http://www.maa.org/devlin/LockhartsLament.pdf


-- 
~Mike

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Mike Raiford <"m[raiford]!at"@gmail.com> wrote:
> http://www.maa.org/devlin/LockhartsLament.pdf

  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. If you change the problem slightly, for example
to "sum of all integers from 1 to 200", most people will still be able
to *guess* how to change the formula to get the result (ie. you just
change the '100' to a '200'). However, change it eg. to "all the even
integers between 50 and 250" and most people are left completely baffled.
No idea whatsoever. Why? Because they don't understand where the formula
is coming from.

  A more logical approach to actually deducing the formula is to think
that summing all the integers between 1 and 100 is the same as summing
the average of those numbers with itself as many times as there are
numbers in that range. (In other words, the product of the average and
the amount of numbers, ie. 100 in this case.) This can be intuitively
demonstrated graphically (if you visualize the integers as vertical
bars of the length of the integer, forming a triangle construct, you
can split this construct in half and mirror the upper half to get a
rectangle).

  Knowing this makes it easy to deduce the answer to a whole lot of other
similar problems. For example "sum of all the even integers between 50
and 250" can be done with the same principle: The average of all those
numbers times their amount. (The average is, obviously, (50+250)/2, and
the amount of values is (250-50)/2 + 1.)

  (Of course you need to be careful to not to apply this idea too far.
For example it cannot be applied to "sum of all squared numbers between
1 and 100". If you visualize it graphically like earlier, you'll see why.)

-- 
                                                          - Warp

Post a reply to this message

On 12/9/2010 11:50 AM, Warp wrote:
> Mike Raiford<"m[raiford]!at"@gmail.com>  wrote:
>> http://www.maa.org/devlin/LockhartsLament.pdf
>
>    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.

Post a reply to this message

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.

Post a reply to this message

Mike Raiford wrote:
> http://www.maa.org/devlin/LockhartsLament.pdf

FWIW, I learned virtually nothing in music class and nothing at all in art 
class. Art class, such as it was, consisted of the teacher trying to come up 
with rules about what you drew to ensure it occupied the entire 45-minute 
class to draw it.

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!"

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

Post a reply to this message

On 9-12-2010 21:29, Kevin Wampler wrote:
> On 12/9/2010 11:50 AM, Warp wrote:
>> Mike Raiford<"m[raiford]!at"@gmail.com> wrote:
>>> http://www.maa.org/devlin/LockhartsLament.pdf
>>
>> 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.

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

Post a reply to this message

On 9-12-2010 22:35, Darren New wrote:
> Mike Raiford wrote:
>> http://www.maa.org/devlin/LockhartsLament.pdf
>
> FWIW, I learned virtually nothing in music class and nothing at all in
> art class. Art class, such as it was, consisted of the teacher trying to
> come up with rules about what you drew to ensure it occupied the entire
> 45-minute class to draw it.
>
> 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!"
>
or trigonometry after encountering POVray

Post a reply to this message

andrel <byt### [at] gmailcom> wrote:
> On 9-12-2010 22:35, Darren New wrote:
> > Mike Raiford wrote:
> >> http://www.maa.org/devlin/LockhartsLament.pdf
> >
> > FWIW, I learned virtually nothing in music class and nothing at all in
> > art class. Art class, such as it was, consisted of the teacher trying to
> > come up with rules about what you drew to ensure it occupied the entire
> > 45-minute class to draw it.
> >
> > 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!"
> >
> or trigonometry after encountering POVray

  Geometry, trigonometry and in some cases even calculus has been quite
useful in graphical-heavy and game programming.

-- 
                                                          - Warp

Post a reply to this message

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.

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

Post a reply to this message

Mike Raiford <"m[raiford]!at"@gmail.com> wrote:
> http://www.maa.org/devlin/LockhartsLament.pdf

tl;tr all yet.

but this:

"Her sheets were impeccable— every note in the right place, perfect calligraphy,
sharps, flats, just beautiful. She’s going to make one hell of a musician
someday."

seems to describe very well a serialist composer working on the sheet alone...
:)

yes, perfect analogy to math education today.

Post a reply to this message

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.

Post a reply to this message

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...

Post a reply to this message

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. :)

Post a reply to this message

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."

Post a reply to this message

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

Post a reply to this message

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"...

Post a reply to this message

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!

Post a reply to this message

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

Post a reply to this message

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. ;-)

Post a reply to this message

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."

Post a reply to this message

Am 10.12.2010 14:39, schrieb Invisible:

> 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?

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.

As for divisions, AFAIK slide rules weren't used back then except for 
quick estimates. Precise calculations would either be done by a single 
computer requiring huge amounts of electricity, or by a room full of 
computers requiring huge amounts of coffee.

Back then, computer was an actual job, requiring highly trained people. 
After division, they would have learned how to interpolate using 
logarithmic and trigonometric tables and stuff.

I'd also say it was good for our generation to know how to add, 
subtract, multiply and divide without a pocket calculator, because you'd 
probably not carry one around with you all the time back then. Remember, 
we didn't have cell phones with pocket calculator apps in our pocketses 
wherever we went.

> 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...

Not really. The books were amazingly popular before the movies. I'm 
pretty sure JKR didn't have to beg - if anyone did any begging, it was 
most likely Warner Brothers.

Post a reply to this message

Invisible escreveu:
> 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

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...

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

-- 
a game sig: http://tinyurl.com/d3rxz9

Post a reply to this message

On 12/9/2010 3:57 PM, Warp 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!"
>>>
>> or trigonometry after encountering POVray
>
>    Geometry, trigonometry and in some cases even calculus has been quite
> useful in graphical-heavy and game programming.
>

Or when dealing with writing software for industrial equipment. I can't 
believe how much trig I actually used when working on machinery. It was 
necessary, some of the stuff I was dealing with were circles that had to 
line up tangent to a board, etc..

-- 
~Mike

Post a reply to this message

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

My theory is that she discovered how to do *actual* magic, and used it to 
make tons of money and fame by writing best-selling books about fake magic, 
just to distract the masses.

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

Post a reply to this message

Am 10.12.2010 22:08, schrieb Darren New:
> clipka wrote:
>> Not really. The books were amazingly popular before the movies.
>
> My theory is that she discovered how to do *actual* magic, and used it
> to make tons of money and fame by writing best-selling books about fake
> magic, just to distract the masses.

Nice one ;-)

Post a reply to this message

> Geometry, trigonometry and in some cases even calculus has been quite
 > useful in graphical-heavy and game programming.

Graphical and game programming has been quite useful in learning 
geometry, trigonometry, and in some cases even calculus.

  - Slime

Post a reply to this message

Slime <pov### [at] slimelandcom> wrote:
>  > Geometry, trigonometry and in some cases even calculus has been quite
>  > useful in graphical-heavy and game programming.

> Graphical and game programming has been quite useful in learning 
> geometry, trigonometry, and in some cases even calculus.

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

-- 
                                                          - Warp

Post a reply to this message

Darren New <dne### [at] sanrrcom> wrote:
> clipka wrote:
> > Not really. The books were amazingly popular before the movies. 

> My theory is that she discovered how to do *actual* magic, and used it to 
> make tons of money and fame by writing best-selling books about fake magic, 
> just to distract the masses.

  It reminds me of the "corollary" to Clarke's third law of prediction:

  "Any sufficiently advanced magic is indistinguishable from technology."

-- 
                                                          - Warp

Post a reply to this message

>    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

Post a reply to this message

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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Orchid XP v8 wrote:
> anybody studies them - set theory, number theory, category theory, etc.)

Set theory is how you prove arithmetic works. Prove 2+3=5.

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

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On 11/12/2010 03:08 PM, Darren New wrote:
> Orchid XP v8 wrote:
>> anybody studies them - set theory, number theory, category theory, etc.)
>
> Set theory is how you prove arithmetic works. Prove 2+3=5.

I didn't say that set theory is not *important*. I said it is not 
*interesting*. Certainly not to me, anyway. The obvious stuff is 
obvious, and the non-obvious stuff is all tedious riddle-like splitting 
of hairs rather than interesting insights.

Also, several sources state that set theory is the fundamental basis of 
all of modern mathematics, but not one single one makes any attempt to 
explain this sweeping statement. Your assertion that set theory somehow 
has something vaguely to do with arithmetic is typical of this type of 
statement. If there *is* some kind of connection between set theory and 
arithmetic, it is highly non-obvious.

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

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Orchid XP v8 wrote:
> I didn't say that set theory is not *important*. I said it is not 
> *interesting*. 

It's interesting if you want to know whether arithmetic is complete and 
consistent, for example.  Godel's theorem certainly is an interesting insight.

> Also, several sources state that set theory is the fundamental basis of 
> all of modern mathematics, but not one single one makes any attempt to 
> explain this sweeping statement. Your assertion that set theory somehow 
> has something vaguely to do with arithmetic is typical of this type of 
> statement. If there *is* some kind of connection between set theory and 
> arithmetic, it is highly non-obvious.

And *that* is what makes it interesting!  Set theory is like the lambda 
calculus - it's tiny and trivial, but you can prove anything with it.

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

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Am 11.12.2010 12:26, schrieb Orchid XP v8:

> 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...)

I've never read them (nor did I see any of the movies). I too heard tell 
that the books were rather bad, but my father insists that while the 
German translation sucks, the English original is actually quite good 
(he's a retired English teacher, and none too easily impressed by any 
"hype", so he may be a comparatively credible witness after all).

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On 12/9/2010 1:04 PM, Mike Raiford wrote:
>
>
> http://www.maa.org/devlin/LockhartsLament.pdf

It's an interesting picture of an establishment.  You get accepted as an 
expert by agreeing with the experts.

Regards,
John

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Am 11.12.2010 13:14, schrieb Orchid XP v8:

> 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.

... despite there existing proof that this desire is insatisifiable 
(which I find an interesting thing to know, too) - and not only in this 
world, but also in any imaginable one (another interesting thing) except 
for pretty boring worlds.

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Am 11.12.2010 14:46, schrieb Orchid XP v8:

> 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.)

Hehe - when I was a teenager, I wrote a raytracer, too - without even 
knowing POV-Ray; all I had was a few printed b/w chrome spheres on 
checkered planes, the desire to do the same, Snell's law, and a 
classmate who took over the task of developing UI components for the 
interface (we didn't have a library for that either) :-P

Needless to say that the thing was awfully simple, and did not even work 
properly - it got the normals of non-uniformly scaled spheres wrong - 
but hey, you can't do everything right all the time, can you ;-)

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Am 11.12.2010 14:48, schrieb Orchid XP v8:
> 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.

You don't need unshakable solid proof for conjectures, so maybe you 
should stick to those. In my eyes that's mathematical enough.

I guess more often than not, proving a good conjecture boils down to the 
mere applying of rules again. Sure, there are famous conjectures out 
there that seem to defy any attempt to prove them, but you don't hear 
about the conjectures that are easy to prove. And after all, /every/ 
proof starts off as a conjecture.

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Am 11.12.2010 06:37, schrieb Slime:
>  > Geometry, trigonometry and in some cases even calculus has been quite
>  > useful in graphical-heavy and game programming.
>
> Graphical and game programming has been quite useful in learning
> geometry, trigonometry, and in some cases even calculus.

Yup ;-)

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Orchid XP v8 <voi### [at] devnull> wrote:
> On 11/12/2010 03:08 PM, Darren New wrote:
> > Orchid XP v8 wrote:
> >> anybody studies them - set theory, number theory, category theory, etc.)
> >
> > Set theory is how you prove arithmetic works. Prove 2+3=5.
>
> I didn't say that set theory is not *important*. I said it is not
> *interesting*. Certainly not to me, anyway. The obvious stuff is
> obvious, and the non-obvious stuff is all tedious riddle-like splitting
> of hairs rather than interesting insights.
>
> Also, several sources state that set theory is the fundamental basis of
> all of modern mathematics, but not one single one makes any attempt to
> explain this sweeping statement. Your assertion that set theory somehow
> has something vaguely to do with arithmetic is typical of this type of
> statement. If there *is* some kind of connection between set theory and
> arithmetic, it is highly non-obvious.

suppose you have a set of 2 things and a set of 3 things.  How many things you
have in total?  Is that not how you even learn basic arithmetic?

functions?  a formula mapping things from a set to things from another set?

yeah, set theory is fucking huge.

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Orchid XP v8 wrote:
> 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"?

are you talking about that calculus?  the one with integrals, 
derivations etc?  Yeah, pretty hard stuff.  I gave up on integrals in 
the complex space...

If you don't believe so, you're clearly gifted... although I'm sure 
povray newsgroups may host a lot of math lovers who put my average math 
skills to shame...

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clipka wrote:
> Am 11.12.2010 13:14, schrieb Orchid XP v8:
> 
>> 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.
> 
> ... despite there existing proof that this desire is insatisifiable 
> (which I find an interesting thing to know, too) - and not only in this 
> world, but also in any imaginable one (another interesting thing) except 
> for pretty boring worlds.

you're talking about that Godël dude, aren't you?

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