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.
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I agree with you about integrals, but I don't know what you mean by "determinant proofs." Still, this question seems to be primarily a matter of opinion. – saulspatz Sep 18 '19 at 15:02
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related : Why do we need to learn integration techniques? – Raymond Manzoni Sep 18 '19 at 15:20
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I've always wondered if there is really no way to teach both. – Sam Sep 18 '19 at 15:24
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@saulspatz Determinant proofs are essentialy proofs where you have to factorize a determinant without opening it up. You can only use row and column operations. It's completely pointless though , as opening the determinants up is often faster. – PhysicsRuinedMySocialLife Sep 18 '19 at 15:37
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@IshaanParikh For a $10\times10$ determinant? If by "opening up" you mean expansion by minors, that's completely impractical for a determinant bigger than about $3\times3$. The theory is everything when it comes to determinants. As a calculation method, they're not very useful at all. – saulspatz Sep 18 '19 at 15:40
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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.
Adrian Keister
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I do agree with your argument, but the way math ends up getting tested like this doesn't actually require one to think much. If one has practiced enough , one will remember the problems and solve it. Getting the integral of root tanx or the surd form of cot 7 , cannot be found very easily without having seen solutions. This leads to rote memorization of math. Often scaring away stude ta away from the subject. – PhysicsRuinedMySocialLife Sep 18 '19 at 15:39
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Rote memorization has gotten a bad rap, in my opinion. While no serious mathematician would argue (and I don't) that rote memorization is all there is to it, I do believe there are facts, theorems, etc., that are simply worth knowing off the top of your head. An expert in any field, in fact, will have the basic facts of that field at his fingertips, so to speak. Without those basic facts at his immediate disposal, he will not be able to do the higher reasoning quickly. See the book Why Don't Students Like School? by Daniel T. Willingham for an explanation. – Adrian Keister Sep 18 '19 at 15:48
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The structure of any subject is like this: basic facts you should know (the "grammar" of the subject), then how do those facts fit together (the "logic" of the subject), and finally how you express yourself or apply the basic facts plus the logic of how they fit together (the "rhetoric" of the subject). – Adrian Keister Sep 18 '19 at 15:50
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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...
Surb
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It's not against learning basic trigonometric formulas or integration. Not against learning how to apply them either. I'm against the part where you get forced to memorize methods you do not understand/can't do yourself. It leads to write memorization of math like a tool to solve probleams. For example, Cramer's rule can easily be used to solve simultaneous equations but what use is it of without understanding the geometry behind it. Rather have a computer do it instead. – PhysicsRuinedMySocialLife Sep 18 '19 at 15:48
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"I'm against the part where you get forced to memorize methods you do not understand/can't do yourself." : At this point I completely join you. But what is interesting is not learn them by heart, but to be able to re-find them. At my opinion, it the problem in many ingeneer program : they see theorems, proposition... without any proofs, and that's of course silly, since not knowing the proof is a real handicap to understand the result. Often, theorem can have quite esoteric hypothesis, and learn they by heart is really idiot. – Surb Sep 18 '19 at 16:01
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But as we see, you can not seprate theory and practice :-) Just an stupid example (but quite relevant) : can you imagine learn by heart the formula of integration by part ? For my part, I cannot, I always re-find it. I admit that the proof is very easy. But when I was in High-school, we just had to learn it by heart, and they never show us the proof... this is of course completely stupid. @IshaanParikh – Surb Sep 18 '19 at 16:03