Saturday, July 12, 2008

Devlin On Multiplication (or What is the meaning of "is"?)

I generally like to be pretty sure I'm right about something before I blog about it here (or at least that I'm not absolutely and irredeemably wrong). Some topics, of course, are relatively safe in that it's pretty hard to determine a definitively correct answer (though perhaps that doesn't prevent some answers from being almost certainly wrong nonetheless): matters of just how to teach something in mathematics or other subjects rarely avail themselves of a single right answer that serves all students equally well for all times and places (regardless of what my antagonists at Mathematically Correct, NYC-HOLD, and similar nests of absolutism may believe or claim). On the other hand, it's easy to be shown to be entirely wrong about specific pieces of mathematics, especially if one is not a mathematician, something I neither am nor have ever claimed to be. Not that members of that priesthood never err or that we laypeople are invariably wrong when we disagree with one of the elect on matters mathematical, of course. But a betting person would do well to give the odds and go with the mathematician when such disputes arise about mathematical particulars. I've known of errors in mathematics problems published by the Educational Testing Service, too, but you'd clean up betting on them against the likelihood they've erred. 

Imagine then my trepidation at taking on someone like Keith Devlin, the popular mathematician who writes regularly for the general public and the profession about mathematics in books, in a monthly column for the Mathematics Association of America, and on National Public Radio. His June 2008 column ("It Ain't No Repeated Addition") came to my attention this morning because several people who blog about home school mathematics teaching (Maria Miller and Denise,  an Illinois-based homeschooling mom who blogs under her first name only at Let's Play Math) were puzzled by what Devlin had to say.

In particular, as his title suggests, Devlin argues quite vehemently that "(m)ultiplication simply is not repeated addition, and telling young pupils it is inevitably leads to problems when they subsequently learn that it is not." It's hard to be more unequivocal than that. Instead, he argues, we need to teach students from the beginning that addition and multiplication are two basic things we can do with numbers that share some similarities in places but are not the same thing (I insert the similarity issue that he mentions in a way that likely gives it more attention than he would like, so responsibility for where I plan to go with that point is mine alone).

By his own admission, Devlin's not a K-12 mathematics teacher and he is perhaps a tad disingenuous when he mentions this at the end of his article, stating, "I should end by noting that I have not tried to prescribe how teachers should teach arithmetic. I am not a trained K-12 teacher, nor do I have any first-hand experience to draw on." But considering the respect both K-12 teachers and the general public have for professional mathematicians, it's hard not to think that his intent and deep hope is that his position might carry enough weight to have a great deal of influence in how arithmetic is taught. This isn't, after all, a major league baseball player endorsing a presidential candidate, something that would carry zero weight with most of us; it's a world-famous mathematician with a good deal of media exposure (more than the vast majority of his colleagues) making a claim about what multiplication is and is not. That would likely influence more than a couple of elementary school teachers, school administrators, and other interested and concerned stake-holders. Or so I would expect.

So now I need to go out on a limb and suggest that while Devlin isn't wrong, he also isn't quite right, and the problem lies with the nature of school mathematics and its teaching, as well as issues of mathematical maturity. His argument reminds me of a similar one I have when I do professional development with K-5 teachers regarding things they tell students that simply aren't true. Well, that's actually not always the case, either. The problem is that so much depends on what set of numbers one is working with. Teachers state without hesitation to lower elementary students that "you can't subtract a larger number from a smaller number" and "you can't divide a bigger number into a smaller number" (alternately on this last one, we hear "six doesn't go into three, so. . . "). And if one is working in the natural or whole numbers, both statements are true. But in the integers, the first statement is false, and in the rational and real numbers, both statements are false. (And one could go on to look at other sets of numbers, but for K-5, this suffices to make the point). So are teachers wrong to make the statements they do?

If you say "No," unequivocally, you're suggesting that you likely agree with the hypothetical teachers Devlin cites who believe that kids simply aren't adequately sophisticated at a given age (or individual point of development not directly attributable to age alone) to deal with more advanced or sophisticated mathematical ideas. It's tough enough to understand subtraction and division of whole numbers without throwing negatives or rationals (let alone irrationals) into the mix. We must walk before we can run, after all.

If you say "Yes," unequivocally, you likely find the reasoning above objectionable, as does Devlin, and conclude that the solution is to eschew making any untrue mathematical statements to kids at any stage of the game (assuming one knows all the requisite mathematics, something perhaps much to be wished for, but not so easy to guarantee in practice, even if we raised not just the bar, but the actual minimal mathematical knowledge of anyone teaching K-5 mathematics in this country in public education). 

Naturally, I wouldn't be writing this if I thought Devlin's analysis was sufficient and proposed solution ("Don't tell kids that multiplication is repeated addition"), such that he's made one , adequate. One thing I find lacking in his piece is a solid example that would communicate well and clearly to K-5 mathematics teachers (based on the ones I've known and worked with) how multiplication differs in some deep way from addition. I does not suffice merely to assert that the two are, for the most part, not the same. Teachers at the K-5 level may be intimidated into accepting an order not to claim otherwise, at least when they're being watched, but if in their hearts they believe otherwise, that belief will surely inform their teaching and the idea is going to be communicated to their students. Similarly, research indicates that college mathematics students struggle to accept fully such concepts as the equality of 0.9999... and 1 even when they can be taught to repeat that this is factually true. This gap between surface knowledge and deep understanding and concomitant belief should not be dismissed.  

But even more importantly, what Devlin proposes presumes that there is one correct model or metaphor for understanding multiplication. This assumption runs counter to vast teacher experience in K-5 mathematics and beyond. The multiplication-is-repeated-addition metaphor fits a number of physical models of multiplication grounded in even more fundamental and deep metaphors of arithmetic discussed extensively by Lakoff and Nunez in WHERE MATHEMATICS COMES FROM. Further, it is clear from the perspective of how arithmetic developed historically that the use of multiplication from a practical perspective can readily be called "fast addition," just as exponentiation can be called "fast multiplication." The associated algorithms that emerged for doing whole number and integer multiplication compress some of the individual steps through the clever use of place value in ways that are sometimes confusing to many students, and it is often helpful to show students how this viewpoint is valid by "unpacking" the algorithms and recreating the "repeated addition" that could be done to obtain the same results. Of course, at the same time, we want students to understand why using multiplication algorithms to find the product of even moderately large numbers is preferable. 

The modern view of multiplication Devlin advocates as "one of the things we can do with numbers" that is different from addition is not necessary for students who are learning arithmetic with any set of numbers up to and including the integers. Therefore, it is questionable that students would be advantaged by being given that more sophisticated idea immediately. Further, Devlin's claim that they are actually disadvantaged by being told something else is something he would need to provide proof or at least strong research evidence for, something that is absent in his article. I'm reminded in part of those who argue that failure to stress or thoroughly teach the long division algorithm by, say, fourth grade, will make students unable to learn and do synthetic division in algebra class. This argument sounds plausible to some people (generally those who know little or nothing about teaching K-12 mathematics), but I have yet to see anyone provide even a shred of evidence to support the claim.

The "obvious" solution to this seeming quandary (lie to students about what multiplication "is," or run the risk of over complicating things for them unnecessarily and losing one of the more effective and understandable metaphors/models for it) is, on my view, the same one I have always suggested would solve the above-mentioned issue of lying unnecessarily to students about the nature of subtraction and division: make clear to students that how we define specific arithmetic and other mathematical operations depends upon what set of numbers we're speaking of and that while they may currently be dealing with numbers that do not allow us to answer questions like 7 - 9 = ? or 3 divided by 5 equals ?, the fact is that there ARE numbers that they will learn about in a few years (if they don't know about them already!) that will allow us to answer such questions. It's not that multiplication "is" repeated addition, but rather that there are contexts in which that is one reasonable way to think about it. However, like all the arithmetic operations, it also "is" many other things, each of which will prove valuable to consider in turn.

In higher grades, teachers constantly tell students that "We can't take the square root of a negative number," but the fact is that this is true only if we are restricted to the set of real numbers or one of its subsets. Using Devlin's argument, that practice should be outlawed because we might be making it "too confusing" for these students when they encounter complex numbers or any other new set of numbers (e.g., quaternions). We shouldn't tell them that multiplication is commutative, because they will by high school encounter matrix methods for solving systems of linear equations and be taught that matrix multiplication with elements drawn from the real numbers is NOT commutative. Surely THAT should be a mind-blowing experience that will destroy students' abilities to progress in mathematics! 

Of course, nothing of the sort happens. And to the extent that it might with some pupils, the same approach I've mentioned previously would smooth the way: simply be honest with students about what restrictions apply to broad generalizations we ask them to accept (or even prove!) in class. This practice would actually raise the level of sophistication students bring to the mathematics they are doing, making them more careful about simply accepting any given statement as true for "all numbers" and any given operation. Instead, we will have impressed upon them the necessity of considering what set of numbers is under consideration and whether statements made about a particular operation are restricted to a given set and situation. Multiplication is NOT always commutative. It also isn't always repeated or fast addition. But it surely is commutative in many circumstances and it is important for students to be taught this and to see that it makes sense. Similarly, it is valuable for students to explore the notion of multiplication as repeated or fast addition, as long as they are given the tools to understand that this might not always make sense as they consider other, more abstract sorts of numbers or applications of the multiplication concept. Students so taught might well develop the valuable mathematical habit of mind of questioning how broadly a particular mathematical idea applies. And that is one of the things upon which they will be able to build a deeper and more mature knowledge of higher mathematics. 

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