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

math.toronto.edu

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

#61

Post by danielbarla »

batbomb wrote: post #41
mightybyte wrote: post #35

I couldn't agree with the last point more. > 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. The bag of tricks mentality is, in my opinion, a defeatist mentality...In an elementary course in differential equations, students should learn a few basic concepts that they will remember for the rest of their lives... I…

I'm not sure what that problem would have been, but an integral part (no pun intended) of DEs is expansions. It's not part of a bag of tricks, it's a very common technique used to solve a problem.

I think the problem is that not enough time is spent explaining these very common techniques - what the rationale behind them is, hot to intuit that it could be applied to a specific problem, and how to get from that intuition to the actual answer / application.

Often, there's a very large skill gap between the student and the teacher (especially at college level, where many of the career mathematicians live and breathe maths), and these things are hand-waved away as obvious. Even worse is when the teacher doesn't actually know, and is just presenting the material straight from a guide. The way most course material seems to be set up is to skip over the middle part, and as a result the best short term strategy is to learn it as a bag of tricks.

I love maths, but I think its universal applicability and beauty get lost due to the way it's generally taught.

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

#62

Post by rorykoehler »

It seems there are lots of posts complaining about how ODE's are taught here. I am planning to study them in the next months. I can only learn maths through applied mathematics and always need to know the why before I can get the how. Can anyone please point me in the right direction for online materials which will help me self-learn, taking into account my learning preferences?

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

#63

Post by antman »

Can anybody comment on that? (page 8 paragraph 1 ):

Professional mathematicians have avoided facing up to density functions by a variety of escapes, such as Stieltjes integrals, measures, etc. But the fact is that the current notation for density functions in physics and engineering is provably superior, and we had better face up to it squarely

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

#64

Post by tomekowal »

Please, write another textbook.

The internet changed drastically the process of writing books. I saw people making profit from books available online for free. I saw books written chapter by chapter with errors found quickly by first readers. I saw systems that allow commenting parts that are not clear enough with comments how to clarify them.

If you promise to deliver a textbook that teaches skills relevant to engineers, they will fund the time it takes to write the book. I would spare couple of dollars even if you said that it will take 5 years. I believe, there will be even companies that will give you funds upfront. If you reach out for help there will be people who will help you collect example problems from different fields to replace couple of "salt tank problems".

I am not a mathematician, so I don't know how mathematics textbooks are written and how much effort goes into them, so feel free to point out that this idea is stupid.

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

#65

Post by yiyus »

kevinr wrote: post #36
rafinha wrote: post #21

"FORGET ABOUT EXISTENCE AND UNIQUENESS OF SOLUTIONS" what? most important thing about differential equations.

As an engineer, I promise you, I give not one single solitary conscious fuck about the uniqueness of solutions to differential equations. Mostly I just hit things with Fourier or Laplace transforms as appropriate until they stop moving.

As an engineer working in some optimization problems where is utterly important if a solution is a local or a global minimum, I assure you I give many fucks about the uniqueness of solutions.

I totally agree with your way of working for most applications, it is what I do most of the time too, and I agree with the article in that this point is given more importance in DE courses than it really is for engineers, but there are perfectly valid use cases for these theorems in engineering.

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

#66

Post by yiyus »

Although I agree that word problems can be somewhat distracting in a DE course, I do not think they should be totally avoided. One of the main reasons it was difficult to grasp the concepts we studied in our first DE courses is that they were too abstract. When later we studied other subjects, such as electric circuits or fluid dynamics, everything started making much more sense.

In my opinion, the ideal way of learning would be to first have very basic (only conceptual) introductory courses of applied fields, where we find some basic equations that we do not know how to solve. And then, we study DE to learn the techniques to solve these problems, avoiding direct references but keeping in mind where we are going with all this.

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

#67

Post by raverbashing »

rorykoehler wrote: post #62

It seems there are lots of posts complaining about how ODE's are taught here. I am planning to study them in the next months. I can only learn maths through applied mathematics and always need to know the why before I can get the how. Can anyone please point me in the right direction for online materials which will help me self-learn, taking into account my learning preferences?

I would say skip studying "pure ODEs" then examine what you need specifically depending on your area

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

#68

Post by kevinr »

yiyus wrote: post #65
kevinr wrote: post #36

Earlier quoted context omitted.

As an engineer, I promise you, I give not one single solitary conscious fuck about the uniqueness of solutions to differential equations. Mostly I just hit things with Fourier or Laplace transforms as appropriate until they stop moving.

As an engineer working in some optimization problems where is utterly important if a solution is a local or a global minimum, I assure you I give many fucks about the uniqueness of solutions. I totally agree with your way of working for most applications, it is what I do most of the time too, and I agree with the article in that this point is given more importance in DE courses than it really is for engineers, but th…

I'm glad someone got something out of that section of the course besides a bad grade and the lingering suspicion that if it had been taught out of the engineering school it would have made more sense.

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

#69

Post by gohrt »

batbomb wrote: post #41
mightybyte wrote: post #35

I couldn't agree with the last point more. > 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. The bag of tricks mentality is, in my opinion, a defeatist mentality...In an elementary course in differential equations, students should learn a few basic concepts that they will remember for the rest of their lives... I…

I'm not sure what that problem would have been, but an integral part (no pun intended) of DEs is expansions. It's not part of a bag of tricks, it's a very common technique used to solve a problem.

The common techniques ARE a bag of tricks. Feynman was famous for being really good at integrals, because he had memorized the huge bag of tricks. Today, we have Mathematica for that, you don't need to be Feynman.

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

#70

Post by fdej »

Kinnard wrote: post #2

"We are kidding ourselves if we believe that the purpose of undergraduate teaching is the transmission of information. Information is an accidental feature of an elementary course in differential equations"

In my opinion, the first course on ODEs shouldn't be concerned with the transmission of information.

That is where the wave equation comes in, in the first course on PDEs.

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