To:
Subject: Sugestions
Date: Sat, 29 Aug 1998 02:34:47 -0400
Mr. Rusin,
Hi. My name is John and I'm a 14 year old in Michigan. I
love music with all my heart. I've played the violin since I was four.
It'll be eleven years on July 18 when I turn 15. My parents gave me the
lessons for my fourth birthday. I also enjoy singing more than anything and
acting is also in my repetoire. In fact, not to brag, but I became a member
of the Screen Actors Guild March of '96. I'm writing because I have a
problem. I've always been a good student with at least a 3.44 GPA until
this year. I started my freshmen year in high school achieving my lowest
GPA ever...the 3.44. However as the year went on and I became my Freshman
class president my grades began to drop. I was always able to get them up
by report card time. Math, though was always hte subject that plagued me.
I'm in attendance at a high school with an accelerated program. In your
first two years of high school you cover the required Freshmen, Sophmore,
and Junior ciriculum. By the second simester of your junior year you start
college level courses. It is called the International Baccalaureate
program. The IB diploma is the only one that is accepted at all colleges
world wide. My question is I cannot continue through high school, no matter
if I stay at my current school (the International Academy) or go back to my
public high school, with my current math grades. What do you suggest? Is
there any way I can play of my love for music to enhance my math skills?
The Math program I'm in is an integrated program with algebra, geometry,
etc. all combined into one book. Our school year started Aug. 4 and ended
June 19. Well now that I've bored you to death I'll go and let you get back
to whatever you were doing earlier. Please e-mail me back if you have any
suggestions or comments. E-mail me at [deleted]. Thanks alot for
taking the time to listen.
John
==============================================================================
From: Dave Rusin
Date: Wed, 2 Sep 1998 23:55:47 -0500 (CDT)
Subject: Re: Sugestions
I'm not an education specialist, exacly, but I'll try to give a couple of
suggestions.
Why, exacly do you like music? I know this is a difficult question to
answer, but perhaps your response would be something like this: it is
through music that you can express some ideas which have no easy
parallel in the world of words. Oh certainly, you can find long
soliloquies extolling the beauty of a Bach invention or the emotional
depth of the Adagio for Strings, but that's just a lot of yadda yadda,
right? You've got to be there, immersed in the music, to really appreciate
what you're hearing.
Indeed, you may have found friends without musical training who can't seem
to understand the music as being more than just a sequence of notes.
The problem here is not their thick-headedness; rather, they simply
haven't worked through the little details over and over (and over!) that
you have had to, as you practice some tricky phrase in a piece -- one
which, perhaps, sounds easy to everyone else! Your reward is more than
just being able to execute that passage -- a machine could do that --
your reward is an intimite familiarity with _what's in that phrase_.
Your friends can never really share that with you if they spent their
time doing something else.
So your diligence has paid off cruelly: now you understand something
really cool but you have no way to share that understanding with the
people around you. You've mastered a part of a complex language which
comparatively few people speak. Frankly, no one who didn't learn this
language as a child will ever even attempt it seriously. So you've
got to enjoy this by yourself. You surround yourself with music, enmeshed
in its details, its power, its intricacies.
Very well, then. Let's consider that Bach piece. Look at one of the
fugues perhaps. Follow the interplay of the three or four voices. What
makes this so enchanting? Do you see how Bach has let _this_ voice climb
just when _that_ voice sank? Here's one which is now silent, and lets
another voice follow its pattern. Here we have each voice skimming its
theme, one more rapidly than the other, so that they arrive at the
final line in unison. Delightful! Fascinating! The patterns, the
interplay, all fitting together just so. Do these words do it justice?
Surely not -- what a pile verbiage. Yet this is the only way I could
evoke those ideas in your mind -- ideas which you recognize, and I only
need to bring forward.
This is the way you must understand your mathematics. Does the quadratic
formula look just like a sequence of symbols? Surely it is not. Have you
never idly moved pennies around the table into interesting shapes? Four
make a square, and so do nine. How big are the squares? You solve
x^2=4, then x^2=9; you're unconsciously watching the interplay between
two very simple voices, here the side lengths and the areas of the squares.
Then you have eight pennies, and you make -- what? Two squares now?
Now you solve 2x^2=8. Tomorrow, maybe 5x^2=45 if you have 45 pennies.
Now you're practicing! Tiny little details over and over, and you watch
the patterns brew. When you see 3x^2=363, your mind is now ready, and it
enjoys the old pattern-- the three floats over, splits the 363 into 121,
then the 121 arrange themselves into a square and you've solved the problem.
Keep at it, and soon this blends into a deeper, more satisfying pattern:
you see 10x^2-400=0 as a variation on the theme. z^2+80=129 is the
jazzman's take on it. Soon you're Beethoven, ready for a tour-de-force:
you weave all these snippets into a grand A X^2 + B = 0. As the performer,
you think: piece of cake!
But wait -- what happens when generations of dedicated musicians play this
piece? Does it not become quaint, and old-fashioned? Isn't it just a
matter of time before some Berlioz sees the same music in an equation
(X+3)^2=16? You're ready now -- you've practiced the other so that it's
no more than scales. But wait: 16 pennies, a square, sides are 3 longer
than X, what's X -- ah, the old standby! The exercises I cut my teeth on!
Yet here's a new way to see it:
* | * * *
--+------
* | * * *
* | * * *
* | * * *
That's the X-by-X square, two 3-by-X's, and a 3-by-3. Scoot the pennies
apart. A new pattern! Yet it just says X^2 + 2*(3*X) + 3^2 = 16. And now
we have a new way to look at those old ideas. All the old sets of pennies
split up in new ways; the old voices are given new shine. We see
(X+4)^2=36 means X^2 + 8X + 16 = 36, then another example and another.
And now it strikes us: we can go backwards, too! How to put perhaps 25
pennies into a pattern, with, say, a 3-by-3 block, and a couple of 3-by-X's
and an X-by-X? Can we solve X^2 + 2*(3*X) + 3^2 = 25? Yes! All that
practice, playing with the Berlioz tune, and we instinctively read this
as (X+3)^2 = 25, which we now can solve. How about X^2 + 4X + 4 = 49?
How about X^2+20X + 100 = 221? Over and over, we play with this theme
until it's not just an exercise, it's a familiar, even comforting, pattern.
And then we dream up our own variations. Why not 3*(X^2+10X+25) = 48?
And then 3X^2 + 30X + 75 = 12. And then 5X^2 + 10X + 5 - 20 = 0? Or
X^2+6X+5 = 0? Slowly but surely, we come to _own_ that idea -- someone
else had an inspiration, yes, but we worked it over and over, thought
about it, visualized it, made our own variations, and now it's ours!
So you have a pattern that you can barely explain to your friends, but
it's completely familiar to you nonetheless. You see the score, maybe
7X^2+7X-14=0, and with some amazement you watch your mind play the tune:
the fourteen moves safely away 7 X^2 + 7 X = 14
the seven comes out 7( X^2 + X ) = 14
and the square forms 7((X + 1/2)^2 ...an unresolved G7 chord?
...and you balance the chord 7((X + 1/2 )^2-1/4) = 14
The voices pull apart 7( X + 1/2 )^2 -7/4 = 14
...and come together 7( X + 1/2 )^2 = 14 + 7/4 = 63/4
The "Great Amen" (X + 1/2)^2 = 9/4
(X + 1/2) = 3/2 or -3/2
X = 1 or -2
So you know what to do, instinctively. As you search for a way to share
your understanding with your friends, you grope around for a language;
words can't really convey what you've learned by playing with pennies,
exploring variations, and practice-practice-practice. But you try:
"Move this number _here_, divide by that one _there_, cut this in half,
hold this squaring theme in soprano and interchange chords in the bass
to make it right, then everyone knows that last do-si-do."
Fortunately for musicians, there's a sophisticated way to say what
must be said so that other musicians can figure out what they have in mind.
Fortunately for mathematicians, there's another such language, and it's
algebra. It's not music, and it's not math, but it's a language for
conveying the bare bones of the mathematical music in our heads. We
can, just as easily as above, write the tune as
A X^2 + B X + C = 0
A X^2 + B X = -C
A(X^2 + B/A X) = -C
A((X + (1/2)(B/A) )^2...
A((X + (1/2)(B/A) )^2) - (1/4)(B/A)^2 = - C
A((X + (1/2)(B/A) )^2) = (1/4)(B/A)^2 - C
(X + (1/2)(B/A) )^2 = ( (1/4)(B/A)^2 - C )/A
(X + (1/2)(B/A) ) = square root of...
X = - (1/2)(B/A) + square root of...
There's no real beauty here, but among the cognoscenti, this is the score
of a real symphony of ideas. Fortunately a little artistry can even make
this score look pretty; that gives the form of the quadratic formula
you see on T-shirts!
I'll be upfront with you: I'm stretching the musical metaphors awfully thin
here; mathematicians don't usually gush effusively about the point and
counterpoint in their discoveries. But the key ideas really are the
same. Mathematics, like music, embodies certain patterns and ideas which
don't translate well into words. We can feel them, see them, understand them
--- but only after we have really worked to lift them off the paper and into
our minds; only after we've tried to see where they come from; only after
considerable practice with the minutiae, gradually adding the trills
(oops -- there I go again) until we have the full spirit of the idea.
Mathematics, like music, is a human adventure: people create and discover it,
they try to share it, and they enjoy it. Maybe with that perspective, you'll
have better luck with your schoolwork.
Good luck,
dave
P.S. -- Mathematics has its Shostakoviches too: someone eventually will
ask, "Why not X^2 + 2 X + 2 = 0 ?" ...