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Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

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Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#91

Post by ianai »

BeetleB wrote: post #82
eli_gottlieb wrote: post #79

Earlier quoted context omitted.

You don't make students take Linear Algebra before differential equations? I thought students were supposed to take Lin-Alg alongside their Calculus 2 or Multivariable Calculus course, then have all that in their heads for DiffEqu.

>You don't make students take Linear Algebra before differential equations? In my university curriculum, linear algebra was taught after. Perhaps my ODE course was watered down, but that wasn't a problem at all for us.

We had to take 2 linear algebra and an ODE course (amongst other things). The first LA class was second year level and the second was senior level. The ODEs were in between. I feel that's a good way to go about it.

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#92

Post by rsfern »

thanatropism wrote: post #75
eiji wrote: post #74

Earlier quoted context omitted.

Ansys, Abaqus CAE, Dyna, Optistruct, FEKO, AcuSolve, NASTRAN, ... The list goes on and on. Each bigger DE or partial DE has a huge ecosystem of numerical software around it to approximate solutions for real world applications. High and low electro magnetics (Maxwell), Navir-Stokes, ...

Engineers should be taught about actual numerical algorithms and their qualitative properties. Cases in which there exist explicit symplectic algorithms, etc.

Most engineering programs do have at least one full semester of numerical methods, even if in practice most engineers use matlab, scipy, abaqus etc...

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#93

Post by _5ysi »

rocqua wrote: post #64
conistonwater wrote: post #10

Earlier quoted context omitted.

Funnily enough, from a mathematical point of view, it's the opposite: integration is a really nice operation that can be applied to basically anything, whereas differentiation is really finicky, sometimes derivatives don't exist and you can't even be too sure when, so you need to be extra careful. This carries over to doing numerical computing: integrating an arbitrary function is easy, for smooth 1d functions it's a…

Integration can definitely not be applied to basically anything. See the entire subject of measure theory, the concepts of Riemann, Riemann-stieltjes and Lebesgue integrals. The class of C(1) functions is quite easy. The class of intergrable functions is much more difficult. All we know is that it is larger. Consider this: to prove a function isn't differentiable, you need only give a single point where the derivativ…

Perhaps he means from a practical point of view. Signals arising from the types of natural phenomena measured in engineering is always real, contiguous, and limited. An integrating op amp circuit is going to be stable up to the limits of the power supply, but a differentiator is likely unstable and unusable due to noise. Fourier transform and freqiency analysis rely on integration. Feedback loops with delay. Etc.

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#94

Post by GeorgeTirebiter »

mathattack wrote: post #45

Apparently he passed 2 years after this was written. RIP. A great mind. https://en.wikipedia.org/wiki/Gian-Carlo_Rota

I clearly remember his class on, basically, Mathematics for Philosophers. As an engineering student, I thought my pal from Urban Studies who suggested taking it was a way to pad one's GPA, which wasn't my style. Nonetheless, I agreed, and, now, I regard it as one of the most important courses I've ever taken. I have never met anyone with that level of enthusiasm, wonder, and love of Mathematics - who also knew how to partially impart it to his students! Mainly, it's the Wonder of it all that remains with me. What a privilege.

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#95

Post by abecedarius »

BeetleB wrote: post #90
abecedarius wrote: post #85

Earlier quoted context omitted.

OK, but the algebra of complex numbers is so much easier than trig -- less to memorize -- that I think it must be inertia that's keeping our trig courses around, not inherently better pedagogy. Has anyone tried teaching this way in high school or middle school?

>OK, but the algebra of complex numbers is so much easier than trig -- less to memorize -- that I think it must be inertia that's keeping our trig courses around, not inherently better pedagogy. Let's not forget the geometric aspect of trigonometry, which will be more intuitive to many than complex numbers. I have a force of 10N applied at an angle of 32 degrees. I need the x-component. You still need basic trigonome…

Yes, you need to tie the algebra to the geometry. I had a friend taking a trig class, and when I showed him how all those trig identities just fall out of multiplying complex numbers, he was pissed.

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#96

Post by LolWolf »

wolfgke wrote: post #72
CalChris wrote: post #32

Earlier quoted context omitted.

Trig is artificially hard in calculus. When you get to complex numbers it all gets easier. But they don't teach that until later. One thing I learned at Berkeley was that there are two kinds of problems: linear problems and problems you can't solve. The trick ( EE 120 Linear Systems ) was always how to transform a complicated problem into a linear problem. Yeah we used complex numbers as part of the trick to get to l…

> Trig is artificially hard in calculus. When you get to complex numbers it all gets easier. But they don't teach that until later. And if Clifford algebras were introduced, it would get even easier. But they typically don't even teach that.

This needs to be mentioned more frequently to be honest. I still am unsure why the hell Gibbs vector calculus is taught when Clifford algebras are so beautiful.

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#97

Post by g10r »

dotancohen wrote: post #62
uoaei wrote: post #5

Earlier quoted context omitted.

Interestingly, children as young as 5 show an aptitude for understanding overarching concepts of calculus.[1] This makes sense: it is much easier to talk about "rates of change" and "accumulation" in simple terms and show how they are related using models that appeal to children. We don't need to dive right in to the notation and algebraic manipulations to get across the basic idea. That can come later when children…

My oldest, while in second grade, learned enough calculus to determine the _location_ of a train given it's acceleration and since it started. That's because she was interested, and asking questions, and luckily I had explained to her the slope of a graph just a few days before the train ride. She didn't learn the formula to memorize, but rather the concepts. Only then did we do the calculations the long way, on pape…

why?

The question my three-year-old son asks over and over each day. It's exhausting, and I love it. I do my best to provide the answer instead of simply stating "because" or "just do it" as one of my greatest fears is to suppress his natural desire to understand as much as possible about the world. Also, as a child I hated memorization, yet loved delving into a subject that intrigued me.

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#98

Post by WhitneyLand »

avs733 wrote: post #83
WhitneyLand wrote: post #24

Earlier quoted context omitted.

Let’s take that up a notch - try both and actually measure the results. It’s accepted to design products and services by trying lots of permutations and measuring the success, is this ever done with teaching? So many people (myself included) have stories of, if only I had been exposed to such and such concept in a different way it would have had a much bigger impact. Why not measure multiple aspects? Efficiency of le…

>It’s accepted to design products and services by trying lots of permutations and measuring the success, is this ever done with teaching? Yes and...of course it is done and tried. The problem is that most parents (especially those whose kids are most likely not to have the support to get past these struggles) are not exactly ecstatic for their children to be used as experiments. They want concrete answers, fixes, sol…

That’s helpful background, thank you. A lot of it makes sense but, hear me out, I don’t believe you’re point on difficulty or definitions has much to do with the the most frustrating parts you mention.

Research is always difficult, in any field. You don’t get a PhD for courses, when you’re expected to have insights that no one ever has had before, that’s just a hard thing. Same with designing research. It’s so difficult and complex across the board people make big mistakes doing it all the time.

Same with the parent problem. Recruiting humans to study is always a huge mine field, medical/psych/sociology/etc. all deal with it.

So what’s different about education? I don’t know your field, but it sounds like things are too tightly (inherently) coupled to public policy controversy, on top on the money thing.

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#99

Post by srean »

gaurav_v wrote: post #16
conistonwater wrote: post #10

Earlier quoted context omitted.

Funnily enough, from a mathematical point of view, it's the opposite: integration is a really nice operation that can be applied to basically anything, whereas differentiation is really finicky, sometimes derivatives don't exist and you can't even be too sure when, so you need to be extra careful. This carries over to doing numerical computing: integrating an arbitrary function is easy, for smooth 1d functions it's a…

More like `from a computational point of view.' Analytically, integration is much more difficult and richer than differentiation!

Finding the closed form of the integral of a closed form, yes then what you say is true (this is different from 'analytic' which in mathematics has a different meaning). Scope of the concept of even baby integration of a function is much much larger, and OP is talking about that.

Note the key word there is a function not a function with a closed form that's a tiny subset.

Re: Lessons I Wish I Had Learned Before Teaching Differential Equations (1997) [pdf]

#100

Post by srean »

burnerOh2125 wrote: post #31

Interesting, I'm a phd-drop out in computational biology, working as a data science consultant: I use mathematics including multi-dimensional statistics, linear algebra, and calculus everyday. Being self-taught, I'm very self conscience about the math I don't know, but so far, not knowing differential equations doesn't seem to have hurt me. I actually just ran into a problem that uses Hamiltonian dynamics, so maybe I…

Think of it this way, most of the people in data-sciences are not familiar with differential or difference equations. So once you have a tool that many others don't have, certain problems (can) become accessible to you in a way it is not accessible to others. Sure, the problems solved by people who do not have differential equations in their tool chest will not need differential equations, that's a tautology.
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