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Q&A #2797

Teachers' Lounge Discussion: Division

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From: Jeff Wilson <wilson@silcom.com>
To: Teacher2Teacher Public Discussion
Date: 2005100418:52:22
Subject: Re: Re: Double Division - ?beter than long division?


Double Divsion: http://www.clubiris.com/DoubleDivision.html

Hi Loyd,

You raise some interesting points - thank you for your thoughts.

I don't know if this should replace long division or not.  It could be
that this would be taught before long division to reinforce how
division works and reinforce the distributive property.

The value of teaching manual division is to give people a method they
can actually use if they have to, and to teach about mathematics.

It could be argued that if students seldom use long division after
leaving school then it might be better for them to have a simpler and
more intuitive method at their disposal.  Would someone ten years out
of school have an easier time doing long division or double division? 
-  I'm not sure, but double division seems simpler.

About teaching math, I'm not sure that long division teaches math very
well.  It teaches people how to follow a long procedure and it gives
some practice multiplying and subtracting.  I'm not sure how many
people understand what is happening when you bring down the next
digit, or understand why you have to add a zero to the answer when the
"number after subtracting" is less than the divisor.

About polynomial division, on the one hand long division is more
similar to polynomial division because in both cases you are choosing
a multiple of the divisor from out of the blue.  But on the other
hand, double division does a better job of reinforcing the idea that
division is really subtracting off multiples of the divisor.

One problem I see with double division is that you have to do more
subtraction.  I personally find subtraction harder than
multiplication.  The other problem is that (not counting the trial and
error) double division will usually have more steps.

Here is how I see it so far:

Reasons To Teach Division:
- to teach mathematics
- a method to actually use in rare instances

Double Division Advantages (compared to long division):
- simpler procedure
- teaches how division works
- no trial and error
- gives practice doubling numbers

Double Division Disadvantage (compared to long division):
- more subtraction
- more steps (see note below)
- requires more space on the paper
- may not lead as directly to polynomial division - arguable

Your thoughts?

Jeff



IS DOUBLE DIVISION LONGER?

If we assume there is an equal chance of all ten digits being in the
answer then on average there will be 1.5X as many "multiple and
subtract" steps.  For example a "7" in the answer requires 3 steps,
and a zero in the answer requires no steps.

Also remember that the multiply part of the "multiply and subtract"
steps is already done for you.  So this part will be faster.  Of
coarse you have to pre-multiple the divisor three times in the
beginning.

In the end I think it is longer, but not as much as you might think
initially.


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