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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)

#121

> What can we expect students to get out of an elementary course in differential equations? I reject the “bag of tricks” answer to this question. A course taught as a bag of tricks is devoid of educational value. One year later, the students will forget the tricks, most of which are useless anyway. That's exactly how I was taught differential equations 20 years ago. And as expected I have totally forgotten all of the…

How often have you used them. It's been decades for me as well, but I still remember how to solve basic linear differential equations as I needed to use them often (in other courses).

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

#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.

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

#123
post #66

Earlier quoted context omitted.

I don't think the line of thinking is directly "lets make this painful on purpose" but there is a sense of - everyone has to go through the same gauntlet and some people are just not cut out of math/physics/etc. It's just the way of the world. There is not direct desire to make the experience better/smoother/easier than it was your generation. I don't know how it is in the Math department, but in Physics there is alm…

I don't know why you're catching downvotes, I think you're completely right. In my engineering curriculum in some cases the teachers were openly hostile to students that had a hard time understanding the material. Professors of undergraduates don't seem to think from the undergraduate perspective. Most undergrads have only ever known school, and are just following directions while fumbling their way to their first in…

I mean, have you ever taught a class? Do you know how it feels when we make sure a topic is covered both in class and recitation with plenty of time for questions and the students still stuff it?

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

#124

>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

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

#125

Earlier quoted context omitted.

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 understan…

We call this "University"

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

#126

Earlier quoted context omitted.

Just out of curiosity, what was that classical mechanics textbook?

Pretty sure it was An Introduction to Mechanics Kleppner, Kolenkow The book was probably okay-ish, the point is that surely someone could have made some improvements in the past few decades

Doesn’t it use cgs only (I think the new editions finally bit the bullet and switched to mks)

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

#127

“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…

Makes me think of Jackson's electrodynamics - yit always seemed to me a hazing ritual. I remember a story of someone asking him a question from the book and he basically said "how the hell would I know?"

I enjoyed the book and class myself but it did not make me a better physicist, just a better mathematician

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

#128

Earlier quoted context omitted.

You need analysis to make sense of infinite series, indefinite integrals, exponential functions, etc.

At the end of the day, analytical Fourier analysis doesn't get nearly as much practical use as numerical approaches, and you don't really need THAT much background to make sense of DFTs. I know that, for myself, revisiting Fourier Analysis after going through DFTs in my numerical analysis classes made a lot more sense, and I kinda wish I had started with that angle in the first place.

Analytical fourier analysis is needed for many other fields of math though. For example in statistics it's very central for dealing with density functions. I'd guess in many fields of physics it's also very crucial.

But for a lot of fields you just need the DFT and the calculus stuff can be mostly a distraction. How I finally figured out FT is something like the infinitesimal limit of DFT.

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

#129
post #45

Earlier quoted context omitted.

You're conflating things. This issue isn't that Classical Mechanics has somehow evolved or changed. It's that people continue to find new and better ways to explain and illustrate concepts. At lest personally I don't know of any field or book where I've felt "Hmm, this is basically perfect and I can't imagine a better way to explain these concepts". Have you looked at for instance Khan Academy's Grant Sanderson (aka…

> Have you looked at for instance Khan Academy's Grant Sanderson (aka 3Blue1Brown) Math videos? I have. I went through Khan Academy, Brilliant and 3Blue1Brown. After spending more than 100s of hours I started getting the feeling that these are all good for elementary level math. But for any serious math (think real analysis, complex analysis, group theory and beyond), all these platforms did was leave me with a warm…

I've given up on these online learning media. Back to textbooks. The difference is like night and day.

Are there people who think this is an "either/or" choice, as opposed to a "use both" thing?? I ask, because it's pretty well established that learning is enhanced by use of multiple media types and it seems self-evident to me that books and videos are complementary.

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

#130
post #61

Earlier quoted context omitted.

There are excellent textbooks on Classical Mechanics, probably because it's a crystal clear subject and you can give a detailed account of the essentials in a single volume without handwaving. Of course everything can be improved, but it also can be muddled. If it works think twice before fixing it. Kind of what happens with Rudin and introductory Real Analysis. On the other hand, there's for instance Optics where yo…

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…

> 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

I think you overestimate the capabilities of students entering university (even 20 years ago), and underestimate how poor high schools can be in preparing said students.

I went to a mediocre university. A 50-80% drop out rate was there for both physics and EE - I don't know how it compares to the other engineering. And I did not even consider it challenging. Almost all the classes were a breeze for someone like me who was well prepared going in. At least in that EE department, the teachers were very dedicated to teaching. They would allocate 3-9 hours a week for office hours, and the pace they taught as was slow (probably only covered 70-80% of the material that is covered in a top university).

Students were given lots of chances.

The reasons they drop out are:

- Poor preparation at the high school level

- Poor discipline. A lot of students didn't transition well to independence, and didn't have an authority figure (e.g. parent) controlling their schedule.

- Realizing too late what it means when courses are built on top of other courses. Thus you'd have people getting an A in Calculus I, but almost failing Calculus III because they didn't realize they needed Calculus I beyond the course.

- In high school you can get far with a cursory understanding of the material. At university, you could get a B, or even an A, with that approach for introductory courses, but that approach will start trending towards an F in junior/senior level courses.

Sure, I agree with you that pedagogy can be improved, but I expect that 80% would at best become 60% if all you focus on is pedagogy.

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