Final Examination
Clare Briggs
Clare Briggs is a famous cartoonist who lived from 1875 to 1930. Poems by Wilbur Nesbitt.
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“You’re not always going to have an encyclopedia in your pocket.”Report
This has become a problem in a lot of disciplines and is probably worthy of a post. Much of the final testing in post-graduate work is complicated memory exercises. No one does actual work under the conditions of a bar exam, or a typical Ph.D. field exam. Calc I got me thinking the last time I taught it. In real life, a person can access Mathematica (or similar) software on their phone from anywhere in the world, and the software is enormously better at indefinite integrals than any human. Why does Calc I put such an emphasis on doing indefinite integrals by hand?Report
Why should someone who’s been through a Calc I course not be capable of doing indefinite integrals by hand?Report
It depends on the integral. I expect any decent STEM student should be able to integrate polynomials in their head. I expect them to understand how to juggle a few sines, cosines, and the exponential. That stuff is foundational. You shouldn’t need Mathematica for that — although as your formula grow, the advantage of software is you won’t make minor mistakes. It turns out Mathematica will never screw up signs. That’s a pretty big advantage.
That said, I wouldn’t drop something like the Dirichlet integral on someone without preparing them for that specific problem. It’s rather non-trivial: https://en.wikipedia.org/wiki/Dirichlet_integralReport
Supposed it’s not that way — integrals for polynomials, trig, and exponentials aren’t pounded into their heads. More emphasis on setting up and much less on solving. Applications like sketching the shape of functions using derivatives is tossed entirely. Go beyond the venerable TI-83 and use Mathematica or similar. Where do the students with a $100 device in their pockets suffer from not being able to do it on paper?Report
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
“Supposed it’s not that way — integrals for polynomials, trig, and exponentials aren’t pounded into their heads.”
Why should we expect that Calc I should not be that way?
What’s the point of Calc I, if not to be that way?Report
There’s definitely a post in there.
The Red Queen shows up in it.Report
The thing that gets to me about this final exam is that it is asking for the student to regurgitate trivia rather than solve problems or do any analysis. Memorizing trivia can be a big pain but it isn’t really a mark of intelligence or that you mastered the subject. A test should show that you can solve problems or analyze the facts.Report