first and (some) second order differential equations during the second semester - students learn integrating factor and separable differential equations in particular. Sometimes, depending on the professor/speed of the class, we teach power series solutions to differential equations. The homeworks that we assign were very long (at my undergrad) and are moderately long at my current university. But the graders at both were hired undergrads, for whatever that is worth.

And every lecturer I know proves everything from the ground up, even though the students are not held directly accountable to the proof material.

In general, I find high school calculus to be a much better course, with more contact hours, required attendance, more and more completely graded homeworks, and most topics (except $\epsilon-\delta$ and some differential equations bits that are covered in some university classes). In my university classes, we cover fewer subjects, but we cover them deeper (i.e. with proof), but there are fewer (though perhaps harder) homeworks that are graded by undergrads. And we omit most numerics.

I would suspect a student who got a 5 on the AP calc BC test to get an A or B at the end of the second semester of calculus at my university. But before one concludes that I think high school better, I find that most of my students (including those who took calculus in high school) have almost no conceptual understanding of calculus coming in. Some do when they pass the course.

• I've noted that trigonometric substitution is not taught in most BC calc classes(it's not tested on the AP test so teachers probably figure why should they). Looking at the calculus courses at my university I would expect someone who did well on BC Calculus to do good in second semester, but not know the majority of it to start. – ruler501 Mar 27 '14 at 6:50
• I mean this constructively: I think this answer is longer than it needed to be. It seems like the main differences you found could have been put into a short bulleted list, and then your own interpretations would appear more front-and-center as the meat of the answer. That said, I especially liked the insight about calculator usage being a driving force behind many of the differences. Thanks for the good read! – Chris Cunningham Mar 27 '14 at 21:15
• Losing trig substitution would be another AP watering down I guess. I remember it well in 80s AP BC – guest Jun 29 '17 at 22:53

Although I've not taught in 4-year universities, I've taught in a small variety of up-scale "research" universities for some decades. Even with that disclaimer, I have observed the math-sociology of nearby 4-year colleges lo' these last 30+ years. In particular, I'd wager that the dominant issue in appraising reaction of calculus students in 4-year colleges is sociological, with the corresponding issue in high schools being (with random perturbations depending on school district) somewhat less so.

As in other comments and answers, and as might be clear for many other reasons, it's not that "calculus" has changed much, ... er, at all, ..., in the last 150, or, really, maybe 300 years. True, our collective official opinions have drifted, and the Cauchy-Weierstrass mythology has been dominant for many decades. Still, that doesn't necessarily mean anything about peoples' understanding of calculus and how to use it in amazing ways.

As in my comment above, in high schools in the U.S., to take "AP calculus" is a significantly sociological thing... and, at the same time, is (in my direct observation) often degraded to a sort of "you've earned an easy ride, based on previous work", so that kids do not actually learn much. (And, sadly, invidiously, I do speculate that, statistically, mathematics is the least-competently-taught high school subject. I say this despite both my parents' being high school math teachers. It's not meant to impune anyone's ethics, etc, either...)

"In college", although quite a few kids don't manage to achieve this minimum, "showing up for the midterms and final" no longer garners high praise. :)

The chief psychological symptom is the expected: kids who were once "the best", have managed to be admitted to situations where everyone was "best in their high-school", and this can be confusing/comforting, depending.

As to teaching of calculus: I do seriously claim that it is a special case of that very general sociological situation and its corollaries. That is, seriously, it's not about calculus per se.

• Paul, you should clarify what you mean by the "Cauchy-Weierstrass mythology". Some editors may take you at face value :-) – Mikhail Katz Jun 30 '17 at 7:17

I took AP BC calculus in early 80s in a top suburban public school. The sequence there was Algebra, Geometry, Algebra 2, Trig (1 sem), Functions (1 sem), Analytical Geometry (1 sem). Then AP Calc.

Most of the GT kids took "Algebra 2, trig" (2 semesters instead of 3). The Functions class was pretty cool: inequalities, precalc (basic differentiation and integration), other basic properties of functions and relations. Analytical geometry was pretty much the standard thing with translation and rotation of coordinates (what a pain in the ass), conic sections, etc. I would note that it did do a good job on how to analyze and graph strange, difficult expressions (looking for critical points, using calc for that, intercepts, asymptotes, etc.). This is pretty standard, but some (even older, classical) texts do not do a good job on curve visualization.

For AP calc We used Thomas Finney, 4th edition. Designed for AP Calculus. We did cover epsilon-delta. If not in AP Calc, we hit it in "Functions". I sorta think that is how we did it. Also remember doing Archimedes method for parabola. I think that was all year before, actually in Functions class. So we covered some of it to get exposed. But we definitely did not jump down the rathole of "baby real analysis" like Caltech does.

I had ZERO problems handling higher level classes at a decent (not Caltech, but not community college) school afterwards. Calc 3, 'zoics, EE, etc. Actually the school was the US Naval Academy.

I have read that AP Calc has had some minor changes to water it down since I took it. (They do a little less rotational solids integration methods, they do a little less series methods but still some, and they cut second order constant coefficient diffyqs.) And they do calculator games (most colleges still don't do this). What can I say, that is ETS, being ETS. They have gotten a little captured by the ed reformers and more interested in moving how things are done than in just copying college (for AP). But still way less damage in Calc than in Chem or Bio.

I really expect that for most people calc is a pretty standard thing (you can just read Granville from 100 years ago and it is almost like a template for how things are done now, still.) I suspect the much bigger factor in the differences amongst your students is native ability along with how many homework problems they did in high school. Calc is calc. Sure there are differences in a course. But student variability is the bigger factor.

 Just read more of your questions asking for differences. Well obviously one is high school and one is college. More hours in the room in HS. More homework. More interaction and feedback. More grading. More going to the blackboard. More frequent testing. Basically (IMO) a superior pedagogical situation than college. Then again, I went to a trade school for college and they taught like high school (and did so because they were more interested in training people than a laid back attitude of sink or swim). But other than that, same content man. Math is math.

I think it is important to avoid lumping all "colleges" (in the broad sense) into a single category but distinguish between small (and expensive) colleges, on the one hand, and large universities, on the other. I had a conversation with a budding dentist a few years ago in Chicago who engaged me in a conversation and I ended up telling him about my interest in teaching calculus with infinitesimals. His reaction was completely unexpected. He said: "How can you teach calculus without infinitesimals?" Apparently at his small (expensive) college in California they have been using infinitesimals all along (without letting the big shots in the AMS know what they are doing).

I would say that the difference between highschool and small college calculus teaching, on the one hand, and university teaching, on the other, is that the former is more likely to be practical-minded whereas the latter is more likely to be ideologically committed to what research mathematicians view as the "right" way of doing mathematics since Weierstrass.