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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)

#81

Post by mnl »

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…

I learnt a lot of physics from Marion's 2nd edition (not SR there though). An older and completely forgotten fine textbook is W. Hauser's Introduction to the Principles of Mechanics. Then you jump into Goldstein (the 1980 one, again not SR there). It's a good idea to buy any Schaum book from Spiegel about this too, also for vector analysis if you can't take a course on that.

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

#82

Post by rudolph9 »

thorin wrote: post #14

If when I started learning about calculus at 14/15 someone had explained why we were doing this it would have made it so much less confusing. Explaining in terms of speed/distance/acceleration seems to make complete sense now (one of several potential examples). However discussing a function and talking about splitting in into small (infinitesimal) strips of delta quantities and a list of equations / proofs made it c…

I had a professor freshman year of college who offered the question “How many times can you split the distance between me and you in half?”. Ultimately I arrived at the conclusion infinity which forever more made the general idea of calculus stick.

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

#83

Post by samvher »

> 4. Teach changes of variables

This has been something that always seems a bit magical to me, when someone comes up with an obscure change of variables that completely simplifies the problem. I'd love to learn this in a more structured/rigorous way. Does anyone have any recommendations for picking this up?

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

#84

Post by tihal »

samvher wrote: post #83

> 4. Teach changes of variables This has been something that always seems a bit magical to me, when someone comes up with an obscure change of variables that completely simplifies the problem. I'd love to learn this in a more structured/rigorous way. Does anyone have any recommendations for picking this up?

People might disagree strongly, but my recommendation for you is to look into lambda calculus for this.

There are two important concepts that can be generalized: - Substitution, the method of renaming and replacing variables - Abstraction, replacing a variable with another term (function) that can also serve as output

This should give you a more natural feel to concepts like parametrization or what it means that the solution to one function may be another function. Perhaps you might even look a bit into the simply typed lambda calculus which has given me yet more perspectives on the fundamentals of mathematics.

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

#85

Post by hoseja »

mnl wrote: post #22
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…

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 the pedagogy, as in, the best most effective way to teach people things, should improve massively in 70 years. Whether it actually does, that is another question.

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

#86

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)

#87

Post by jampekka »

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 obvious what's going on (and understanding the phase part after this was no problem). Such practical examples seem to be almost banned from university math, I'm guessing to make sure everything is very abstract and rigorous (e.g. you have to work with finite length and discretized signals where the maths don't strictly apply). And after getting this intuition the formal math started to make sense too.

But when one begins to teach, it becomes quite easy to see why things are like this. All of these things are so obvious to the teacher that it's hard to understand how one thinks before these are obvious and the standard notation/vocabulary is typically a good way to work with these, but only after understanding the stuff.

What I try to do is to probe out something that the student already knows, which may be from a totally different field, and find a simple example in the new topic so I can say that "this is exactly the same thing, but with this different notation/abstraction". This very often causes the things to "click" for them.

This is of course very hard to do with a textbook or a mass lecture. And probably the main thing why we need humans to do teaching instead of just giving out material.

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

#88

Post by housecarpenter »

hoseja wrote: post #85
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 the pedagogy, as in, the best most effective way to teach people things, should improve massively in 70 years. Whether it actually does, that is another question.

Should it really though? Maybe pedagogy is just hard and we can't expect constant progress.

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

#89

Post by rawgabbit »

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 took thermodynamics over 30 years ago. I remember having the feeling of learning a different language using a text book whose explanations also needed translation. I remember the book explained Entropy using chaos theory or randomness and talked about popular philosophy during the 19th century. After a bit of mental torture, I realized by Entropy they really mean Thermodynamic Stability. It is just that heat usually dissipates when materials touch, they were using words like chaos or randomness to describe the process. But their description was vague and poorly conceived.

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

#90

Post by marcosdumay »

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…

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.

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