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Topic: Algorithms, long division in particular
Replies: 3   Last Post: Feb 24, 1995 6:47 PM

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Edward S. Miller

Posts: 15
Registered: 12/6/04
Re: Algorithms, long division in particular
Posted: Feb 24, 1995 6:47 PM
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This is a long posting, so if you are not interested or in a hurry,
perhaps you might want to skip it or save it for later.

On Thu, 23 Feb 1995, Tad Watanabe wrote:

> On Thu, 23 Feb 1995, Edward S. Miller wrote:

> > Now for my personal spin on algorithms. Humans have spent countless
> > effort to develop efficient means for accomplishing all sorts of tasks.
> > In mathematics, we see dozens or thousands of algorithms put to use,
> > depending on our level of immersion and experience. The long division
> > algorithm and the multiple digit multiplication algorithm (again, the
> > one I use based on the distributive law) are the earliest instances I
> > recall of iterated algorithms with multiple _different_ operations;
> > division is the first with a nontrivial ending condition.

> Could you expand on this a little?

Certainly. In the algorithm I call long division, when approaching 345
divided by 16, my first step is to compare 16 to 34 and decide (usually
by estimation, sometimes by emulating the next specified step) that 16 is
about half of 34. I need then multiply 16 by 2. Then I subtract32 from
34. I carry down a digit. Then I iterate the process. I stop when the
remaining number is actually less than my divisor.

I have to start with the appropriate bookkeeping technique to preserve
digit alignment. Sometimes, but not in this particular problem, the carry
down still doesn't make the current step dividend large enough, so I
carry down another digit; I don't just stop. I also have to place a
zero in the appropriate position of the quotient because of our
positional numeration system.

There are multiple other arithmetic operations involved in the algorithm
as well as estimation skills. The decision to terminate is unlike the
multiplication algorithm stop point in the sense that you can't just keep
on until you have nothing to work with.

As for understanding the algorithm, in the case of no remainder problems,
the algorithm is literally the reverse of the multiplication algorithm
except that the rows would be added one at a time instead of all at
once. Each single digit multiplication is present and executed. Try
doing 42 times 37 and then 1554 divided by 42. The remainder, of course,
complicates things a little but divisions with remainders can be viewed
as the reverse of a compound multiplication and addition exercise.

> >
> > Throughout subsequent mathematics we encounter such algorithms of all
> > types. Sometimes, in "real life," for physical, temporal, or economic
> > reasons, it is necessary to use algorithms for single or repeated
> > operations. We do need to teach our students how to apply algorithms. I
> > happen to prefer long division because if we have to wait for
> > factoring or Euclid's algorthm or Horner's algorithm or L'Hopital's rule,
> > we're dead before we start.
> >

> But you are not advocating teaching in the sense of imposing algorithms
> on students whether or not they make sense, are you? Or, are you saying
> that there are contexts in schooling (which I happen to believe is a part
> of "real" life), there are occasions we should?

In reverse order, I meant "real life" as opposed to objects and actions
which only exist in thoughts and discussions without being actualized.
For example, many fine discussions about strategies for and consequences
of colonizing other planets have happened, but planets have not been
colonized (by humans, at least) in "real life". Of course "real life"
includes schooling. Sorry if that wasn't clear.

An example of an algorithm used in "real life" schooling for physical and
economic reasons is when I actually, in the middle of some other problem,
/ 2
evaluate |x sin x dx by hand using integration by parts twice because
I don't happen to have a computer handy in class that day. Honest, it
happens on Mondays and Wednesdays in calculus.

Furthermore, there are times when it is necessary to just learn when an
algorithm is applicable and how to use it for the sake of expedience (in
non-schooling situations, at least). I learned the braking algorithm for
my car, I know it doesn't often work on ice, but I certainly didn't
design a brake nor did I invent the concept of braking independently.
Sometimes it is necessary to go with thousands of years of accumulated
human knowledge and methods in some areas so that you can think in
others. An interesting account along these lines is contained in the
book "Planet of the Apes" by Pierre Boulle.

And finally, I never "impose" an algorithm on a student or anyone else
for that matter. I do demonstrate algorithms which I think make sense
(or perhaps more accurately, which I think will make sense to the
audience) and try to help the students understand it at some level.
I rarely try to prove that an algorithm works in a rigorous way until the
precalculus/calculus stage of the game.

On the other hand, if a student uses a self-developed algorithm or one
learned from someone other than myself with which I am not familiar, I
try to prove it rigorously for myself. If I do, I allow the student to
use it. If I find a counterexample I either provide it to the student or
give a good hint, depending on the nature of the example. Sometimes I am
stuck without proof or counterexample, in which case I caution the
student that the algorithm may be unreliable and ask her to convince
me that it is. Sometimes she does and sometimes she doesn't. If she
does, I allow its use; If she doesn't, I do not allow its use.

After ten years in the classroom, not a single class goes by where I do
not improve one of my own algorithms and where no student shows me one I
haven't seen before which works as well or better than the one I use. On
the other hand, some 20 years after learning long division, it is still
the algorithm which makes the most sense to me.


Edward S. Miller

Division of Natural Sciences VOICE 208-799-2810
Lewis-Clark State College FAX 208-799-2064
500 8th Avenue
Lewiston ID 83501-2698 USA

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