Q&A #1733

Teachers' Lounge Discussion: Long division

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From: Pat Ballew

To: Teacher2Teacher Public Discussion
Date: 2000111613:13:36
Subject: A suggestion on helping students understand

	There are lots of algorithms by which one can obtain the answer to a
division problem, but to "understand" any of them begins with
understanding what division is all about, and that isn't necessarily
an easy task.
    We use division in several ways in applied and abstract
mathematics, but at the middle grades I would think it could be
translated generally into "how many sets of this much can be formed
 from that much?".  The word quotient for the answer to a division
problem is literally the Latin for "How many?".

   From that approach, if students had difficulty with division, I
would begin by asking them to describe situations that might entail
85/23 or other division problems.  If the student quickly sees this as
"How many classes of 23 can I make from a student body of 85?" or some
equally clear explanation, then we can begin to ask them how they
would find that answer.

At this point I would let them come up with algorithms that met their
understanding of "division".  If they wanted to use repeated
subtraction until less than 23 remained, I would encourage them to do
so, and describe the result in the context of the problem, i.e." There
will be three classes of 23 and a group of 16 to be otherwise assigned
(what you do with the remainder is a social decision, not necessarily
a mathematical one)
  If they have a good understanding of the division process, and you
want to work them toward the traditional long division algorithm, at
this point you can point out that if we had much larger numbers, it
would be necessary to incorporate some bookkeeping into our process to
help keep track of what we are doing.

If you are just beginning as a teacher, you might keep in mind that
many of the algorithms for computation were invented to do arithmetic
quickly and in a small space (like a student slate, or a sheet of
paper which was expensive in the 16th and 17th century).  They most
certainly were NOT created to teach the ideas behind the mathematical
operations.  If understanding is important to you, you may want to
seek other algorithms for these, or guide students to create efficient
ones of their own.  After a while working with their own, if you show
them the traditional space conserving algorithms, you can ask the
students to explain how it works (a very good question for student
teachers to ask each other, "Why does this process produce the right

If this is way off the direction you wanted to go, I apologize, but
this is how I would approach the problem you have.  In any event,
whether you use this approach or another, good luck.

 -Pat Ballew, for the Teacher2Teacher service

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