How rigorous should high school calculus be?

In UK 40 years ago the calculus was really rigorous first studying limits and series, we did virtually proved all results, except perhaps some integrals but we also had to memorize all formulas as we did not have formula sheet in the exams like they have today in UK A level. In Spain it is still like UK traditional syllabus from back then. I was shocked to see the huge changes in UK A level nowadays compared to back then. There has been even more radical change in UK physics eliminating almost all math. BUT after teaching the modern “international” UK syllabus (Pearson and Cambridge) I am a partly converted. The formula sheets and getting to the point skipping some tricky proofs is more fun. And usually there is a simple way to intuitively “get” the answer Eg. Looking at the x cubed graph can anyone imagine which power might work as the gradient function? Lots of kids can do that!

Speaking as an A Level physics teacher having seen what maths students do, they do stuff like quotient rule and chain rule. Hell, I even teach exponential decay with caclculus if I have a mathematically inclined class. But to get a C/4/5 grade in the edexcel maths papers you needed 21%. To get a 6/7/B you needed about 45-50%. In the UK schools I've worked at, they start the GCSE content in year 9 because of the extra content and rigour to cover. Add to all of this the peverse incentive to achieve good inspection grades and GCSE results, you have a factor in the teach to the test mentality and reduced time to teach important mathematical foundations. There are wider issues with recruitment itself. Being so desperate for maths teachers, I've seen economists and computer scientists recruited as head of maths. Sometimes I've even seen no pure mathematicians in a department. As much as I love calculus, any curriculum changes need to be thought out carefully.

I think that high school calculus is not supposed to be too rigorous. Calculus is a subject introduced in high school, and if it is too theoretical, then students can have a hard time understanding it.

It is pretty reasonable to think that proofs of rules of differentiation(like chain, quotient, product etc.) must be told. But as far as course is concerned one is not required to do much with the proofs of the rules. Important thing is to understand their applications in various domains of mathematics( and physics too) and being able to solve problems.

Also its not like that proofs aren't taught for any rule. They are certainly taught and sometimes derivation is more focused upon than the rule itself, especially in integration and differential equation part.

Also proofs are meant to be actually studied, if one goes on to major in mathematics in their college years. But at high school levels it isn't very much needed.

The question really is: how individualized is your high-school calculus course ?

If you have a kid who reads Godel Escher Bach and is fascinated by formal logic etc then probably you should offer a pretty rigorous calculus with $epsilon delta$ and basic topology and attempts at proofs of most theorems.

If on the other hand, your student intends to be an engineer then your focus might be more on how to apply Theorems and how to interpret derivatives and integrals.

All students of calculus should be taught the major theorems of calculus (IVT, MVT, EVT and FTC I and II etc.) Furthermore, I think all students of calculus should be taught how calculus extends what we knew from algebra alone. For example, the binomial series vs. the binomial theorem. Compare and contrast. I suspect graphing and interpreting the derivative is largely short-changed in our current calculus because interpreting derivatives requires a mastery of inequalities we're afraid to test in the general population. Calculating a derivative is easy. Finding which $x$ make $df/dx >0$ is almost always much harder.

Some of the problems I just described are not what we would consider "proofs" formally speaking, but they require logical analysis which is more difficult than epsilon-delta proof $lim_{x rightarrow 1}(3x-7) = -4$.

So, what is the audience ? Can they factor a polynomial without flinching ? Can they solve a quadratic inequality ? Do they have a clear understanding of basic function graphing ? What about trigonometry, do they have mastery of identities and domains and solution sets to reasonable trigonometric equations ? Depending on these questions we may or may not be able to offer a real version of calculus. We may have to be careful to not offend the audience by teaching them prerequisite material in the midst of this wild new world of limits, derivatives and integrals.

As far as teaching to the test goes, it would be nice if there was a lot less predictability in the test. It ought to be a goal to make the only way to prepare for the test is to teach the whole of the subject. This includes proofs. There ought to be a reward for students who do more than just learn routine calculations. But, until that day, it is crucial for teachers to teach as if there was no test. Otherwise, well, look at the schools in America...

Students who go on to be math majors will get a later course in theoretical calculus. The vast majority of science and technology students (i.e. non math majors) will NEVER have such a class. And not need it, either (to support their mechE, chemistry, etc. majors courses).

So, I think the current approach is fine. The math majors get taken care of with rigor, later. The others don't, but don't need it. This is more efficient than cramming rigor into people who will never need it. And it preserves option flexibility for students not sure if they will study math or science, after high school.

I would also add that it is not pedagogically clear that rigor prior to manipulation is the best way to learn a difficult topic, given the inherent imperfections of "meat computers". For instance, would you try to force first graders to learn formal properties of numbers before arithmetic? Would you make high school algebra students learn Galois theory before solving quadratics or factorable higher polynomials? Do I have to have a perfect handstand (which is non-trivial on rings, try it) before I'm allowed to just swing circles on the apparatus (as a beginning gymnast).

So I think the current approach is just ducky. They even really get SOME exposure to the theory topics, but are not required to master them. Maybe a little bit like how LaPlace transforms are handled within the time restrictions of a typical ODE class. This is done as an exposure, with some basic translation back and forth. However, transform mastery or derivations are really only done by the EEs and systems engineers, who use that topic a lot. And they do that in more specialized classes, later. But for the mechEs, at least they have briefly seen it. So if some minority of them (say doing controls work) needs to dive in more later, they'll at least have sort of heard of it before before diving into more difficult/detailed work.

All that said, I took AP BC in the early 80s at a rather competitive public US high school. So we did see the epsilon delta. And I was fine with it, but I was spending an inordinate amount of time on calc class. But most of the (pretty strong) class hated it and had other demands on their time from tough courses in chem, English, etc. And the calc teacher said, you won't need any of this for the rest of the course when we do partial fractions and related rates and all that jazz. And she was right.

OK, yeah, yeah, evidence by anecdote. But the point really is that very few people will need/benefit from the rigor push, when learning high school calculus. And it is important to think of the overall audience. NOT "well, I could handle it" or "well, I like it", but instead think about the audience and their needs/desires, which may be DIFFERENT from yours.