Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

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

web.williams.edu

Post reply on HN
Print view View original

151–160 of 261 posts

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

#151

Post by phkahler »

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 think I had any ideas one what was going on until it was mentioned in physics several years later.

This is where kids with an engineer parent have an advantage. Such a parent can offer examples of how some of the math is used in the real world. Once relevance is established it's OK to point out that this stuff is often buried in software but someone had to put that math into code form so we can all benefit from it. "You may never use it, but you use tools that use it under the hood" can be motivating when they're wondering if it's all just weird busy work.

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

#152

Post by SpaceManNabs »

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 tricks is perfect. The students that need more will learn that theory later anyways in a second or even third class.

You don't have to sacrifice that much rigour either.

Not every class needs a narrative either. I agree that concrete examples to go with the abstract is more needed in mathematics, but differential equations isn't a victim of that.

I see other comments say that differential equations are victim to :

> why we are we learning this?

I don't think this really applies to differential equations at all to be honest. And people say the same complaints about learning scales in music or wtv. Just learn shit and figure out meaning later works too.

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

#153

Post by jampekka »

marcosdumay wrote: post #140
jampekka wrote: post #118

Earlier quoted context omitted.

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…

In fact, there is a lack of linear algebra in infinite-dimensional spaces on the general curriculum, and it is an important subject for physics and a few engineering areas besides the relevance for mathematicians. It wasn't clear to me how the dependency between calculus and algebra was supposed to be, but your comment makes it crystal clear. The discrete transformation is quite fitting for finite-dimensional algebra…

Infinite-dimensional spaces do indeed come up in e.g. Gaussian processes and the kernel trick. I have to admit I've never really understood them the rigorous form. For ML for example it would be likely more useful than e.g. most matrix decompositions.

I'd guess math uses continuous forms because it's where the mathematical tools are and many things tend to get simpler in mathematical sense when you let something go infinitesimal or infinite. This could maybe be different if digital computers would have been invented before calculus.

I've learned to appreciate that mathematicians think of maths quite differently to engineers or scientists. To them the interest is in the "mathematical objects" and their (provable) properties, not the applications or relationships to "the real world" (and this is probably a good thing in itself) and for engineers/scientists it's the opposite. Maybe something like how linguists vs novelists approach language.

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

#154

Post by zozbot234 »

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.

Only a small subset of O/PDE's can be solved analytically in the first place, and few of those are anything but theoretical curiosities. Linear w/ constant-coefficients are the one class that's of big practical interest, which is why OP argues that a well designed course should be focusing on these.

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

#155

Post by pif »

jlokier wrote: post #102
pif wrote: post #78

Earlier quoted context omitted.

???

For what it's worth, I was a researcher at CERN with colleagues from multiple countries, where I turned down an offer to do a PhD in particle physics :-) (A foolish career decision in retrospect.) I think they are saying that if you were clever enough to do physics at that level, you wouldn't understand the problem being discussed. So when you said you don't understand, you gave a live demonstration. When you emphasi…

What I meant is that I think I know something about academics in science in general, and in Physics in particular.

Getting my PhD was not easy for me, as it wasn't for the most people I met, because none of us is a so-called genius. Rather, all of us had to put a lot of effort and discipline in order to move on.

And never did we met any artificial obstacles or deliberate difficulties. Physics is hard; it is as hard as many other academic fields. You need focus, you need discipline, and yes, you need passion to be able to keep focus and discipline.

But you will be welcome if you try, and you will receive a lot of help. Maybe you will find out it was not the right choice for you and you will change your target, but you will not be screened out and rejected just because.

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

#156

Post by elteto »

da39a3ee wrote: post #143
threatofrain wrote: post #72

Earlier quoted context omitted.

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.

> If you want anything past Analysis 1 I think you'll find that universities guard their content. Not so; there's an absolutely vast amount of freely available undergraduate mathematics resources available at all levels. Honestly, so much that it makes it confusing to choose and not get distracted by the options -- perhaps AI-mediated distillation could be helpful in the future.

Really? Can you link some for Analysis and beyond?

I wanted to find good analysis video lectures from a real university complete with problem sets, homeworks and their solutions. I couldn’t. I think MIT OCW now has one analysis course like that, but it’s relatively “recent”.

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

#157

Post by elteto »

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…

Taylor’s is fairly “introductory” and it’s fantastic. Very well written.

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

#158

Post by ink_13 »

ajkjk wrote: post #32
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.

Yeah it's not rigorous. The tragedy is that the organizations responsible for teaching math, like how to do it and how to understand things mathematically, for other purposes , would rather waste their time on rigor. No problem with there being a rigorous math department. For mathematicians. But, quite literally, no one else cares. Mathematicians and everyone else are at crossed purposes, and the mathematicians are w…

> No problem with there being a rigorous math department. For mathematicians. But, quite literally, no one else cares

As someone who did an undergrad in mathematics, I have to disagree with you here. People think math is about numbers and computation, but that's like saying literature is about letters and composition. Fundamentally, mathematics is the study of things that are provably true. Without rigor it's not math.

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

#159

Post by ducttapecrown »

When I was a kid, one of my parents told me diff eq was a course they had not even gone to class for and passed, which set me up for disappointment in diff eq pedagogy for sure. It didn't help when I read this article, and then took sophomore diff eq and it was exactly as described. Later in grad school I'd actually teach the course to my complete chagrin. I think I mentioned this essay to my students, because how could I not? Of course I did not try to redo the diff eq pedagogy, but I did put a lot of emphasis on linearity.

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

#160

Post by ducttapecrown »

mrkeen wrote: post #134
maweaver wrote: post #119

> 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…

The further I got into maths (not very far mind you, but I think I at least did a few differential equations,) the more it felt that maths really was just a bag of tricks, moreso at the advanced end. * Keep deriving circular functions until they cancel out. * Completing the square * Use the quadratic formula to solve degree-2 polynomials ... No formula for degree-5 or higher. * Use the Laplace transform here... for r…

Many famous mathematicians have quotes along the lines of "it's not the result that mattered here, it's the new method/trick that is cool!" and essentially it's because solving math problems requires a bag of tricks and a new one can open new doors in every field it hasn't been applied to yet.
Post reply on HN