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 (1997) [pdf]

web.williams.edu

Post reply on HN
Print view View original

21–30 of 112 posts

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

#21

Post by conistonwater »

eximius wrote: post #17
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…

Only numerically? Analytically, differentiation is much easier and integrals are tough or impossible, unless I'm missing something.

I think you mean analytically in the same sense as the comment I replied to, of having a closed form you can write down by hand, but that's not right. But that's not the really important thing from a mathematical point of view: you really want to know useful properties about what you get out of integration, and integration gives you nice mathematical objects with good properties, very much unlike differentiation. It's wrong to think of getting a closed form as the goal, that's nice, but not as important as the other stuff. The numerical consequences just follow from the mathematics, so they can illustrate the difference.

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

#22

Post by chestervonwinch »

conistonwater wrote: post #10
rocqua wrote: post #9

Earlier quoted context omitted.

I feel the same, especially with integrals. Integration is way to arbitrary and 'most' integrals cannot be done in a closed form.

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…

One way to interpret why noise = bad for differentiation: The Fourier transform of f'(x) is iw F(w). So differentiation is essentially a high-pass filter.

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

#23

Post by ben_jones »

My differential equations class was the only course I ever took with over 30% of the final grade being derived from homework assignments. Unsubstantiated - but my peers and I all believed this was because everyone was failing and dropping the course.

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

#24

Post by WhitneyLand »

stephengillie wrote: post #2

Please teach calculus before teaching trigonometry. There's no prerequisite to learn trig first, and forcing people to learn trig-calc excites many mathophiles but is a major turn off to other students. Calculus can be taught using just basic algebra, and most students will benefit from already understanding calculus, when they are learning trigonometry.

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 learning, motivation, inspiration, relevance...

Maybe it’s being done and I don’t see it. Maybe it’s not being done, because companies will pay six figures to have people A/B test a different button location on a web site, but the business case for optimizing learning curriculum at traditional institutions is piss poor.

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

#25

Post by hprotagonist »

you can fake knowledge of undergraduate differential equations if you know about 3 hours worth of linear algebra.

More profound understanding is difficult but not terribly impossible.

The best diffeq text i've found is by Blanchard, DeVaney and Hall, and it remains the only math textbook i've ever been able to read.

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

#26

Post by analog31 »

jessaustin wrote: post #7
inbox wrote: post #6

I wish I came across the so called proof-based math before calculus and trigonometry. It would have grabbed me instantly. High school math (and especially physics) classes would leave me with very uneasy feeling that something crucial is left unmentioned, something important is swept under the rug and something important is hidden for whatever reason. Turns out I was wanting for proofs, but couldn't articulate it - I…

Your high school didn't teach geometry?

My kids are in high school right now. I was getting my daughter psyched up for geometry, by promising her that she'd get to do proofs.

The geometry class completely glossed over proofs. It was much more oriented towards solving problems. I don't know if it was because of standardized testing, but I have my suspicions. Fortunately, my daughter worked on the proofs herself, outside of class.

I was saddened for many reasons, one of which is that lots of people I've talked to -- especially women -- loved high school geometry because of the proofs. That was where math came alive.

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

#27

Post by analog31 »

stephengillie wrote: post #2

Please teach calculus before teaching trigonometry. There's no prerequisite to learn trig first, and forcing people to learn trig-calc excites many mathophiles but is a major turn off to other students. Calculus can be taught using just basic algebra, and most students will benefit from already understanding calculus, when they are learning trigonometry.

While we're talking about turning school math on its head... I suggest teaching vectors before trig. You can get the cosine and sine from the dot and cross products.

Learning about vectors in physics finally got me comfortable with trig.

Now, one thing we did learn in trig was to get much more proficient with algebraic manipulation.

Amusingly, with complex numbers, proving trig identities becomes a trivial algebra problem.

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

#28

Post by stephengillie »

jacquesm wrote: post #3
stephengillie wrote: post #2

Please teach calculus before teaching trigonometry. There's no prerequisite to learn trig first, and forcing people to learn trig-calc excites many mathophiles but is a major turn off to other students. Calculus can be taught using just basic algebra, and most students will benefit from already understanding calculus, when they are learning trigonometry.

I found trigonometry a lot easer than calculus, which even today usually takes me forever even for what I'm sure are simple problems.

I found sine and associated concepts to be arcane and Byzantine when learning then through algebra and geometry alone.

Seeing sine, cosine, etc as merely each other's derivative was astonishing and eye opening. So elegant. It made me love math again.

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

#29

Post by analog31 »

tempw wrote: post #20

It's sad one needs to reinforce the teaching of concepts rather tricks. Even worse is when you have all the proof based and concept oriented course and are tested on trickeries on exams.

I believe this is a huge shortcoming of how math is taught. You can bet your last dollar that the teacher doesn't think it's about tricks. But the students are convinced that it is. Students and teachers are both exposed to the exact same material but end up with diametrically opposing conclusions.

Disclosure: I taught college freshman math for one semester, long ago. It was a course where I was supplied with a syllabus and exams, and the students could buy a packet of exams from previous years.

The tricks are what you remember from doing problems over and over, and recognizing patterns. There is also a higher level pattern that isn't mentioned in class, but is vital to solving problems: You learn to identify each problem with a particular chapter or section in the textbook, and then solve the problem by recalling the methods in that section. This is of course a grotesque distortion of what math is, but will get you through the lower level college math courses with good grades.

The other skill is being able to perform the manipulations quickly enough that you can try one or two before hitting on one that works.

Disclosure: I taught college freshman math for one semester.

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

#30

Post by adamnemecek »

TheRealPomax wrote: post #18
adamnemecek wrote: post #12

Earlier quoted context omitted.

Geometry proofs aren't real proofs.

They most certainly are real proofs. However, they are a different _kind_ of proof if you mistakenly believe that only algebra can yield proofs. Provided you have the algebraic rules for the type of geometry you are working with, any geometric proof can be expressed as algebraic proof and vice versa, but the trick is to appreciate that depending on what needs to be proven, one can represent in a single step what the…

Ofc, geometrical proofs are technically algebraic proofs (due to Homotopy type theory) but that's neither here nor there.
Post reply on HN