As far as the discussion of what is simpler in terms of communication, many
times an approximation is simpler with little or no loss of accuracy. For
example, if a line regression curve comes out to be y = 489.7656x -
80.3567876, we lose no significant accuracy by finding a "nice number"
approximation, y = 490x - 80. Although 75/150 in simplest form (as a
fraction) is 1/2, when discussing ratios, communication might indeed be
improved stated as 75 out of 150 instead of 1 out of 2. The unreduced form
might give us information about the size of the group polled. If 76 out of
150 hold a particular opinion and we would like to project or predict with
the ratio, the ratio might best be understood rounded as 0.5 to 1.
Estimates and round-offs are sometimes as accurate as the "exact"
ratio--when there are intervals of error in the data--which there
gennerally is.
In simplifying radicals, the carpenters I know do not have either the
cutting or measuring tools to distinguish the difference between sqrt(75)
in. and 8.66 in. and certainly do not see 5sqrt(3) in. as a simpler
representation. Besides that, they look at you strangely if you ask them
to cut a board 5(3)^(1/2) inches.
Students do need to know about equivalent representations of numbers and
the difference between approximations and exact values, however, the
student who can relate fractions and radicals to decimal approximations as
well as recognize equivalent and exact representations of a number has a
leg up on the number sense issue over the student who just does the pencil
and paper simplifications.
When we teach mathematics with an "every trick in the book" approach to
manipulation skills, sometimes we lose track why we historically needed to
know all of the tricks for factoring and for simplifying radicals. Knowing
the tricks helped scientists (or carpenters) get a decimal approximation to
a numeric problem in the fastest way possible. Now that we can get that
answer in seconds, maybe the tricks can be a fascinating study for
interested parties (I have to admit I was interested in divisibility rules)
but let's not torture every college student in the world by making this the
focus of our classes. When we predominately focus on pencil and paper
skills and exact answers we necessarily limit our students to working with
"pretty numbers" and contrived numeric expressions that simplify to some
nice concise form. Limiting our understanding of numbers to these
simplistic, managable numeric expressions precludes any chance of
understanding the federal budget, the speed of computers, or even reading
the newspaper intelligently.
Once again, this is my $ 2/100 worth.
Martha
----------
> From: Phil Mahler <mahlerp@admin.middlesex.cc.ma.us>
> To: mathedcc@archives.math.utk.edu
> Cc: mahlerp@admin.middlesex.cc.ma.us
> Subject: [MATHEDCC] tithing and abaci
> Date: Monday, October 06, 1997 1:51 PM
>
> Enjoyed the site at
> http://www.ee.ryerson.ca:8080/~elf/abacus/
> on the abacus. Worth the detour.
>
> Someone on the list once mentioned they thought that Richard Feynman once
> wrote about finding cube roots with an abacus. I don't recall seeing
that, but
> did find the passage below in "Surely You're Joking, Mr. Feynman!" A
great
> read for many reasons but one for this list is that he talks about doing
> mental arithmetic. Much of what he says would not apply to today's
audience -
> memorizing key logarithms, for example, but it's fascinating reading. It
> starts slow, but persist. I was sorry when I got to the end. It's
available in
> inexpensive paperback.
>
> Phil Mahler
> Middlesex CC
> Bedford, MA
> ----------------------------------------------------------------
>
> Lucky Numbers (Richard Feynman)
>
> The first time I was in Brazil I was eating a noon meal at I don't
know
> what time--I was always in the restaurants at the wrong time--and I was
the
> only customer in the place. I was eating rice with steak (which I loved),
and
> there were about four waiters standing around.
> A Japanese man came into the restaurant. I had seen him before,
wandering
> around; he was trying to sell abacuses.
>
> He started to talk to the waiters, and challenged them: He said he could
add
> numbers faster than any of them could do.
> The waiters didn't want to lose face, so they said, "Yeah, yeah.
Why
> don't you go over and challenge the customer over there?"
> The man came over. I protested, "But I don't speak Portuguese well!
> The waiters laughed. "The numbers are easy," they said.
> They brought me a pencil and paper.
> The man asked a waiter to call out some numbers to add. He beat me
hollow,
> because while I was writing the numbers down, he was already adding them
as he
> went along.
> I suggested that the waiter write down two identical lists of numbers
and
> hand them to us at the same time. It didn't make much difference. He
still
> beat me by quite a bit.
> However, the man got a little bit excited: he wanted to prove himself
some
> more. "Multiplicao!" he said.
> Somebody wrote down a problem. He beat me again, but not by much,
because
> I'm pretty good at products.
> The man then made a mistake: he proposed we go on to division. What he
> didn't realize was, the harder the problem, the better chance I had.
> We both did a long division problem. It was a tie.
> This bothered the hell out of the Japanese man, because he was
apparently
> very well trained on the abacus, and here he was almost beaten by this
> customer in a restaurant.
> "Raios cubicos" he says, with a vengeance. Cube roots! He wants to do
cube
> roots by arithmetic! It's hard to find a more difficult fundamental
problem in
> arithmetic. It must have been his topnotch exercise in abacus-land.
> He writes a number on some paper-any old number-- and I still remember
it:
> 1729.03. He starts working on it, mumbling and grumbling:
> "Mmmmmmagmmmmbrrr"-he's working like a demon! He's poring away, doing
this
> cube root.
> Meanwhile I'm just sitting there.
> One of the waiters says, "What are you doing?"
> I point to my head. "Thinking" I say. I write down 12 on the paper.
After
> a little while I've got 12.002.
> The man with the abacus wipes the sweat off his fore- head: "Twelve!"
he
> says.
> "Oh, no" I say. "More digits! More digits' I know that in taking a
cube
> root by arithmetic, each new digit is even more work than the one
before.
> It's a hard job.
>
> He buries himself again, grunting, "Rrrrgrrrrmmmmmm ...," while I add
on
> two more digits. He finally lifts his head to say, "12.0!'
> The waiters are all excited and happy. They tell the man, "Look! He
does
> it only by thinking, and you need an abacus! He's got more digits!''
> He was completely washed out, and left, humiliated. The waiters
> congratulated each other.
> How did the customer beat the abacus? The number was 1729.03. I
happened to
> know that a cubic foot contains 1728 cubic inches, so the answer is a
tiny bit
> more than 12. The excess, 1.03, is only one part in nearly 2000, and I
had
> learned in calculus that for small fractions, the cube root's excess is
> one-third of the number's excess. So all I had to do is find the fraction
> 1/1728, and multiply by 4 (divide by 3 and multiply by 12). So I was able
to
> pull out a whole lot of digits that way.
> A few weeks later the man came into the cocktail lounge of the hotel I
was
> staying at. He recognized me and came over. "Tell me," he said, "how were
you
> able to do that cube-root problem so fast?"
> I started to explain that it was an approximate method, and had to do
with
> the percentage of error. "Suppose you had given me 28. Now, the cube root
of
> 27 is 3..."
> He picks up his abacus: zzzzzzzzzzzzzzz-- "Oh yes," he says.
> I realized something: he doesn't know numbers. With the abacus, you
don't
> have to memorize a lot of arithmetic combinations; all you have to do is
learn
> how to push the little beads up and down. You don't have to memorize 9 +
7 =
> 16; you just know that when you add 9 you push a ten's bead up and pull
a
> one's bead down. So we're slower at basic arithmetic, but we know
numbers.
> Furthermore, the whole idea of an approximate method was beyond him,
even
> though a cube root often cannot be computed exactly by any method. So I
never
> could teach him how I did cube roots or explain how lucky I was that he
> happened to choose 1729.03.
>
>
****************************************************************************
> * To post to the list: email mathedcc@archives.math.utk.edu *
> * To unsubscribe, send mail to: majordomo@archives.math.utk.edu
*
> * In the mail message, enter ONLY the words: unsubscribe mathedcc
*
> * Words in the Subject: line are NOT processed! *
> * Archives at http://archives.math.utk.edu/hypermail/mathedcc/
*
>
****************************************************************************
****************************************************************************
* To post to the list: email mathedcc@archives.math.utk.edu *
* To unsubscribe, send mail to: majordomo@archives.math.utk.edu *
* In the mail message, enter ONLY the words: unsubscribe mathedcc *
* Words in the Subject: line are NOT processed! *
* Archives at http://archives.math.utk.edu/hypermail/mathedcc/ *
****************************************************************************