What is the point solving hard integrals, trigonometric proofs and determinant proofs

Question

There is this culture among highschool boards where they would rather spend time teaching you how to integrate $\sqrt{\tan(x)}$ instead of helping you develop intuition on how integration by parts or product rule works. While I totally understand that you must be proficient in things such integral solving, they rather teach you 'types' of integrals. These are so hard to the point where people learn what substitution must be made. Shouldn't time be spent on developing intuition and application rather than learning methods to solve trig identities or integrals. Especially cause the really hard ones are things one would never tackle in real life.

Answer 1

I actually couldn't disagree more. In the 1990's in the USA, there was a Calculus Reform movement to do exactly what you suggest: instead of spending time learning how to calculate, the idea was to make sure the students had an intuition about the concepts. The result, in my opinion, was rather disastrous: students can't actually do the calculations in real life! Contrary to many students' opinions, there are no prophets in math classes. Students really don't know the future, and they don't know what integrals they might come up against.

They might have access to a computer algebra system (CAS) like Mathematica (quite a good one, for sure), but without understanding the basics of what the CAS is doing, they're at the mercy of whatever answer the CAS decides to spit out. There's one quite good calculus book (Stewart, I believe) that, for a long time, had a section called "Lies My Calculator Told Me". Exactly! As one physicist put it, "You should never calculate anything unless you already know the answer." He meant that you should be able to verify independently anything you tell a computer to do.

I'm not arguing that students shouldn't have an intuition about the concepts; what I'm saying is that such intuition is not a substitute for being able to calculate. Indeed, I have found often that being able to do the calculations (and especially doing enough of them) produces the desired intuition, anyway. Then you have mastery of the concept!

Yes, integration is difficult. So is life! You do not usually solve hard problems (pretty much the only remaining problems) without a lot of hard work.

Answer 2

Honestly, I completely disagree separating mathematics and calculus. Could you imagine being a mathematician without knowing computing technical integrals ? I personally do not. How could you develop intuition on something that you are not able to compute. Even if you see how to compute something, often, when you do calculations, surprises arises. For me, spending more time on theory without practicing is exactly the same as learning the theory of driving without driving. Theory is good, but you learn by practicing. And trust me, stupid/long/hard calculation is a very good way to get used to the material.

By the way, proofs are not only pure theory, there are a lot of tricks, subtly calculations... How will you proof a result if you are not confortable with technical calculations ? Just a stupid example : you need convergence of a series to deduce a result. And imagine your series is upper bounded by $\sum_{k=1}^\infty (1-\cos\left(\frac{1}{n}\right))$ ? What will you do if you don't know basis trigonometry formula ? Wolfram will not always be here to help you...