Lessons I wish I had learned before teaching differential equations [pdf] (1997) • Hacker News Board

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

#171

Post by fn-mote » Mon, Nov 13, 2023, 6:12 PM UTC

cyberax wrote: post #148

contrarian1234 wrote: post #41

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? it's really apparent there is a LOT of room for improvement in pedagogy. There is a study showing that you actually understand material better, if you use the most primitive methods: chalkboard and a lecture. Because you are forced to visualize the material yourself, instead of being presented with a ready-made animation.…

1. I think the research shows that increasing the cognitive load increases the retention. So in general, when it is harder to learn something, you retain it better.

2. When I struggle with books it is because they do not present the motivation behind what they are doing. Videos and "more popular" articles can both provide the big-picture motivation and overview. Sometimes, you have to construct a motivation for yourself, based on what you read. That's hard. Maybe you even invent something new in order to understand a concept better. This approach is slow, though. It's easier if someone explains to you why a certain concept is "hard" or a point of view from which the concept is "easy".

3. I think students who build on a partial understanding are not going to have a better time with videos. They are in greater need of learning how to learn something than they are of facts, but school does not teach that skill (afaik).

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

#172

Post by anon291 » Mon, Nov 13, 2023, 6:12 PM UTC

Xcelerate wrote: post #24

I went to grad school for chemical engineering a decade ago and part of my frustration with the courses is that the math was never rigorous enough; when I asked for clarification, the inconsistencies were glossed over and hand-waved away. The linked article mentions differential forms, which is a perfect example. In engineering courses, these are often introduced without any rigor or formality. They just “appear” as…

My experience as a math major is that most non-mathematicians, including engineering and chem/phys professors, have a working understand of math and don't think too deep about it.

> Is there some kind of axiomatic basis for how these symbols can be manipulated in a consistent way?

Yes.

> This was a guy whose research career was built on dozens of computational chemistry papers that heavily utilized DFT by the way (and by “research”, I mean plug a file with atom coordinates and atom types into the DFT software, run the software, and publish the results).

Again, large portions of academia simply stick to their small subfield, publish there, and don't think much about the broader implications, if they think about them at all. This is why so many discoveries are made by new entrants to a field with a slightly different background.

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

#173

Post by jrgoff » Mon, Nov 13, 2023, 6:12 PM UTC

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 don't know how or where you learned calculus, but when I learned it and when I taught it, the classes were filled with physical world examples to motivate and train the material. That said, my experience as a student reflected on as an instructor, was that a lot of the explanation and example stuff didn't have a lot of fertile ground in my mind yet when I was first learning the material. I'm sure it helped, but there were definitely times when as a grad student I reviewed the foundational material and thought "This makes so much sense, why didn't they teach me this when I was in high school?" only to realize that almost certainly they had taught me that in high school and I just didn't have the mathematical maturity yet to retain it. I think I was also hampered in my chances of learning the conceptual fundamentals because I was able to do most of the work through a solid ability at algebraic manipulation. While solid skills in algebraic manipulation is quite important, I do think it would be a good idea to restructure those classes so that a solid conceptual understanding is also more necessary to pass the class. Of course, easier said than done.

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

#174

Post by mnl » Mon, Nov 13, 2023, 6:13 PM UTC

Folcon wrote: post #166

mnl wrote: post #120

Earlier quoted context omitted.

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?

You seem to have a very individualistic notion of education. Even at the research level we are not independent islands of learning and discovery. People collaborate, some pickup certain concepts better than others and vice versa. So we teach and aid each other. It seems you're firmly against this notion? Or if not please clarify your position?

I mentioned peer learning, collaboration is that.

I think everyone should be capable of working alone as well, and that has been the general assumption around as far as I've noticed. Of course collaboration is usually way more productive and also unavoidable.

But we were talking about education. Theses are individual for a reason.

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

#175

Post by anon291 » Mon, Nov 13, 2023, 6:15 PM UTC

gbjw wrote: post #164

kriops wrote: post #142

Earlier quoted context omitted.

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…

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, 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?

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

#176

Post by mindcrime » Mon, Nov 13, 2023, 6:19 PM UTC

distcs wrote: post #149

mindcrime wrote: post #129

Earlier quoted context omitted.

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.

> it seems self-evident to me that books and videos are complementary Can't speak for others but for me it is more about efficient utilization of time rather than complementing multiple learning methods. I've found that time spent in learning math from videos have poor return of investment. That time is better spent re-reading a chapter or that thing that I couldn't fully understand the first time and doing more exer…

Fair enough. For me personally, I find great value in jumping back and forth between different modalities, where the different presentations reinforce each other. But what works for me may not work for everyone, and vice-versa.

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

#177

Post by [deleted] » Mon, Nov 13, 2023, 6:19 PM UTC

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)

#178

Post by ajkjk » Mon, Nov 13, 2023, 6:19 PM UTC

anon291 wrote: post #170

Workaccount2 wrote: post #101

Earlier quoted context omitted.

Many math teahcers (and likely mathematicians) could not care less about the real world applications of math. My college calculus teacher had no interest in physics or science. She liked math in the same way someone would like doing a crossword or Sudoku. A puzzle game with a well defined rule set and the challenge of finding answers. I cannot overstate how destructive this is, because it basically presents math as t…

> I cannot overstate how destructive this is, because it basically presents math as this arbitrary logic game rather than as a fundamental language of the universe. You are making a clear philosophical assumption yet don't realize it. Is math really the 'fundamental language of the universe'? or is it an arbitrary logic game that, in its most common interpretation, describes the universe well? Given that we have no f…

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, and generalize it. The proof is a tool for not being wrong, and that is a perfectly fine thing for mathematicians to do and a waste of everybody else's time (unless they really need convincing).

It is an absolute travesty how many people enjoy math until they get to their first college class which is taught by a disconnected mathematician that ruins the subject for them. At my school everyone stopped at multivariable because that's the level at which mathematicians systematically ruined math -- by teaching and testing the wrong stuff. Personally, I survived by the skin of my teeth, and then later found physics and actually learned multivariable calculus and learned to love math again. Despite the efforts of the people who were paid to teach it.

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

#179

Post by WalterBright » Mon, Nov 13, 2023, 6:25 PM UTC

cyberax wrote: post #148

contrarian1234 wrote: post #41

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? it's really apparent there is a LOT of room for improvement in pedagogy. There is a study showing that you actually understand material better, if you use the most primitive methods: chalkboard and a lecture. Because you are forced to visualize the material yourself, instead of being presented with a ready-made animation.…

I've suggested to college students that they leave their laptops behind and attend lecture with pen and notebook, and take notes. It makes things a lot more sticky in the mind.

And do the homework problems. You'll never understand the material without doing the problem sets.

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

#180

Post by dboreham » Mon, Nov 13, 2023, 6:27 PM UTC

threatofrain wrote: post #72

distcs wrote: post #45

Earlier quoted context omitted.

> 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…

Note that 3B1B often warns you that his videos bring perspective for a book/class that you're doing or are already done with. And Khan Academy's focus is for K-12. If you want anything past Analysis 1 I think you'll find that universities guard their content.

> I think you'll find that universities guard their content.

Hmm. All the way back to when I was in college there was advanced content available from the Open University. You had to be awake at 2am and it was in black and white, but it was there.