What Can Division Tell Us About Multiplication? (Part I)

In my tireless pursuit of the question raised this past summer by Keith Devlin in a series of columns in the MAA MONTHLY ("It Ain't No Repeated Addition" (June 2008); "It's Still Not Repeated Addition" (July-August 2008); and "Multiplication and Those Pesky English Spellings" (September 2008)) regarding the potential harm of leaving students with the belief that multiplication is nothing more than repeated addition, I am doing extensive reading about how children learn about, think about, and are taught to think about mathematics and the connection amongst and differences between the operations. One leg of this journey brings me to consider division. There are a number of ways to think about division, but the two "big ideas" looked at in school mathematics are the partitive or "fair sharing" model and the quotative or "measurement" model.

Partitive Division With Whole Numbers

As a quick overview or review, let's take some simple examples of both. Consider, for example, a simple sharing situation:

"Alexa has 30 cupcakes and wants to share them equally amongst six people. How many cupcakes does each person get?"

Note that this situation has several key elements, perhaps most important being the idea of "sharing equally" or "fair sharing." Without that stipulation, there's no clear-cut mathematical solution and instead any distribution one chooses that results in 30 cupcakes being shared in any manner whatsoever is "correct." But in this situation, there is only one right answer, which of course is that each person gets five cupcakes.

The other things to notice are that we have a fixed total (30), a known number of equal shares (six), and an unknown SIZE of each fair share which is to be determined.

How might a lower middle school student solve this problem? One way would be to model the problem with 30 counters of some sort. The student could keep placing one counter into each of six piles until all the counters are exhausted. She would then count how many there were in one pile and thus determine the size of the equal shares.

A similar approach would be for the student to count "one" during the placement of each counter on the first round, "two" during the second round, and so on through the fifth and final round, whereby the student would know that the last number that is repeated, "five," is the desired answer.

Students might all try subdividing strategies in which the original pile of 30 is subdivided into three equal groups of ten, and then each of those groups could be split into two equal groups of five, resulting in the final answer of five (in six groups). There are several variations on this sort of scheme depending on the factors of 30.

A more seat-of-the-pants approach would be to take a rough estimate of the share size, say "four," give that many to each of the six piles, and note that there are enough left to give one more to each resulting in a share of five. Or if one guessed too high, some chips could be redistributed until the piles were equal. Clearly, this sort of approach becomes less and less viable as the number of items to be shared and/or the number of shares goes up.

Quotative Division With Whole Numbers

Turning to the quotative or measurement model, let's take an example similar to the previous one, with appropriate differences:

"Zane has 30 cupcakes and wants to give six each to a number of his friends. How many friends can he do this for?"

Note that here we still have a known total (30) but now the size of a fair share is given (six). What is not known is the NUMBER of such shares possible of the given size with the original total. Though the solution here (five) is the same as that of the previous problem and the numbers given are the same, they do not represent the same things (in the case of the two smaller numbers). In the partitive case, we know the number of fair shares and seek to find the size of one fair share given the total; in the quotative case, we know the size of a desired share and need to find how many such shares can be made from the given total.

Possible solution strategies include taking a pile of 30 counters and subtracting six at a time until the pile is exhausted, then counting the number of piles to get the desired answer. Thus, it may be clear why the quotative or measurement model relates to the idea of division as repeated subtraction: one can repeatedly subtract a given number (the divisor) from the dividend until either zero or a number smaller than the divisor is reached. The number of such subtractions is the quotient, and any left over is the remainder. This relates the quotative model to the so-called division algorithm (more accurately a theorem):

given two integers a and d, with d ≠ 0

There exist unique integers q and r such that a = qd + r and 0 ≤ r < | d |, where | d | denotes the absolute value of d. The integer q is called the quotient, r is called the remainder, d is called the divisor and a is called the dividend.

The relationship and difference between partitive and quotative models of division is not always made clear in elementary mathematics education, and often their existence comes as news to elementary school teachers and education students. When asked to create division word problems, teachers and students often find it difficult to do so for the quotative model, even though the natural approach to solving such problems, repeated subtraction, would appear to be "obvious" and to link to the widely-held belief that multiplication is "nothing more than" repeated addition. The imbalance between ease of constructing partitive examples of real-world problems and difficulty of construction quotative examples becomes exacerbated for many teachers and students when dealing with rational numbers in fractional form. There is also an interesting conceptual break-down of the partitive and quotative models when dealing with division of integers, as we shall see shortly.

A Side Trip To Division of Fractions

One notable point about division of rational numbers in fractional form must be made before turning to integer divisions. It has become increasingly well-known to those interested in elementary mathematics learning and mathematics teacher education that many Americans have difficulty with the concept of division of fractions. While fraction multiplication is a fairly straight-forward proposition (the product of a/b and c/d is equal to ac/bd, so that a standard error in fraction addition and subtraction, which is to add/subtract numerators and denominators from left to right, is correct when it comes to computing the product of two fractions: simply multiply numerators and denominators, respectively, from left to right.

When it comes to dividing fractions, however, there is often both procedural and conceptual difficulty (not that the simplicity of computing products with fractions is an indication that conceptual understanding is common): the old standby rhyme, "Ours is not to reason why, just invert and multiply," not only indicates that many teachers and learners either give up worrying about students' understanding of the mathematics or never cared about it to begin with, but also isn't even a guarantee that students will compute correctly: after all, it matters enormously WHICH fraction is inverted before doing the multiplication.

From a purely symbolic perspective, one approach is to write a/b divided by c/d as a complex fraction - a fraction that has one or more rational expressions in the numerator and/or denominator. One then shows how to simply this fraction by multiplying both numerator and denominator by the reciprocal of the latter. Thus, (a/b) / (c/d) = (a/b)(d/c) / (c/d)(d/c) =
(a/b)(d/c) / (cd/cd) = (a/b)(d/c) / 1 = (a/b)(d/c). Thus, we show a justification, though informally, for the "invert and multiply" procedure for fraction division.

A second commonplace is to ask students to construct real world situations that would be correctly addressed by solving something like 2/3 divided by 1/2, and/or to simply compute the correct quotient. A very significant percentage of both elementary students and elementary teacher education students will produce problems that represent 2/3 * 1/2 and give 1/3 as the desired answer. These failures have been cited as proof that the groups in question lack procedural knowledge of rational number division, conceptual knowledge of that process, or both.

I suggest that while it is true that many people have difficulty with rational number division, it is important to look at how the above sorts of questions are posed before drawing conclusions about any given individual or group. It is particularly risky to draw to strong of a conclusion if the question is posed orally with 1/2 as the divisor. It seems plausible and even likely that a significant number of people are so used to being asked to find 1/2 OF a number that they will mishear/misconstrue a question asking them to divide by 1/2. I think it would be important for formal research, classroom assessment, or instruction to be sure that students have the opportunity to respond to written as well as oral questions, and that unit fractions, particularly 1/2, are not the only divisors they are asked to work with. I would be more confident that there is a significant conceptual and/or procedural problem if someone were asked to construct a problem that would be solved by dividing 3/4 by 2/3 for which they gave something like, "Dominique has 3/4ths of a pizza and wants to give 2/3 of it to his brother. How much should he give him?" or gave the answer for the computation as 1/2, than if s/he gave similar responses to the previously mentioned problem, particularly if the first one were only given orally.

Division of Integers

It is heuristic to have in-service and pre-service elementary teachers, as well as their students, to try to apply on their own the partitive and quotative interpretations of division that they employed successfully with whole numbers to division of integers.

Take the simple example of dividing the four combinations of +/- 6 by +/- 2. First, consider dividing +6 by +2. It is to be hoped that individuals will conclude that both the partitive and quotative interpretations can be used sensibly. Six balloons can be shared equally by two friends, with each getting three balloons. Or six cups of flour can make three batches of cookies if it takes two cups of flour per batch.

Consider next -6 divided by +2. It does not make sense to use the quotative interpretation here. There are no positive 2's to be taken from -6. However, the partitive interpretation can be used sensibly, as there is no difficulty dividing -6 into two equal groups of -3. So, for example, if two friends wish to share equally a deficit of 6 dollars (-$6), each gets a share of -$3.

Next, consider dividing -6 by -2. Here, the partitive interpretation fails because it does not make sense to talk about -2 groups or sharing equally among -2 friends, etc. However, the quotative interpretation can be used sensibly: If you want to share a deficit of six dollars (-6) so that each person gets a share of -$2, then 3 people are needed.

Finally, we come to +6 divided by -2. If one attempts to use the partitive interpretation of division, the same problem mentioned in the previous example arises. If one tries the quotative model, we run into the problem raised in the second example: there are no -2's to remove from +6. So how does one interpret 6 / -2 intelligibly? One obvious solution is to talk about division as the inverse of multiplication. Thus, x = 6 / -2 asks the question, "What number times -2 equals 6?" and the answer is -3, which for the moment we will assume is something already established with students. There is no problem finding the correct answer.

Does this mean that partitive and quotative interpretations of division are wrong and should not be taught? Well, it's clear that they make perfect sense in whole number arithmetic. And they cover 3 of the 4 possibilities for integer division where the dividend is an integer multiple of the divisor. It seems reasonable to believe that by the time students reach integer arithmetic (typically 5th or 6th grade), most will be ready for some of the necessary algebraic concepts needed to make sense of all cases, with notions of inverses, identity elements, and so forth coming into play. It doesn't seem like too much of a stretch to think that many will be comfortable with the question given in the previous paragraph as a way to think about six divided by negative two.

No model or interpretation of a mathematical idea is likely to be perfect. But that should not be shocking. The model/interpretation is NOT the mathematics. It's a way of helping students to think about the mathematics. A model or interpretation may in fact be precisely what a given student will carry away with him from his education, though this is not necessarily "the" goal of mathematics teaching: certainly most educators hope that every student will have the opportunity to experience the beauty and power of mathematical abstraction for its own sake, as well as a way of thinking that can deepen as one encounters increasingly complex and challenging mathematical ideas. We should not leave students with the mistaken impression that the map is the territory, that the model or application or interpretation IS the mathematics. But absent evidence that exploring models and interpretations is harmful to student learning or that it inevitably limits their long-term mathematical growth, it seems reasonable for teachers to continue to use them thoughtfully.

What next?

The title of this piece suggests that there is something in all this talk about division, interpretations, models, and applications that can inform us about multiplication and whether it suffices to leave students thinking of it as "no more than repeated addition." In the next part of this piece, I hope to help shed some light on just that question.

Does This Mean That The Phone Company Did Kill Kennedy?: Even More on Keith Devlin, Multiplication, and Repeated Addition

A recent post on the continuing conversations, fights, and debates inspired by Keith Devlin's recent trio of columns on problems he and some mathematics educators (in England, if not elsewhere) see with teaching multiplication-as repeated-addition (or, variously ONLY teaching it that way and never leading students to see that multiplication either is not repeated addition or is a lot else besides repeated addition) to elementary students, let me to write a response that gets at some of what I've been thinking about as I read those British researchers. Below is a re-edited version of my reply:

A crucial point I've tried to make previously in various places: there is a serious problem with the absolutist language a lot of folks in this debate insist upon using and which Prof. Devlin falls into at times (and hardly just at math-teach: check the comments at lets play math; good math, bad math, and any number of other blogs where this issue has been debated for several months. Some folks on those blogs make the argument on this thread at math-teach seem tame. See in particular comments by Joe Niederberger). However, as I've come to appreciate Prof. Devlin's style and intentions, I'm not so worried about his habit of overstating his case as I was initially. I think he wanted to draw attention to a number of important issues in the teaching of school mathematics, perhaps most broadly the likelihood (or at least the possibility) that sloppy teaching of elementary mathematics from a perspective that doesn't consider things from only slightly higher mathematics causes difficulties for a lot of kids as they progress into college mathematics courses. That would certainly be worth thinking about if true, and given the many posts we get on this list about how the sky is falling, too many kids aren't ready for college math, and so forth, one might expect that Prof. Devlin's columns would meet with approval from some of the educationally conservative thinkers here. Indeed, I was caught off guard at the anti-Devlin sentiments from Wayne Bishop and, later, Haim Pipik (though perhaps given that Devlin is on speaking terms with mathematics educators and doesn't find all math education theory or research to be garbage, I should have expected their knee-jerk dismissals).

I tried initially in my first blog post on Devlin's columns (before his third one appeared on this issue) to call into question his very strong assertion that there would be inevitable problems for students if they were taught that multiplication was repeated addition. But by now, anyone who is really paying attention to what he's said knows that he modified the claim near the end of his third column in a way I find acceptable:

Meanwhile, now you know why (or at least you know where to start finding out why) it is crucially important that we not teach children that multiplication is repeated addition. (Or, if you prefer, why we should not teach them in a way that leaves them believing this!)

That last parenthetical remark is precisely what I think most reasonable thinkers on this issue would likely conclude if they had taken the trouble to read the references. And the language in those articles and books, if you look at what Devlin himself quotes, is actually very modest. It's only Devlin who goes a bit heavy-handed for much of his three columns (though he gives plenty of explanation for both his perspective and his concerns), but ultimately I'm unable to fathom how anyone could argue vehemently and unrelentingly against the very last sentence in the quotation above.


For example (but many other examples are similar),

Research SUGGESTS that such teaching MAY BE at the root of later misconceptions. Alternative models for teaching have been shown more effective in experimental studies "(Clark & Nunes, 1998) but EVIDENCE IS STILL LIMITED AND MORE RESEARCH IS URGENTLY NEEDED. [emphasis added]

I'd add that this research would likely be most profitable if it were of the kind Ball describes when she and her colleague talk about research "inside education." The interaction between a number of factors needs to be considered in judging whether there really is a meaningful disadvantage in even including, let alone teaching exclusively, the multiplication-as-repeated-addition idea, or a meaningful advantage in the "alternative models" just mentioned. It's not merely a question of materials or merely of teacher moves or MERELY any one thing in isolation.


I think Prof. Devlin has gone on to point to a lot more than what he raised in his first two pieces. I doubt, for example, that he meant that we MERELY say that addition and subtraction are just two things we do with numbers, but rather that among other things we teach students, we make clear that two main things we do with numbers (or "to them") are to add them and multiply them and that the two are not identical. I think he's right that students should see that and continue to be exposed to the differences, just as we can't be satisfied with tell students that exponentiation IS repeated multiplication.

I would argue that students will, as they gain mathematical maturity, be increasingly able to see different ways in which there are deep differences between addition/subtraction and multiplication/addition, even if they ALSO consider some similarities. One issue Devlin raises that I think is worth noting is his concern that we're hobbling kids by teaching them that "multiplication makes things bigger," and I agree that it's a problem for a lot of students that arises repeatedly (one glaringly obvious example is on the typical SAT question where students are asked: for x > 0, which of the following is greatest - A) 1/x B) x C) x^2 D) sqr(x) or E) the relationship cannot be determined. Even when the problem is modified to exclude x = 1, many students are all-too-quick to pick "C" and I think this reflects the "multiplication (and exponentiation, which is "merely repeated multiplication") makes things bigger" idea. But I would point out that we see the same confusion about addition/subtraction: students get stuck at the whole number (or even natural number) assumption that addition always makes things bigger; subtraction always makes things smaller. So I believe we need to address a related though probably less complex problem with how we present addition and subtraction to students as if they are NEVER going to have to deal with integers. If no one else things any of this is important, the folks who write the SAT, ACT, GMAT, and similar test where math is at issue obviously do, because they exploit student misconceptions about all this relentlessly.

Returning to Devlin, he repeatedly states that one can arrive at the correct answer to many (though not all) multiplication problems through the act of doing repeated addition, but that doesn't mean that the former IS the latter or is "merely" the latter. I would have balked at the second notion, without ever having read Devlin, but I think he was trying to break through a lot of inertia amongst K-12 teachers, especially in the lower grades. And clearly he was right, given the reactions his columns engendered, as many teachers did (and do) seem to think that multiplication IS repeated addition and that's all ye know or need to know. Or at least all THEY want to know or teach. And that is really not acceptable.


Frank, the fellow whose post inspired this entry wrote,

The rest of the first article and the second article were mainly devoted to explaining why equating multiplication and repeated addition is mathematically incorrect (without providing any clear examples of why it was incorrect or important for K-12 education.) At the end of the second article, Devlin pleads:

Let me stress again that I am not suggesting we teach children arithmetic the way professional mathematicians view it. Rather, my point is that, however you teach it (and I defer to professional teachers in figuring out the how), don't do anything that is counter to the way the mathematicians do it .... some of your pupils may well end up in universities where they will HAVE to do it the right way.

In other words, don't create any misconceptions in K-12 that might make my job harder.

That doesn't seem to be Devlin's point, but rather not to make those students' lives harder. I don't think he is worried about the difficulty of his job, but the intractability of some student conceptions.


Frank continues:

In his third article (Multiplication and Those Pesky British Spellings), Devlin finally gets around to explaining why K-12 teachers and students (as opposed to university mathematicians) should care whether or not multiplication is exactly the same process as repeated addition: It may make it difficult to separate the concepts of addition and multiplication when students do proportional reasoning. Devlin states that

"Arguably Nunes and Bryant know more about multiplication and how to teach it than any other researchers in the world", so he quotes them on the subject of repeated addition:

"The common-sense view that multiplication is NOTHING BUT [Frank's emphasis] repeated addition, and division is nothing but repeated subtraction, does not seem to be sustainable after a careful reflection about situations that involve multiplicative reasoning. There are certainly links between additive and multiplicative reasoning, and the actual calculation of multiplication and division sums can be done through repeated addition and subtraction. [DEVLIN NOTE: They are focusing on beginning math instruction, concentrating on arithmetic on small, positive whole numbers.] But several new concepts emerge in multiplicative reasoning, which are not needed in the understanding of additive situations.

After reading the portion of their book that is available on the Internet, Nunes and Bryant make a great deal of sense to me. Notice that Nunes and Bryant say that multiplication is MORE than repeated addition, while Devlin says that "It Ain't Not Repeated Addition". Nunes and Bryant refer to repeated addition a "common-sense view" (that is incomplete), while Devlin equates it with the level of ignorance represented by the phrase "ain't not".

Do you think Professor Devlin has done a good job of educating the mathematics community about this issue?

I think Devlin was citing sources earlier than in the third article, but I won't quibble. He clearly was informed by that research, and it's interesting that it's pretty much all BRITISH research. He pokes some fun at the fact that it seemed to be news to a lot of the blog respondents that this body of work existed. I'm not sure many of them would be any more aware of it had it been American research. We have a few members of this list who don't seem to much care what research is out there if it doesn't confirm their prior beliefs.

And yes, I think Prof. Devlin has done everyone a wonderful service with these columns. Was he absolutely right in every particular? Probably not, though I think all three columns bear rereading with great care: he answers a lot of questions that folks claim he does not, and he certainly provides support for his ideas, though as he's told me, that didn't stop some some folks from excoriating him for NOT providing support for his ideas. Go figure.

I know that there aren't many people in mathematics or mathematics education who can reach as many people as Keith Devlin can, and as the issue he's raised is clearly of great interest and likely importance, he deserves everyone's thanks and respect, though that hardly means that everything or anything he says should be accepted. He meant to provoke thought and has done so, though unfortunately he has also provoked a lot of thoughtlessness. That's the nature of things these days, of course, and I have no doubt he can handle the heat.


Postscript: for those of you puzzled about the title of this post and the accompanying photo: the actor Tom Bower played a character name Cecil Skell in a terrific movie with James Woods and Robert Downey, Jr. called TRUE BELIEVER. Skell is a schizophrenic who believes that the phone company killed JFK (because he wanted to break up the phone company). He happens to be a key eyewitness in a murder who is never called by the defense in the original trial of Woods' client, a Korean who has now killed an "Aryan Brother" in self-defense while in prison for the first crime. Skell insists that the murderer "wasn't Chinese, because Asians have a different aura." When Woods and Downey, Jr. discover proof that in fact Skell is right and the murderer was not Asian but merely looked like he might have been, Downey, Jr. says to Woods, "Does this mean that the phone company DID kill Kennedy?" To me, it's the all-time greatest movie line to use when you realize that something that initially struck you as a bit crazy is in fact probably true.