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Mathematicians are chronically lost and confused

j2kun.svbtle.com

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Re: Mathematicians are chronically lost and confused

#61

Post by simias »

nly wrote: post #20

Secondary math education, for me in the UK, didn't deal with anything outside of elementary algebra, Euclidean geometry, some statistics, and relatively simple calculus. Nobody talked to us about imaginary or complex numbers, or bayes theorem, decision theory, or non-trivial mechanics problems until I was in college (age 16+). Nobody mentioned matrices, broader number theory or discrete transforms until I was in univ…

For me the problem is that math papers/articles lack a proper API documentation. By that I mean that it's very hard for me to understand what a lot of symbols mean because mathematicians (and physicians) love to use single letters to name various concepts and functions.

Worse, they also like to use the same symbol to denote different things in different fields. I'm sure it's extremely convenient to have a shorthand when working and sharing "code" with some peers in the know but for stuff like wikipedia articles it makes things appear more complex and obscure than they should.

I don't mind if people call a local function variable "n" in some code, it's usually non ambiguous. But if you export a variable "n" in an external API you will be screamed at. Why is it ok for maths?

To give a quick example, the letter R in maths can mean the set of real numbers. It can also (with a different typeface!) mean Ramanujan summation. Oh wait, you're doing physics? Then R is the gas constant, silly.

Also, i is the imaginary unit. Except in physics, then it's j, because i is used for currents. Makes sense.

Re: Mathematicians are chronically lost and confused

#62

Post by j2kun »

jorleif wrote: post #58
makmanalp wrote: post #55

Earlier quoted context omitted.

I've argued the same with a mathematician friend of mine. I hate academic papers because of their seemingly convoluted and backwards way of explaining things. His answer was that papers were not made to convey thoughts to laymen, they were made to communicate facts and proofs with as little ambiguity as possible, optimized for reading by other mathematicians. It's meant to be high bandwidth (hence the terse style and…

I think the process of peer review often causes papers to be optimized for being hard to critique rather than easy to understand. I have myself participated in writing papers where we decided to leave out some non-crucial but very useful detail just because it opens up too many questions and opportunities for critique.

Is this in mathematics or another field?

Re: Mathematicians are chronically lost and confused

#63

Post by tokenrove »

nly wrote: post #20

Secondary math education, for me in the UK, didn't deal with anything outside of elementary algebra, Euclidean geometry, some statistics, and relatively simple calculus. Nobody talked to us about imaginary or complex numbers, or bayes theorem, decision theory, or non-trivial mechanics problems until I was in college (age 16+). Nobody mentioned matrices, broader number theory or discrete transforms until I was in univ…

> So many ideas seem closed to mortals because of the nature of mathematics.

Which is why everyone, but especially programmers, should learn a lot more math.

Re: Mathematicians are chronically lost and confused

#64

Post by ronaldx »

nly wrote: post #20

Secondary math education, for me in the UK, didn't deal with anything outside of elementary algebra, Euclidean geometry, some statistics, and relatively simple calculus. Nobody talked to us about imaginary or complex numbers, or bayes theorem, decision theory, or non-trivial mechanics problems until I was in college (age 16+). Nobody mentioned matrices, broader number theory or discrete transforms until I was in univ…

> Secondary math education, for me in the UK, didn't deal with anything outside of elementary algebra, Euclidean geometry, some statistics, and relatively simple calculus.

I've been a secondary math teacher in the UK and I want to defend this point a little.

The job of a secondary math teacher at this level is to teach everyone math, particularly including a majority who don't have a strong interest and won't go on to study more mathematics. You probably underestimate how difficult a job this is.

With that in mind, this is already quite a long list of diverse topics. You neglected to mention an introduction to number, up to the real numbers, perhaps because you now think it is obvious. You were taught that.

I personally try to teach 'looking forward': explaining how these concepts link towards what direction you might take in the future.

However, it's very difficult to cover the whole scope of mathematics and mathematical subjects. For example, I personally knew only a little about what's relevant to EE (although I've learned over time). But, it's not that you could skip anything from the school curriculum anyway - my general advice is that potential EE students need to show interest enough to study independently outside of school.

Something like algorithmic complexity, you should be learning from Knuth. Well done for that: there are not many educational experiences better than learning independently from someone who has devoted his life to making his subject accessible.

Re: Mathematicians are chronically lost and confused

#65

Post by pygy_ »

cs702 wrote: post #22
nly wrote: post #20

Secondary math education, for me in the UK, didn't deal with anything outside of elementary algebra, Euclidean geometry, some statistics, and relatively simple calculus. Nobody talked to us about imaginary or complex numbers, or bayes theorem, decision theory, or non-trivial mechanics problems until I was in college (age 16+). Nobody mentioned matrices, broader number theory or discrete transforms until I was in univ…

Often I find I spend days or weeks deciphering mathematics in compsci papers only to find the underlying concept is intuitive and plain, but you're forced to learn it bottom up, constructing the authors original genius from the cryptic scrawlings they left in their paper... and you realise a couple of block diagrams and a few short paragraphs could have made the process a lot less frustrating. This is SO TRUE. The sa…

> The purpose of most academic papers is not to explain (let alone teach!) ideas in an intuitive manner, but rather to express them in formal, correct, unambiguous terms -- that is, to make them as accurate and critique-proof as possible for publication in some journal. Their intended audience is subject matter experts.

Their purpose really is to let the authors show off how smart they are, impress their peers, and advance their careers. The other properties derive from that.

;-)

edit: I don't want to disparage research in general, BTW, but specifically, the scientific paper redaction process.

Re: Mathematicians are chronically lost and confused

#66

Post by acjohnson55 »

j2kun wrote: post #46
japhyr wrote: post #35

Earlier quoted context omitted.

as the math truck barrels on ahead I've been teaching math to at-risk high school students for the last 10 years. I have spent more time helping students understand that they are not stupid, that something just got in the way of their learning at one point, and they never understood anything after that. I'm going to use your quote in some of these conversations now. What most of my students think: "I could never do m…

I think part of the problem is that the math track is way too linear. It doesn't need to be that way, as I've written about a few times in the past.

I'm not sure how addressable that is. While math could be modeled as a DAG globally, I think it is inherently linear locally (no smooth function pun intended) and incremental. Sure you could jump around, but I think at the end of day, if a student is going to progress to advanced math, they can't dodge tricky concepts.

But maybe I'm misinterpreting your point. Do you have links to what you've written?

Re: Mathematicians are chronically lost and confused

#67

Post by nbouscal »

graycat wrote: post #37
nbouscal wrote: post #27

Earlier quoted context omitted.

You misread the article. He is not recommending that students solve every exercise; he's recommending the exact opposite.

No, I'm correct: He set up an extreme straw man to knock it down. I clearly agreed that his extreme straw man is foolish. There is a common reason students fall for his straw man: They are concerned that if there is an exercise they can't work they are missing something important. My advice was, instead, for a very diligent student, to solve 90-99% of the exercises and just let go of the last few as illposed, stated…

This is a good example of how being correct is completely irrelevant if you can't communicate it well. That said, I still maintain that you're thoroughly misunderstanding the position the OP was arguing for.

Re: Mathematicians are chronically lost and confused

#69

Post by j2kun »

acjohnson55 wrote: post #66
j2kun wrote: post #46

Earlier quoted context omitted.

I think part of the problem is that the math track is way too linear. It doesn't need to be that way, as I've written about a few times in the past.

I'm not sure how addressable that is. While math could be modeled as a DAG globally, I think it is inherently linear locally (no smooth function pun intended) and incremental. Sure you could jump around, but I think at the end of day, if a student is going to progress to advanced math, they can't dodge tricky concepts. But maybe I'm misinterpreting your point. Do you have links to what you've written?

I mean this about the typical subject matter of high school (which is what this branch of the comment thread concerns). Nobody needs to learn how to graph accurate ellipses and the various facts about congruent triangles before doing calculus. You also don't need excellence in algebra to do geometry. There are some fundamentals, like being able to work with fractions, but largely high school education is a lot of parallel topics that they make you think are linearly dependent (no linear algebra pun intended).

Advanced math, on the other hand, is a different matter. And as far as HS education is concerned I believe the focus should be on building mathematical thinking skills and not worrying about preparing students for a particular subject they're unlikely to ever use.

For example, here is a lecture that I give to HS math students on graph theory [1]. You'll notice there's no algebra, no geometry, no calculus, almost nothing is required except the idea of a function (and even that is technically not required, and I tell them not to worry if it's confusing). What is in this talk is a whole lot of mathematical thinking, and I do believe (though this sounds like bravado) that if I were to put my mind to it I could model a year's worth of HS education around developing this kind of mathematical thinking. It would also have some highly nonlinear components to it, organized instead primarily around proof techniques.

[1]: http://jeremykun.com/2011/06/26/teaching-mathematics-graph-t...

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