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comment by Devac
Devac  ·  731 days ago  ·  link  ·    ·  parent  ·  post: An Axiomatic Approach to Algebra and Other Aspects of Life

    "we can rearrange the order of these terms because of the commutative property". At the end, several of my students thanked me for doing this!

and

    It's incredibly frustrating to watch students struggle to understand the material because they're being taught a process instead of why that process works.

Plus after looking through some of the books (both Polish and American) with maths for engineers I can only muster "why?!". It's so idiotically rushed at places that I don't know how anyone could make any sense out of them. Here's an abbreviated 'contents' page:

1. Limits and continuity of functions (not a word about fucking sequences)

2-5. Derivatives of one and multiple variables, applications. Implicit functions.

6-7. Antiderivatives and integrals (practically no applications!).

8. Approximations and numerical methods.

9-10. ODE, 2nd order ODE, 1st order PDE.

11. Series. (Yes, after approximations!)

12-13. Multivariable integration.

14. Complex functions.

15. Integral transforms (Laplace and Fourier).

And believe it or not, that's Calculus 1! I don't even want to look what's on Calculus 2, you probably have one week of numerical PDE followed by Hankel's transform followed by Green's functions. No wonder that 1st year engineering has something close to 75% drop-out rate.

I can't fathom which is more frustrating. Teaching or being taught this curriculum. And that shit was squeezed into the 15-week long semester. I think that you can imagine how I was feeling when I got a few people from mechanical engineering to tutor and realised that most of their current knowledge of maths is pretty much the first two weeks from about six 101 courses. They only lacked a week of probability to make it a complete gibberish :/.

    Furthermore, teaching focused on the process does little to discourage the "I can just do this on a computer in the real world" mindset. I could go on for hours on this topic, since it's hardly specific to teaching math...)

If you have time and energy, do it. There's always a #rant tag and it's there for a purpose ;). Although I have to say that while I'm strongly opposed to people going to the computer to get their answers… it's at least useful to know the basics of both approaches. Maxima and Cadabra are just such amazingly useful tools for error catching or simplifying some of the really wacky formulae that I don't even know where to start singing their praises. But I'm not going to advocate people substituting linear algebra or calculus requirements with a phrase "can't I just run it through Mathematica?". CAS is something that you should earn as a privilege, just like in mr Harkey's class ;).




lm  ·  731 days ago  ·  link  ·  

Here, it's a little different:

Calc I:

- Limits

- Derivatives, antiderivatives for single-variable functions

- Riemann sums

- Basic integration, up to u-substitution

Calc II:

- Trigonometric substitutions

- Partial fraction decomposition; applications to integration

- Natural logarithm/Euler's constant

- Logarithmic differentiation

- L'hopital's rule

- Indefinite integrals

- Sequences & series

- Taylor series

- First-order ODEs

The most infuriating part of Calc II is that students are expected to be able to do basic convergence/divergence proofs for the series & sequences part without ever having learned how to write a proper proof in their lives. Grading those exams is painful because the class uses online math homework, so for many, the first time they have to write a proof for another human is on that exam. Each answer you have to read closely to see if they understand the ideas but can't explain them well or if they just wrote random words on the page.

So many students give up on that section since it's not well connected to any other material in the class and some of the concepts (especially the various remainder theorems) cannot easily be rotely applied to a problem.

And, yeah, "I can do this on a computer" does you no favors when you're trying to understand why some technique (even in another field of study) works, rather than just "oh I have this technique, let me apply it to some problems".

Anyway, you and I should be thankful that we've had something of a nonstandard education in math, and do our best to help others see what we see.

kantos  ·  731 days ago  ·  link  ·  

    The most infuriating part of Calc II is that students are expected to be able to do basic convergence/divergence proofs for the series & sequences part without ever having learned how to write a proper proof in their lives. Grading those exams is painful because the class uses online math homework, so for many, the first time they have to write a proof for another human is on that exam. Each answer you have to read closely to see if they understand the ideas but can't explain them well or if they just wrote random words on the page.

First day of Calculus II happened Tuesday. Professor stated we are going through series and sums first. It's going to be a pain on top of all homeworks being online. This is my first time taking a math class in two years as well which is a bit of a pain having to go back and relearn (read: brush up on) many rules. Thanks for the article. I couldn't help but relate to this section of the comments. Thankfully, I'm also taking Proofs as a class, hired a tutor and plan to attend all the extra supplementary instruction sessions I can. As you can see, Devac, it really can be an odd system [from the outside].

Devac  ·  731 days ago  ·  link  ·  

    Thankfully, I'm also taking Proofs as a class, hired a tutor and plan to attend all the extra supplementary instruction sessions I can.

Regardless of the tutor, feel free to pester me if you would happen to have any problems! And even if your class about proofs does not follow Daniel Solow's How to Read and Do Proofs it's not a bad book to have on hand. Aside from one treatise on vector bundles, it was the sole English-only book that seemed to actually benefit my studies (at least so far). And from me, that's practically a glowing recommendation of Western textbooks. ;)

kantos  ·  730 days ago  ·  link  ·  

I'll keep you in mind. :) My main crux is just refreshing on math/logic symbols. Of course, that's our first chapter thus far, though.

lm  ·  731 days ago  ·  link  ·  

I think sequences and series are quite cool, and they tend to pop up all over the place in other fields, including probability, combinatorics, computer science, and communications.

I'm interested: what all does your proofs class cover? Is it something like a foundations class, where you start off with a few definitions and axioms and prove a bunch of stuff, or does it focus on logic and different proof techniques, or a mix?

Perhaps the best thing I can recommend for this whole "why are things true" business is to find someone else in the class to compare and critique proofs with.

kantos  ·  730 days ago  ·  link  ·  

According to the syllabus, we'll be going through Logic, Direct Proof and Proof by Contrapositive, Existence and Proof by Contradiction, Mathematical Induction, Prove or Disprove, Equivalence Relations, and Functions. Some Proofs in Calc.

Bless my HS Calc teacher for getting our toes wet early on since some of this sounds familiar. Just a matter of dusting of memories. My tutor has gone through the class so he's my first line of defense in the comparison department. If not, I'd like to start a study group (especially so if some from my Calc class are in the Proofs as well).

lm  ·  730 days ago  ·  link  ·  

Cool, sounds like a good all-around useful class. (Well, useful as far as general mathematical knowledge goes...)

Devac  ·  731 days ago  ·  link  ·  

    and they tend to pop up all over the place in other fields

I would not be myself without adding "you forgot science". I think that I have seen a series or two in physics. ;)

By the way, one of my favourite uses of sequences in physics is a proof that no matter how many times you will boost your speed (ν) by some constant (Δν) you are still going to be restricted by the speed of light. That's a pretty cool exercise to solve in class.

    Perhaps the best thing I can recommend for this whole "why are things true" business is to find someone else in the class to compare and critique proofs with.

That's a really cool way to learn proofs. After getting some practice it gets better when you can play the "I may or may not have deliberately made an error, can you spot it?" reversal.

The critical point of no return is when you can do this with category theory :P.

lm  ·  731 days ago  ·  link  ·  

"Category theory is the result of taking concrete, easy to understand ideas and abstracting them until they're incomprehensible." --Philip Wadler

Devac  ·  731 days ago  ·  link  ·  

    Grading those exams is painful because the class uses online math homework, so for many, the first time they have to write a proof for another human is on that exam. Each answer you have to read closely to see if they understand the ideas but can't explain them well or if they just wrote random words on the page.

Damn, my most sincere wishes of good will toward you should manifest in some sort of alcohol in front of you about now. Not to mention that this format just feels wrong. Making an exam on something that most people never attempted before must look to students completely arbitrary on top of that. Like a medical school that ends with a pie eating contest to determine who gets to become a doctor.

    Anyway, you and I should be thankful that we've had something of a nonstandard education in math, and do our best to help others see what we see.

The thing is that I wasn't all that aware about the gap. But I do agree, the least we can do is to show that maths is not about learning some schema but about actually deducing solutions and formulating proofs. Hell, now I feel like I should rewrite most of the complex numbers primer that I made since I have assumed that the reader is prepared to do some heavy lifting instead of (at least that's how it seemed to me) being 'spoon-fed'.

On that note, it's probably for the best that I didn't share most of those Maths for Project Euler sheets. After about second part I have started shortening proofs and fell into the mathematicians mindset of "but that's a trivial step, removed". XD

lm  ·  731 days ago  ·  link  ·  

    Like a medical school that ends with a pie eating contest to determine who gets to become a doctor.

This is a disturbingly apt comparison.

    Hell, now I feel like I should rewrite most of the complex numbers primer that I made since I have assumed that the reader is prepared to do some heavy lifting instead of (at least that's how it seemed to me) being 'spoon-fed'.

I say leave it as-is. At least part of the problem I see is that bad math education tends to hand people properties that seem to have appeared from thin air. On the other hand, if you say, "why is X true?", then that gives an explicit clue to people that they should (if interested) sit down and try to work that out.

Proofs are tricky things to write, but I do prefer a proof written to explain, rather than simply to derive the conclusion. A little exposition here and there can go a long way towards showing others the relationships you see. I am hardly innocent here, though!

I do need to work out a good way of writing proofs in LaTeX. It feels like mine always get compressed into a paragraph of math symbols. Maybe I just need to focus on writing a sentence or two of exposition per step and make one paragraph per conceptual step.

Devac  ·  731 days ago  ·  link  ·  

    I do need to work out a good way of writing proofs in LaTeX. It feels like mine always get compressed into a paragraph of math symbols. Maybe I just need to focus on writing a sentence or two of exposition per step and make one paragraph per conceptual step.

I assume that you want it for notes and not publications, right? I have been using this style for student physics club handouts and it seems pretty well received. Sometimes with margin notes, if I had some general remark about the proof as a whole but could not figure out how to fit it into the text.

lm  ·  731 days ago  ·  link  ·  

Oooh, alignat looks nice! I'll keep that post in mind next time I write a proof (which will hopefully be soon--I've got a little side tangent that I've been working on here and there that I'd like to get online sometime soon). Thanks!