[Phys-L] Re: long division

[John Denker <jsd@AV8N.COM>]

Wes Davis wrote:
> I can't say that any "real-world" scenarios come to mind to
> justify the need for division, but I certainly see the effects
> in my Calculus classes.  Because math teachers have opted to
> teach calculator skills, few of my calculus students can do
> any algebra that involves fractions - complex or otherwise.
>
> Isn't knowledge of fractions, multiplication and long division
>  necessary for doing algebra (and therefore calculus)?  I find
>  that I have to teach long division and complex fractions in
> my Calc classes so that students can learn to do such
> operations as integrating certain functions and finding
> oblique asymptotes, among others.
>
> I agree that these are not, for most people, necessary.
> Nonetheless, I don't think that students should come into
> advanced classes without a firm foundation in these skills.

That's a good argument in favor of long division.  To summarize
it in my own words: Everybody (not just students) ought to
understand how the decimal number system works.  And if they
understood that, then as a natural corollary they should be
able to do long division.

Note that the convserse does not hold:  Due to the execrable
way long division is commonly taught, there are lots of
people who do long division without understanding it, and
without understanding how the number system works.

Along these lines, Ludwik Kowalski wrote:
> > An interesting question about the long division, and about
> > other algorithms, is 'How come that I got the correct answer
> > if I do it in that way?"  Or "is this the only way to get the
> > correct answer?" Or "how was this method discovered?"

!!!!!

The bright kids are particularly likely ask this question.
The not-so-bright ones learn the algorithm by rote.  (I'm not
sure whether aversion to rote is the cause or the effect of
brightness, but that's a topic for another day....)

The way long division is usually presented drives me nuts:
you take a guess at this, and then multiply by that, and
bring down that, then you do the hokey pokey and you turn
yourself around.

I have my own way of teaching this that seems to work well,
especially with bright kids who recoil from the hokey-pokey
approach.

1) First point:  Start by making a multiplication table.  That
is, supposing you are dividing X by Z, make a table of the
first nine multiples of Z.  You can do this in less time than
it takes to talk about it.  You can do it using successive
additions, so you don't need to do any multiplications at all.
(If you're smart, you'll fill in the tenth row of the table
as well .. if it comes out to 10*Z it's a valuable check on
the accuracy of your additions.)

The main point of this table is to eliminate guessing.  Why
should "guessing" be part of the long division algorithm?
It rubs me the wrong way, and it rubs some of the kids the
wrong way.  Guessing on what basis?  Bah humbug!

The table is used as follows:  just look down the table to
find the biggest number that "fits into" the dividend X (or
remaing part of the dividend).  If the Nth row of the table
fits, you can with confidence write N as the next digit of
the quotient, and proceed with the algorithm.

Without the table, you would have to guess what the next
digit should be.  It's hard to teach (and hard to learn)
how to make such guesses, and if you guess wrong it doubles
or triples your workload, since you have to multiply out
(N-1)*Z, erase it, and then multiply out (N)*Z.

To repeat:
   1a) The table is a labor-saving device.  The more digits
    there are in the quotient, the more dramatic the savings
    become.
   1b) It removes "guessing" from the algorithm.

2) I also emphasize that long division is not some hokey-pokey
ritual that came out of nowhere.  It is motivated rather directly
by the basic axiomatic properties of the number system.

To wit:  Suppose we have a four-digit decimal number ABCD.  By
definition of decimal place value, we have
   ABCD == A*10^3 + B*10^2 + C*10^1 + D*10^0

Now suppose ABCD is the quotient, Z is the divisor, and X is
the dividend.  By definition of division, we have
    X = Z * ABCD
and plugging in we have
    X =  Z*A*10^3 + Z*B*10^2 + Z*C*10^1 + Z*D*10^0

The basis of the algorithm is to find A, subtract off Z*A
with appropriate place value, and then iterate to find  the
remaining digits BCD.

This tells you why numbers like Z*A appear in the long-division
calculation.  It also tells you why they are written _where_
they are:  they are genuine, meaningful numbers, with their
own place value.

As part of this approach, where the algorithm calls for
"bringing down" the next digit of X, it is helpful, at
least at first, to bring down _all_ the low-order digits
of X.  That creates an explicit representation of the
number
   X - Z*A*10^3
which is of course just
   Z*B*10^2 + Z*C*10^1 + Z*D*10^0
so clearly we are in a position to find the remaining digits
BCD by making another trip through the main loop of the
algorithm.

Also: bringing down all the digits reduces the chance of
errors due to not keeping things liked up in columns.

More generally: sloppiness is fatal to the long division
algorithm.  Place value plays a huge role in the algorithm,
so kids who are sloppy with their columns are doomed.  Also
it is a multi-step algorithm, so kids who are sloppy with
their basic subtraction skills have an exponentially small
chance of getting to the end unscathed.

Another thing that helps more often than you might think:
get 'em to do their work on graph paper.  Typical store-
bought graph paper has squares waaay to small for little
kids, but you can make your own on the computer.  There
are a gazillion programs to do this.  I've been happy with
   http://www.farm.kuleuven.ac.be/pharbio/gpaper.htm
It's free and does the job ... everything from plain grids
to polar coordinates to music staves.

For arithmetic, the best thing is to make the boxes
taller than they are wide, by a ratio of 4:3 or so.

PS:  There's a psychological/pedagogical angle:  I don't
insist that the kids be able to rederive the algorithm or
explain it back to me in terms of the place-value axioms.
The main issue IMHO is that students can smell fear.  If
I don't trust the algorithm, the kids will pick up on that
immediately, and they will not trust the algorithm either.
To say the same thing in more positive terms:  it is always
important to know the limits of validity of an algorithm.
In this case it's good to be able to tell the customers
that long div is pretty much guaranteed to work, without
limitation, because it is so firmly rooted in the axioms
of the number system.