This blog is really not meant to be a running commentary on math education items from the New York Times, but I can’t let the latest go by without a response. In “The Faulty Logic of the ‘Math Wars,'” Alice Crary and W. Stephen Wilson make the claim that a “progressive” approach to mathematics education should regard “algorithm-based calculation” as “elegant and powerful” and central to mathematical thought, and hence worthy of regard rather than “de-emphasizing.” They look to philosophy for support of this argument (Crary is a professor of philosophy; Wilson is a mathematician).
Philosophy majors (you know who you are) can help me figure out whether the authors have conveyed the ideas of Wittgenstein and Wilfred Sellars faithfully. The algorithms I learned for addition, subtraction, multiplication, and division are indeed elegant and powerful. So how do we help students appreciate both their elegance and their power? According to the authors, we should “teach both that these algorithms work and why they do.” Why do they think that’s not happening?
In fact it is happening in at least one fourth-grade classroom. When I visited in March, the class was working on problems involving an area model for the distributive law. The teacher gave an exit question — something like 7 times 17, that they were to illustrate as the area of two adjacent rectangles (such as 7 by 8 and 7 by 9). After class I watched her sort the graded papers into three piles: completely correct, conceptually correct but weak on math facts, and not clear on the concept. There were a few kids who had partitioned 17 into three numbers, rather than two, ending up with 7 x 5 + 7 x 6 + 7 x 6, say. I thought that was clever, but she pointed out that they just didn’t know their times tables for 9 yet. In this classroom, at least, investigation of a concept is intertwined with a “traditional” expectation about procedures. I am convinced that once these students got to the “standard” algorithm for multiplying two-digit numbers together, they had a solid understanding of why it works, and hence a better chance of remembering how.
What struck me first about the Crary and Wilson piece is that teachers are mentioned explicitly just once. Students are given problems, and procedures are offered to them, but the emphasis is on the “programs.” Math instruction teaches (or does not). To borrow the doctor analogy from the previous post, I’m trying to imagine an essay about health care that avoids use of the word “doctor.” “Diagnostic procedures are ordered, the procedures provide a diagnosis, and medications are prescribed according to protocol.”
The one place we see “math teachers” is here: “The staunchest supporters of reform math are math teachers and faculty at schools of education.” If it wasn’t obvious from their title and the paragraphs leading up to this part of their piece, we can now be sure that despite the mention of the Common Core, we’re going to rehash the arguments of the 1980’s and 90’s as if we’ve learned nothing since. There’s the drawing of lines between “staunch reformers” and “college teachers of mathematics” — never mind the fact that the NCTM 1989 Standards that the “champions of preparatory math” loved to hate were replaced by the 2000 NCTM update after consultation with more mathematicians, a.k.a. people with “special knowledge of the discipline,” and that the lead authors of the CCSSM are a university mathematics professor, a mathematics educator and a former professor of math and physics. There’s the focus on “programs” (with Investigations standing in for all of them), with no mention of the much more important role of teachers in student learning. There’s the absence of recent findings from people with special knowledge of how children learn. Then, of course, there’s the caricature of those “staunch reformers,” none of whom are actually named.
“The standard algorithms should be avoided because, reformists claim, mastering them is a merely mechanical exercise that threatens individual growth.” Who, exactly, said this, and when? “The reformist’s case rests on an understanding of the capacities valued by mathematicians as merely mechanical skills that require no true thought. The idea is that when we apply standard algorithms we are exploiting ‘inner mechanisms’ that enable us to simply churn out correct results. We are thus at bottom doing nothing more than serving as sorts of ‘human calculators.'” Again, citation, please? My recollection is that reformists at the end of the last century objected to what they saw as an emphasis on procedural knowledge with scant attention given to conceptual knowledge, regardless of what those old textbooks might say. Ask a non-math friend who went to elementary school before 1985 what it means to “carry the one,” and you might be dismayed by the response. As a mathematician, I value a balanced approach that allows procedural and conceptual knowledge to support each other.
Even the recurring terms “reform math” and “reformists” feel outdated. These days (post-NCLB), when I hear the word “reform” in connection with education, I think of high-stakes testing and “value-added” teacher evaluation systems and the like.
Suppose, then, that the authors had skipped the “math wars” bit and gone directly to the notion that algorithmic computation by the standard algorithms deserves a privileged place in a progressive classroom (though the word “classroom” never appears at all). First, as a mathematician and hence lover of definitions, I’d want to know what they mean by “standard” algorithm for all of the basic operations. In multiplication of multi-digit numbers, I assume that either the way we add the partial products in this country or the different order used in Japan (for example) would be okay with them. They might go to this discussion at the Mathematics Teaching Community website, including:
In our view, the essence of an algorithm is the steps and the logic and reasoning behind those steps. With that perspective, we think that different systematic written methods for recording the steps of an algorithm as well as minor variations in those steps should qualify as being the same algorithm.
“We” in the quote means Karen C. Fusman and Sybilla Beckmann. I found that discussion and their article on the subject by going to the site’s main page and searching for “algorithm.”
In any case, the authors want us to appreciate the “centrality to mathematical thought of the standard algorithms.” But does such centrality require that we reject a program that asks students to explore and, yes, investigate?
Yesterday I asked one of my 21-year-old sons how he would add 127 and 398. I lined up the numbers vertically, assuming he’d do it the way the authors describe (starting in the ones column). Instead, he said, “one hundred plus three hundred is four hundred. Twenty plus ninety is one hundred and ten, plus four hundred is 510. Seven and eight is fifteen, so the total is 525.” Would Crary and Wilson tell him he did it the wrong way? I say if he can keep that stuff in his head (a steel trap, apparently), why not? It seems to be the “same algorithm” according to Fusman and Beckmann. Then we both noticed that it would be most efficient in this case to move 2 from 127 to 398 and get 125 + 400. (His brother, a product of the same school system, did it right to left, complete with clear explanation.) I’m now remembering a mathematician at a meeting years ago who complained about a teacher who wouldn’t let his son use alternate but effective methods. Is that what “privileging” the standard algorithm means? (My colleague’s fear, not unfounded, was that the teacher couldn’t discern a valid alternative from an invalid one. That teacher, of course, would have been a product of a “traditional” math program.)
As Rob Root pointed out to me in an email, the “numerical reasoning” supported by the Investigations curricular materials includes the kind of reasoning you use in everyday life. To quote Rob (with his kind permission):
For the daily mental calculations I need to perform, I’m either estimating: my mileage when I fill up my car, what I can still eat in a day and stay on my diet, how fast I need to finish my bike ride to achieve a certain average speed; or I’m doing exact calculations, like totaling scores on an exam. Either way, I’m using context-specific algorithms. Sure, I occasionally find that I need to do a long division problem, and break out the standard algorithm, but for every problem like that, there are probably a dozen that I’ll do some other way. In other words, my experience is that the TERC Math Investigations approach fits my daily use far better than a “standard algorithm” focused curriculum would.
As someone trained in pure mathematics, let me hasten to add that of course I hope that all students get practice in the “distinctive kind of thought” that mathematics demands. This is why I keep quoting Uri Treisman: “the [Common Core] Practice Standards are exquisite.” I’m particularly fond of the “contextualize” and “decontextualize” language in standard #2 (Reason abstractly and quantitatively). The ability to “manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents” is a lovely thing in its own right, and speaks to efficiency and elegance. It’s even more lovely, though, when students are capable of “attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.” Of course students should learn standard algorithms; I don’t know of a single teacher who “avoids” them. The CCSSM, along with the teachers I know, want more: a robust enough understanding of the logic behind the algorithms to allow students to make wise choices about when and how to use them.
- Steve Wilson was part of the Mathematics Department at Johns Hopkins when I was a graduate student there, but I never had him for a course.
- I’m unclear on the etiquette of quoting oneself, but here goes: “A math specialist at an elementary school in a threadbare Vermont town once explained to me exactly how she and the other teachers supplemented their district’s curricular materials to make up for deficiencies. Never again will I assume that teachers slavishly follow whatever texts are on their shelves.” This is a small part of a longish piece meant for a more general audience.