Final Examination

veronica d in reply to Michael Cain says:

That’s a good question.

First, I’m not a professional educator, so I’m just kind of speculating here. There are people who research these kinds of questions as their job, so maybe I’m wrong.

I’ll say this, if you’re reading through a journal article, and you have to fire up your symbolic integration engine every time you see a basic integral, that is a kind of “cognitive load.” By contrast, if you just “see” the solution, without having to think much, then you’ll be able to hold more of the “higher level” concepts in your head at once.

For example, if you show me a weird double integral that includes the sinc function, multiplied by some other function — well if I’ve played with those integrals enough I might recognize that sinc behaves like a delta function, and what I’m seeing is a kind of Fourier transform.

Or I can just put it into Mathematica and it will spit out some answer with no obvious connection to the formula I put in, because there is a very weird path from sinc to delta to Fourier.

So now I’m thinking, “What the fuck? Why does that work?” By contrast, I could be thinking, “Oh yeah, that a delta function. Okay, now what are they doing with it?”

In theory, an instructor can show students how the delta function works, and how it leads to the Fourier transform. However, will they remember it? Will it get “baked in”?

Speaking for myself, I remember topics better when I’ve actually had to do hard problems, compared to ones I only saw briefly in lecture.

Moreover, I suspect there is a great deal of value in learning to “juggle layered, complex, abstract ideas.” It’s something that requires practice. It doesn’t just happen by magic. It takes work.

I’m sure there are many specific abstract tasks that one could use to train such mental faculties, but symbolic calculus is a pretty good one. We have a lot of experience teaching it. It’s useful in a lot of fields.

(That said, proving binomial identities, just to pick a random example, also exercises those “thinking muscles”, and is perhaps more useful in compsci. I’m open to many ideas.)

One might say that students can learn these skills by “juggling” the “layered, complex, abstract ideas” from their particular domain, rather than pure math. Perhaps. I don’t know. I think it depends on the domain. Physics — yeah probably. Economics — OMG yikes.

I have an analogy. When learning piano, you learn scales. Lots of scales. However, you don’t simply play scales in actual music. You do other stuff. I suspect, nevertheless, there is still value in drilling scales. I think “pure symbolic math” might play a similar role. It’s building a foundation of abstract problem solving that will be useful in any technical field.

That said, I also think graphical calculator are god-tier awesome. Moreover, I see Mathematica is a brain-force-multiplier. I’m very pro-tools.

I want both the “drilling basics” and the leveraging of awesome tools. Each has their place.Report