>    Or perhaps more like: Programming has been a good incentive to 
learn math?

Pretty much. I don't think it's coincidence that I got a lot better at 
math a couple of years after my family got our first computer. POV-Ray, 
Visual Basic, and JavaScript all pushed my math skills, because there 
was a real, immediate payoff to every thing I learned.

  - Slime

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On 10/12/2010 05:55 PM, nemesis wrote:

> You should really have gone to a special school of another kind. Your
> parents didn't realize your math brain was running faster than those of
> other kids of your age...

My parents followed the advice of the experts - experts who are supposed 
to know what they're talking about. And hey, I can now read and write, 
which is kind of crucial...

> calculus is tough for anyone. I wouldn't ever be able to grasp it while
> a teen...

In what sense is calculus "tough"?

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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>> 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?
>
> Hollywood lied to you about the moment in the Apollo 13 mission when the
> astronauts asked the ground grew to double-check their maths: They did
> /not/ use slide rules in that situation, but indeed pen & paper, because
> slide rules don't help with simple addition of numbers.

OK, fair enough.

> Back then, computer was an actual job, requiring highly trained people.

I gather that this is the original meaning of the word "computer", yes.

> I'd also say it was good for our generation to know how to add,
> subtract, multiply and divide without a pocket calculator

I'd say it would be good for *every* generation to know how to do this. 
Given that it's not exactly complicated, it still astonishes me how many 
people cannot do this. I mean, hell, I haven't memorised the 
multiplication table, but I know the *algorithm* for all of these 
operations.

(I do not, however, know of any way of computing a square root that 
isn't horrifyingly torturous...)

>> 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...
>
> Not really. The books were amazingly popular before the movies.

Yeah, I know that. I was just trying to inject a little humour.

In spite of the insane popularity of her books, there are apparently a 
significant number of people who assert that they aren't very good. (And 
not just the radical Christian sects who assert that any magic without 
reference to God is the work of Satan...)

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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> This paper seems like a pretty accurate description of everything that
> is wrong with mathematical teaching, and society at large.

Although it did leave me wondering for a moment: Am I actually "good at 
math"? Or am I just good at following directions?

> 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.

By contrast, this paper seems to assert that mathematics is about 
constructing abstractions and building theories out of them as a matter 
of creativity. Obviously I've never seen any cutting-edge mathematics 
(and I never will), but I've always thought of mathematics looking at 
interesting systems and discovering their properties, out of simple 
human curiosity. Certainly that's why *I* explore mathematics; it's the 
desire to know everything about everything.

(Well, "everything" is an exaggeration. There are plenty of fields of 
mathematics that are so utterly boring that you have to wonder why 
anybody studies them - set theory, number theory, category theory, etc.)

The paper, on the other hand, likes it sound like doing mathematics is 
like modern art - an essentially empty and pointless exercise designed 
to impress art critics and other artists, in the style of the Emperor's 
Magical Suit. And here I was thinking it was about furthering mankind's 
knowledge...

Seriously. I enjoy exploring mathematical ideas just for the hell of it. 
But you just never know when this stuff will end up being useful.

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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Orchid XP v8 <voi### [at] devnull> wrote:
> > This paper seems like a pretty accurate description of everything that
> > is wrong with mathematical teaching, and society at large.

> Although it did leave me wondering for a moment: Am I actually "good at 
> math"? Or am I just good at following directions?

  Perhaps it could be compared to computer algorithms: If you are given
a computational problem for which you need to create an efficient algorithm,
how good are you at coming up with such an algorithm? (Of course this
requires quite a bit of both knowledge and experience in algorithmical
programming. Eg. you need to have knowledge about computational complexities,
have developed a kind of "computational complexity instinct", have experience
on different problems and their solutions, have the experience and talent to
come up with either modified or brand new solutions to the given problem,
and so on.)

  If you routinely can devise efficient algorithms for given computational
problems, then one could say that you are good at algorithmic programming.
Likewise I'd say that if you routinely can solve mathematical problems,
devise mathematical formulas to describe things and so on, you are good
at math.

  Anybody can calculate 2+2 or even solve an equation, in the same way
as anybody can write a "hello world" program or a program which calculates
the sum of a list of numbers, but that doesn't make that someone *good* at
math/programming.

-- 
                                                          - Warp

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>> Although it did leave me wondering for a moment: Am I actually "good at
>> math"? Or am I just good at following directions?
>
>    Perhaps it could be compared to computer algorithms: If you are given
> a computational problem for which you need to create an efficient algorithm,
> how good are you at coming up with such an algorithm?

To be honest, I've yet to discover a computational problem which hasn't 
already been solved 25 different ways (usually before I was even born). 
The art of writing efficient computer programs appears to be the art of 
figuring out where the **** all these algorithms are written down. 
Because, let's face it, any algorithm that I can invent will be several 
complexity classes worse than what the greatest minds of the 20th 
century could think up.

>    If you routinely can devise efficient algorithms for given computational
> problems, then one could say that you are good at algorithmic programming.
> Likewise I'd say that if you routinely can solve mathematical problems,
> devise mathematical formulas to describe things and so on, you are good
> at math.

Well, when I was a teenager I wrote my first ray tracer. Which doesn't 
sound that impressive, until you realise that I have to figure out how a 
ray tracer actually *works* in the first place, using nothing other than 
the documentation for POV-Ray. (This is *before* Warp added the SDL ray 
tracer, I should point out... That would have made it significantly 
easier, obviously.)

In short, I had to deduce that you perform ray intersection tests by 
solving simultaneous equations. (Usually if you want to draw a shape, 
you want a parametric representation of it. But POV-Ray seems to talk a 
lot about equations rather than formulas, which was a big tip-off.) And 
then I had to work out how you actually solve simeltaneous (non-linear) 
equations. (No, I didn't know how to do that.) And then I had to shift a 
bunch of algebra around to make it happen. (That, arguably, is just 
blindly applying rules. Mathematica(tm) could have done it just as 
capably as me.)

FWIW, I was utterly astonished when I ran the program and it actually 
freaking *worked*!

>    Anybody can calculate 2+2 or even solve an equation, in the same way
> as anybody can write a "hello world" program or a program which calculates
> the sum of a list of numbers, but that doesn't make that someone *good* at
> math/programming.

Actually, I suck at arithmetic. (Fortunately, I have a computer to do 
that for me...) What I consider myself to be modestly good at is 
mathematical reasoning.

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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On 11/12/2010 12:14 PM, Orchid XP v8 wrote:

> By contrast, this paper seems to assert that mathematics is about
> constructing abstractions and building theories out of them as a matter
> of creativity. Obviously I've never seen any cutting-edge mathematics
> (and I never will), but I've always thought of mathematics looking at
> interesting systems and discovering their properties, out of simple
> human curiosity. Certainly that's why *I* explore mathematics; it's the
> desire to know everything about everything.

Thinking about it... No, I'm not very good at math. I never bother 
proving things.

According to the paper, the proof is far more important than the result 
(at least as far as mathematicians are concerned). I always considered 
the proof to be the most boring part.

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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Orchid XP v8 <voi### [at] devnull> wrote:
> >> Although it did leave me wondering for a moment: Am I actually "good at
> >> math"? Or am I just good at following directions?
> >
> >    Perhaps it could be compared to computer algorithms: If you are given
> > a computational problem for which you need to create an efficient algorithm,
> > how good are you at coming up with such an algorithm?

> To be honest, I've yet to discover a computational problem which hasn't 
> already been solved 25 different ways (usually before I was even born). 
> The art of writing efficient computer programs appears to be the art of 
> figuring out where the **** all these algorithms are written down. 

  There is an infinite number of possible variations to any given problem,
so you often can't just use a pre-defined algorithm as-is to solve the
problem, but you have to adapt it. Also, you can't know every single
algorithm in existence, and many of the most obscure algorithms may be
extremely hard to find (especially if you don't now the proper terminology),
and hence you have to come up with the solution yourself.

> Because, let's face it, any algorithm that I can invent will be several 
> complexity classes worse than what the greatest minds of the 20th 
> century could think up.

  Not true. There are many, many algorithms where you can *know* how fast
it can be solved (in terms of computational complexity) and you can come
up with an algorithm that solves it that fast. The difference is in details
(basically, speaking in terms of computation complexity, the constant
factor of the complexity function).

  Also, in many cases even if you know that a problem can be solved faster
(in terms of computational complexity), but the optimal solution is extremely
complicated and hard to implement, a less-than-optimal solution may still be
perfectly suitable for the problem at hand (eg. a completely naive solution
would take 10 minutes on a given input, your not-completely-optimal but
good-enough solution takes 1 second, and the known-to-be-optimal would
take 0.9 seconds, it doesn't really matter all that much). If the
less-than-optimal solution is suitable for the problem, and its
implementation is considerably simpler than the most optimal solution,
it might even be better to use the simpler solution from a programming
development point of view (simpler source code, easier to understand, less
chances for bugs).

  If, however, you indeed can only come up with a solution that is
"several complexity classes worse" than the currently known (or proven)
optimal solution, then I'd say you are not all that good at programming
and need more experience.

-- 
                                                          - Warp

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Orchid XP v8 wrote:
> In spite of the insane popularity of her books, there are apparently a 
> significant number of people who assert that they aren't very good. 

They aren't. They're fairly trite, at least the first three or four are 
fairly obvious, the magic is lame and without rules, the stereotyping is 
stereotyped, and it's all fairly predictable.  That's part of what makes 
them popular, tho.

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

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Warp wrote:
> extremely hard to find (especially if you don't now the proper terminology),

*Especially* before google.

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

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