Wording Division Problems

Date: 08/10/98 at 16:42:48
From: phil
Subject: Wording division problems

I have told my students that when you have a problem such as 36 
divided by 6, you cannot always read it as how many times does 6 go 
into 36. It depends on whether you are asking how many groups of 6 are 
in 36, or whether you want to know how many would be in each group 
of 6. If it's the latter, you are actually taking 36 into 6. 

If you agree, can you help me justify this to parents?

Date: 08/11/98 at 12:01:29
From: Doctor Peterson
Subject: Re: Wording division problems

Hi, Phil. 

Sometimes I think the hardest part of math is English. I myself tend to 
avoid the word "in" when I talk about division, because it can lead to 
confusion. My own six-year-old daughter confronts me if I accidentally 
fall into the habit of saying "let's divide 6 into 36," saying "you 
can't divide 6 into 36 parts!" 

She's partly right. It is a typical inconsistency of English that "in" 
is used with almost opposite meanings. (In most languages prepositions 
in general are the hardest words to define carefully, and there are 
many other instances of phrases used in superficially contradictory 
ways - there are just too many meanings and not enough prepositions!)

Of course, you do have to talk about these phrases and use them in 
problems. My approach is to distinguish carefully between math-talk 
and ordinary English. One of the features of math is to define 
everything carefully so there is no ambiguity (as far as possible). 
So when we are talking about the math, I use careful language like 
"divide 36 by 6" and avoid "into". At that point we are talking about 
division as an abstract operation on numbers, and whether the problem 
deals with groups of six or six groups is irrelevant. (In fact, for 
students still learning what division means, you can even switch to a 
different concrete model, such as rows and columns in a rectangle, 
unrelated to the original problem being solved.) 

When we move into word problems or applications, I can say "divide 
these 36 beans into six groups" or "into groups of six" or "how many 
groups of six can I fit into a box of 36?", but I would emphasize that 
we need to think carefully about what we are dividing by what. The 
fact that "into" can be ambiguous is worth pointing out, to emphasize 
that we have to think, not just follow some formula that "A into B" 
should always be written as "A/B" or as "B/A". There is no one-to-one 
correspondence between English and math.

In other words, if you are saying that you teach children to read the 
mathematical sentence as "divide 36 into 6" I would object (and 
perhaps you have children or parents who are objecting because they 
think you are teaching that). That just isn't correct mathematical 
language. If, however, you are teaching them to be aware of ambiguity 
in everyday language, and to remember that when they hear a problem 
described as "divide 36 into 6" they have to think carefully about 
what it really means, then I fully agree. 

You could also point out that even the word "divide" in a problem 
doesn't always mean you have to divide to get the answer. If I divide 
each of 36 pies into six pieces, I have 36 times 6 pieces, not 36 
divided by 6 pieces in all!

My one warning would be to avoid confusing students (or their parents, 
who may have to help out) by overemphasizing the confusing variety of 
ways a concept can be expressed. It's best to emphasize the meaning 
behind it, and focus not on the words but on the entities being worked 
with: what is the whole, what are the parts, how many of them are 
there? Then even if they have to solve problems written in Russian or 
Navajo, they will have the tools they need - not language, but math.

- Doctor Peterson, The Math Forum
Check out our web site! http://mathforum.org/dr.math/