What Can Division Tell Us About Multiplication? (Part 2)

Preface

This post has been sitting in the hopper since the beginning of November. Due to a number of conflicting writing projects and a major presentation I've been preparing to give on Dec. 1., I'm at a point where I need to either fish or cut bait with this now, pending a lot of reading that I don't have time to do immediately. My decision today is to post this "as is," with the major investigation of existing research left hanging fire until I have time to do it real justice. So for some readers, this second post may disappoint, not getting to what I personally think is the nitty-gritty that must be explored to determine whether Keith Devlin's heuristic columns on multiplication and repeated addition are simply provocative pieces that fail to lead to urgent rethinking of how we present multiplication to students in K-12, or if in fact he has put his finger on something mathematics teacher-educators and elementary and middle school teachers need to take very seriously indeed. I continue to try to occupy a semi-moderate middle ground, not happy with some of the inflammatory language that has been used by various participants in the debate (including Devlin himself, of course, though I believe he did so with the intention of stirring up things; regardless of intention, he clearly succeeded in provoking a lot of discussion, some of it actually quite productive), but unwilling to buy the simplistic viewpoint that there's nothing underpinning Devlin's ideas that need to be looked at carefully from the perspective of what we teach kids and how regarding multiplication.

Part 2

Previously, I explored two major interpretations of whole number division, the partitive (sharing) and quotative (measurement) "models." My purpose was to see how well these interpretations work for whole numbers (though they do not exhaust possible interpretations), and also how these interpretations are almost, but not quite adequate for integers. The problem comes when one tries to interpret dividing a positive integer by a negative integer using either of these ideas: there's simply no way to make sense of them. Of course, looking at division as the inverse of multiplication solves that problem readily. And there are other interpretations of multiplication and division that might do equally well.

But my initial purpose for looking at division and interpretations of it was to see what light, if any, it could shed on meanings for multiplication. In particular, my concern was to see whether "repeated addition" sufficed as a definition for multiplication, given Keith Devlin (and other's) insistence to the contrary. Minimally, he asserted in his most recent column on this continuing controversy that teachers should not leave students with the impression that multiplication is "nothing more than repeated addition."

One hint that multiplication can't simply be repeated addition comes from the above-mentioned breakdown of the two interpretations of division of whole numbers when applied to integers, particularly the failure of the measurement model, which can also be viewed as "repeated subtraction." Though I'm very fond of using that viewpoint to help convince people that division by zero cannot result in a real number, it's clear that repeated subtraction will not result in any meaningful interpretation for 6 / (-2) = -3. There are no -2's to subtract from 6. The colored chip model, which works so well for the same examples that partitive and quotative interpretations work for with integer or whole number division, will not make any sense for this problem. Adding zero pairs and then subtracting (removing) pairs of negative (red) chips gives the wrong result. It seems that division "ain't no repeated subtraction" after all, or it least it isn't JUST repeated subtraction.

I'm not sure that I mind all that much that the repeated addition and repeated subtraction ideas don't work all the time; ditto for the chip model and the partitive/quotative interpretations. After all, these aren't definitions so much as they are interpretations or physical models: the map is not the territory. And it's pretty clear that while it's possible to stretch the repeated addition/subtraction ideas to positive rational numbers, the same difficulties will arise for signed rationals, and the entire thing will collapse when one looks at irrational numbers. I won't try to address complex numbers here, as they are well beyond the scope of K-5/6 arithmetic.

By saying that I don't mind, what I mean is not that I would abandon these notions or not use them in elementary mathematics teaching, but rather that I would insist that kids need to see OTHER ideas about multiplication and division, other sorts of models, with every reasonable effort made to ensure that: A) elementary teachers understand that "repeated addition/ subtraction" does NOT suffice for discussing multiplication/division with students; and B) that students will NOT be left with the mistaken belief that these operations are nothing more than repetitions of addition and subtraction.

So What The Heck IS Multiplication?

Don't continue if you're expecting a definitive or single answer to the above question. For one thing, I'm not a mathematician, professional or otherwise, and I would never claim to have a sufficiently deep background in mathematics to offer 'the' answer. For another, a major part of my point of view is that there are rarely, if ever, single definitions of that type, even in mathematics. Perhaps Keith Devlin is right in that regard to ask:

Why not say that there are (at least) two basic things you can do to numbers: you can add them and you can multiply them. (I am discounting subtraction and division here, since they are simply the inverses to addition and multiplication, and thus not "basic" operations. This does not mean that teaching them is not difficult; it is.) Adding and multiplying are just things you do to numbers - they come with the package. We include them because there are lots of useful things we can do when we can add and multiply numbers. For example, adding numbers tells you how many things (or parts of things) you have when you combine collections. Multiplication is useful if you want to know the result of scaling some quantity.

But then again, this quotation might just be Professor Devlin's cry in the wilderness, or perhaps it's an example of much ado about nothing. Is there really such a difference between multiplication and repeated addition?

Consider the following, offered at "TheMathPage.com," a page maintained by a former colleague of mine at Borough of Manhattan Community College, Lawrence Spector:

Here is the most general definition of multiplication:

Whatever ratio the multiplier has to 1
the product shall have to the multiplicand.
Consider this multiplication of whole numbers:

3 × 8 = 24

The multiplier 3 is three times 1; therefore the product will be three times the multiplicand; it will be three times 8.

Similarly, to make sense out of

½ × 8,

the multiplier ½ is half of 1. therefore the product will be half of 8.

Proportionally,

As the Multiplier is to 1, so the Product is
to the Multiplicand.

Or inversely,

As 1 is to the Multiplier, so the Multiplicand is
to the Product.

The Product, then, is the fourth proportional to 1, the Multiplier, and the Multiplicand.

To be perfectly fair, Spector's earlier discussion of multiplication (of whole numbers) is grounded in precisely the "multiplication is repeated addition" interpretation that Devlin decries. But in his "most general definition," he clearly sees multiplication as deeply related to proportional reasoning, not repeated addition. And this is exactly what Devlin and others see as a serious matter for concern and inquiry.

Given the critical importance of proportional reasoning in the US middle school math curriculum (or so the NCTM Standards and many mathematics educators and state curriculum frameworks suggest), and given that there is disturbing evidence that American students as a group do less and less well in mathematics as they get older, it's important to consider their performance on proportional reasoning in particular. Here, I'm not just talking about by comparison with students in other countries, since there are reasons to question whether such comparisons are accurately comparing similar groups; some critics of TIMSS results have claimed that some of the higher-scoring countries are testing a narrower band of students than does the United States. I'm talking about the general sense that the percentage of mathematically proficient students declines in America as students continue through middle and high school and into post-secondary education.

It seems reasonable to suppose that if such a decline is real, that one reason that middle school students underperform compared with elementary students is that they are not well-positioned to tackle the central curriculum focus for their grade band: proportional reasoning. (I would suggest that a continuing weakness in integer arithmetic goes along with and exacerbates this shortcoming). And I suspect that part of that weakness in handling proportional reasoning can be found in how multiplication is taught and (mis)understood.

What remains to be done, then, is to look carefully at the research that is already out there on children's mathematical thinking, as well as that of older students (middle school, high school, and college students in, say, freshman calculus) and see if and how their thinking about multiplicative situations, proportional reasoning, etc., is impacted, for good or ill, by their introduction to whole number multiplication. Is there in fact evidence that some/many/all students who are introduced to multiplication as repeated addition and left continuing to think of it solely or primarily as repeated addition are hamstrung when it comes to doing and/or understanding multiplication as they move to other kinds of numbers?

And, in due time, I hope to explore some of that research here. More to come. Perhaps much more.