From: Claire (for Teacher2Teacher Service)
Date: Dec 12, 2008 at 09:27:32
Subject: Re: Division language

Hi, Michelle --

Thanks for writing to T2T. These two ways we describe division are very
confusing to children. They mean the opposite, even though the numbers are in
the same order. The "divided by" language is the form I prefer, because of
the fraction connection, and its more prevalent use in higher math. However,
it's important for students to recognize and understand both, since they'll
encounter them.

I'd suggest making a visual representation for each, connecting the language
as you did above with the calculation of each and a picture illustrating
each. Better yet, have kids make their own in a math journal, or as a poster.
If this is available, on the wall and/or in their journal, so they can refer
to it, they will gradually sort out the difference. 

"4 divided by 12" can be represented as a fraction, 4/12, and signifies that
4 of something are being shared equally 12 ways. Pizzas, cookies, and candy
bars come to mind.

"4 divided into 12" can mean "How many groups of 4 are there IN 12?" OR "If I
make 4 equal groups out of 12 things, how many will be IN each group?"
Caution: I don't like to teach kids to rely on key words, since they can
mislead. We really want them to understand the meaning of problems and
language, and conceptualize what's happening. I emphasize the IN here to help
in understanding these two kinds of division.

Then the critical idea is for students to get lots of practice using the
correct language, in both spoken and written form.

I hope this is helpful. Please write again if you have more questions.

 -Claire, for the T2T service

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