Date: Feb 23, 1995 7:01 PM
Author: Edward S. Miller
Subject: Algorithms, long division in particular

I am going to defend the teaching of the long division algorithm I 
learned (which I egocentrically assume is the long division algorithm
everyone else is talking about) in the early grades.

On page 47, in the grades 1-4 section, the standards document says:

"It is important for children to learn the sequence of steps -- and the
reasons for them -- in the paper-and-pencil algorithms used widely in our
culture. Thus, instruction should emphasize the meaningful development
of these procedures, not speed of processing. . . problems with
remainders should be integrated throughout division. This approach is
more efficient and eliminates some misconceptions that often occur."

At the end of the next paragraph, the standards document continues,

"Although the exploration of computation with larger numbers is
appropriate, excessive amounts of time should not be devoted to proficiency."

It is clear the the standards document advocates teaching a (the)
division algorithm, but _repeated_ exercises on the order of 142748 divided
by 372 is wasteful. I wholeheartedly agree.

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.

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.


Edward S. Miller

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