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

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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.

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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.

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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."

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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

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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

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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

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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."

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

tl;tr all yet.

but this:



someday."

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

yes, perfect analogy to math education today.

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