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In the UK, calculus taught in secondary school focuses mainly on computation of derivatives and integrals and solving simple differential equations. There is a small amount of discussion about limits and the definition of the derivative, but students are not required to know things like: proofs of the product, quotient and chain rules, riemann integration, FTC, MVT, etc.

Examination questions (national exams) usually test ability to perform calculations rather than deeper understanding, so from a results standpoint, there is not much incentive for teachers to spend a long time motivating concepts or discussing proofs.

I also realise there has to be a balance between a rigorous presentation of the material and student enthusiasm and interest. There is a risk that students might become discouraged if things are done too formally. For example, I do not think there would be much value in showing students the $\varepsilon-\delta$ definition of a limit.

How rigorous should high school calculus be? Which things are better left for later and why?

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    $\begingroup$ What is the goal of high school calculus? $\endgroup$ – Xander Henderson May 10 at 12:50
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    $\begingroup$ Students who will specialize in mathematics in university should do proofs (at some point). Other students, not. $\endgroup$ – Gerald Edgar May 10 at 12:59
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    $\begingroup$ Hi friends; please consider putting answers to the question (even short ones) in an answer instead of a comment. Comments that are mini-answers or partial-answers don't work well since they bypass the voting system, can't be accepted, don't have comment threads of their own, and suppress other answers. $\endgroup$ – Chris Cunningham May 10 at 13:21
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    $\begingroup$ @Chronocidal No, I mean 6th form. While C4 no longer exists, the vast majority of C4 questions from the most popular exam board (Edexcel) were just plug-and-chug. The current syllabus is a slight improvement, but not much. $\endgroup$ – A. Goodier May 11 at 15:27
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    $\begingroup$ "Rigour" means different thing in different communities. For mathematicians, rigour is in proofs. For computer science, rigour is in correct programs. For physicists, rogour is in knowing that the numerical results are correct, to so-and-so many decimals! I have seen physicist sad of failing rigour of mathematicians, because they just did numerical integration without bothering of to-how-many-decimals the result were correct. $\endgroup$ – kjetil b halvorsen May 11 at 16:34
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Not very rigorously at all, but that doesn't (and shouldn't) mean just memorizing calculations. (I should add that I'm basing this on my experience teaching calculus to non-major college students, but I think the relevant issues are similar.)

Mathematicians have a bad habit of conflating rigor with conceptual understanding. A lot of this seems to come out of mathematical training, where we internalize that something hasn't been said clearly until it's been said in the rigorous language of formal mathematics.

But for non-mathematicians (and, secretly, most mathematicians) that's not really how understanding works, and it's really not true for high school students. Most high school students don't have the background to understand or make sense of proofs, so teaching calculus "rigorously" isn't really a step to conveying the concepts: it's play-acting, where the teacher goes through some steps to satisfy the teacher's internal sense of what a justification is, while the students watch. (Witness what happens to 𝜀−𝛿 in a lot of non-major calculus courses which keep them: students do fill-in-the-blank "proofs" where they "prove" specific values of limits from a small class of functions, because they don't actually understand how to work with the quantifiers, but they can memorize how to do the specific calculations.)

That doesn't mean the only thing left to do is plug-and-chug calculations.

  • There are non-rigorous conceptual explanations. These have real explanatory value, even when they're not precise or might not cover cases which aren't going to be considered in the course. (People did calculus with informal infinitesimals for centuries!) Even for students who are laser focused on the exam, these can be useful because understanding at this level makes it easier to memorize, problem solve, and catch one's own mistakes.

  • There's a lot of space for laying the foundations of more rigorous math for students who might go on to study more later. I'm always astounded at how much we talk about covering proofs in calculus for students who don't even know what a theorem is: students at this level often struggle with applying theorems with the form "if ... then ...". They treat the conclusion as true in all situations, or they use the converse instead, or they decide they can't use it in a situation where the variables have different letters.

One of the reasons there's not much point in dealing with rigorous proofs at this level is that students don't know what they're for. Mathematicians care about proofs because they solve a problem; but students don't know what the problem is, so they're not likely to care about the solution. One of the most useful things a calculus course can do to prepare students for later math is to introduce the problem: that we have a lot of intuition for very nice functions like low degree polynomials, and only some of it extends to more complicated functions (inverse trig functions, rational functions, and so on). Theorems like IVT, MVT, and FTC which apply to many but not all of the functions students know, provide a chance to begin thinking about those issues.

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    $\begingroup$ The second paragraph is a great observation, thank you. And as an example of something I would highly recommend for giving that conceptual understanding, the 3blue1brown 'Essence of Calculus' series on YouTube is tremendous: youtube.com/… $\endgroup$ – dbmag9 May 11 at 13:39
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    $\begingroup$ This is a brilliant observation. As a student in some of these non-major college classes I can confirm that "more rigorous == more understanding" really does not hold for non-mathematicians. The term "play acting" describes what we did in the most rigorous courses fairly well. We memorized how the teacher wanted stuff done without having any notion of what we actually did, or why. $\endgroup$ – xLeitix May 11 at 16:21
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    $\begingroup$ As a professional mathematician, I wish I could upvote each paragraph on its own, not just +1 for the whole thing. But +1 for sure! $\endgroup$ – Greg Martin May 12 at 7:43
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    $\begingroup$ +1 for the reasons already given. Too much rigour in high school calculus lessons could lead to this on steroids. $\endgroup$ – J.G. May 13 at 19:05
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    $\begingroup$ @J.G.: I put that comic on my door right before the calculus final one semester and never took it down. $\endgroup$ – Henry Towsner May 13 at 21:49
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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...

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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.

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  • $\begingroup$ I remember that in first year physics we were introduced to Schrödinger's Equation, and by "introduced", I mean it was basically: "here's Schrödinger's Equation; here's the equation for a particle in a box; here's what they mean but you aren't really expected to understand it yet; now let's prove that this particle in a box equation is a valid solution to Schrödinger's Equation" $\endgroup$ – user253751 May 14 at 13:01
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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.

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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.

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  • $\begingroup$ I don't disagree, but my question is more about whether there is value in presenting things in a more rigorous way than is required. In the context of recruitment challenges and low standards, it cannot be expected that all teachers teach A-level mathematics students proofs of results from calculus. $\endgroup$ – A. Goodier May 13 at 9:34
  • $\begingroup$ I do agree with the rigour point. Teaching attention to detail is an invaluable skill that this can bring. Unfortunately this sort of "rigour" has fallen victim to a climate with very perverse incentives of teaching to a test. $\endgroup$ – Kishan Bhatt May 13 at 9:44
  • $\begingroup$ Being so desperate for maths teachers --- Given this comment, I think a slightly broader job search would take care of this, at least unless almost every other hiring manager (or the equivalent) tries the same thing, but I think this is not likely. $\endgroup$ – Dave L Renfro May 13 at 19:41
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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!

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    $\begingroup$ The bit about what happened 40 years ago is interesting, but I am not sure that it is relevant. Regarding the rest of your answer, I am a little bit confused---can you please edit it to give a little more detail about what educational system you are describing (is it Spanish or British?). Can you explain in more detail what is meant by the "UK syllabus"? Please do not assume that we are all familiar with what this means, as most of us are not in the UK. $\endgroup$ – Xander Henderson Jun 30 at 23:41
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    $\begingroup$ I’ve edited hope it explains better but I wasn’t at all sure which things I was saying that were unclear. I don’t understand why you want me to explain what I mean by the “UK syllabus” for math A level Pearson. Isn’t that well known or anyone can search that. Pearson is the worlds biggest education giant? Also the previous responder talked freely about A level without defining them and they are the UK thing. $\endgroup$ – blanci Jul 2 at 23:43
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    $\begingroup$ You should strive to write answers which are self-contained. I am not in the UK, and I have no knowledge of the syllabi for classes taught there. I should be able to understand your answer without resorting to Google. $\endgroup$ – Xander Henderson Jul 2 at 23:59

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