As a high school teacher, I struggle routinely with getting students to understand that \(x/0\) is undefined. Students don’t seem to understand that division with a remainder is incomplete.
I have long attributed this to the way that division is taught in the elementary school years. For instance, \(17\div 5 = 3R2\) is considered perfectly acceptable, even though \(3R2\) isn’t a valid number (\(11 \div 3 = 3R2\) as well, but \(11 \div 3 \ne 17 \div 5\)). The remainder structure is ostensibly used so that students don’t have to work with fractions, but in order to understand the significance of \(R2\), we need to know what the original divisor was.
The other day, I was reading this blog entry, which responds to a middle school teacher suggesting that elementary school teachers stop using this notation, which led to the question: What should they do instead? My brain went one step further: I want elementary school teachers to stop acting like division can have a remainder. Division with a remainder, I have said in the past, is incomplete division.
But then I got thinking. What if there were a concept where the remainder were relevant? Is there terminology for such a thing?
I realized there are two concepts of division. The standard one, the one we hope high school students graduate fully grasping, is that division has no remainder. This is why we need fractions, after all. Rational numbers close a gap in the number system created by dividing integers. \(5 \div 0\) does not equal \(0R5\) because \(R5\) is not a valid part of a number; it’s a temporary fix we create in elementary school to postpone fractions.
But there is a branch of mathematics that deals with that remainder in a meaningful way: Modular arithmetic. In modular division, both the floor of the division and the remainder are used, and often they’re used separately. For instance, when dealing with the trigonometric functions, we don’t care how many times around the circle we go: \(\sin(\pi) = \sin(3\pi) = \sin(5\pi) = \sin7\pi\) because \(\pi \equiv 3\pi \equiv 5\pi \equiv 7\pi \mod 2\pi\).
The key here, though, is that modular arithmetic indicates the divisor. The statement \(17 \equiv 2\) is not wrong, but it’s incomplete: \(17 \equiv 2 \mod 5\), but \(17 \equiv 3 \mod 7\). That is, \(17 \div 7 = n + 3/7\), where \(n\) is an integer. In more useful number theory terms, \(17 = 7n + 3\). If we’re focused on the remainder, we don’t care what \(n\) is.
I am not recommending teaching fourth graders about the modulus, at least not explicitly. The initial focus should be on the floor. Another thing that’s often missing in these discussions (as is pointed out at the blog entry) is that sometimes we want the floor, sometimes we want the ceiling, and sometimes we want an understanding that we’re between the two. If we’re talking about how many pizzas we need to order for a party, we need the ceiling: If 18 students will eat an average of 3 pieces, and there are 10 pieces per pizza, how many pizzas should we order? \(18 \times 3 = 54\), so we need six pizzas (not five!).
What I’m saying, though, is that the remainder is a useful concept, but an incomplete one. There’s nothing wrong with teaching that the remainder of \(17 \div 5\) is 2… if we emphasize (a) we’re not really done, because standard division doesn’t have a remainder and (b) we have to mark what we’re dividing by.
But we could do that in a way that immediately readies students for fractions, for seeing fractions as being related to division, for understanding improper fractions, and (ultimately) for modular arithmetic. We could use a notation that is essentially equivalent to the remainder notation while marking what we’re dividing by: \(17 \div 5 = 3 R 2 D 5\). If that’s too much to write, consider an abbreviation: \(17 \div 5 = 3 + 2/5\).
Why can’t we teach fraction notation without teaching fractions? Why is this considered an inferior route to teaching a notation that no serious mathematics uses beyond middle school? Beyond which, the R notation is misleading: When we divide 17 into 5 piles, we do get three in each pile, with two left over, but what do we do with those two? We still need to figure out how to distribute them over the piles. Why not explicitly note that?
Teaching fractions means teaching what it means to distribute parts of things into piles. Do students need to understand what that means in order to use the notation, though?
A rebuttal: We don’t teach children in a vacuum. Children who have never seen division before won’t know the difference between R-notation and fraction notation. Parents, on the other hand, will be at the ready to criticize fractions, and to reinforce math anxiety in their children. “Oh, fractions!” they’ll say, “I can’t do those. Already, fourth grade, you’re doing math I can’t do!”
Plus, I learned R notation, and I have a fine understanding of fractions. It’s clearly possible to do it.
Those are both fair enough points. I don’t particularly care so much about the specifics of what happens in fourth grade math class as I do about the fact that, like many high school teachers, I’m teaching teenagers who don’t know how to add fractions with different denominators. There is a way to teach R notation and still keep both number sense and the concept of division solid.
I started by pointing out that many students don’t understand why \(x/0\) is undefined (except when \(x = 0\), in which case it’s indeterminate). I know that division and multiplication are taught as inverse operators, and yet somehow students struggle with seeing the problem with \(x/0 = a \Rightarrow x = 0a\) having a non-zero value for \(x\). One thing that the R notation blurs, I think, is the idea that the remainder must be less than the divisor. For instance, it’s also true that \(17/5 = 1 R 12\). I doubt that a teacher would allow a student to write that without comment, but it’s mathematically true. If we don’t have the restriction, then \(17/0 = 0 R 17\), no problem. Conceptually, this ties directly into how we teach division at the earlier years: You have 17 things, and you have to distribute them into no piles at all. Well, that’s easy, there won’t be anything in any pile because there are no piles. Since we’re allowed to set the remainder aside and basically ignore it, no problem.
That’s a problem.
If we stress that the remainder has to be less than the divisor, though, the problem becomes immediately clear: How can our remainder be less than zero? There’s nothing less than zero (if we’re in the realm of counting numbers). So we can’t do that. We can use the remainder notation to our advantage now: It doesn’t make any sense.
Unfortunately, though, this argument relies on a nuanced understanding of zero that we tend to avoid telling fourth graders about. So we paint ourselves into a pedagogical corner of having to use a wonky, quickly outmoded structure because we want to postpone key concepts.
Which brings me back, again, to: There is no perfect solution here, but I don’t like the R notation, and I think there are some good reasons for my feeling that way. If the costs outweigh the benefits, perhaps it’s time to retire, or at least modify, it.