Models and Delving Deeper Into Division
Last month, we unpacked the long-division procedure taught in American K-6 classrooms, tied it into the standard “long multiplication” algorithm, discussed issues with the loss of information traded for an increase in speed and efficiency, and also explore some alternative methods for doing long division.
What was taken for granted was that we knew what division “is” (I use scare quotation marks here advisedly: it’s dangerous to try to be too restrictive about the underlying meaning of “simple, elementary” mathematical operations, and I do so here without wishing to suggest that what I am offering is or could possibly be definitive). My sense is that to a large extent, K-6 teachers have at best a very limited grasp of what division means, particularly when extended to the integers. This piece proposes to help flesh out that understanding, particularly for those who have struggled with the requisite ideas or not given the matter much consideration.
First, it is reasonable to state that division is the inverse operation for multiplication (I recommend avoiding or minimizing the use of “opposite” in this context), in the same sense that subtraction is the inverse operation for addition. In fact, it’s not at all unusual to define subtraction as “adding the inverse (or negative)”; similarly, we may define division as “multiplying by the inverse (or reciprocal).” However, some caution is required. If children haven’t learned about integers, teachers will have to restrict subtraction to cases of a – b where b ≤ a. This raises issues about whether to state that “You can’t take away a bigger number from a smaller number,” which is false once students learn about negative numbers. I recommend stating something along the lines of, “Given the numbers you know about so far, questions like “What is 3 – 5?” don’t have an answer; however, you will learn about numbers that allow us to answer questions like that in a few years.”
Similarly, before they have access to the rational numbers, questions like 16 ÷ 5 = ? have to be answered using remainders, and something along the lines of 3 ÷ 7 =? might not make much sense at all! Stating flatly, “You can’t divide a smaller number by a bigger one” is not only misleading but also leaves students and teachers alike in a serious quandary when faced with dividing -6 by -3. Does no one teaching K-6 realize that while the answer is POSITIVE 2, the problem shouldn’t be possible within the integers if that proscription about division is true?
Real-World Models for Division
It’s not my contention that students learn arithmetic better if they can ground it in their real-world experiences, or that they’ll enjoy mathematics more if they can relate it to their everyday concrete experiences. That might well be true for some children and irrelevant for others. Be that as it may, I believe teachers benefit from having various models at their disposal as they teach wide-ranging students mathematics. And good models won’t take away from understanding as long as teachers understand and communicate to their charges that models are ways to aid understanding the mathematics; they are not the mathematics itself.
The two main models for division in elementary school mathematics are the partitive (aka, fair-share) model and the quotitive (measurement) model.
Most young children have a reasonably good intuitive/experiential understanding of what fair shares are by the time division is introduced in school, even if they are not all that comfortable with formal arithmetic operations (there is a good deal of research with children in Brazil and elsewhere that suggests many children who work with money in market stalls and other non-school settings develop facility with complicated calculations that are not mirrored in their school performance with arithmetic).
Given a fixed number of items, x (say, gummy bears), and a fixed number of friends k (including herself), students have effective ways to determine how much each person gets, q (share size), and how many are left over, r (remainder) if any. What gets done with the remainder varies, of course. Local customs are not standardized.
What most children know is that, remainder aside and assuming that all gummy bears are equally desirable, fair-sharing requires that each child gets the same number of bears. (However, be careful if you’re trying to figure out if given kids really get the “fair-share” idea enough to connect it accurately with fractions (i.e., rational numbers). It’s not unusual for younger children sharing a candy bar with one other person to ask for “the bigger half.” That makes perfect sense as long as we’re using ordinary English rather than mathematical terminology.) A typical strategy for figuring out the quotient is to distribute one bear at a time to each person, repeating the process until there are no more gummies or not enough for everyone to get an additional bear.
Note that in the partitive model of division, there is a fixed total/whole, x, a fixed number of shares, k, and an unknown share size, q, which is to be determined. For simplicity, we’ll not discuss how the remainder, r, is “disposed of. “
With the quotitive model, there is again a fixed total/whole, x, but the size share is determined, and what is unknown is how many shares/groups of this size can be made before exhausting the supply or not having enough left to make another group.
A typical situation for quotitive division is cooking, where there might be 12 cups of flour, and a cookie recipe that calls for 1 1/3 cups of flour per batch. The question would now be, “How many full batches of cookies can be made with 12 cups of flour?” (Assumed for this example is that there are no other constraints: adequate amounts of all other ingredients are available).
Generally, this is a more difficult situation for many students (and teachers) to grapple with. Research has indicated, for example, that many elementary teachers and teacher education students have a real struggle writing word problems that involve division by proper fractions. Asked to write a ‘real world problem’ that would be solved by dividing by ½, a significant number of those asked will instead provide one that represents division by 2, even when the dividend and divisor are explicitly provided in writing.
Modeling Division With Integers
Consider how partitive and quotitive models apply to signed number division, keeping in mind, too, that multiplication of real numbers and their subsets is commutative, but division is not.
For integers p, q, r, with r ≠ 0, consider what happens for various combinations of p & q being positive or negative. With p & q both positive, we can easily imagine both partitive and quotitive division situation and already have mentioned such examples.
If p & q are both negative, say -12 and -3, we can ask meaningfully, “Into how many groups of size -3 can we divide a total of -12?” The answer, positive four, makes sense arithmetically, but it can also be viewed as distributing a debt of negative $12 into equal groups of size negative $3, then asking how many partners would be needed to absorb the debt equally. This is an example of quotitive division.
If p is negative and q is positive, we can also make a meaningful model. Let’s say that p is again –12 and that q is 3: then we might ask, “How much debt must each of 3 partners take to cover a debt of $12?” Here, the answer, -4, makes sense because we’re talking about sharing a fixed debt into a fixed number of groups and each share contains the same negative number of dollars. The model is partitive.
Now, what happens when p is positive and q is negative? Can we make a sensible partitive or quotitive model? A bit of thought suggests that we cannot. We can’t have a partitive model with a negative divisor because a negative number of groups simply makes no sense in the real world. On the other hand, a measurement model doesn’t work either. Trying to divide a positive total into shares of negative size won’t fly.
Nonetheless, such computations pose no actual difficulty: 12 ÷ -3 = -4 and this is consistent with our notions about the relationship between division and multiplication, since -4 • -3 = 12 Furthermore, the rules students are taught about signed-number multiplication and division hold up: no contradiction is introduced into arithmetic thereby. We should all be happy.
But what about our nice models? The answer is, they break down here. And perhaps that’s a good thing. Mathematics does not depend on a correspondence with the “real world.” It depends on logical consistency from the objects, rules for working with them, and the laws of reasoning. If we do not arrive at contradictions, we’re happy. Finding one or more models or metaphors to help understanding may be desirable, but it is not necessary.
So students who are in upper-elementary or middle school who are learning to think about sense making with signed-number division should have a chance to play and grapple with these issues. In many cases, they should be able to tackle the more abstract idea of division as multiplication by the reciprocal. Eventually, we would like all students to be able to think more abstractly in terms of mathematical objects and rules for working with them. The interplay between models and mathematics is ongoing even as the abstraction is ramped up, but models are a kind of scaffolding much of the time that we should be prepared to abandon when necessary or convenient, and the inability to find a model for a particular bit of mathematics should not be an insuperable barrier to tackling it.