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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)

#231

Post by bethekind »

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…

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

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

#232

Post by ryncewynd »

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…

What is TA'ing?

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

#233

Post by sundarurfriend »

ryncewynd wrote: post #232
analog31 wrote: post #229

Earlier quoted context omitted.

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…

What is TA'ing?

TA = Teaching Assistant, usually a student who helps the main teacher of a course by teaching some units of it, evaluating tests, etc.

TA'ing = being a TA

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

#234

Post by heisenzombie »

SpaceManNabs wrote: post #152
whatever1 wrote: post #146

Just like solving analytically integrals, (O,P)DEs require tremendous skill of pattern matching. You need to know all of the available tricks, and pick the one that seems to fit. You test it and it just works. No wonder there is super-high barrier of entry. And that is why us mortals are just using numerical methods instead.

This is why I found this article kinda pointless. At the end of the day, most practitioners will work with just a bag of tricks, so if that is what the class teaches most of the time, I really don't mind at all. Yeah there is deeper theory to changes of variables and all that, but for an intro class that is aimed at a wide audience (physicists and any sort of applied math), then yeah some light theory and a bag of tr…

I think the underlying question here is: "Is it better to spend lecture time learning the bag of tricks, and then the people who need more can learn theory? Or is it better to teach theory and intuition and then people who need to do it can later look up the bag of tricks?".

Reasonable people can probably disagree, and this might be biased by their personality and learning style.

I personally forgot every trick from my bags mere moments after walking out of my graduate exams. Later, when I got a job that required me to actually /do/ things, I (re-)learned the theory and tricks I needed for the task at hand. The courses that have had a real impact on me were not the "bag of tricks" ones, they were the ones that planted a small grain of understanding and/or appreciation in my brain. I think these are like seed crystals: They make later learning orders of magnitude more efficient and are far less tangible but far more important than "bags of tricks" learning.

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

#235

Post by jimmaswell »

GeorgeTirebiter wrote: post #217
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'?

In the olden days, a lot of problems were solved numerically by methods such as weighing a printed graph of a function

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

#236

Post by ianai »

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…

"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?

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

#237

Post by hyperthesis »

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…

Aren't most DEs not solvable analytically? i.e. only a tiny subset are neatly solvable - like precisely manufactured puzzles. In general, for any real problem you encounter "in the wild", numerical methods are the only way to find solutions (which are approximate).

DE theory is still useful for designing frameworks within which numerical methods can operate.

At least, that's my understanding.

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

#238

Post by rendaw »

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.

Then shouldn't the professor have said that it isn't rigorous? Saying it's perfect implies that it is rigorous, just that they took issue with the association.

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

#239

Post by JadeNB »

denotational wrote: post #224
JadeNB wrote: post #221

Earlier quoted context omitted.

> My grad school was basically working through F=Ma-Cv-Kx in all it's expanding varied glory, working up to full blown finite element analysis. If I'm understanding correctly, you want another = there for a standard damped spring: F = ma = -Cv - Kx.

F is a forcing function, not the resultant force. It’s often arranged this way with all the derivatives on one side (as opposed to having the resultant force, ma, on one side of the equality by itself) so that it matches the general form of a non-homogeneous second-order linear differential equation. (At least I assume this is what the original commenter meant!).

Ah, that makes more sense. Thanks!

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

#240

Post by SpaceManNabs »

heisenzombie wrote: post #234
SpaceManNabs wrote: post #152

Earlier quoted context omitted.

This is why I found this article kinda pointless. At the end of the day, most practitioners will work with just a bag of tricks, so if that is what the class teaches most of the time, I really don't mind at all. Yeah there is deeper theory to changes of variables and all that, but for an intro class that is aimed at a wide audience (physicists and any sort of applied math), then yeah some light theory and a bag of tr…

I think the underlying question here is: "Is it better to spend lecture time learning the bag of tricks, and then the people who need more can learn theory? Or is it better to teach theory and intuition and then people who need to do it can later look up the bag of tricks?". Reasonable people can probably disagree, and this might be biased by their personality and learning style. I personally forgot every trick from…

I completely agree that theory is incredibly important to undestand and retain things. I just disagree that DE doesn't teach the necessary theory, at least the ones at HYP (for physicists) didn't.
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