After a lifetime of approaching math the wrong way, I took two college math courses this quarter with a newfound zest for math. These classes are integral calc and multivariable calc.
Integral calc started out okay, learning about Riemann sums and the Fundamental Theorem of calculus. But instead of spending a great deal of time gaining the intuition behind these things, we jumped into integration technique after integration technique.
Why on EARTH would we need to memorize and regurgitate a bunch of integration methods on toy problems for 6 weeks? It’s absolutely bizarre. I’m not taking this at some random JC either, this is at a top research university. This class is single-handedly destroying my enthusiasm for calculus.
When we have software that can do much more difficult integrals than we can with pencil and paper, why would we waste time memorizing Trig substitutions or integration by partial sums? Doesn’t that just make it a glorified algebra class?
This is, in my opinion, a common feeling after a “lifetime of approaching math the wrong way”. People are taught math in a very rigid rule-based formula/pattern method, and then when they contrast this against mathematical proofs they have a knee-jerk reaction against anything which looks even remotely like what they did before. The fact of the matter is, however, that you will need to be able to do some of this without aid of a computer.
When reading a proof, it is easy to take for granted that you are able to fill in the details between steps of the proof, when really all these steps are able to be filled in precisely because you have the understanding of solving equations (basic algebra) and working with inequalities (basic arithmetic), for example. The same thing holds true for proofs involving integration and derivatives.
Perhaps it’s best to leave it to those who really know what they’re talking about – Spivak writes in his chapter on integration that our motivation should be that:
- Integration is a standard topic in calculus, and everyone should know about it.
- Every once in a while you might actually need to evaluate an integral, under conditions which do not allow you to consult any of the standard integral tables.
- The most useful “methods” of integration are actually very important theorems (that apply to all functions, not just elementary ones).
He emphasizes that the last reason is the most crucial.
I would personally advocate that students should be wary of falling into the trap of thinking that such pedantic methods are beneath them. It is often easy to think you understand something at a high level, but you don’t truly learn what it is all about until you really get your hands dirty with it.