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)

#161

Post by BeetleB » Mon, Nov 13, 2023, 5:43 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…

Although I studied calculus in a pure math perspective, I knew enough physics to tie the two concepts together and it may have benefited me some. For the most part, though, all I needed to know was "integration = area under curve" and "differentiation is computing the slope at a point on a curve" and most of the first 3 semesters in calculus made sense from that perspective.

However, my concern with your approach is that those examples help only if you are interested in physics (or whatever field those examples come from). I worked at a math tutoring institute for a few years and often saw that physics examples in textbooks like Stewart's confused a lot of students who were not that interested in physics. Not only do they have to learn the math, they now have to learn physics concepts just to understand the examples! There were plenty of students who could do differentiation/integration, but struggled at the questions involving physics.

Likewise, the department had a separate calculus course for people in finance, economics, etc. That textbook had applications in finance. All the tutors struggled to help those students because we had to learn basic finance concepts to understand the problems. And on the flip side, the students ended up useless at solving calculus problems on their own - but they could do the ones involving finance concepts.

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

#162

Post by bigbacaloa » Mon, Nov 13, 2023, 5:47 PM UTC

contrarian1234 wrote: post #41

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…

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…

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

#163

Post by BeetleB » Mon, Nov 13, 2023, 5:48 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…

For those not in the know: DFT = Density Functional Theory

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

#164

Post by gbjw » Mon, Nov 13, 2023, 5:48 PM UTC

kriops wrote: post #142

meroes wrote: post #139

How math relates to the physical world is not at all clear. Like yes 2 apples clearly makes sense, but non physical quantities like 10^241 of anything start to break down the relationship. It’s not the abstraction that gets me usually, it’s the lack of an explicit relationship between math and physics. I’m sorry but personally I am not satisfied by what I learned in my math and math/science classes on this. It starte…

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).

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

#165

Post by dustingetz » Mon, Nov 13, 2023, 5:50 PM UTC

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…

No Bullshit guide to Math and Physics https://minireference.com/

(serious reply, if it's insufficient for a freshman course i propose following up with Feynman https://www.amazon.com/Feynman-Lectures-Physics-boxed-set/dp..., any objections?)

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

#166

Post by Folcon » Mon, Nov 13, 2023, 5:51 PM UTC

mnl wrote: post #120

BobaFloutist wrote: post #103

Earlier quoted context omitted.

Why should pedagogy ever end? That's like saying at some point in health care medicine must end and you're responsible for your own treatment. What are the professors for, doing research and abusing their grad students?

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?

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

#167

Post by ducttapecrown » Mon, Nov 13, 2023, 6:01 PM UTC

scythe wrote: post #35

ianai wrote: post #13

Earlier quoted context omitted.

They talk (rightly) about mathematics replacing infinitely small but not 0 numbers with limits there. Infinitesimals were later reintroduced with rigor through https://mathworld.wolfram.com/NonstandardAnalysis.html I have long thought such would be easier to work with on modern computers.

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)sin(e)-sin(x)) = st(1/e (sin(x)(1-e^2)+cos(x)e - sin(x))) = st(1/e (sin(x) - sin(x) - e^2 sin(x) + cos(x)e)) = st(e sin(x) + cos(x)) = cos(x)

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

#168

Post by JadeNB » Mon, Nov 13, 2023, 6:04 PM UTC

quacked wrote: post #138

JadeNB wrote: post #67

Earlier quoted context omitted.

> In my engineering curriculum in some cases the teachers were openly hostile to students that had a hard time understanding the material. While there's no excuse for it, I think faculty see so much apathy on such a regular basis that sometimes it's easy to mistake sincere struggle for a lack of desire to engage. For precisely that reason, I try very hard to recognize and reward my students who are willing to take th…

> This is a beautiful description. I know as a teacher that I try to communicate this to my students, but also that I surely fail much more often than I succeed. Thanks. There's an element of hopelessness in trying to explain the immensity of human knowledge to someone who's lived their entire life captured by mandatory schooling. It's just not an interesting thought to most of them. Their worldview has been so artif…

> At the undergraduate/introductory level, the only important question is "what awareness do you have of the breadth of this field, and what mastery do you have over the concepts that most agree are its "most fundamental"? You and I both agree that class is a "playground", and any "grade" is in fact meaningless. Rather than assigning As, Bs, Cs etc. as a percentile of subjective completion of arbitrary problem sets, As-Fs should be assigned at the discretion of the instructor as a holistic assessment of oral and written examination, completion of problems and problem sets, participation in the class, and general wisdom.

I'm not completely sure I agree that this is the solution—if we're re-inventing grades anyway, then I'd like to do something more radical than using the same old A–F and just interpreting them differently (although, even if given free rein to do whatever I liked, I don't know what I would do!)—but I definitely agree that grades, and the standard approach to them, are the most pernicious part of "education" (in the sense of the current schooling system). If there were any way to get away with it, then I would be happy to—indeed, I would prefer to—have all evaluative exercises be diagnostic and informative, only for the students' benefit, and to assign no grade at all, or an A for everyone; but this seems incompatible with a modern university structure (and anyway is essentially forbidden by university administration).

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

#169

Post by frognumber » Mon, Nov 13, 2023, 6:09 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…

I have a lot of expertise in this domain:

> I had a lot of hope for things like Khan Academy, but the issue is video is not text and it's hard to iterate and improve on.

Not the issue.

> I really wish textbooks with open licenses would take over and they could be reworked and improved year after year by different people

The issue.

There actually isn't even a good open content license, analogous to the GPL-style licenses. Improving on video means having access to the source files. Ditto for interactive activities. Khan Academy is designed to look as open as possible, while withholding just enough and being just mean enough with license to make any sort of reuse a hopeless endeavor.

With the proper piece in place, video is very possible to iterate upon.

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

#170

Post by anon291 » Mon, Nov 13, 2023, 6:09 PM UTC

Workaccount2 wrote: post #101

ajkjk wrote: post #16

Earlier quoted context omitted.

Physics doing the work of actually teaching math is a universal embarrassment to math departments. Or, well, it should be.

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 full formulation of the universe in terms of mathematics (no theory of everything), the claim that mathematics as it is today is the 'fundamental language of the universe' seems laughably wrong.

On the other hand, there are thousands of formal mathematical systems that don't describe anything 'real' as far as we can tell. I mean, the entirety of the lambda calculus was developed before there were any computers that could realize any it in any physical system. Yet, its development has proven incredibly important in the entire foundations of mathematics (many theorem provers, etc are based on some variant of lambda calculus).

I'm not denying that some people learn calculus, algebra, etc (i.e., standard high-school/early college maths) with physical systems in mind. However, mathematics is much more than that and limiting the field only to that which we can sense is actually detrimental to mathematical progress as a whole. It seems best to me to encourage purely mathematical thinking itself in the hope that the arbitrary systems humans create may one day be used to describe something. History is littered with 'useless' subfields of math later becoming fundamental to the economy (number theory and elliptic curves to cryptography, lambda calculus to computation, group theory to quantum physics, etc)