- Next story Apologia of Human Nature: Charlatans, Saints, Hoffer, and Burt Likko’s Query
- Previous story The People’s Court: Live-streaming the Supreme Court
Search
TEN SECOND BUZZ
- The FTC has banned (nearly all) non-compete agreementsApril 23, 202418 Comments
- More Campus Palestine Protests, More Arrests, More Viral VideoApril 23, 202417 Comments
- Open Mic for the week of 4/22/2024April 22, 202440 Comments
- Open Mic for the week of 4/15/2024April 15, 2024232 Comments
- OJ Simpson: Football Great, Murder Suspect, and Convicted Felon, Dead at 76April 11, 2024115 Comments
Features
Hot Posts
A Message From Devcat
We have been experiencing some system resource issues. We believe the problem may be resolved, but if it is not please bear with us.
Recent Comments
- LeeEsq on Free Speech, But No Freedom to HarassOne of the things that I don't understand from Pro-Palestinian Westerners is how they keep on insist…
- Dark Matter in reply to LeeEsq on Free Speech, But No Freedom to HarassThink of all the bricks that get thrown at BLM. The theory of building a minority organization is di…
- Dark Matter in reply to DavidTC on Free Speech, But No Freedom to HarassWhen I read the wiki on the alleged sex crimes during the attack we seem to have a large volume of c…
- Dark Matter in reply to DavidTC on Free Speech, But No Freedom to HarassOne of the issues with "subtract the situation" is "everything we know could be wrong". So for examp…
- LeeEsq in reply to Jaybird on Free Speech, But No Freedom to HarassThe time frame is more complicated. Even after the Six Day War, it was seen as a larger Israel-Arab…
- Jaybird in reply to LeeEsq on Free Speech, But No Freedom to HarassIf you haven't seen this, you should. It's only a few seconds and it's really a treat. (Note: You sh…
- Jaybird in reply to LeeEsq on Free Speech, But No Freedom to HarassThe underdog status ended after the Six Day War over there. It's wobbly over here but there are stil…
- LeeEsq in reply to Jaybird on Free Speech, But No Freedom to HarassI also expect that the period where Jews were seen as an underdog was short and not entirely univers…
- Michael Cain in reply to North on The Shifting Politics of AbortionThe two big stories of political geography in the US over the last 30 years is the huge swing from b…
- LeeEsq in reply to Chip Daniels on Free Speech, But No Freedom to HarassAnd what really pisses me off isn't necessarily the anti-Zionism per se but just treating Jews as di…
Comics
-
April 24, 2024
-
April 23, 2024
-
Friend Husband At The Ballpark
April 22, 2024
-
Good Morning! Are You An Amateur?
April 21, 2024
More Comments
- LeeEsq in reply to Jaybird on Free Speech, But No Freedom to Harass
- North in reply to Michael Cain on The Shifting Politics of Abortion
- Dark Matter in reply to DavidTC on Free Speech, But No Freedom to Harass
- Jaybird in reply to LeeEsq on Free Speech, But No Freedom to Harass
- Chip Daniels in reply to LeeEsq on Free Speech, But No Freedom to Harass
- LeeEsq in reply to Chip Daniels on Free Speech, But No Freedom to Harass
- Michael Cain in reply to North on The Shifting Politics of Abortion
- Chip Daniels in reply to LeeEsq on Free Speech, But No Freedom to Harass
- Dark Matter in reply to DavidTC on Free Speech, But No Freedom to Harass
- North in reply to Michael Cain on The Shifting Politics of Abortion
- Dark Matter in reply to LeeEsq on Free Speech, But No Freedom to Harass
- Michael Cain in reply to Chip Daniels on The Shifting Politics of Abortion
- LeeEsq in reply to Chip Daniels on Free Speech, But No Freedom to Harass
- LeeEsq in reply to Saul Degraw on Free Speech, But No Freedom to Harass
- Chip Daniels in reply to Michael Cain on The Shifting Politics of Abortion
“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