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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)

#31

Post by pastage »

EsportToys wrote: post #10

Speaking of 2nd-order linear ODEs w/ const factor (a.k.a. mass-spring-damper systems), I wrote a blog post[0] deriving all possible general solutions in a concise matrix that makes it easily implementable in code. The following is the complete solution in Lua: function sprung_response(t,pos,vel,k,c,m) local decay = c/2/m local omega = math.sqrt(k/m) local resid = decay*decay-omega*omega local scale = math.sqrt(math.a…

This is a good example. That was math heavy, it took me too long to parse for basic understanding and honestly I am not sure I got it right. The LUA code was easy to understand even though those variable names force you to read all the code, and guess what is what.

It has been a long time since I did diff eq at work, and I agree with the PDF the more I knew the less I understood. I dont know why I need to have the math tainted by the unclean reality to understand, and if that hinders my understanding of them.

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

#32

Post by ajkjk »

DoughnutHole wrote: post #26
f1shy wrote: post #19

Earlier quoted context omitted.

I remember my first calculus exam at the University: I made a lot of explanations and justifications based of physics. Of course I did not pass. As I asked why? It was all correct, right? The prof. said “is perfect reasoning, but this is the math department, physics is one floor below”

He’s right - intuitive physical reasoning is useful (and often valid in the applied sciences) but it isn’t rigorous mathematical proof. Some areas of mathematics are applicable to physical realities, but they aren’t defined by those realities.

Yeah it's not rigorous. The tragedy is that the organizations responsible for teaching math, like how to do it and how to understand things mathematically, for other purposes, would rather waste their time on rigor.

No problem with there being a rigorous math department. For mathematicians. But, quite literally, no one else cares. Mathematicians and everyone else are at crossed purposes, and the mathematicians are wrong to choose their purposes over their students' needs.

Uh... Not that I'm defending the previous poster. Just griping about rigor in general. Physical arguments for mathematical problems are rarely sound, and I'd assume that an undergraduate who thinks they have made a sound one is wrong until reading it to see.

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

#33

Post by SaintRomuald »

siftrics wrote: post #15
ajkjk wrote: post #6

Earlier quoted context omitted.

The problem isn't writing a better text, it's having departments willing to teach it, or to try alternate approaches at all.

You have to have a pretty helpless attitude to blame someone else (teachers) on your inability to learn from a good textbook. How are you ever going to learn anything truly novel in the world if you always delegate teaching to a teacher?

I shouldn't need to read a 500page textbook to understand a class that a teacher has been 'teaching' for 20 years. He's there to give me an understanding of the subject. I have at least 5 classes each semester which are very technical and difficult, it's impossible for me to read a textbook on each while also reading a 20-30 page lab instruction each night, all written in a technical language which I barely understand. I can try but so many times constant consumption of this literature lead to burnout

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

#34

Post by ajkjk »

siftrics wrote: post #15
ajkjk wrote: post #6

Earlier quoted context omitted.

The problem isn't writing a better text, it's having departments willing to teach it, or to try alternate approaches at all.

You have to have a pretty helpless attitude to blame someone else (teachers) on your inability to learn from a good textbook. How are you ever going to learn anything truly novel in the world if you always delegate teaching to a teacher?

You are just wrong. There is a lot more that goes into teaching a subject better than 'generating a better textbook'. There are thousands of hours of work that need to happen to figure out how to make the subject make more sense. It is not a question of 'writing down known result sin a different order'. The subject needs to be reworked in its entirely. Advanced results need to be made elementary. Simple formulations of complex ideas need to be discovered, articulated, and developed into exercises. Out of every available model the best models need to be picked and refined and tested. Etc. And then you have to get everyone on board with the project!

Almost none of the work is 'rewriting the book.'

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

#35

Post by scythe »

ianai wrote: post #13
itronitron wrote: post #2

Previous discussion from 2022... https://news.ycombinator.com/item?id=32530035

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.

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

#36

Post by TheOtherHobbes »

tux1968 wrote: post #5
contrarian1234 wrote: post #3

“Why is it that no one has undertaken the task of cleaning the Augean stables of elementary differential equations? I will hazard an answer: for the same reason why we see so little change anywhere today, whether in society, in politics, or in science“ I see a lot of parallels with the Physics department and I think the reason is much more depressing. Both fields embrace a sort of masochism and active desire to keep…

> I had a lot of hope for things like Khan Academy, but the issue is video is not text and it's hard to iterate and improve on When the entire video is generated by an AI, that dynamic changes. It's no issue to improve "the script", and have it seamlessly regenerated. We can't be that far away from this capability, and it could have a profound impact on available learning resources.

The video won't be generated by AI, because there will no video.

The mid-game is personalised, conversational, interactive tutoring.

The end-game will be deciding how much humans need to know about any of this, because AI will be far better at 99% of it.

If this seems unlikely, consider that Cauchy, Gauss, Laplace, and so on were all super-elite 0.001%-ers in terms of ability.

No one knows whether AI will eventually have the creative ability to work at their level of insight.

Personalised training could select for prodigies, and - whatever we decide to do with them - we'd have more super-talents than we do now.

Currently a lot of talent is wasted because of political and economic inefficiencies.

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

#37

Post by monista »

f1shy wrote: post #19
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.

I remember my first calculus exam at the University: I made a lot of explanations and justifications based of physics. Of course I did not pass. As I asked why? It was all correct, right? The prof. said “is perfect reasoning, but this is the math department, physics is one floor below”

And they were right imho. Having a clear notion of a subject is good for both physics and mathematics, but what's good for physics or engineering, is not enough for math, in math, you also need rigorous reasoning, or you'll fall into subtle mistakes.

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

#38

Post by zozbot234 »

monista wrote: post #28
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.

Note though, that nonstandard analysis isn't compatible with more "intuitionistic" https://en.wikipedia.org/wiki/Axiom_of_determinacy (in place of axiom on choice), which free you from Banach–Tarski paradox and have some other appealing properties.

On the contrary, in an intuitionistic setting you can use https://en.wikipedia.org/wiki/Synthetic_differential_geometr... and https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis which aren't nearly as easy to phrase classically. (Sure you can phrase things in terms of toposes, but that's just a roundabout way of saying that you're operating in a constructive setting.)

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

#39

Post by [deleted] »

contrarian1234 wrote: post #3

“Why is it that no one has undertaken the task of cleaning the Augean stables of elementary differential equations? I will hazard an answer: for the same reason why we see so little change anywhere today, whether in society, in politics, or in science“ I see a lot of parallels with the Physics department and I think the reason is much more depressing. Both fields embrace a sort of masochism and active desire to keep…

[deleted]

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

#40

Post by eigenket »

ajkjk wrote: post #32
DoughnutHole wrote: post #26

Earlier quoted context omitted.

He’s right - intuitive physical reasoning is useful (and often valid in the applied sciences) but it isn’t rigorous mathematical proof. Some areas of mathematics are applicable to physical realities, but they aren’t defined by those realities.

Yeah it's not rigorous. The tragedy is that the organizations responsible for teaching math, like how to do it and how to understand things mathematically, for other purposes , would rather waste their time on rigor. No problem with there being a rigorous math department. For mathematicians. But, quite literally, no one else cares. Mathematicians and everyone else are at crossed purposes, and the mathematicians are w…

I quite strongly disagree with this. Mathematicians aren't rigourous just for the hell of it. We have examples where a lack of rigour caused a bunch of people to waste a bunch of time working of stuff that ultimately didn't work at all.

For example the Italian school of algebraic geometry caused the waste of legitimately decades of work by many people because the foundations weren't right: https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge....

Part of the purpose of a maths degree is to teach the students the rigour required to be a mathematician. If you don't want to learn that rigour then thats completely fine, you can use physical or intuitive arguments, but the place to go and do that is in the physics or engineering departments.

I agree mathematicians would be wrong if they were forcing their rigour on physics students or whatever, but I think they're emphatically right to teach it to maths students.

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