clipka <ano### [at] anonymousorg> wrote:
> Am 09.12.2010 21:29, schrieb Kevin Wampler:
>
> >> That reminds me vividly of something which I have noticed and realized
> >> all by myself: I have never seen *anywhere* the *logical* explanation of
> >> how you can easily calculate, for example, the sum of all the integers
> >> from 1 to 100. It's *always* just the raw and sterile formula, and that's
> >> it. No explanation, no intuitive nor logical way of deducing it. Just the
> >> formula and that's it.
> >
> > I can't say I've had the same experience. I've seen plenty of
> > explanations of that formula, probably more so than any equation other
> > than the Pythagorean theorem.
>
> Same here. Maybe I was lucky in having maths teachers who loved maths
> just as much as I do; maybe it's the difference in educative system;
> maybe it has changed over time.
>
> In my case, our teacher did...
>
> - show us how to deduce the formula for that sum;
>
> - give us historical background of how the basic idea behind the formula
> was discovered by later mathematician Carl Friedrich Gauss when he was a
> kid, to the surprise of the teacher who had thought he had found a way
> to keep his pupils busy; and
>
> - use it as an introductory example for mathematical induction, which
> they then trained us to apply to other problems (and by that I mean
> types of problems, not just the same problem with different parameters).
>
> So I think I did get a truly mathematical education. I might add that
> this was German Gymnasium in the 80's. I'm pretty sure pupils at
> Hauptschule weren't that lucky; maybe even today's Gymnasium pupils
> aren't either, but I hope things haven't changed that dramatically.

Precisely my same experience, the Gauss story and all.  Perhaps the education
system just needs more such motivated teachers.

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Darren New <dne### [at] sanrrcom> wrote:
> And while I took and understood a great deal of math, it wasn't until
> physics class that I suddenly said "Oh, *that* is what an integral is for!"

indeed application should come before ready-given formulas...

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Darren New <dne### [at] sanrrcom> wrote:
> Warp wrote:
> >   Geometry, trigonometry and in some cases even calculus has been quite
> > useful in graphical-heavy and game programming.
>
> But in a sense, that's because you're simulating physics. None of those
> apply to programming a board game, for example.

they do for your game interface, unless you go for 70's ASCII interface. :)

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nemesis wrote:
> Darren New <dne### [at] sanrrcom> wrote:
>> Warp wrote:
>>>   Geometry, trigonometry and in some cases even calculus has been quite
>>> useful in graphical-heavy and game programming.
>> But in a sense, that's because you're simulating physics. None of those
>> apply to programming a board game, for example.
> 
> they do for your game interface, unless you go for 70's ASCII interface. :)

I didn't use any of those for my puzzle game. Sure, the graphics card was 
doing that stuff, but that's only because it's doing 2D with a 3D engine. 
Mostly you don't really need that sort of thing outside of simulating 
(however approximately) physics, methinks.

-- 
Darren New, San Diego CA, USA (PST)
   Serving Suggestion:
     "Don't serve this any more. It's awful."

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Darren New <dne### [at] sanrrcom> wrote:
> Warp wrote:
> >   Geometry, trigonometry and in some cases even calculus has been quite
> > useful in graphical-heavy and game programming.

> But in a sense, that's because you're simulating physics.

  I'm not so sure I would call, for example, the problem "is this point
inside this polygon?" a problem of physics. It sounds mostly a problem
of pure mathematics.

> None of those 
> apply to programming a board game, for example.

  If you need to program an AI opponent for such a board game, it certainly
requires knowledge on several sub-branches of mathematics. Perhaps not eg.
trigonometry (unless the board game is more heavily based on the actual
geometry of the board), but certainly others.

-- 
                                                          - Warp

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On 09/12/2010 07:50 PM, Warp wrote:

>    That reminds me vividly of something which I have noticed and realized
> all by myself: I have never seen *anywhere* the *logical* explanation of
> how you can easily calculate, for example, the sum of all the integers
> from 1 to 100. It's *always* just the raw and sterile formula, and that's
> it. No explanation, no intuitive nor logical way of deducing it. Just the
> formula and that's it.
>
>    The raw formula is rather useless by itself when you don't understand
> where it's coming from.

I do recall seeing in my sister's maths book the infamous quadratic 
solution formula. The book then went on to explain how to derive it...

...by starting with the quadratic equation and applying a seemingly 
arbitrary sequence of exotic algebraic manipulations to it. Sure enough, 
the final result *is* the standard formula. But surely no person would 
have thought of plucking this exact random series of transformations out 
of thin air.

In other words, it demonstrates that the formula *works*, but offers no 
intuitive insight into *why* it works. I got /that/ a bit later that 
day, after I discovered something called "factorisation"...

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On 09/12/2010 09:37 PM, andrel wrote:

> mildly related http://www.cs.utexas.edu/users/EWD/ewd05xx/EWD538.PDF

You can't tile this room because there are more black squares than white 
squares, and every tile covers one black and one white square.

Genius!

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On 09/12/2010 09:35 PM, Darren New wrote:

> And while I took and understood a great deal of math, it wasn't until
> physics class that I suddenly said "Oh, *that* is what an integral is for!"

I read my dad's "elementary calculus" textbook when I was a teenager. 
The only thing I discovered was Pythagoras' theorem (which, 
astonishingly, I had never even heard of before). The rest of the book 
made absolutely no sense to me at all. It talked a lot, and it had some 
formulas that gradually got more and more complicated, but I had no idea 
what the hell it was trying to *do*.

Fast forward a few years to a quants ("quantitative methods") class some 
day at college. (I'm not exactly sure what this has to do with 
"computing", but anyway...) We were working on graphing the income, 
profit, etc. of a fictional company. Rather than invent some fictional 
data, DKJ invented some low-degree polynomial that was supposed to 
represent income, and another that represented costs, and we were 
sitting there computing this stuff with pencil and paper.

I don't know if you've ever tried it, but tabulating even a low-degree 
polynomial by hand is tedious, even with a calculator. You tabulate the 
income polynomial, tabulate the cost polynomial, subtract one column 
from the other, and that gives you monthly profit. You then compute a 
running total of that to determine how much money in total this company 
has made or lost.

I figured out that we only actually *need* the final column. Well, you 
can subtract one polynomial from the other and do some simplifications. 
That enables you to compute the profit with fewer operations. But now 
how the hell do you compute the running total? By observing the general 
shape of the graph I was plotting (with pencil and paper, mind you), I 
was able to fit a polynomial to it. And you know, it's the damnedest 
thing... the coefficients seemed to be somehow *related* to the original 
polynomial.

At this point DKJ peered over my shoulder, and pointed out that what I 
had just done was "integral calculus", and some guy called Laplace had 
invented it 200 years ago. He showed me the general formula for 
integrating a polynomial. I was kind of suspicious at the exact integral 
coefficients; of all the possible real numbers in existence, that seems 
like a rather large coincidence. None the less, I went home and managed 
to find my dad's old calculus book.

Now it actually made perfect sense. :-P

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On 09/12/2010 06:04 PM, Mike Raiford wrote:
> http://www.maa.org/devlin/LockhartsLament.pdf

Now they need to make a film about a society where musicians are taught 
without being allowed to play or listen to music. And then a renegade 
faction breaks off and starts teaching people by, you know, *playing 
music*. Possibly lead by Robin Williams. That would probably work quite 
well.

Oh, wait a second... I just described the Dead Poets Society, didn't I?



As some of you may recall, I went to a school for stupid people. Perhaps 
unsurprisingly, our "maths lessons" consisted *only* of filling out 
countless billions of long-division sheets.

I mean, seriously. Knowing how to add, subtract, multiply and divide 
(not to mention *estimate*!) is important. But dividing 6-digit numbers 
giving an exact result and a remainder? When the **** am I *ever* likely 
to need to do /that/? And *if* I do, I'll use a calculator. Obviously.

I understand that people need to know how to do division. I have no idea 
why they need to practise double-sized sheets with 40 quotients per side 
featuring 6-figure numbers. Hell, even NASA used a slide rule instead of 
pencil and paper! WTF?

Still, I guess it keeps the demented kids quiet for a while...



It wasn't until nearly the end of my time at school that I discovered 
that "mathematics" was something *other than* arithmetic. There's 
actually more to it than that.

That's actually kinda shocking, when you think about it. It's like 
saying that being an author is about good spelling and grammar. Oh, 
sure, that's *part of* being a good author. A pretty friggin' /tiny/ 
part, though. JKR didn't get to where she is today by using punctuation 
correctly. She got there by convincing Warner Brothers to make her books 
into big-budget films. Oh, wait...

I think, for me, the turning point was where I read an issue of the 
Guinness Book of Records. On one page, it had an image of the Mandelbrot 
set, "the worlds most complex mathematical object". It all went downhill 
from there, really. I discovered complex numbers, and algebra, and I got 
books from the library and read them end to end, trying to discover what 
commands I needed to type into my computer to make it produce these 
amazing fractal patterns. (Usually this was an entirely futile pursuit; 
such books like to include pretty pictures and fail to explain how they 
are made.)

Then I went to college, and I met a man named DKJ. Brilliant 
mathematician. Utterly incapable of comprehending the real world. Quite 
strange. Welsh. Anyway, I spent 2 years camped in the college library, 
absorbing just about every textbook that wasn't so far advanced that I 
couldn't even understand what it was talking about.

At uni, they didn't have any maths books. (Or, they did, but they were 
all at the Leicester branch. I'd have to wait a week to get them before 
I could find out if they were actually worth reading.) They did have GP 
though. More of a programming buff than a maths nut, but all the same... 
Nice sandals, BTW. :-P



This paper seems like a pretty accurate description of everything that 
is wrong with mathematical teaching, and society at large. People seem 
to think that mathematics is about moving symbols around on a sheet of 
paper according to a set of complex and technical rules. Which is a bit 
like saying that writing a best-selling novel is about arranging special 
blobs of black ink in the correct sequence according to a set of rules. 
Strictly speaking, that's what writing a book *is*. But no sane person 
actually thinks about it in those terms.

I've recently seen a few interesting programs on TV about mathematics. 
One featured some guy who's apparently a mathematical genius and another 
who's a comedian, and general hilarity ensued. But I doubt the viewing 
public got much out of it.

Another featured a guy explaining the history of mathematics. 
(Apparently India and China between then invented most if not all of 
modern-day mathematics.) The trouble is, every concept they explained 
had to be watered down so much that even I, as a person who knows the 
underlying mathematics, would be hard-pressed to figure out what they're 
getting at.

The fact that you can compute the result of an infinite sum in a finite 
number of steps is amazing and almost unbelievable. But I'm not sure 
that some computer graphics of a boat sailing forwards and backwards 
adds anything to the understanding of /how/ such a feat is possible.

The whole attitude of society seems to be like "oh, mathematics. Yeah, 
it's *so complicated* that you can't possibly understand it. Better not 
even try. They have *other* people to sort that out anyway. So long as 
you can count, who needs to know more than that?" It's almost like it's 
not "mathematics" if it doesn't look cryptic and incomprehensible.



Some while ago, I read a (fictional) quote that I thought hit the mark 
quite accurately: "You're not a director. You know what the problem with 
people like you is? You don't have anything to say, you just want to BE 
A DIRECTOR."

Well, now suddenly I feel like I have something to say...

Now, if only I didn't completely suck at POV-Ray, I could go make an 
animated short, now that I have found my muse. ;-)

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Warp wrote:
>   I'm not so sure I would call, for example, the problem "is this point
> inside this polygon?" a problem of physics. It sounds mostly a problem
> of pure mathematics.

That's a fair point.  I hadn't really been thinking along those particular 
lines.

>> None of those 
>> apply to programming a board game, for example.
> 
>   If you need to program an AI opponent for such a board game, it certainly
> requires knowledge on several sub-branches of mathematics. 

True, true.  But only "mathematics" because that's what we call things like 
alpha-beta pruning and graph running and so on, if you see what I mean. Lots 
of the stuff that AI does isn't what I'd call "mathematical" even tho you 
use math to describe it. It's just algorithms that the inventors happened to 
describe as math first, I'd say.

-- 
Darren New, San Diego CA, USA (PST)
   Serving Suggestion:
     "Don't serve this any more. It's awful."

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