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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)

#191

Post by nuancebydefault »

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.

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?

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

#192

Post by lordnacho »

mnl wrote: post #120
BobaFloutist wrote: post #103

Earlier quoted context omitted.

Why should pedagogy ever end? That's like saying at some point in health care medicine must end and you're responsible for your own treatment. What are the professors for, doing research and abusing their grad students?

The idea is becoming intellectually independent, arriving in the stage of self-pedagogy if you like. Peer learning when there's the chance. You can't realistically expect that there will always be someone up the ladder to explain things to you. I mean, who explains stuff to the professors if it worked like that?

When you get to university, the lesson is very much that you have to learn things yourself. I found that the more decorated the professor, the worse he was at explaining anything, due to some mixture of being unable to go back to a state of ignorance and being in a seat where his main responsibilities are elsewhere (grant applications!). I'm talking about 1or2-to-one tuition here, done several times a week to kids who did very well in high school.

Yes, you have to shed the expectation that others will teach you, I agree with that. In the end, people slogged through by doing a bunch of reading from various sources. It is maybe the main lesson of university for everyone: you're not in high school anymore, you won't just learn whatever the guy says while talking to you. It's quite the shock if you had actually good teachers at school.

The thing is though, you can still demand good teaching materials. Textbooks have to explain things in the clearest way possible. They shouldn't be confusing, especially considering they end up being the main source for just about everything. In this modern world where there are online lectures and textbooks, there's no reason we can't all have the very best explanations of every relevant concept. Yes, of course as a student you still have to put in the time, but the materials ought to be the very best.

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

#193

Post by nuancebydefault »

ink_13 wrote: post #158
ajkjk wrote: post #32

Earlier quoted context omitted.

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…

> No problem with there being a rigorous math department. For mathematicians. But, quite literally, no one else cares As someone who did an undergrad in mathematics, I have to disagree with you here. People think math is about numbers and computation, but that's like saying literature is about letters and composition. Fundamentally, mathematics is the study of things that are provably true. Without rigor it's not mat…

> Study of things that are provably true.

Why would you study such trueness if whether you know it is true or not does not have any added value in the physical world?

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

#194

Post by sudosysgen »

jampekka wrote: post #153
marcosdumay wrote: post #140

Earlier quoted context omitted.

In fact, there is a lack of linear algebra in infinite-dimensional spaces on the general curriculum, and it is an important subject for physics and a few engineering areas besides the relevance for mathematicians. It wasn't clear to me how the dependency between calculus and algebra was supposed to be, but your comment makes it crystal clear. The discrete transformation is quite fitting for finite-dimensional algebra…

Infinite-dimensional spaces do indeed come up in e.g. Gaussian processes and the kernel trick. I have to admit I've never really understood them the rigorous form. For ML for example it would be likely more useful than e.g. most matrix decompositions. I'd guess math uses continuous forms because it's where the mathematical tools are and many things tend to get simpler in mathematical sense when you let something go i…

It also comes up a lot in the foundations of RL - the basis of how it is justified (or in some cases proven) to work is contraction mappings and functional operators.

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

#195

Post by gbjw »

anon291 wrote: post #175
gbjw wrote: post #164

Earlier quoted context omitted.

Well, based on a quick Google, there are approximately 10^186 Planck-length volumes in the known universe. So 10^241 is not physical in the sense that there is no way to 'realize' a set of 10^241 things. Another related notion is that there is no way to 'realize' an irrational 'number' -- sqrt(2) can be defined but we can never draw a length of sqrt(2).

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. > Another related notion is that there is no way to 'realize' an irrational 'number' -- sqrt(2) can be defined but we can never draw a length of sqrt(2). Again, i…

> 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 anyone means by things like 'sqrt(2)'. Why is drawing a thing of length sqrt(2) any different than drawing a thing of length 1? If I draw a thing of length 1 and say it's a line of length 1, then why is that different than my drawing the same line, claiming its length is sort(2) and then pointing out that it's actually now impossible to mark where 1 would appear along its length?

Perhaps a more precise way to describe the situation is that one can define one or the other as a base 'unit' but you can never get one from the other (they are 'incommensurate', as the greeks would say). Irrational numbers can be defined but not 'realized' (or 'constructed') in the same way that the rationals can.

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

#196

Post by Raidion »

commandlinefan wrote: post #112
washadjeffmad wrote: post #74

Earlier quoted context omitted.

I had a 1930s edition of a "radio physics" textbook as a child, and because there could be no prior assumptions about exposure or familiarity, it was filled with very complex ideas explained so cohesively and coherently that, well, I could understand. The author knew that this book was for people who might be as involved in the business of radio as its science or engineering, so they wrote as much about the applicati…

> complex ideas explained so cohesively I wonder if the same thing isn't happening with computer science. When I started studying the topic in the late 80's, I was part of the earliest generation that actually did, and everything seemed to be explicitly written with the goal of making sense. Some things (like recursion and pointers) were fundamentally complicated, but they were made as simple as they reasonably could…

I feel like teaching is mostly a one size fits all endeavor, but people learn and think in different ways, just like a processor is optimized for some operations but not others.

Take simple arithmetic like 12x17. Some people do the long form multiplication (carry the one..), some people say it's 12x10+12x7. Some remember 12x12 from times tables and go 12x12+12x5. Some people make it 24x8+12 => 48x4+12 => 50x4-8+12 etc. Some do it on the abacus in their heads.

All valid, though some are slightly more optimal than others. Good teachers empower alternative solutions and try to help people connect what they already know to what they already understand.

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

#197

Post by eternauta3k »

not2b wrote: post #185

Lots of good criticisms of the traditional way of teaching in here. The only issue I would raise is his claim that Bessel functions should be dropped. In physics and engineering they come up all the time when the geometry is cylindrical. I suppose he could argue that they could wait for a second course, perhaps.

I agree Bessel functions are not excessive, they are just enough to scare you straight.

And yeah, physics students need to be able to solve QM and EM equations, can't get away from (some) special functions.

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

#198

Post by dang »

Related. Others?

Lessons I wish I had learned before I started teaching differential equations [pdf] (1997) - https://news.ycombinator.com/item?id=32530035 - Aug 2022 (177 comments)

10 lessons I wish I had learned before I started teaching differential equations - https://news.ycombinator.com/item?id=19005798 - Jan 2019 (2 comments)

Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf] - https://news.ycombinator.com/item?id=15163979 - Sept 2017 (108 comments)

Ten lessons I wish I had learned before teaching differential equations (1997) [pdf] - https://news.ycombinator.com/item?id=11207183 - March 2016 (118 comments)

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

#199

Post by ink_13 »

nuancebydefault wrote: post #193
ink_13 wrote: post #158

Earlier quoted context omitted.

> No problem with there being a rigorous math department. For mathematicians. But, quite literally, no one else cares As someone who did an undergrad in mathematics, I have to disagree with you here. People think math is about numbers and computation, but that's like saying literature is about letters and composition. Fundamentally, mathematics is the study of things that are provably true. Without rigor it's not mat…

> Study of things that are provably true. Why would you study such trueness if whether you know it is true or not does not have any added value in the physical world?

I don't even understand the question. Is there in truth no beauty? Why do anything that doesn't add value to the physical world?

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

#200

Post by ink_13 »

ajkjk wrote: post #188
ink_13 wrote: post #158

Earlier quoted context omitted.

> No problem with there being a rigorous math department. For mathematicians. But, quite literally, no one else cares As someone who did an undergrad in mathematics, I have to disagree with you here. People think math is about numbers and computation, but that's like saying literature is about letters and composition. Fundamentally, mathematics is the study of things that are provably true. Without rigor it's not mat…

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 mathematician, doing calculations is not mathematics, no more than spelling is doing poetry. Which is not to say that doing calculations is without value! I think what we have here is (ironically) an unrigorous definition.

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