One of the great benefits of being where I am is that I get to be part of two mathematical communities. One is the community of research mathematicians, for whom teaching is interesting, but the source of the fascination that drives professional lives is mathematics itself. The other is the community of mathematics educators, including K-12 teachers and the people who teach them and study how mathematics learning occurs, for whom mathematics is interesting, but the source of the fascination that drives professional lives is teaching itself. Some topics are internal to one or the other of the communities, but others span the two, and produce intense conversations in both, with very different perspectives. Me, I am fascinated by those conversations.
One such topic of which I have recently been highly aware is algorithms. The background of the conversation is as follows: for many years -- many, MANY years -- the format for teaching elementary and secondary mathematics has been presentation of an algorithm, followed by multitudinous problems on which to practise that algorithm, followed by presentation of the next algorithm,... . Word problems have been objects of dread because it is not always instantly clear how to apply the algorithm, or sometimes even (horrors!) which algorithm to apply. Those for whom this is a highly successful method of instruction (alternative description: those with an indestructible appetite for mathematics) have gone into research mathematics or perhaps science. High school mathematics teachers have come from the ranks of those for whom it is at least not totally catastrophic. Elementary teachers, to an appalling extent, have come from the ranks of those sufficiently traumatized by this presentation to have become convinced that the algorithms are the entire of mathematics and that the mandate of a teacher is to instruct students in the absorption and manipulation of the algorithms, orchestrated by some mysterious motivation which the student can clearly do without because the teacher has always done without it.
Lone voices have sporadically protested this procedure. In the past few decades the protest has become much more general and coherent, and been backed by serious, solid research. It came to general consciousness with the publication and widespread acceptance of the NCTM Standards (expanded version: the Curriculum and Evaluation Standards for School Mathematics, published in 1989 by the National Council of Teachers of Mathematics.) The ensuing period has brought more and more voices into the conversation, with many insights from a wide variety of perspectives. Also with some misapprehensions, some concerns and a great deal of flat out bewilderment. This newsletter was inspired by recent conversations exhibiting all four of the above characteristics. As should by now be clear, I make no pretense of presenting any sort of resolution. All I am is another voice in the conversation, with the slightly odd feature of internal two-part harmony (or sometimes dissonance!)
I'll start with what I hear from and between my colleagues in the mathematical sciences. At the phrase "not teach by algorithms" there flashes into their minds the image of a class of undergraduates adding by counting on their fingers. This is not a pleasant image, and must be supplanted before any progress can be made. An initial step towards defusing the situation comes easily, because most of us teach international students. Whereas American schools, to an extent which is amazing when you stop and think about it, are uniform in their teaching of the algorithms for each of the basic operations, close observation of someone from another continent carrying out a subtraction problem frequently reveals some funny little subsidiary numbers appearing in places where none of ours do, and it doesn't even take close observation to realize that long division often looks wildly different. It's hard to argue that our international graduate students, much less international colleagues, have been hampered by the lack of the American algorithm.
That's the first step. After that comes the stickier question: "Yes, but why not give them some algorithm so they can be efficient?" What speaks to me as a reply to that is a series of experiences involving addition. One comes from the Developing Mathematical Ideas seminars, where participating teachers are asked to put down their pencils and add 27 and 18 in their heads. They are then asked to say just how they came up with their answers. A huge variety of tactics invariably emerge, frequently prefaced by "I'm afraid I didn't do it the Right Way -- I started off by adding the 20 and the 10 ." [That's at the first seminar. By the end the apologies come from the ones who did use the standard algorithm -- but that's another story!] Left to her own devices, a first or second grader will come up with one of those tactics, usually adding the existing tens first, and then inventing a method for keeping track of the tens that emerge from adding the units.
Neat, but that still doesn't answer the "Why not...?" question. For that, there is both a general answer and a specific one that is attached to the previous situation. The general one has to do with an aspect of the student-teacher relationship to which the field of Didactique assigns the title of the didactical contract. Only an exceptionally intellectually feisty child is immune to the impact of privileged information: a student will accept an idea that comes from the teacher as one which must be used even at the cost of abandoning an idea that he actually understands. A non-understood algorithm, no matter how slick, is a poor substitute for an understood one even if that one is a little crude. On a more specific level, we return to the issue of multi-digit addition. Clearly, before tackling that children must have a grasp on place value. A child who has that grasp can, as I said above, invent a method for carrying out addition. And if instead, or thereafter, she is taught that the way to add is first to add the numbers in the right hand column and write a little 1 above the 2 in 27, etc., what is the consequence? Frequently, documentably, a total disappearance of the concept of place value, lost in the struggle to figure out whether it's the little 1 or the little 5 that goes above the ... now where was it supposed to go?
All right then, suppose we accept that the learning of mathematics is not well served by the presentation of a sequence of neatly packaged algorithms, what then? That's where the tenor of the conversation becomes quite different in my two communities. Amongst university mathematicians the reaction, accompanied by a skeptically cocked eyebrow, is "If they're not given algorithms, what are they supposed to do? Suppose their invented algorithms are crude, or insufficiently general, or even false -- what will become of them?" Amongst elementary teachers the reaction has far more anguished overtones: "If I'm not supposed to hand them algorithms, what am I to do? What is it OK to tell them? How do I know if they are on the right track? Suppose I blow it -- what will become of them?"
Legitimate anxieties, both, and predictably enough we have now reached the point where a comprehensive answer is not merely implausible, but impossible. So I'm not about to attempt one. But I do, of course, have my own take on the matter, and that is what you are about to get -- with apologies if a few bits of soap box seem to be emerging beneath my feet.
It seems to me that the fundamental, key, indispensable element to resolving this is a belief by all of us, from pre-kindergarten teachers through university faculty, that children -- ordinary, non-genius, meet-them-on-the-street-corner children -- have both the desire and the capacity to make sense of their world, and in particular of mathematics when it is part of their world. With that belief, the job of the teacher is to foster the desire and nourish the capacity, and not to be the one who supplies the sense. That means providing scaffolding for learning (not simple to do) and support for doing so, and (yes!) telling a student something when she herself has expressed a need to know it. It also means hanging back and letting a student muddle around, groping his way towards an intellectual idea which the teacher can see so clearly that it is painful not to simply pick him up and plunk him down in front of it, because the muddling time is essential to the ownership of the idea. And, if the idea in question takes the form of some sort of algorithm, it means recognizing when the student needs the chance to celebrate his idea by practising a bit with it.
That's a huge demand to make of teachers, especially those who, as I said above, have been traumatized out of believing they can do anything other than reproduce their own disastrous mathematical past. Fortunately, there are now teaching materials out (notably some of the NSF-sponsored ones) which support that kind of teaching. In this area, thanks to the ECML project, teachers also are supported through the Developing Mathematical Ideas seminars. Still, with all that, the demands remain high enough so that I would wince at pushing them if it were not for two things: 1) teachers are in the profession because they want to teach well, and though they may groan a bit, they can and do appreciate something that unambiguously improves their students' learning and 2) although the process of learning to develop students' mathematical ideas is far from speedy, one of the rewards begins very early on: tuning in to student thinking produces absolutely fascinating results.
I seem to have concentrated on the anguished K-12 overtones, so I shall finish by returning to the professorial cocked eyebrow. "So you're telling me not to expect the techniques and algorithms I have always felt I should be able to require. What then are you offering me in recompense?" Fair enough question. Let me preface my reply by pointing out that we are not talking about the class of 2004. The specific changes I have been focusing on need to take place at the elementary level, preferably starting in kindergarten at the latest. Apply your own addition algorithm to figure the timing. My answer to the question itself would be that what we should see will be students with the perspicacity to recognize when they are lacking an algorithm, the gumption to set out to find one, and the perseverance to stick with the search until it's found. I don't think many of us would question that that is a net gain. --