I think the valuable discussion came from Math Mom asking: “so what should we do instead?”

In fact, who was Devlin’s audience? Not teachers, for the most part. It’s not nice taking a shot at people who are not around to hear what you are saying. And teachers take enough shots, we are half the country’s punching bag, without this, from someone who should have known better.

But once the question was out there, we should have focused on the mathematics and the questions related to teaching. I still don’t get why you wrote that last post about Devlin. And I haven’t stopped asking Denise’s question, “So, what should we do?”

I tried in an earlier post:

What should we do?

Teach multiplication through repeated addition, skip counting, counting arrays, finding unit areas of rectangles, Cartesian product, scaling… That’s too many to start, isn’t it? Pick one, then one more. And tell the kids that multiplication will manifest itself in many other ways as well… Add more… and remind them… No big deal here.

Did I get it? Maybe. Maybe not. But it was an attempt at a contribution. (you may find the comments on that post of some interest)

There is a much, much harder question. Who teaches kids their first math? And how have they been trained? Not a knock on ed schools, though I am not delighted by you folks, in general. It’s a question of living in a society that treats elementary education as babysitting with a few books. How do we make sure that 5 and 6 and 7 year olds are learning math from people who understand it?

Either we figure a way to train math-phobes to understand, then teach math, or
we find a way to bring math specialists into their classrooms, or
we find a way to recruit math-strong folk into lower grades, or
we rejoice in procedural-without-understanding mathematics being a permanent fixture in the American landscape.

But at least as important and disturbing is that even where I see teachers offering different algorithms and models for multiplication, very few (approaching none) either explain the underlying connections amongst these models or seem to care to UNDERSTAND how the models and algorithms work. I’ve blogged about teachers presenting the lattice multiplication algorithm as merely another “black box” approach. All of this does a huge disservice to students and reflects the mechanistic view of mathematics that seems all-too-pervasive in our schools.

Finally, for the Devlin-haters: he did a brilliant job of provoking more conversation about a specific issue in elementary school mathematics teaching than anyone I can think of in the last 16 years. I’d slice off a finger or two to be able to get that much discussion going about how to teach a topic in basic mathematics. Further, he cited a good deal of research, some of which is enormously provocative and bears careful consideration. I was in touch via e-mail with Terazinha Nunes on Friday and hope to be able to glean more from her about the research she and others in England have done on basic arithmetic and kids.

Unlike some of those who seem passionately focused on “proving” that their understanding of all this is right, I’m trying to learn some things, not the least of which is what those who’ve conducted extensive studies of kids have discovered. This isn’t an argument that’s going to be settled by yelling, epithets, and the usual internet baloney: it’s going to take many studies and careful reflection. Those of you who are primarily interested in rhetoric and abstract “proofs” of some theory of arithmetic have completely missed the boat: this is about kids and how they think and learn about mathematics. Sorry, but I have no interest in providing a forum (on my blog) for people who seem pissed off at Devlin but aren’t focused on kids and their teachers.

“The arrays, by the way, were of relatively fat dots, meaning, I hope, that half rows or quarter rows could be introduced at some point in the future.”…

Well, Devlin is vague in his articles ( he talks a lot on all manner of concerns, but comes up short on specifics) — so it is absolutely NOT clear whether showing an array of dots to illustrate multiplication (which is the way it was presented to me at age 5) would meet with his approval or not.

Certainly if the array of dots approach was “Devlin Approved” he should have said so, because that way has been in widespread use for a long time.

But its not clear, because he makes a big deal about wanting something that fits very naturally and effortlessly with real number multiplication. His suggestion is “scaling”. Granted fat dots can be coerced into serving the purpose of rational or real multiplication, but arguably less clear how natural and effortless the fit is.

I think the Devlin’s biggest sin is not so obvious – inconcsistency.
He wants real number multiplication to be taught from the beginning.
But the same problems he points out for going from integer multiplication to rational to real – well, those problems just don’t stop at reals. Complex multiplication means building on real multiplication with a new definition and is maybe a little harder to understand. Matrix multiplication is a little more convoluted still. He’s inconsistent to argue for an approach that has the exact same problem that he says he’s attacking.

There is nothing intrinsically wrong with teaching students one definition that works for integers, then another slightly more complication one that works for rationals and so on. (Nor is there anything wrong with the “multiple models” approach – I like that myself.) *That method* (call it progressive extension if you like) is a pattern that they will eventually have to come to terms with over and over again in mathematics, and not just in the realm of operations like multiplication and exponentiation. It is a vast meta-pattern that permeates mathematics.

I say there’s nothing *intrinsically* wrong with that approach – but it is a separate issue whether there are pedagogical concerns with that approach and whether there are tried and true methods for addressing those concerns. At that point his musings become purely speculative without much evidence to support his preferences.

not suggesting that NY State knows the best way to teach math (in fact, i would argue it’s pretty far from it!), but it is interesting to note that the NY State 6th Grade Standards instruct teachers to teach exponents exactly as Devlin would not have us do:
6.N.23 Represent repeated multiplication in exponential form
6.N.24 Represent exponential form as repeated multiplication

in 2nd grade, teachers are instructed by NY State to introduce repeated addition to facilitate understanding of multiplication:

2.N.20 Develop readiness for multiplication by using repeated addition

however, by third grade, teachers are to expand on this basic understanding:

3.N.21 Use the area model, tables, patterns, arrays, and doubling to provide meaning for multiplication

and by fourth grade, students are expected to understand that multiplication is more than simply repeated addition:

4.N.16 Understand various meanings of multiplication and division

i think the reason we teach multiplication as repeated addition and exponents as repeated multiplication is because it’s easy then for the students to grasp the basic concept and apply it.

anyway, just wanted to throw those standards into the mix.

I mean, it’s not like second- or third-grade teachers are marching into their classrooms and saying, “All right. Listen up, bitches. Multiplication is repeated addition for now and evermore. So, go ahead and do today’s warm-up while I chisel that definition into the front wall of the classroom. Does anyone have a sledgehammer?”

And, as you point out, most textbooks don’t make–or even border on making–that kind of ridiculous statement either.

Instead, in that first (or second) post, I just suggested that Devlin was wrong to frame the issue as a classroom-level problem–i.e., one that can be quickly picked up and solved by anyone who makes a living explaining math to 9- or 10-year-olds. (Textbook publishers are sort of in the same boat.)

If we’re going to explore it to some kind of meaningful depth, the issue is not an “at-hand,” “right-now” kind of issue. It is a higher-level curricular issue. It may or may not involve moving around or changing bits of curricula here and there; it involves some new time management, perhaps new materials, etc.

But for most, it’s just about expediency:

If we’re going to spend one, two, three? class periods making sure that students understand the distinction between multiplication and repeated addition, then that’s time taken away from their getting a quick, down-and-dirty explanation that’s really helpful in computing products. Computing products makes the kids happy because they see that they’re actually capable of instantly DOING something in math. Making kids happy makes teachers happy. Making teachers happy makes administrators happy.

So who is Devlin to come along and suggest that we make all the happy go away?

I think you can look at the Devlin articles and call out a dozen places where he’s wrong about pedagogy.

So what?