#### (Archive question of the week)

For Wednesdays, I plan to find a single interesting question from the past and discuss it. One of the things I’ve enjoyed about Ask Dr. Math is getting questions I’d never have thought of on my own, but which lead to some fascinating ideas. Here is an old favorite from 2000:

Fundamental Idea of 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.

What do you think? Is there a definitive answer?

There are two parts to the question, which go in opposite directions:

- First, what operation was the daughter doing? Can you tell from her actions what arithmetic operation she had in mind?
- Second, what action did the husband want his wife to take? Can you tell from the arithmetic operation he stated, what action he expected?

What is happening here is confusion between arithmetic operations and models of them. The reality is, there are two models of the same operation, so you really can’t equate either model with the operation. What you think (about numbers) and what you do (with objects) are two different things. Moving things around is not arithmetic.

As I said,

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.

At this point in my life, I hadn’t yet taught *Math for Elementary Teachers*, or researched the names given by educators to the two models; I later learned that the first is called “quotitive division” (finding the number of parts of a given size), and the second “partitive division” (finding the size of each of a given number of parts). (See Dividing Bit by Bit … But What Are the Bits?)

[Note: sometimes you’ll see it as “quotative”; but the noun form is definitely “quotition”, not “quotation”, so I think “quotitive” is right! But neither word is in my dictionary.]

Two different actions could correspond to the same operation (as the wife learned in trying to carry out her husband’s request); and two different operations could correspond to the girl’s actions. What she is doing is not really arithmetic at all.

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.

The husband evidently thinks of division only *quotitively* (finding the number of sets); and this is not too surprising, since this is the basic model in our language. Our word “quotient” comes from the same source, Latin for “how many”: *how many times* does 3 go into 12, for example. So he pictures his daughter as trying to make piles of 3, with 4 as the answer. And his idea of “dividing 48 by 8” fits the same model, making sets of 8 (and finding that 6 sets can be made). But the reality is that even he does *partitive* division all the time, too. Knowing that 6 people were coming to the meeting, he may have divided 48 by 6 to find that each set had to have 8 items. (Well, quite possibly what he knew was that each of the 6 people needed 8 items, and he multiplied to get 48. This is the third model of division: the *inverse* of multiplication.)

It’s interesting that the daughter, asked to divide 12 by 2, made piles of 2, which is again the quotitive model. Why would she do that? Because that’s the easiest way to carry out the division in a model! 2, 4, 6, 8, 10, 12, and we’re done: 6 piles. We might count them out this way: 2 in the first pile, 2 in the second pile, …

But you can do the very same counting to divide the items into 2 equal piles: “dealing out” one for me, one for you; another for me, another for you, …

So if you were listening to her as she did the “division”, you still would not know whether she had made 6 piles (counting two into each), or two piles (alternating between them). The arithmetic operation does not determine the physical actions that are done.

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.

So what is the answer to the question? First, the daughter wasn’t doing a division at all; she was just arranging necklaces. Second, the husband wasn’t stating what he wanted done at all.

Someday we’ll look at the other side: What is multiplication? There’s a lot more to say there, and it’s generated its share of controversy. One could almost say I’ve received hate mail about it. But that’s for another time.