Fundamental Idea of Division

Date: 03/24/2000 at 21:11:34
From: Tess Ammerman
Subject: Division

My 8-year-old daughter had 12 bracelets that she arranged into 4 piles 
of 3. Did she divide 12 by 3 or by 4?

Most people I've asked can only offer an opinion. What I'm looking for 
is a definitive answer, if there is one.

My husband says the number of sets is the answer: 12 divided by 3 = 4.
Most others I've asked said that the number in each set is the answer: 
12 divided by 4 = 3.

I asked my daughter to do a different problem, 12 divided by 2, to see 
what she did. She took 2 bracelets and made a pile and continued until 
she ran out of bracelets, then counted the piles to come up with the 
answer, 6. This is the way my husband would have done it but not the 
way the others I've asked or I would do it.

I realize the answer is the same either way and thought it didn't 
matter until my husband gave me 48 items that needed to be divided for 
a meeting. As he left the room he asked me to divide the items by 8. 
When he returned I had made 8 neat piles of 6 items each. 
Unfortunately there were 6 people in attendance at the meeting and 
each needed 8 items.

Thank you in advance for you help in this matter,
Kelly Allen and Tess Ammerman

Date: 03/24/2000 at 23:18:56
From: Doctor Peterson
Subject: Re: Division

Hi, Kelly and Tess.

We were asked a similar question recently, so you're not alone. The 
question then wasn't phrased in so thorough a fashion, so I had to 
play with ideas for a while before I came up with a clear answer. This 
time I think I'm ready, though you've added some interesting twists.

I think you have to distinguish between the mathematical operation of 
division and the physical act of dividing into groups. Just as 
multiplying 6 by 8 can be equally well represented by 6 rows of 8 or 8 
columns of 6; the division of 48 by 8 can represent either making 
groups of 8 until you run out after making 6 groups, or "dealing out" 
into 8 piles until you run out with 6 in each pile. Neither action has 
a monopoly on the name of division.

So, first, your daughter was doing ALL of these:

    3 x 4 = 12
    4 x 3 = 12
    12 / 3 = 4
    12 / 4 = 3

Or rather, I could say she was doing NONE of them. She was arranging 
bracelets! She divided 12 by 4 if her goal was to make 4 piles; she 
divided 12 by 3 if her goal was to make piles of 3; she factored 12 
into 3 x 4 if her goal was just to make the piles come out even. If 
she simply enjoyed the pattern, she wasn't dividing anything by 
anything, because she wasn't paying attention to the numbers. And in 
any case, what she did with her hands was not division of numbers; 
that's what her mind might have been doing.

Second, if your husband asked me to divide 48 items by 8, when he 
returned he would have found a piece of paper with the number 6 
written on it. He didn't say what action he wanted performed, which 
was to divide 48 items INTO SETS OF 8. Might he have actually said 
"Divide them by 8's"?) If I had recognized that he actually wanted me 
to form sets, though, I probably would have done what you did, simply 
because the emphasis on the 8 would suggest to me that there are 8 
people coming. In other circumstances, where the size of a set is the 
important thing (say, dividing people "by 5" to assign them to cars 
for a trip), I would have taken it the other way.

The root idea of division is to ask "How many times does 8 go into 
48?" That is, how many 8's can I form? So the presumed action might 
be to form 6 sets of 8, counting out 8 at a time; I can understand 
your husband's seeing it that way. On the other hand, when your 
daughter "divided 12 by 2" by forming sets of 2, that seems odd to me 
because I think of dividing by two as forming halves. Again, it's the 
context that suggests the interpretation, not a fixed definition. You 
can count 8's equally well by dealing the items out into 8 piles, 
counting out 8 repeatedly. Someone listening to you couldn't tell 
which way you are grouping them; the difference is only in whether the 
items you put down are all in one pile after you've counted to 8, or 
are in 8 separate piles. The division (48 / 8 = 6) is the same, but 
the distribution of objects is different.

I have to admit all this is just my opinion, not a definitive answer. 
I hope it makes sense and prevents arguments, rather than starting 
more!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/