*Elmer has 29 marbles and 4 vases. He puts the same number of marbles in each vase. How many marbles does he put in each vase? How many marbles are left over?*

I saw a problem like this in a math textbook, and the intended answer was, "7 marbles in each vase, with 1 left over."

I wondered why the answer couldn't be "1 marble in each vase, with 25 marbles left over." After all, \(4\times 1 + 25\) equals \(29\).

Or "2 marbles in each vase, with 21 marbles left over." \(4\times 2 + 21\) equals \(29\), too.

If the intent of the problem was to have a unique answer, then it ought to have said something like this:

*Elmer had 29 marbles and 4 vases. He put the same number of marbles in each vase. What is the largest number of marbles each vase could have received? In that case, how many marbles would be left over?*

Such problems are often presented to students as division problems; students are taught to record the answer as \(29 \div 4 = 7 R 1\), or perhaps \(29 \div 4 \rightarrow 7 R 1\). Notice that neither of these jottings is an equation: in the first case because "7 R 1" isn't a number, and in the second case because \(\rightarrow\) isn't an equals sign. Couldn't students be taught to represent a situation like this using an honest-to-goodness true equation, namely \(29 = 4\times 7 + 1\)?

The equation \(29 = 4\times 7 + 1\) shows clearly what is going on: the 29 marbles can be grouped into 4 equal groups of 7, with 1 left over.

I know it is difficult to break with a tradition like "\(29 \div 4 = 7 R 1\)." Maybe there are even reasons for that tradition that make it inevitable. Perhaps, in that case, students might be taught to represent the problem

*sometimes*using an actual equation like \(29 = 4\times 7 + 1\).

It does bother me, though, this approach of presenting problems as division problems when division doesn't solve them. I mean, the answer to Elmer's problem isn't \(29 \div 4\), so in what sense is this a division problem?

One might wish to say that the reason this is a division problem is that "we are sharing equally." Well, sort of yes, sort of no. We start out with 29 marbles, and in fact we

*don't*share those 29 marbles equally. We share 28 of them equally. And we throw the other one away.

Here's another way to think about it. Take the (true, actual) equation \(29 = 4\times 7 + 1\), divide both sides by 4, and simplify. The result is

\(29\div 4 = 7 + \frac{1}{4}\).

But what is the meaning of "seven and a quarter marbles"? That notion doesn't make sense in the context of the problem.

Division breaks a quantity into equal parts. That's what it does. If the stated number of objects in your problem can't be broken into the stated number of equal parts without the situation turning silly, then that could be a hint that your problem isn't really a division problem.

***

I thought I'd keep this space for additional thoughts that might accumulate over time.

If Elmer's problem isn't a division problem per se, then we could still see it as a problem with division in it. Along those lines, here is another way to represent the problem and the answer: \((29 - 1)\div 4 = 7\). This equation is another nice account of what's going on: it's an (actual, true) equation that correctly uses division, and moreover it uses subtraction to show the "throwing away" that I alluded to above. 29 marbles, throw away 1, share the rest equally among 4 vases, yields 7 marbles in each vase.

UPDATE 11/14/15 - A one-page PDF of this post, formatted nicely for printing, is available at tinyurl.com/notdivision.

## 3 comments:

There's something very interesting going on in this problem that could be considered analogous to mixed numbers, unlike terms, or complex numbers (i.e. quantities composed of unlike terms: 1+3/4, 2x + 9, or 3i - 5).

Though no one would write it this way, the standard way we teach this problem plays out something like this:

(28/4) + 1 = 7 + 1

The problem with this logic is that is should be equivalent to 8, but in the context of the problem that makes no sense; There are not 8 marbles in each group, nor 8 groups of marbles, nor 8 marbles left over. In fact there are 8 of nothing. That's because, though the equation I've written is true, it doesn't label units. If it did we would get something like:

(28 marbles / 4 vases) + 1 marble = 7 marbles/1 vase + 1 marble

In sentence form that would read "You have 7 marbles in each vase and 1 marble not in a vase."

In essence what we have is unlike units. The first quantity, 7, is measured as something like a rate (marbles per vase), where is the second quantity, 1, is a number of marbles. An expression like 7R1, which is how kids are often taught to write their answer, does very little to illuminate for children why these two units are different in nature and why they therefore cannot be combined.

I'd argue that the introduction of remainders and their logic, is going to be confusing and counterproductive to teach students if they don't have a very deep grasp of the concept of units, and ingrained habit of attributing units to quantities. Many of the kind of errors that are so painfully common around division seem to relate to this murkiness around units.

For example when students divide 14 by 5, end up with the answer 2 and 4/5 and then combine them to get 6/5, they have fundamentally misunderstood the nature of the 4 and the 2 in that expression. They are failing to see that there are 2 wholes and 4 fifths and that wholes and fifths are not like units and can't be combined.

Likewise, when students start moving from remainder based approaches to division to fractional answers it's common for them to say things like:

17 / 8 = 2 R1 = 2 and 1/2.

Where did the 1/2 mistakenly come from? Well the students know they are supposed to "do something" with the R1, and that somehow they will turn it into a fraction, using one of other numbers in the expression as a denominator. So they just grab the nearest one (which is the 2) and voila!

But add a scenario with units attached and the error becomes much more clear. Say we have 17 candy bars and 8 kids and we want to divide them equally among 8 students. We would quickly note (especially if our multiplication facts are fluent) that we could easily divide 16 candy bars equally among 8 students by giving them 2 candy bars EACH. The key word here is EACH, because the 2 really means something like 2 candy bars PER 1 STUDENT. The natural next question for someone doing this task (in real life or in a math class) is "Well, I can divide the 16 easily enough, but about about that one extra candy bar? How could I divide that one extra candy bar up among those 8 students?" The conversion of 2 R1 to 2 + 1/8, only makes sense if we understand that the 2 was candy bars PER STUDENT. How do we understand that last 1 leftover candy bar as a number of candy bars, PER STUDENT? There were 8 students so we divide 1 evenly among 8 to get 1/8 PER STUDENT. Dividing 1 candy bar by 2 candy bars would not yield candy bars per student.

Having said all this, what I'm driving at is that underneath all of these confusions (both in teaching and student error patterns) is a problem of understanding the Story of Units. (Tip of the hat to Scott Baldridge).

So punchline: you must help students develop a really rich understanding of the story of units before you start asking students to write numbers composed of mixed units, and then convert those units so that they are like units. In other words the idea of division with remainders is a rather severely advanced topic within division. (This probably also explains why the idea of long division is so poorly taught, and miserably misunderstood by students.)

Dewey, thanks for these thoughtful remarks - the role of unit thinking here is really interesting. Your comment also reminded me that when my wife first read this post, she told me that in school she was never taught "what to do with the remainder" but eventually figured out on her own how it relates to the divisor...probably by thinking in unit terms. Looking through the unit lens, one sees that in both of the equation models 29 = 4x7 + 1 and (29 - 1)/4 = 7, we are only ever adding or subtracting quantities with the same units (marbles with marbles - the "extensive quantity" as we would say in physics). And in both equations, the "EACH" belongs to the 7 (what we would call in physics the intensive quantity).

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