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Lessons I wish I had learned before teaching differential equations [pdf] (1997)

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Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#211

Post by gbjw »

anon291 wrote: post #202
gbjw wrote: post #195

Earlier quoted context omitted.

> It really depends what you consider 'real'. It is actually pretty straightforward to realize a representation of 10^241 (you just did). Why does that make it any less real than the digit 2, which represents two things, but is actually just itself one thing. One is a map, and the other is the territory. Both 'real' in some sense but a map without the territory feels less 'grounded' (pun?). > Again, it's unclear what…

> Irrational numbers can be defined but not 'realized' (or 'constructed') in the same way that the rationals can. Once again, I'm not 100% sure what you're saying. If you have something that's of length 1, then you can easily construct the line with a ratio sqrt(2):1. Draw another line of length 1 (use compass and straightedge) at a 90 degree angle. Repeat 4 times until you have a square. Now draw the diagonal. You h…

Yes, apologies -- using the term 'construct' muddied the point as sqrt(2) is a 'constructible' number as you point out. The term 'real' is what's at issue here and I am arguing for a distinction between a 'map-like' real and a 'territory-like' real, the latter of which has some sort of spatiotemporal grounding.

> There are some numbers that are not, and perhaps these can truly be said to not exist.

So then we have a real issue because the vast majority of the real line is composed of these uncomputable numbers which you've suggested don't exist.

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#212

Post by sheepshear »

mnl wrote: post #190
sheepshear wrote: post #182

Earlier quoted context omitted.

I actually completely agree with everything you wrote. I'm also very familiar with the ongoing battle between education and training. What I'm talking about is expanding options to meet additional education needs. Since universities are a shared resource, any solution must be carefully designed to preserve the ability to continue providing existing services. That's difficult to achieve, so I understand the obstructio…

I'm very adaptable nowadays. It's just that I think I know which things should work, but I'm not fighting society, particularly not on this. The thing with giving the public what they want and being too much of a pragmatist is that we've seen it before. Consider Western universities in the 17th century, they were still there churning out degrees, but modern science, mathematics and technology developed elsewhere.

The old one-size-fits-some approach is as pragmatic as it gets. Society's education needs have grown beyond what the old model can adequately service. We need new solutions that don't cause regressions on the old solutions.

You're right to protect your existing solution against regressions, and there's value in revisiting old topics in the new discussions, but you're not going to constructively contribute much if you're unwilling to engage with why so many people feel the need for something different in the first place.

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#213

Post by [deleted] »

jampekka wrote: post #118
marcosdumay wrote: post #90

Earlier quoted context omitted.

I'd say that the Fourier Transformation doesn't belong in the calculus program at all. It should be taught in linear algebra, where it does not only make sense, but is a non-trivial example of application that the textbooks have so little of.

Why should it be in linear algebra? You mean as in doing linear algebra with functions in infinite-dimensional spaces? What I recall mostly (not) learning from linear algebra were various classifications of matrices and their properties (which made very little sense to me at the time too). I'm not exactly sure where FT should belong in the math syllabus. It's heavily related to trigonometry of course but it is an int…

[deleted]

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#214

Post by selimthegrim »

Xcelerate wrote: post #24

I went to grad school for chemical engineering a decade ago and part of my frustration with the courses is that the math was never rigorous enough; when I asked for clarification, the inconsistencies were glossed over and hand-waved away. The linked article mentions differential forms, which is a perfect example. In engineering courses, these are often introduced without any rigor or formality. They just “appear” as…

DFT uses the Kohn-Sham approximation which is single particle and is a ground state theory. There’s definitely been research along the lines you bring up in quantum chemistry but not in DFT. You want to look at the Levy-Lieb density and maybe some of Mazziotti’s papers

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#215

Post by tux1968 »

robertlagrant wrote: post #21
tux1968 wrote: post #12

Earlier quoted context omitted.

The OP referenced the Khan Academy as an attempt to displace the ossified educational institutions we have. But it has its own limitations and challenges, some of which will soon be rendered moot by the coming wave of AI technologies, perhaps giving it the final push it needs to truly take hold. Can imagine a world where it doesn't matter where or how you get your education, all that matters is that you can have your…

This might work well for knowledge/IC-based attainment. There are still other groups, who are possibly the majority when combined: Some people are there for "the experience", no matter how much it costs, nor how little benefit it results in. They will still be paying $80k for an experience that qualifies them for nothing, even though they had that qualification already before they started. Some other people are there…

Agreed. One size does not fit all. It's just the hope that options like Khan Academy are made truly viable for those who are well served by such options.

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#216

Post by yuppiemephisto »

jampekka wrote: post #128
Chabsff wrote: post #108

Earlier quoted context omitted.

At the end of the day, analytical Fourier analysis doesn't get nearly as much practical use as numerical approaches, and you don't really need THAT much background to make sense of DFTs. I know that, for myself, revisiting Fourier Analysis after going through DFTs in my numerical analysis classes made a lot more sense, and I kinda wish I had started with that angle in the first place.

Analytical fourier analysis is needed for many other fields of math though. For example in statistics it's very central for dealing with density functions. I'd guess in many fields of physics it's also very crucial. But for a lot of fields you just need the DFT and the calculus stuff can be mostly a distraction. How I finally figured out FT is something like the infinitesimal limit of DFT.

Related to the DFT view: https://www.sciencedirect.com/science/article/abs/pii/S00492...

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#217

Post by GeorgeTirebiter »

aj7 wrote: post #137

I had him for second term calculus, which was the first math course I took in college. So I get to listen to him speak it in my minds eye —- his voice is unforgettable. He was a great teacher. 2. I have an interesting take on this subject. I became an engineer at 50. Out of personal choice. What I found was that engineering had changed and what mattered was your savvy in manipulating expensive programs on the compute…

I, too, had Rota for his "exploring higher mathematics" seminar. What a remarkable human being! I retain excitement for hierarchies of infinity to this day.

As to your second point - yes, programs solve DE's numerically. I suppose it's still nice to know a bit about how they were solved in the 'olden days'?

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#218

Post by JadeNB »

__rito__ wrote: post #184

Here's the most intuitive introduction to Differential Equation that I have ever seen: https://www.complexityexplorer.org/courses/31-introduction-t... Explains DEs from scratch, physical significance, one or two traditional methods, then goes on to numerical methods. If you ever want to learn DEs, I HIGHLY recommend this tutorial. This is short, too. I have never found a resource where the concepts behind DEs are exp…

For a course-level approach by some of the early pioneers of the pedagogical approach of using numerical methods to understand differential equations, I strongly recommend Blanchard, Devaney, and Hall's book (http://math.bu.edu/odes).

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#219

Post by eszed »

Workaccount2 wrote: post #101
ajkjk wrote: post #16

Earlier quoted context omitted.

Physics doing the work of actually teaching math is a universal embarrassment to math departments. Or, well, it should be.

Many math teahcers (and likely mathematicians) could not care less about the real world applications of math. My college calculus teacher had no interest in physics or science. She liked math in the same way someone would like doing a crossword or Sudoku. A puzzle game with a well defined rule set and the challenge of finding answers. I cannot overstate how destructive this is, because it basically presents math as t…

I think you put your finger on a challenge fundamental to teaching itself. Speaking as a (former, at least for now) teacher, the concepts that I find most intuitive were always the most difficult to teach, whereas skills I struggled to master were the ones I was best able to instruct. For problems in the second category I could diagnose the gaps in my students' understanding or application - likely because I'd faced them, or similar, myself - and strategize an approach that would help. Conversely, the subjects that I "got" without effort, left me floundering (at least at first) to find an alternate way to present to students.

(I suppose that's where the terrible canard of "those who can't, teach" comes from. There's a kernel of truth to it, of course, but the more generous and relevant point is that success through effort is better preparation for teaching than success through natural ability alone.)

My guess is not that your teacher would disagree that math is "a fundamental language of the universe", just that she spoke it so fluently that she wasn't well able to relate to people who don't.

Re: Lessons I wish I had learned before teaching differential equations [pdf] (1997)

#220

Post by yuppiemephisto »

scythe wrote: post #35
ianai wrote: post #13

Earlier quoted context omitted.

They talk (rightly) about mathematics replacing infinitely small but not 0 numbers with limits there. Infinitesimals were later reintroduced with rigor through https://mathworld.wolfram.com/NonstandardAnalysis.html I have long thought such would be easier to work with on modern computers.

Nonstandard analysis is one of those things that sounds nice in the abstract, but even though I first learned about it more than ten years ago, I have yet to see a worked didactic-level example of how you would, for example, obtain the derivative of sine using nonstandard analysis, or derive the natural logarithm.

https://alok.github.io/2022/09/12/y539043/
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