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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)

#241

Post by contrarian1234 »

mnl wrote: post #69
contrarian1234 wrote: post #65

Earlier quoted context omitted.

I think you have the perspective of somehow who's succeeded in Physics. In your typical introductory physics class less than half the students will walk away with a very solid understanding of classical mechanics. To my mind, if the textbook was actually excellent then that would be 80%+. We're nowhere near there. I think there is LOT of room for improvement But sure.. Thermodynamics.. things could be worse :) Someti…

I almost wrote above that, to my knowledge and IMVHO, no one has succeeded at writing a book on Thermodynamics yet. I self-censored because that would be too flippant, wouldn't it? Lmao

I think Thermo can only be taught if you have a solid foundation in statistics. And stats... is not really taught in the US? I tried to selfstudy a bit, but the textbook situation with stats make math look amazing. For very intro practical things, there are stuff like John Taylor's book... but past that anything rigorous - I actually have no idea how people learn anything

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

#242

Post by RoyalHenOil »

Koshkin wrote: post #204
Workaccount2 wrote: post #101

Earlier quoted context omitted.

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 would not call this "destructive." Solving problems is fun in its own right, and things would start making sense in their interconnection, regardless of whether they can be applied to sciences or not.

It is destructive to people who do not find abstract math problems inherently fun, which is (I would guess) the great majority of people. They come to the conclusion that they hate math or that they are bad at math, even though they would have grasped it better and enjoyed it had it been presented to them differently.

If your goal is teach math to the bulk of people, including those who will not go on to be mathematicians, it makes sense to tie math to something that the bulk of people can relate to. And most people DO enjoy thinking about physical objects and physical space (because we ourselves are physical beings who evolved to interact with our physical environment), so this is a really great starting point for introductory math for the average person.

If your goal is to only teach the subset of the population who prefer highly abstract puzzles, and to alienate all others, then our current methods are working fine I guess. But I don't think this is a good goal for general math education.

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

#243

Post by idontpost »

bethekind wrote: post #231
jampekka wrote: post #118

Earlier quoted context omitted.

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…

Is there a good book for learning ML Linear Algebra? I've taken linear algebra math courses in uni and most of it went in and out one ear. I'm looking for a book that helps me understand ML algebra, so that when I read research papers I'm not just lost and nodding my head aimlessly. Such a book may not exist, maybe it's a group of books, but if ANYONE would have pointers in this direction, I would be in your debt

[dead]

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

#244

Post by scythe »

ducttapecrown wrote: post #167
scythe wrote: post #35

Earlier quoted context omitted.

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.

I literally saw the sin one yesterday, so I'll rewrite it!! Let e be an infinitesimal. We write st(x) for the function dropping the infinitesimal part of a number. Then f'(x) = st(1/e (f(x+e)-f(x))) Now use the angle sum identity and cos(e) = 1 - e^2, sin(e) = e. I don't know how to justify these values other than the power series identities for sin and cos... So st(1/e (sin(x+e)-sin(x))) = st(1/e (sin(x)cos(e)+cos(x…

You can get the relationship:

1 - x^2 from ordinary trigonometry, although you can't just do this with nilsquare infinitesimals; you need a more sophisticated setup.

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

#245

Post by __rito__ »

JadeNB wrote: post #218
__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 ).

Looks really good. Thanks for sharing it.

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

#246

Post by Jensson »

ink_13 wrote: post #200
ajkjk wrote: post #188

Earlier quoted context omitted.

That's just false. What physicists (and engineers, etc) are doing is also math and it's way more useful and insightful.. and (imo) the world would be a better place if that's what everyone else was also learning in college. The idea that math is only about rigor isn't intrinsic; it's a historical accident that people seem to not realize is optional. It is also about understanding things and being able to wield concep…

Novelists and lawyers both do writing, but I wouldn't call their output the same thing. You might call legal writing more useful or novelists more insightful, but that's a matter of opinion. One is not intrinsically more valuable than the other. > The idea that math is only about rigor isn't intrinsic; it's a historical accident that people seem to not realize is optional I mean, it is by definition. To a mathematici…

> doing calculations is not mathematics

I hope you realize that physics isn't about calculations either? You can put numbers or types in the equations, just like you can put numbers or types in the expressions or results you get in math.

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

#247

Post by raffraffraff »

analog31 wrote: post #229
jampekka wrote: post #87

There seems to be something similar in other branches of mathematics too (and lots of other fields). A clear example I recall was studying Fourier transforms in math and I couldn't make any sense of it. To me it was just some by-rote algebra with integrals of exponentials of complex numbers. But then I happened to do some audio signal analysis, and when I saw a magnitude spectrum of a waveform, it was instantly obvio…

It's hard if you don't know any practical examples. Anecdote: I taught a college math course for a semester while I was between jobs. I shared an office with some other teachers including a bright grad student who was TA'ing differential equations. My degree was in physics, and I had worked in industry. I told him that I wished the math courses had included some engineering applications of differential equations. He…

Lol. That's every teacher I had, including in university. It wasn't until I dropped out and years later resumed reading physics and maths for my own interest that I properly understood three point of some of the maths they were teaching.

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

#248

Post by The_suffocated »

sampo wrote: post #205
nuancebydefault wrote: post #191

Earlier quoted context omitted.

If some mathematics are not applicable to physical realities, to what are they applicable? Is there any un-applicable math and if so, why does it even exist?

Up until about World War II and use in cryptography, number theory was pretty much an un-applicable field of mathematics.

I may be wrong but this seems inaccurate. In WWII, the cryptanalysis of the Enigma (https://en.wikipedia.org/wiki/Cryptanalysis_of_the_Enigma) was related not to number theory, but to group theory. Number theory became relevant to cryptography only since the 1970’s/1980’s when Diffie and Hellman published their paper on public-key cryptography. Although number theory was related to cryptography in the past (e.g. in shift ciphers and block cyphers), it was the Diffie-Hellman paper that placed number theory in the central role.

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

#249

Post by karatinversion »

ianai wrote: post #236
ducttapecrown wrote: post #167

Earlier quoted context omitted.

I literally saw the sin one yesterday, so I'll rewrite it!! Let e be an infinitesimal. We write st(x) for the function dropping the infinitesimal part of a number. Then f'(x) = st(1/e (f(x+e)-f(x))) Now use the angle sum identity and cos(e) = 1 - e^2, sin(e) = e. I don't know how to justify these values other than the power series identities for sin and cos... So st(1/e (sin(x+e)-sin(x))) = st(1/e (sin(x)cos(e)+cos(x…

"cos(e) = 1 - e^2, sin(e) = e. I don't know how to justify these values other than the power series identities for sin and cos..." Isn't that from the definition of cos and sin, even geometrically?

The version of rigorous infinitesimals I've seen didn't let you have these identities, because e^2 != 0 (because it still maintains the field axiom xy = 0 => x=0 V y=0); but the argument still goes through by carrying the whole power series in place of these simplifications – you can write the higher order terms as e*f(x, e) and drop them when you evaluate st(...).

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

#250

Post by onetimeuse92304 »

jampekka wrote: post #87

There seems to be something similar in other branches of mathematics too (and lots of other fields). A clear example I recall was studying Fourier transforms in math and I couldn't make any sense of it. To me it was just some by-rote algebra with integrals of exponentials of complex numbers. But then I happened to do some audio signal analysis, and when I saw a magnitude spectrum of a waveform, it was instantly obvio…

> Such practical examples seem to be almost banned from university math

For background, I studied theoretical math. I am also amateur electronics engineer.

I agree, theoretical math students and staff are a bit dismissive about everything else. At least where I studied. Where I studied, we would regularly joke about other students claiming that math is hard saying, "what they are learning isn't even math, it is just a bunch of formulae and rote learning". And this included physics or applied math students. Don't even get started about CS students...

It is unfortunate, because math concepts were frequently invented to solve a real physical problem. Or they were invented and we found them extremely useful for a physical problem. Historically people would not put much distinction and would usually do both physics and math and there would be a lot of cross-pollination.

Just read a bit about how Einstein came up general relativity -- it is fascinating story about how Einstein wasn't good at math but needed a problem solved so he reached out to some friends and those friends basically gave him some private lessons until Einstein clicked and figured it out.

For me, I pretty much understood Fourier analysis before I started doing electronics. But the usefulness and practicality of it only clicked after I moved to higher frequency problems and started using frequency domain when working with my circuits. You don't get much significance of it just studying theoretical math.

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