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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)

#251

Post by TwentyPosts »

selimthegrim wrote: post #124
hiAndrewQuinn wrote: post #122

>LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS ARE THE BOTTOM LINE Yes! Jesus. People so often forget you can literally take variables and just fill in easy constants to get a sense of how they work. I can't believe this isn't, like, the first thing they teach in DE.

OK, then wait until you run into the students who somehow passed algebra two but don’t understand the difference between a constant coefficient and a variable coefficient

Clearly they should not have passed linear algebra without this understanding, and you're free to fail then if they didn't personally manage to correct for it by the end of the term.

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

#252

Post by DoughnutHole »

nuancebydefault wrote: post #191
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.

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?

The biggest example I can think of is Fermat's Last Theorem. Uncounted mathematicians devoted years of their lives to the theorem, and the developments they made along the way may have some applications. But I have yet to hear of any physical truths derived from the fact that a^n + b^n != c^n for n > 2. Most math is like this - beautiful, and useless.

The motivation of most non-applied mathematicians is aesthetic, much like the motivation of someone in the arts. It's done for its own sake ("mathematical beauty"), not because it has any practical application.

Mathematics might turn out to be "useful" more often than the average novel - but that doesn't mean the math was done with any use in mind. For example many developments in differential geometry made in the 19th century were inapplicable to anything physical for decades until Einstein applied them to create general relativity. Plenty of math becomes useful long after its development, and plenty more never finds any practical application.

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

#253

Post by jampekka »

onetimeuse92304 wrote: post #250
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…

> I agree, theoretical math students and staff are a bit dismissive about everything else.

I think this does go somewhat both ways. E.g. in the current discussion there are quite a few people complaining about "useless maths". I think a big problem is that different areas of studies don't appreciate that they are, well, different, and it's not trivial to transfer ideas across fields.

> Just read a bit about how Einstein came up general relativity

Also an interesting story about special relativity: Minkowski (Einstein's math professor) proposed very early on to Einstein that special relativity should be formulated geometrically (with what's now called Minkowski spaces), but Einstein deemed it too mathy and unintuitive, only to accept later that it was the right way to go. Without the geometric formulation it's very possible Einstein wouldn't have developed general relativity later on.

> 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.

One argument for teaching FT more abstractly is that the "useful parts" are quite different in different fields. E.g. in many applications of sound and acoustics (and probably in EE) the main interest is in the spectrum. But in e.g. statistics the main thing is the convolution theorem and the spectrum has practically no use.

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

#254

Post by selimthegrim »

TwentyPosts wrote: post #251
selimthegrim wrote: post #124

Earlier quoted context omitted.

OK, then wait until you run into the students who somehow passed algebra two but don’t understand the difference between a constant coefficient and a variable coefficient

Clearly they should not have passed linear algebra without this understanding, and you're free to fail then if they didn't personally manage to correct for it by the end of the term.

They switched to a more forgiving section that didn’t do variable coefficients

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

#255

Post by jasomill »

Raidion wrote: post #196
commandlinefan wrote: post #112

Earlier quoted context omitted.

> 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 et…

As someone who only learned multivariable calculus successfully via independent study of exterior differential forms as a result of off-handed comments from the differential geometry professor who taught my first-year calc course, I wholeheartedly agree.

Oh, and 12×17 = 10×17 + 2×17 = 10×17 + 2×10 + 2×7 = 170 + 20 + 14 = 204.

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

#256

Post by jasomill »

marcosdumay wrote: post #75
mnl wrote: post #22

Earlier quoted context omitted.

What's the problem with a good book, age? Baby Rudin's first edition is 70 and the latest one is from 1976. It's still widely used and will be for a while. Honestly your problem was that you didn't know any Classical Mechanics yet and you were assuming that the volume of recent developments made old books obsolete. Maybe in Biology, in Physics getting to recent developments would mean that you're familiar with Goldst…

The problem is that there is no good textbook for Classical Mechanics out there. At least not on an introductory level. It's funny that for the 19th century and the first few decades of the 20th one physicists were so eager to simplify and generalize their knowledge. With the side-effect of learning quite a few surprising things from the work. And yet for almost a century the goal is explicitly the opposite. (It's al…

While not exactly "introductory" in terms of mathematical prerequisites, Spivak's Mechanics I[1] is an interesting take on the subject, with extensive historical references if you're in to that sort of thing.

[1] https://archive.org/details/physics-for-mathematicians-mecha...

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

#257

Post by RoyalHenOil »

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…

When schools focus on only teaching the most rigorous and abstract definition of math, at the cost of teaching students how to apply math to real-world problems, the result is a lot of students who can do neither.

Likewise, when schools focus only on teaching literature by the most artistic definition, at the cost of teaching them basic day-to-day reading and writing skills, the result is a lot of students who can do neither.

Let's treat rigorous math and lofty literature like the specialized skills that they are, and offer them to students who show particular interest in those areas. For the bulk of students, let's teach them skills that will be useful and relevant.

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

#258

Post by ajkjk »

Koshkin wrote: post #207
ajkjk wrote: post #178

Earlier quoted context omitted.

Personally I don't care that it's about physical things. But I care that it's about doing things: understanding concepts intuitively, solving problems, being adept at thinking mathematically. The thing that is not useful is tedious definitions and tedious proofs. A simple example is: nobody needs a proof of how differentiation or integration work. They need to know how to do it, think with it, apply it to problems, a…

> nobody needs a proof of how differentiation or integration work This "nobody" includes a lot of industry workers, e.g. those in finance and economics, not only "pure" mathematicians.

They... don't? The only reason they need proofs is to write papers proving other things. Not to do the actual work of their fields.

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

#259

Post by ajkjk »

nonameiguess wrote: post #227
ajkjk wrote: post #178

Earlier quoted context omitted.

Personally I don't care that it's about physical things. But I care that it's about doing things: understanding concepts intuitively, solving problems, being adept at thinking mathematically. The thing that is not useful is tedious definitions and tedious proofs. A simple example is: nobody needs a proof of how differentiation or integration work. They need to know how to do it, think with it, apply it to problems, a…

I think you're generalizing far too much from limited experience. Not all mathematics, even applied mathematics, corresponds neatly to intuitive physics models you build naturally by being an animal that interacts with the world. Quantum mechanics was already given as an example, but I had the opposite experience with cryptography. So much of the training as a CS student was along the lines of "accept that this works…

My stance in general is not that we shouldn't teach students why things work; just that rigorous proofs aren't at all a good way to do it. Indeed, as far as I could tell, most of the mathematicians who taught math classes I took had at best a rudimentary knowledge of how to use any of the theory they were teaching to do anything.

I studied physics in undergrad and the general method was, yes, it involved detailed understanding of all the theory, far better than anyone learned it in the relevant math class (differential equations, multivariable calculus, and linear algebra, in particular --- a 1-semester course in QM has to teach everyone linear algebra because the math course on the subject is garbage; a 1-semester course in E&M has to teach multivariable calculus also because the math course was garbage, etc.)

But yeah, absolutely you have to really learn e.g. group theory and algebraic number theory to do cryptography. I actually quit engineering and switched to physics precisely because I was so annoyed that an intro electrical engineering class used the Fourier transform as black magic without bothering with why it worked.

In particular many of the exact theorems that are used in formal math are totally irrelevant and historical accidents. Calculus could have been developed in a thousand different ways; we happened to pick one, the particular names of the theorems or what order the results are constructed in is, IMO, totally irrelevant to anything outside pure mathematics (particular examples: which construction of the integers you use; which construction of the real numbers you use; which definition of the derivative; which definition of the integral, stuff like that. In the case of integrals, yes, different constructions have different theoretical properties in terms of which functions can be integrated --- but, in any application, the answers intrinsically can't be affected based on your choice of definition, so the actual definition can't matter.

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

#260

Post by meroes »

kriops wrote: post #142
meroes wrote: post #139

How math relates to the physical world is not at all clear. Like yes 2 apples clearly makes sense, but non physical quantities like 10^241 of anything start to break down the relationship. It’s not the abstraction that gets me usually, it’s the lack of an explicit relationship between math and physics. I’m sorry but personally I am not satisfied by what I learned in my math and math/science classes on this. It starte…

Can you elaborate on why you describe 10^241 as non-physical? Just because it's too large to describe anything physical? Anyhow, Aristotle had another take on the problem of universals, and taking his view that every concept abstract or otherwise can only exist as part of our physical world (I'm not a philosopher, and there are variations on both themes that different people subscribe to), then your supposition would…

Sorry for not replying.

10^241: I can't find the paper from years ago I thought I would be able to re-find (if it even exists). But, Seth Lloyd has written about there being a hard limit of 10^120 operations on 10^120 quantum bits as a max computational capacity of the entire universe up to today. Whether one just multiplies the two to get 10^240 or there was another paper I can no longer find where this is explicitly done, I'm not sure. The number stuck though. Since it uses plank areas and times, it is a maximum "quantity" for a human to say physically exists according to their best scientific theories. No other physical quantity in this modest metaphysics is larger than this. So we can and must get a little more creative in how we think math relates to the world since I can talk of 10^241. It represents the universe at max capacity since inception to today, what could be greater. (If you want to get really pedantic, we could say well a person thinking of 10^241 is physical too, and science is probably somewhat wrong, ergo 10^241 is physical as is any mathematical object we can think of. What about infinity? Does it take infinity of something to think of infinity? It couldn't as there aren't infinite physical anything at our disposal up to now according to science. So if it doesn't take infinite resources to think of infinity, where is infinity in the physical. We can't just say in our thoughts. This seems hard to argue).

Regarding Aristotle, is 10^241 in the physical world? How? I personally don't want to have to argue that side. Plato's division between physical and abstract (math) say, allows me to better understand math as its own thing, and leave the mystery to how it relates to the physical world as less of a scientific question a la Aristotle.

For something like Aristotle's potential infinities existing, I'm not very knowledgeable on how he envisioned they actually do exist. Maybe it can explain how to find vast quantities in the physical. Maybe. But then is 10^241 only potential?

Funnily enough I am not a platonist about math, nor however do I believe in psychologism which I would attach to Aristotle. I'm more of a fictionalist by default but always trying to challenge this. Plato I see as setting math free from physical conceptions, which is why I gravitate toward him vs. Aristotle.

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