Who Determines "THE" Standard Algorithm for Subtraction?

One popular complaint amongst anti-reform pundits is that so-called reform/"fuzzy" math advocates and the programs they create and/or teach from "hate" standard arithmetic algorithms and fail to teach them. While I have not found this to be the case in actual classrooms with real teachers using EVERYDAY MATHEMATICS, INVESTIGATIONS IN NUMBER DATA & SPACE, or MATH TRAILBLAZERS were being used (the so-called "standard" algorithms are ALWAYS taught, and frequently given pride of place by teachers regardless of the program being taught), the claim begs the question of how and why a given algorithm became "standard" as well as how being "standard" automatically means "superior" or "the only one students should have the opportunity to learn or use." It strikes me that it is almost as if such people are stuck in some pre-technological age in which we trained low-level white collar office workers to be scribes, number-crunchers who summed and re-summed large columns of figures by hand, etc. The absurdity of seeing kids today as needing to prepare for THAT sort of world is evident to anyone who spends any time in a modern office, including that of a small business. Desktop and handheld calculators are commonplace. So are desktop and laptop computers. There is a need for people to understand basic mathematics, but not to be fast and expert number-crunchers in that 19th century sense.



Thus, it seems reasonable to ask what should be an obvious question: if the goal is to know what numbers to crunch and how (what operations need be used) to crunch them, and, most importantly, to correctly interpret and make decisions based upon the results of the right calculations, and further if it is glaringly obvious that the actual number-crunching itself is done faster and more accurately by machines than by the vast, vast majority of humans can reasonably expect to do, why would any intelligent person be obsessing in 2008 over the SPEED of an algorithm for paper and pencil arithmetic? For the big argument raised for always (and exclusively) teaching one standard algorithm for each arithmetic operation seems to be speed and efficiency.



I have argued repeatedly that the efficiency issue is only reasonable if one fairly assesses it. And to do that is to grant that a student who misunderstands and botches ANY algorithm is unlikely to be performing "efficiently" with it. Compared with a student who uses even a ludicrously slow algorithm (e.g., repeated addition in place of any other approach to multiplication) accurately, the student who can't accurately make use of the fastest possible algorithm is going to be taking a long time to arrive at the right answer, which will be reached, if at all, only after many missteps and revisits to the same problem. For that student, at least, the "algorithm of choice" is not efficient at all. So finding one that the student understands and can use properly would by necessity be preferable. But not, apparently, in the mind of ideologues. For them, there's one true way to do each sort of calculation and they are its prophets.



Of course, I'm not favoring teaching alternate algorithms because I dislike any particular standard one or feel the need to "prove" that, say, lattice multiplication is "better" than the currently favored algorithm. On the contrary, I'm all for teaching the standard algorithm. But not alone and not mechanically, and not at the expense of student understanding. Indeed, from my perspective, it's difficult to understand why it is necessary to mount a defense for alternative algorithms in general, though any particular one may be of questionable value and might need some justifying or explaining. If anything, it is those who hold that there is a single best algorithm that is the only one that deserves to be taught who need to make the case for such a narrow position. In my reading, I've yet to encounter a convincing argument, and indeed most people who hold that viewpoint seem to think it's glaringly obvious that their anointed algorithms are both necessary and sufficient.



What compounds my outrage at the narrower viewpoint is the fact that it is based for the most part on woeful historical ignorance. Elsewhere, I've addressed the question of the lattice multiplication method, which has come under attack from various anti-reform groups and individuals almost certainly because it has been re-introduced in some progressive elementary programs such as Everyday Math and Investigations in Number, Data, and Space. The arguments raised against it are very much in keeping with above-mentioned concerns with speed and efficiency. Ostensibly, the algorithm is unwieldy for larger, multi-digit calculations. The fact is that it is just as easy to use (easier for those who prefer it and get it), and while it's possible to use a vast amount of space to write out a problem, it's not required that one do so and the amount of paper used is a social, not a pedagogical issue. But please note that I said RE-introduced, and that was not a slip. The fact is that this algorithm was widely used for hundreds of years with no ill effects. Issues that strictly had to do with the ease of printing it in books with relatively primitive technology and problems of readibility when the printing quality was poor, NOT concerns with the actual carrying out of the algorithm, caused it to fall into disuse. Not a pedagogical issue at all, and with modern printing methods, completely irrelevant from any perspective. Yet the anti-reformers howl bloody murder when they see this method being taught. The only believable explanation for their outrage is politics. They simply find it politically unacceptable to teach ANY alternatives to their approved "standard" methods. And their ignorance of the historical basis for lattice multiplication as well as their refusal to acknowledge that it is thoroughly and logically grounded in exactly the same processes that inform the current standard approach suggests that bias and politics, not logic, is their motivation.

Subtraction algorithms

I raise all these questions because I've recently had my attention drawn to a "non-standard" algorithm (actually two such algorithms and some related variations) for subtraction. Tad Watanabe, a professor of mathematics education whom I've known since the early 1990s both through the internet and from meeting him at many conferences, posted the following on MathTalk@yahoogroups.com, a mathematics education discussion list I've moderated for the past seven years:

Someone told me (while back) that the subtraction
algorithm sometimes called "equal addition algorithm"
was the commonly used algorithm in the US until about
50 years ago. Does anyone know if that is indeed the
case, and if so, about when we shifted to the current
conventional algorithm?

I couldn't recall having heard of this method, and so I was eager to find out what Tad was talking about. Searching the web, I discovered an article that repaired my ignorance on the algorithm: "Subtraction in the United States: An Historical Perspective," by Susan Ross and Mary Pratt-Cotter. (This 2000 appearance in THE MATHEMATICS EDUCATOR was a reprint of the article that had originally appeared several years previously in the same journal. It draws upon a host of historical sources, the earliest of which is from 1819. And there are other articles available on-line, including Marilyn N. Suydam's "Recent Research on Mathematics Instruction" in ERIC/SMEAC Mathematics Education Digest No. 2; and Peter McCarthy's "Investigating Teaching and Learning of Subtractions That Involves Renaming Using Base Complement Additions."

The Ross article makes clear that as far back as 1819, American textbooks taught the equal additions algorithm. To wit,

1. Place the less number under the greater, with
units under units, tens under tens, etc.
2. Begin at the right hand and take the lower figure
from the one above it and - set the difference
down.
3. If the figure in the lower line be greater than the
one above it, take the lower figure from 10 and
add the difference to the upper figure which sum
set down.
4. When the lower figure is taken from 10 there
must be one added to the next lower figure.

In fact, according to a 1938 article by J. T. Johnson, "The relative merits of three methods of subtraction: An experimental comparison of the decomposition method of subtraction with the equal additions method and the Austrian method," equal additions as a way to do subtraction goes back at least to the 15th and 16th centuries. And while this approach, which was taught on a wide-scale basis in the United States prior to the late 1930s, works from right to left, as do all the standard arithmetic algorithms currently in use EXCEPT notably for long division (which may in part help account for student difficulties for this operation far more serious and frequent that are those associated with the other three basic operations, it can be done just as handily from left to right.

Consider the example of finding the difference between 6354 and 2978. Using the standard approach, we write:

6354

-2978

------

and work as follows: 1) 8 is greater than 4 so we "borrow 1" from the 5 and then subtract 8 from 14 and get 6. We cross out the 5 and write 4 to account for the borrowing; 2) 7 is greater than 4 so we borrow 1 from the 3 and then subtract 7 from 14 and get 7. Again, we scratch out the 3 and write 2 to account for the borrowing; 3) 9 is greater than 2, so we borrow 1 from the 6, subtract 9 from 12 and write 3. Again, we scratch out the 6 and write 5 because of the borrowing; 4) finally we subtract 2 from 5 and get 3, leaving us with the answer 3, 376.

(Although it may not be obvious, we could do that subtraction from left to right using an approach similar to what I will show below for the equal additions algorithm).

Equal additions

The equal additions method works as follows for the same problem above: 1) "8 from 14 is 6; 2) 8 from 15 is 7; 3) 10 from 13 is 3; 4) 3 from 6 is 3"

That is to say, each time the digit in the subtrahend is greater than the digit in the same place in the minuend, 10 (or 100 or 1000, etc.) is added to the digit in the minuend and also to the digit in the next largest place in the subtrahend. However, for example, in step #1 above, adding 10 to the units digit in the minuend appears to be "compensated" for by adding only 1 to the 7 in the subtrahend. In reality, they both represent additions of 10 because of place value. The algorithm really does involve equal additions at each step as necessary. And of course, because adding equal quantities to the minuend and subtrahend does not change their difference, the resulting computation is correct.

Left to right subtraction?

Could the calculation be performed correctly operating from left to right? Consider the following approach: 1) beginning on the left, take 2 from 6 and write down 4; 2) since 9 is greater than 3, add "10" (really 1000) to the 3 and take 9 from 13 getting 4 which we write down. Take "1" (really 1000) from the previous 4; 3) since 7 is greater than 5, add "10" (really 100) to the 5 and subtract 7 from 15 and write down 8. Take "1" (really 100) from the previous 4; 4) since 8 is greater than 6, add 10 to the 4 and take 8 from 14 and write down 6. Take "1" (really 10) from the previous 8. The answer is 3,376.

Is this "better" than the standard ("compensation") algorithm? Is it worse? I only mention it because research has suggested that many young students, left completely to their own devices, are likely to develop similar left-to-right strategies, correct or flawed. It seems highly likely that this is a natural outgrowth of the fact that we read English from left to right, and we teach students to read numbers the same way. It seems almost bizarre, once one thinks about it, that the standard algorithms for addition, subtraction, and multiplication, as well as some alternative methods, insist upon working from right to left. I suspect, too, that this both causes some students a lot of difficulty (there is plenty of evidence of kids who have later problems TRYING to do arithmetic from left to right and getting into difficulty)

I will not discuss or describe in detail the Austrian algorithm other than to say that it doesn't feel "right" to me. That's not saying it's "wrong," but rather that I can't see it as one I would use. And here is one major difference between me and the reform-haters: that doesn't mean I wouldn't revisit it, wouldn't show it to teachers, and perhaps if I saw a particular student or class for whom it might prove helpful, I'd teach it. My "taste" isn't the issue, but rather keeping a large number of options available for my practice and for my students. I suppose that's just not very "efficient" of me.

Finally, it bears noting that there is reference in the above-mentioned articles to research on the use of these algorithms, and at least some reason to think that equal additions needs to be looked at again very seriously by mathematics teacher educators and K-5 teachers. If you read the historical treatment of subtraction algorithms in the US, you'll likely note how much chance and arbitrariness there can be in how one particular algorithm comes into fashion while others fall into disuse. I see no firm evidence for the "superiority" of the current most commonly-taught algorithm, and there is clearly a history of it's causing difficulties for particular students. Would the universe collapse if we were to teach both? Even more, would it collapse if we didn't rush to teach it right away, but rather, as has been proposed by more than a few researchers and theorists on early mathematics education, let students play and invent their own algorithms first, before trying to steer them toward one or another of our own? Sadly, the anti-reformers amongst us, the activist educational conservatives who are constantly trying to narrow rather than open up K-12 education, believe that there's always a best way to do everything. And not coincidentally, that way always turns out to be the one they learned as a child. That, more than anything, is why I think it reasonable to call the not-so-traditional math that they push on everyone "nostalgia math." It's not that what they learned is better. It's just what they learned back in simpler times when life was easy and there were no Math Wars and no one like me to suggest that their emperor is stark naked.

Quirky Investigations: More Nonsense From an Old Source (Part 2)

In the first part of this look at the questionable notions of William G. Quirk about the TERC elementary math curriculum, INVESTIGATIONS IN NUMBER, DATA, AND SPACE, I analyzed the interesting deceptions in the title of his recent propaganda effort, "2008 TERC Math vs. 2008 National Math Panel Recommendations." Now the time has come to look at some of the specific charges against INVESTIGATIONS leveled by Mr. Quirk and expose them for what they are.

First, Mr. Quirk asserts:

A major objective of elementary math education is to provide the foundations for algebra, the gateway to higher math education. Although we call it "elementary math," K-5 math content is quite sophisticated and not easy to master. But constructivist math educators believe that concrete methods, pictorial methods, and learning by playing games are the keys to a stress-free approach.

It can't be easy to squeeze so much inaccuracy into so small a space, but Mr. Quirk is adept at doing so. While it is arguable as to whether the correct focus for elementary math is or should be "the foundations for algebra," it's even more questionable as to what is meant by algebra. If Mr. Quirk's other writing is indicative (and I'm sure it is), algebra to him is just as mechanistic as is his take on all other school mathematics: a series of rules and definitions to be memorized to "mastery," such as: theirs not to reason why, just invert and multiply, etc.

On the other hand, we see in the writing and talks of Liping Ma the notion that for Chinese students, the goal of arithmetic is to have a profound understanding of arithmetic. Algebraic facility flows from them rather naturally, according to her, because students have a great familiarity with how numbers are "made": the idea of "composing" and "decomposing" numbers (here, I think is meant more than just factoring, but also what I've seen elsewhere called "number bonds" - the different ways to make a given number by adding two numbers (of course, we're mostly talking about positive integers here for young kids). In some curricula to which Mr. Quirk vehemently objects (as do some more progressive writers, such as Van de Walle), this is explored through fact families or similar approaches. I like the idea of exploring in a progressively focused manner, letting kids start to realize that a given number can be the sum of lots of pairs of whole numbers (and when negative numbers are introduced, this becomes an infinite number of pairs of integers).

I'm unconvinced that what is being done with INVESTIGATIONS (and other progressive programs) is drastically different from one or more of the above ideas. Van de Walle's notion of developmental arithmetic is a sound one that seems better reflected in INVESTIGATIONS and EVERYDAY MATHEMATICS than in the more rote-based approaches that Quirk and other anti-progressives advocate.

Math Wars and Literacy Wars: similar rhetoric, similar tactics

I suspect it's no coincidence that such folks are often found to have been in the forefront of scurrilous attacks on whole language. The tactics there have been very much like those in the Math Wars: claim that the reformers are "destroying" traditional educational methods which are alleged to have been effective in the past (the key word is, of course, alleged; there's no sound data that supports the idea that a higher percentage of kids were effectively taught math or literacy "back in the day" (and exactly when that day was depends directly on the age of the critic. Having been educated back in the '50s & '60s, I'm a bit less sanguine about any so-called golden age of phonics and times tables. I know too many people who didn't learn to read or write adequately (if at all) who went through the same sort of schooling I did. I taught far too many kids at the U of Florida c. 1975 who had high school diplomas from districts that weren't exactly on the cutting edge of reform literacy education. They couldn't write a meaningful sentence about their day at the zoo, let alone an actual college paper. A current article in the ATLANTIC MONTHLY, "In the Basement of the Ivory Tower" by 'Professor X,' discusses an adjunct English professor's concerns about the disservice he believes is being perpetrated on many adult and non-traditional students at various community colleges and four-year institutions, by allowing them to enroll in courses they are unqualified for and cannot reasonably hope to pass given the skills they enter them with. Having taught TRADITIONAL college students at the "flagship" public university of Florida in the '70's, I can attest to the fact that there's nothing special at work in what Professor X observes, criticizes, and bemoans. The only difference is that he's seeing it today with students he feels shouldn't be going to college. I saw it 30 years ago not only with undergraduate students in the most selective public university in one of our most populous states, but with fellow graduate students of English who couldn't write a passable undergraduate piece of literary analysis.

So is the sky really falling more today than it was in the '70s? If not (and I think it is not), when was the golden age of teaching literacy? The students I taught were not much younger than I was and finished high school during the "back to basics" era. My less competent colleagues in graduate school were my contemporaries. How did they manage to get into decent (if not great) colleges and graduate programs and still be embarrassingly unfit in their chosen field? Whole Language cannot be blamed for these failures or for the students in remedial English in the 1960s. We had phonics in my school district. We also had the famous "Dick and Jane" readers that were about as engaging as a poke in the eye with a sharp stick. Whole Language advocates called for the use of more relevant, more challenging literature as part of the basis for literacy education. They didn't call for the death of phonics instruction. Every elementary school teacher I've spoken to who supports Whole Language tells me that s/he teaches phonics as part of balanced instruction and always has. Where are the alleged extremists who aren't teaching phonics that we hear so much about?

Similarly, I have a hard time finding elementary math classrooms that don't teach math facts, that don't test kids under time pressure on addition, subtraction, multiplication, and division of single digit numbers starting by third grade if not sooner. Every district in Michigan with which I've worked in any capacity (and that's quite a few as either teacher, coach, university supervisor, university instructor, or consultant) does the same. While there is some variation in the approach individual teachers use, everyone stresses knowing basic math facts.

On my view, the problem cuts across curricula, both reform, traditional, and what-have-you. Unless there's a strong district commitment to the sort of developmental approach that Van de Walle and like-minded mathematics educators advocate and to employing and supporting elementary teachers and other staff who have all three components of effective teaching in place, there's going to be less than optimal mathematics teaching and learning in K-5. Once students are allowed to emerge from elementary school with significant holes in their mastery of basic arithmetic (I'll leave alone for now other things that students should be given an opportunity to learn and - dare I say it? - play with in mathematics while in K-5), algebra is going to prove far, far more difficult than should be the case and that is, reportedly precisely what many mainland Chinese students are able to avoid suffering through because of their facility with and intimate understanding of how arithmetic works.

Quirky Lies and Misdemeanors

Mr. Quirk asserts correctly that elementary math content is quite sophisticated. He also claims that it's not easy to master. There I think he errs in make an all-too-typical over-generalization. Is it difficult to master in all its particulars? What, exactly, comprises mastery in his view? And do all students suffer equally in trying to master it? One doesn't get the sense from Liping Ma that Chinese students are suffering as they learn. Constance Kamii, one of the mathematics education researchers much-despised by the anti-reform crowd, certainly makes it sound like students she studies enjoy playing with mathematical activities and ideas. Well, maybe that's the rub: the folks who seem to think there's more to math then computation are often the ones who think mathematics isn't torture and need not be taught as if it were. And then they are accused of a rafter of sins: dumbing down the curriculum, promoting "self-esteem" over real learning, making math into fun and games (instead of torment, I suppose), and so on. The more traditional view is what I refer to as the fraternity hazing perspective: math is hell and damn it, my (kids/students/successors) will suffer just as much as I did (if not more). No Pain, No Gain.

However, all that is the least of my concerns with Mr. Quirk. It's the next claim that wins the prize for setting off major alarms: "constructivist math educators believe that concrete methods, pictorial methods, and learning by playing games are the keys to a stress-free approach. " If you thought my analysis in the previous paragraph was speculative, Mr. Quirk here has removed all doubt. It's not, of course, that games are designed to remove stress: they're designed to enhance understanding AND in the case of two of the curricula he despises (EVERYDAY MATH and INVESTIGATIONS), to do the work of some of that much vaunted drill and rote learning. Of course, one would have to actually read the curricular materials on the games with an open mind to see that. It's so much easier to simply used the word "games" and link it with some sort of Dionysian hedonism.

On the other hand, why are folks like Mr. Quirk so enamored by stress, especially in the case of K-5 students? Captain Quirk is, like so many of his fellow traditionalists, a great lover of high stakes tests. And there is little that has been used to create MORE stress for young children and their teachers and parents than these tests since the passage of No Child Left Behind. I'm sure I'm not the first person to note that putting such emphasis on tests for young kids is a wonderfully effective way to destroy the pedagogical relationship between students, teachers, and administrators. One merely need spend a little time in most schools, especially inner-city and rural poor schools, in the month before and the month during a high-stakes test to see schooling at its worst, and kids at their most tense and miserable. And not just the kids. For a progressive, humanistic educator like me, nothing could be more demoralizing, upsetting, and depressing than to hear administrators tell principals and teachers that it's okay to lie to THIRD GRADERS about the impact of the local state assessment on the PERMANENT RECORDS of those kids. I am not making this up: I witnessed it in 2005 in an inner-city elementary school northwest of Detroit.

Quirk Conflates Apples and Oranges

Let's return to the substance, such as it is, of Quirk's complaints about INVESTIGATIONS. First, he offers what he calls the NMP's "2008 View of the Foundations of Algebra." However, what he quotes in fact is a definition of school algebra:

The report first defines 'school algebra' as the 'term chosen to encompass the full body of algebraic material that the Panel expects to be covered through high school, regardless of its organization into courses and levels.

Quirk then goes on to complain:

NMP carefully defined 'school algebra.' TERC counters with 'algebra is a multifaceted area of mathematics content.'

Yes? Is this supposed to be some sort of ruinous critique? Aside from the ploy of inventing a "TERC" that is trying to "counter" statements that were made by a panel of no clear authority to dictate national policy or content AFTER the publication of what TERC published. Nice try, Mr. Quirk, but this is deeply flawed.

More significantly, it is completely absurd to compare a definition of "school algebra" with a definition of "algebra." I don't wish to argue whether the NMP "carefully defined" school algebra, nor whether that definition is at all satisfactory. But it is clearly a definition of "school algebra." That is a far cry from what algebra actually is, as Mr. Quirk knows full well, being a Ph.D in mathematics. Is he really so naive as to believe that "algebra that 'should' be taught in schools" is the same as what algebra IS? Or is he simply being conveniently hazy on what's being discussed in the two quotations?

I believe it's glaringly obvious that the latter is the case. This is simply the classic rhetorical methods of the anti-reform crowd. They don't play fair, they don't worry about truth, they simply do whatever they think will win. And to "win" is to fool the public and anyone not paying close attention to the specifics of the debate. Shoddy, but so often effective.

One might ask Mr. Quirk whether he contends that algebra is NOT a multifaceted area of mathematics content. If so, to what, exactly, would he restrict it? And would his restrictions be acceptable to working mathematicians? If not, then really, of what use is it to compare apples to oranges in this context other than as a cheap method for attacking a program he hates on general principles (and I use that last word quite loosely).

What is "too much" emphasis on patterns, and who says?

Another "devastating" complaint from Mr. Quirk is that INVESTIGATIONS over-emphasizes patterns. His authority is none other that another Mathematically Correct hack, David Klein. According to Quirk, Klein states that "the attention given to patterns is excessive, sometimes destructive, and far out of balance with the actual importance of patterns in K-12 mathematics." One might well ask what comprises "excessive" emphasis on patterns, how exactly focusing on patterns could be "destructive" and in what ways, and who - other than himself and his like-minded reform opponents - legitimated Klein's claim to be an expert on mathematics teaching and learning. Similar questions apply to Quirk, of course. There is a fascinating and utterly false assumption on the part of far too many people that expertise in mathematics (to the extent that having a Ph.D in the subject makes one such an expert) automatically qualifies someone as an expert on the learning and teaching of the subject, and at any and all grade levels.

Those of us who have suffered through pedestrian mathematics teaching and worse on college campuses (which isn't to say that there aren't inspired mathematics professors who understand and care deeply about high-quality teaching), might wish to suggest that there are a lot more folks who know mathematics at a higher level who haven't a clue about how to teach it to anyone except perhaps another mathematician. The true horror, however, is when those who are at best indifferent classroom teachers for college mathematics presume that because K-12 mathematics is so "simple," their ability to breeze through such trivia makes them highly-qualified to speak about what those of us who actually teach and work in K-12 mathematics classrooms should be doing. Not surprisingly, many of such people strongly believe in one-size-fits-all instruction (what could be simpler) regardless of reality. They justify this, in many cases, with phony arguments about racism, equity, and the like. For the most part, they don't even care about these issues, but recognize the necessity of paying lip-service to them. Klein is a relative exception in the Mathematically Correct/NYC-HOLD camp in that he is officially a socialist. This seems to lead him (and his more conservative associates) to believe that these anti-reform organizations represent a "diverse" cross-section of the political arena. However, without wishing to go into an in-depth analysis of the modern American Left, I will simply assert that there are ways in which some strains of that Left are as regressive as many aspects of more conservative or reactionary political thinking, just as there are ostensible conservatives who can come up fairly progressive on some issues. Further, being left-wing in politics appears not to be a guarantee of being in favor of progressive education (and as Alfie Kohn has recently pointed out, progressive education in theory isn't always terribly progressive in any given instantiation of it, sad to say): not only is it hard to find truly progressive schools, it's well-nigh impossible to find politicians who favor progressive education or have a clue about any meaningful aspects of classroom life. Suffice it to say that David Klein is no progressive educator and that his views of K-12 mathematics teaching and learning could hardly be more conservative or wrong.

Returning to the question of patterns, we see well-respected mathematicians like Lynn Arthur Steen arguing that mathematics very much is about pattern.

The rapid growth of computing and applications has helped cross-fertilize the mathematical sciences, yielding an unprecedented abundance of new methods, theories, and models. Examples from statistical science, core mathematics, and applied mathematics illustrate these changes, which have both broadened and enriched the relation between mathematics and science. No longer just the study of number and space, mathematical science has become the science of patterns, with theory built on relations among patterns and on applications derived from the fit between pattern and observation.

I'm sure David Klein and William Quirk know better. Certainly better than mathematician Keith Devlin, author of MATHEMATICS: THE SCIENCE OF PATTERNS. Yes, it's just wrong of the authors of K-5 mathematics books to stress pattern. Where DO they get such wild ideas? Well, apparently from renowned mathematicians.

The Myth of "Standard" Arithmetic Algorithms

Much of the Quirk critique of programs like INVESTIGATIONS hinges upon a single crucial untruth: that there are somewhere enshrined a list of "standard" arithmetic algorithms from which one must never depart in K-5 education and beyond. This stance allows Quirk and like-minded people to bash reform-oriented programs without having to ever make a coherent argument as to why a particular algorithm is "bad," while another is inherently "good." It suffices from this perspective to argue from tradition regardless of the fact that so-called traditions may be relatively recent or limited to particular countries and cultures. France and countries like Haiti that were once French colonies, write long division in a way that would seem "upside-down" to those of us who learned the "standard" algorithm in the United States, yet the workings of the algorithms are identical. Would Quirk & Co. contend that the French are all confused, all "hostile," to use his term, to "standard language, standard formulas, and standard arithmetic"?

In fact, "standard language" is itself another red herring. Mathematically-knowledgeable people are well aware that it is hardly unusual to find multiple terms for the same mathematical concept, as well as multiple notations. This is true in calculus, in abstract algebra, and in other areas of mathematics. Not unlike the squabbles about Macs vs. PCs, the dispute between supporters of one set of symbols or terms may be more about which one the individual learned first than about truly substantive issues. Regardless, however, it is difficult to make the case that mathematics as it has evolved depends on one standard set of terms or symbols, however awkward that reality may be for folks like Quirk. He might wish that were the case, but mathematics appears to have thrived despite that fond hope of his.

It's unlikely that Quirk would have the temerity to suggest that Leibniz didn't know what he was talking about because he used a different notation for calculus than did Newton, or vice versa. Yet he has no compunction about trying to dismiss INVESTIGATIONS for similar "sins." He also fails to make any reasonable distinction among terms as to what might be vital to know and what might not, especially given that INVESTIGATIONS is aimed at K-5 students. Should they have to know "subtrahend," "minuend," and "difference," or does the latter suffice? Should their teachers know all three? Is it fatal for kids and/or teachers to use the term "borrowing" or "carrying"? Just how anal does one need to be about such matters?

It's hard not to wonder if Quirk and his friends understand the difference so perfectly highlighted by Richard Feynman between knowing the name of something and knowing something. If I have students who understand how to do division, what it is, and what the results of doing division mean, I'm not likely to lose sleep over whether they can correctly tell me what the dividend is. Nor do I care whether they use the terms "partitive" and "quotitive," as long as they understand that sharing and measuring are two important ways to think about division. (Of course, being terms from mathematics education, rather than from typical mathematics coursework, these words are likely of no value to Mr. Quirk whatsoever). What wasn't taught to him in his own K-5 classroom is "non-standard," of course.

When it comes to actual algorithms, Quirk, like many other reform critics, resorts to an appeal to "efficiency and speed." However, given the reality of modern computational devices, it's hard to imagine that either of these matters very much to the vast, vast majority of people. What does matter and should matter is understanding of both mathematical procedures and what results from following them, as well as which to use for a given situation. There is a kind of narrow thinking seen far too frequently among some K-12 teachers of mathematics, as well as some professional mathematicians, who most certainly should know better, that reduces school mathematics to "right procedures." It is hard to justify, however, a "My way or the highway!" approach to teaching mathematics to school children. This is not, as reform critics are sometimes quick to falsely suggest, an invitation to "anything goes." Obviously, some procedures are not justifiable. Some are inefficient to the point of being almost useless. However, there is a vast middle ground, and it is there that programs like INVESTIGATIONS invite students to explore, but which Quirk and others refuse to grant ANY legitmacy whatsoever. I have yet to see a single argument from him that holds water against letting students invent their own algorithms (many of which, if not all, are likely to be well-known, possibly still in use in some cultures, and in any event methods that the students UNDERSTAND and can explain). The red-herrings of "speed and efficiency" don't hold up when weighed against utter confusion and error on the part of students. If a student consistently errs with a "standard" procedure, how can it sanely be argued that this procedure is faster or more efficient? Clearly, for the given student, it's quite the opposite. And this point is hardly restricted to elementary arithmetic, of course.

To my thinking, some teachers refuse to consider alternative algorithms for one simple reason: their own grasp of mathematics is so tenuous that they are afraid to think outside their comfort zone. I doubt that is the issue for Quirk, however. He is chosing consciously to close off alternatives because he thinks he has a rhetorical club with which he can beat the authors of reform programs, regardless of how valid and sensible their methods are. In the mind of many US parents who fear and loathe math, they may not know how to DO arithmetic, but they know it when they see it, no matter how poorly they grasp its workings. And so if it was bad enough for them, that, and ONLY that, is bad enough for their kids. The last thing they want to see, sadly, is an approach that the kids might just be able to understand but which confuses the parents even further if and when they look at it. Little is more humliating than not being able to help one's kids with homework, but if the homework doesn't even look vaguely familiar, well, that's sufficient grounds to become roused rabble. And Quirk is just the sort of fellow to provide a battle-cry.

The Bottom Line

I don't argue that INVESTIGATIONS, EVERYDAY MATH, or any other elementary, middle school, secondary, or post-secondary textbook or series is a panacea. But I do not see firm or even plausible evidence from Quirk or his cronies that these books are not offering sound mathematical content, thinking, or problems on the whole. They are not flawless, of course, but neither are those alternatives Quirk and company recommend. And they offer many valuable things one would never find in, say, a textbook from Saxon Math, not the least of which would be challenges to develop mathematical habits of mind, problem solving skills, and an ability to think outside the box. While all of these are prerequisites for real mathematicians, Quirk and his allies continue to decry programs that promote such thinking, all the while falsely contending that their only concern is to see all kids learn authentic mathematics. Were that really the case, they would at the least speak honestly about the positive things these programs offer, not the least of which is the ability to reach many students who are NOT reached or engaged by more "traditional" texts and pedagogy. What lies at the heart of the Quirk/MC/HOLD cabal is a pot pouree of mendacity and misdirection. I have on more than one occasion in this blog and elsewhere suggested some of the motivation for such consistent dishonesty. But regardless the truth of my suspicions in that regard, what matters is that fair-minded people look past the shrill dismissals Quirk and others offer of a host of progressive reform texts, methods, and approaches. Mathematics education is too important to allow a few ideological liars, few of whom spend any time working with real kids in K-5 or even K-12 classrooms, to call the shots for everyone.

NCTM BLOWS THE BIG ONES: Technology Position Paper, MRP response (and so much more) Inadequate

What can you say about an eighty-eight year old organization that died? That it was arrogant and complacent? That it loved power, money, and influence (or at least the illusion of them) more than being effective, and blew it all anyway?

In March, NCTM released its latest position paper, "The Role of Technology in the Teaching and Learning of Mathematics." This paper also appears in its entirety in the May/June NCTM News Bulletin, along with a bland piece on the National Math Panel Report (NMP), and an equally bland first "President's Message" from newly-installed Henry S. Kepner, Jr. 

Put them all (and more) together and you may get a sense of why I am so disenchanted with NCTM as an effective organization we need to lead us towards meaningful change in mathematics education. Though it us the favorite target of the various anti-reform spokes-holes, nay-sayers, and prophets (profits?) of doom, both individuals and groups who decry every new idea about math content, pedagogy, curriculum, and especially technology, NCTM just doesn't get the job done. When I contemplate the waste of time, energy, and resources that has occurred during just the last 15 or so years of the MATH WARS, I'm increasingly tempted to state: "A plague on both your houses." Stick a fork in these turkeys: they're done. And yes, I mean NCTM at least as much as its critics.

Let 'em dangle

Despite my reputation in some quarters as an apologist for NCTM, I have long been privately and publicly critical of the organization (especially its for the most part shamefully conservative board). Some of my strongest criticisms have centered around its continued failure to support districts, schools, administrators, teachers, parents, and students where the very kinds of reform NCTM started calling for in 1989 were implemented and then came under scurrilous attacks from the educational right wing. Rather than taking both defensive and proactive measures to support districts that had the courage to try to reform their mathematics educational practices, NCTM first stood around like a bunch of effete prep school boys who were shocked to discover that their critics weren't going to play by gentlemen's rules. 

The first shots were fired by the late John Saxon, who bitch-slapped NCTM repeatedly in combative advertising "position papers," in which he claimed to have the one true implementation of the 1989 Standards; he then began trashing that same series of documents in no uncertain terms. The fun part was seeing these ads appear in NCTM publications for teachers, who often don't know what all the philosophical and political shouting is about, but do tend to like books that bring up test scores, don't require that they learn new ways of thinking about and teaching mathematics, and which make them look good to administrators and parents without having to work too hard. If Saxon had the goods (and from a certain perspective, he did), and made a lot of noise to get everyone's attention, he was sure to gain a hard-core following among the rank-and-file in the classroom. And of course, he did so, quite profitably.

"Non-Political" is political

How did NCTM, ostensibly the most powerful and influential organization of mathematics teachers in the Americas, allow a poseur like Saxon to repeatedly steal its thunder? And more importantly, why? Purportedly, NCTM is "apolitical." But as most people whose heads aren't stuck firmly in the sand (or somewhere else dark and warm) are aware, there's nothing that isn't political to some extent. Every non-decision is in fact a decision. Every passive response to attacks from the educational right is a choice that is complicit with their demands. And NCTM, which let itself be bullied by an educational "maverick" like John Saxon (the John McCain of mathematics education, perhaps?) acted the 98-lb weakling when confronted with tiny but openly political groups like Mathematically Correct, NYC-HOLD, and their many smaller versions that arose (and continue to pop up) locally throughout the country. Had NCTM decided to put its (or should I, as a paying member since 1992, say, MY and OUR) money where its mouth was, I think it would have been trivially easy for it to find a principled way to support those who had the vision (or gullibility) to take the 1989 Standards seriously by using innovative curricula and reform pedagogy. Instead, despite repeated warnings from myself and others going back to the mid-1990s that these anti-reform groups were deathly serious, well-funded, well-organized, and playing for keeps, NCTM stood by twiddling its thumbs (well, maybe those thumbs were elsewhere) as district after district was picked off by "parents-with-pitchfork" groups, backed by MC & NYC-HOLD, as well as by folks affiliated with right-wing think tanks and foundations such as the Hoover Institution. Visionary districts were forced to offer "choice," which set the machinery in motion for the progressive programs to be marginalized and slowly suffocated, or simply to do away completely with programs like EVERYDAY MATH, CONNECTED MATH, CORE-PLUS, and many others. Even the rare district that made long-term commitments to these programs received nothing from the very organization that had supposedly called for just such things to be the wave of the future for mathematics education in the US in the way of pro-active strategies on how to implement and support reform methods, materials, and tools effectively. 

Why was NCTM so unwilling to offer any sort of direct support? Simple. The board constantly asserted its "apolitical" nature. It would not let NCTM be perceived as endorsing (or proscribing) any curriculum. That would, after all, be (horrors!) commercialism! Of course, concerns about commercialism never stopped NCTM from selling its own materials, generally at prices well out of reach of the average teacher. It never stopped it from demanding that speakers at its national and regional conferences pay full or at best mildly reduced registration fees. Or raising rates for membership every year. Talk to anyone at the management level and one hears nothing but pleas of poverty. Yet the group never listened to calls from within to reduce the $100,000+ cost of buses to take perfectly healthy members a block or two from hotels to conference venues, outlandish wastes of member dollars that could have been drastically cut by limiting such service to the infirm and elderly. While I don't want to suggest that anyone was living in the lap of luxury feeding at the trough of NCTM membership dollars (though I frankly won't rule out the likelihood of waste to make sure that the board and executive members never got less than the best), I'd say it's remarkable that NCTM has been taking a public relations beating for well over a decade from small groups who at least in theory do not have the financial resources of an organization with a huge membership. If what we are supposed to be getting with the dollars saved by not "going political" is high-quality PR, I'd say we've been collectively screwed. 

Curriculum Focal Points fiasco

A glaring example of the above is the execrable handling of the roll-out of the NCTM Curriculum Focal Points in the fall of 2006. It borders on the incomprehensible that this document could have received such completely inaccurate coverage not only by THE WALL STREET JOURNAL, but by major newspapers throughout the country. If there was a major media story about this important document that got things right, I missed it in the wave of "me, too!" coverage that followed the WSJ lead, from the NEW YORK TIMES right down to the more predictably right-wing outlets. What message was given over and over to the public? NCTM has retreated on its basic position from 1989 to 2005 and gone "back to basics." Of course, nothing could have been further from the truth, but no amount of counter-claims from then president Skip Fennell or many other folks who were involved with the authorship or supported the basic purpose of the CFPs could undo the damage. 

Of course, perhaps I am giving NCTM too much credit here in believing that it really didn't intend to back away from the Standards volumes. My sense is that the CFPs were a kind of preemptive strike meant to stave off the impact of what leadership correctly anticipated would be contained in the NMP's report(s). I don't mean to suggest that the authors didn't believe that the fundamental ideas in the Standards were well-founded. And I had personally help develop a similar "bare-bones" content guide for some Michigan elementary school teachers I was coaching who were trying to figure out what absolutely needed to be covered by the end of the 2004-5 school year given the state math framework, curriculum focal points, and the anticipated content of the following October's high stakes tests. The curriculum wasn't being changed (this was the first year the district was using EVERYDAY MATHEMATICS), but the game plan needed to be refined under time constraints). 

Given that a "back to basics" spin was very likely to emerge from the NMP, was NCTM simply trying to head 'em off at the pass? That's the story I've heard and maybe it's even completely true. But by not keeping the tight control on how the CFP was released, NCTM made an enormous miscalculation.

This wasn't just a case of Mathematically Correct & Company spreading a few silly lies: this was every paper that carried the story getting it wrong. How could that possibly have been allowed to happen? Why is that false viewpoint still the one most people believe and will continue to believe? At best, it's due to simple incompetence on the part of NCTM's management and public relations personnel. At worst, it reflects the inertia and conservatism that lies at the heart of NCTM and its board. Again, despite the claim by its enemies that NCTM is too radical for what American mathematics education needs, the truth is far more likely that it is too conservative, too cautious, and too incompetent. It either accidentally or purposefully shot its own foot off with the CFP release, and there's no sign of correction on the horizon. 

Wimpy non-response to NMP report

Another abject failure on the part of NCTM's leadership is its tepid response to the recent release of the report by the National Mathematics Panel (NMP). This clearly politically-skewed group essentially has attempted to roll back the clock to the 1950s or perhaps the last big period of "back-to-basics," the 1970s. With its emphasis on skills and a narrow view of what school mathematics should be (basically NOT what real mathematics IS), the NMP has clearly intended to codify and enshrine a national mathematics policy that is completely in synch with the phony interpretation of NCTM's CFP. And this is not a coincidence. The same forces that tried to attack every reform effort over the past 15 years were well-represented on the NMP, and they were the same voices that were quick to spread and second the WSJ lies about the intent and content of the CFP. 

Yes, you really can fool some of the people all of the time. But should that include the very organization that purports to lead America towards a more enlightened approach to math teaching and learning? Wouldn't you expect a fiery response to the NMP's attempt to turn back the clock and do in the NCTM Standards (1989 through 2000)? Here is what we were given instead:

NCTM appreciates the focus the work of the National Math Panel has brought to mathematics education. In addition to these six Principal Messages, the Panel’s final report includes numerous findings and more than 40 recommendations. More extensive reports of the panel’s task groups on Conceptual Knowledge and Skills; Learning Processes; Instructional Practices; Teachers; and Assessment are also available online.

The report highlights the differences between the American public’s view of the importance and utility of learning mathematics and those of other countries. Ensuring America’s stature as a world leader hinges on its ability to educate all students as we move into an era with increased technical demands.

The Panel’s recommendations are a next step in preparing America’s students not only to excel in mathematics, but to become leaders in the fields of science, engineering, and research. NCTM urges the administration to provide the funding to put the Panel’s recommendations into action and to identify and develop expanded, rigorous research needed to guide future actions.

I'm not sure what to say in response to this remarkably passive reply except to ask what brand of lubricant NCTM's leadership asked for prior to publishing that surrender. I'm struck in particular by the sell-out to the right-wing notion that what mathematics education in this country is about is "ensuring America's stature as a world leader." Really? I always thought it was about educating students to give them the ability to achieve life, liberty, and the pursuit of happiness. Could we see a more naked expression of NCTM's willingness to play along to get along? To kowtow to capitalist, anti-democratic values? To swallow whole a hegemonic view of the role of education in the United States? I know I didn't go into mathematics education, commit to teaching at-risk students, and deepen individual and collective teacher understanding of mathematics and its pedagogy, in order to serve the cause of keeping America ahead in some idiotic, jingoistic competition with people and governments and corporations from other countries. That wasn't what I signed on for when I decided to become any sort of teacher 35 years or so ago. 

Does this mean I don't want to see Americans succeed? Of course not. On the contrary, I'd like to see us succeed individually and as a people. As long as its part of a HUMAN striving for improving the lives of everyone on the planet, rather than some absurd nationalistic xenophobia that informs the anti-NCTM, anti-progressive, anti-reform rhetoric that spews forth from Mathematically Correct, et al. These folks have been trying to scare us with bogie men from China, Singapore, India, Japan, and other "inscrutable" Asian countries that are ostensibly beating our tails through superior mathematics education. Maybe we'll be hearing the likes of Wayne Bishop and other MC ideologues joining with Lou Dobbs to call for an "intellectual" wall to be built around our borders to keep out the invaders. 

Does any of this sound at all familiar? It should if you recall not only the tenor of the post-Sputnik panic that informed the New Math fiasco, but a host of similar documents that appeared from various conservative reform groups going back at least to the early decades of the last century. But when one wades through the naked classism of many of those documents, it is clear that the goal wasn't anything vaguely akin to "math for all." It was always about real mathematics for the few, the elect, the elite. For everyone else, maybe how to use a checkbook. If they were lucky. To steal from CADDY SHACK, "The world needs ditch-diggers, too, you know."

Let's not forget, too, that a mathematically literate populace is more difficult to mislead. Its one that is capable of reading and understanding statistics, of making informed judgments about huge public expenditures (and wastes), of the probability that some idiotic claim is true (be it from corporations, scam artists, or governments). It's one that can weigh more intelligently the real costs of "holy wars" in Iraq and Iran, the plausibility of global warming, the reasonableness of pseudo-scientific claims about medicine, diet, and much, much more. Is that what conservatives generally fight for? Is it what the National Math Panel is calling for? Or is it just business as usual? And is the above the best response we should expect from NCTM's leadership?

Technology is a lot more that calculators and calculating

Finally, I want to address NTCM's most recent failure: its position paper on the role of technology in mathematics education. For quite some time, I've been aware of what might be fairly called "criticism from the left" of NCTM's apparent focus on calculators as the main issue when it comes to technology and mathematics teaching and learning. At times, for a group that is "apolitical," NCTM seems overly-cosy with Texas Instruments, the company that has managed to dominate quite thoroughly the use of calculators and related software and hardware that is supported in reform and many traditional textbooks from elementary school through calculus. It's just about impossible to find evidence that any company except for TI even makes calculators for K-16 mathematics. Of course, we all are impacted by technological competition (competing operating systems in computing make a lot of software available only to those who use a given platform; most of us recall the VHS-Beta war, and we've recently seen Blu-Ray win the high-capacity DVD war). Some of us are aware of what's been lost because TI took the easier route of eschewing reverse Polish notation. Perhaps others recognize the advantages color graphing gives students, an existing technology long offered by Casio,  one of the lesser players in the US calculator market, but which the giant, TI, has eschewed, to the detriment of countless students. One might argue that if color were so vital, Casio would have a  bigger market share, but that's assuming level playing fields. In education, once one company dominates a market, it's nearly impossible for competitors to draw even: everyone wants to be using what is already the norm, especially at the high school level, where college use has a definite impact on what teachers are prone to employ, especially in higher-level courses. TI rules college campuses and is likely to corner this market much as Microsoft has with Windows.

Which takes us to a much more serious issue: while NCTM pays lip-service to technology other than calculators, it's only in terms of specific software such as dynamic geometry and statistics packages. And it is here, in part, where NCTM most horribly fails to take advantage of something truly innovative and important when it comes to American mathematics education. As Kirby Urner, among others, has been pointing out for at least a decade, calculators are far too limited to be what is needed to help push students forward in mathematics. I don't fully concur with him about their limitations, but I agree that the way they're being implemented, even programmable graphing calculators are not being used nearly as effectively as they could be. In many ways, they're already an "old" technology, but they have a lot more power than most teachers or students have any idea of. Urner is only one fellow touting a mathematics curriculum grounded in programming (particularly with the language Python), but with a major focus on both applied and abstract mathematics - from sphere packing to polyhedra to abstract algebras to the work of Buckminster Fuller, and much more. 

Another group that has been looking at computer science mathematics, discrete math, and other powerful, modern mathematical ideas is the group in Australia and New Zealand that produced COMPUTER SCIENCE UNPLUGGED. It's a wonderful collection of materials that can be used even with lower elementary students, yet gets into important mathematical ideas that will engage high school students. As I've been (all-too-slowly) documenting elsewhere in this blog, I'm mentoring a colleague in Saginaw who is getting inspiring results using these materials with very low-achieving, at-risk students in grades 7 to 10 in an alternative school located in Saginaw, MI. For those who don't know this economically-ravaged city, suffice it to say that the program where she teaches is in the worst part of town and that Saginaw is considered by many folks here to make Flint, Pontiac, and even Detroit look idyllic.  Yet she's got kids who never previously earned a credit in mathematics really engaged with a host of important and challenging mathematical ideas. As the title suggests, no computers are required (Kirby Urner finds this ominous, but I think he's wrong in that regard, and given the situation in which my colleague teaches, she has no option to use computers right now). But perhaps the most amazing part of CSU is that it is now entirely free: every module is available as a pdf that can be downloaded at no charge from the above-linked web site.

Michael Fellows, one of the developers of CSU, has another more broad-based curriculum package, also available for free download (THIS IS MEGAMATHEMATICS: The Los Alamos Workbook) that covers such areas as graph coloring, transfinite arithmetic, knot theory, etc. My colleague and I are looking into working some of the modules from that material into what she's offering her students. 

The above is, of course, just the tip of the internet iceberg when it comes to powerful use of not only technology hardware, but the underlying mathematics of computer science, discrete mathematics, etc. NCTM and many states continue to pay scant attention, if any, to the importance of discrete, finite, and related mathematics (one notable exception is New Jersey, where the contributions of Joe Rosenstein, Fred Roberts, and others affiliated with DIMACS at Rutgers University have gotten discrete math firmly embedded in the state standards). My own state, Michigan, makes some small reference to discrete mathematics in its curriculum framework, then fails to have a single mention of it in the Grade Level Content Expectations (GLCEs), the document that most teachers and districts use as a guide to what will actually be tested by the state. Guess what doesn't get taught?

So where is NCTM in all of this? Where is any mention of computer science, discrete mathematics, and a host of alternative branches on the tree of mathematics   (kudos to Dan Kennedy) that might make both a motivational and meaningful career-path difference to countless students who have not been reached by the same old math content and teaching on the road to Holy Calculus Mountain? Yes, discrete math gets mentioned in various Standards volumes, but it's really the red-headed step-child of traditional curricula, lucky to get a mention, rarely taken seriously. I would suggest that a real commitment to technology as a partner in mathematics education demands a very close look at discrete and computer science mathematics as one step on students' mathematical journeys. NCTM could take a meaningful leadership role in this regard by being more critical of the "calculus uber alles" mentality that we've been sold for over a century by various folks. But once again, NCTM fails to rise to an important occasion: another opportunity to do something radical yet utterly sensible goes by the boards (and, of course The Board). 

I don't claim to be an expert on computer science, discrete mathematics, or much else. I've got more questions than answers, and I'm well aware that even the most successful things I've seen relating to computer science, discrete math, and other topics mentioned above, including my own practice and that of those I've been privileged to observe and, on occasion, influence in small ways, have limitations. Something that seems great one day may go down poorly the next. No content, method, tool, or model is flawless and sure-fire. But one thing I do know for certain: some people can and do make a difference. And many of the most exciting, innovative, inspiring people out there don't seem to be making much of an impact on NCTM or in some cases bothering to try. Kirby Urner has written NCTM off as stuck in the past, and from where he sits, I'm sure he's right. The problem is that from where *I* sit, I'm increasingly convinced that he's right, and I've been in NCTM for 16 years. As I said back at the beginning, NCTM looks more and more like the proverbial dead shark. And quite a toothless one at that. 

In this election year in which the question of what comprises authentic change is central, I feel compelled to ask the same thing about mathematics education. Has the time come for those of us who have had our fill of frustration and failure, of timorousness, of the politics of "non-politics," of conservatism and inertia, and of the avoidance of risk-taking, vision, and commitment to making a difference for those who need meaningful alternatives, to leave NCTM behind? I'm sure this will annoy both friends and critics alike, but perhaps it will move them to to think, and to do something without waiting for the NCTM board's imprimatur. Our kids have waited far too long already.

Quirky Investigations: More Nonsense From an Old Source (Part 1)

In the Math Wars, when it comes to hating progressive reform there are pack animals and solitary beasts. There are the hyenas of Mathematically Correct and NYC-HOLD, and then there is the lone, if rabid, wolf, William G. Quirk. Our Bill has long been a voice of prejudice and extremism in the face of efforts to improve the quality of mathematics teaching and learning in the USA. His latest screed comes predictably on the heels of the all-but-useless political tract spewed forth by the National Math Panel: our Willie's (ahem) contribution to the Math Wars is "2008 TERC Math vs. 2008 National Math Panel Recommendations."

This is a political tract so execrable that it deserves to be taken apart piece by piece and exposed for the ugly propaganda it is. Let's start with the title. There are three fundamental bits of idiocy in it. First, there is no such thing as "TERC Math." TERC is not a textbook or a series of textbooks. It's "a non-profit research and development organization whose mission is to improve mathematics, science, and technology teaching and learning. TERC, founded in 1965, is located in Cambridge, Massachusetts. TERC staff includes researchers, scientists, and mathematicians, and curriculum and professional development specialists who ground their work on inquiry-based approaches that deepen all learners’ understandings." One thing they have done is to produce a set of texts for elementary mathematics called INVESTIGATIONS IN NUMBER, DATA, AND SPACE. This is a widely-used "reform" curriculum that, along with EVERYDAY MATHEMATICS, a K-6 program created by the University of Chicago School Mathematics Program (UCSMP) and a few others, has been both praised and attacked over the past decade and a half as part of the Math Wars.

However, virtually any time INVESTIGATIONS is mentioned by a vehement anti-reformer, "TERC" is substituted for the actual name of the program, as if somehow the authors and publishers hadn't given it an actual name. Of course, there are at least two factors at work here, on my view. The more obvious one is sheer laziness. It's so much faster and easier to type "TERC," after all. And maybe some of these critics aren't able to spell "Investigations." But the more subtle effect, one that may be unconscious but which is consistent with people who call any reform math program, method, text, author, or advocate "fuzzy," and a host of similarly prejudicial epithets (and yes, I'm well aware that I return their fire in kind. However, I didn't start the mud-slinging, cheap name-calling, etc. The Mathematically Correct page that lists a host of such names was up before I'd ever heard of them. You can't make this stuff up), is that in the ear of the average parent, this math program sounds like "Turk Math." Not that any political conservative would want to trade on American fear and suspicion of Muslims, of course.

The second lovely bit of sophistry in Mr. Quirk's title is the use of "vs.": the absurd suggestion is that TERC created INVESTIGATIONS to somehow oppose the National Math Panel (the PRESIDENTIAL Math Panel!!! What sorts of Communists are we dealing with here?) when in fact INVESTIGATIONS began in 1990, when the current fraud's FATHER was in office. As we shall see, Quirk repeatedly attempts to suggest in his article that somehow the NMP report came out and TERC replied by creating INVESTIGATIONS as a counter-offer. Yes, that's ridiculous, but lies, absurdity, and mendacity are the order of the day when anti-reform attack dogs like Quirk are let loose to try to terrorize parents, politicians, and school officials.

Finally, there is an implicit assumption in both the title of Quirk's essay and throughout its body that the NMP has widely-accepted legitimacy and that any sane person in the field of mathematics would agree with its report unquestioningly. Nothing could be further from the truth, of course. The NMP was appointed by a Republican administration that has systematically eschewed reality and science for politics and cronyism. This is an administration that backed READING FIRST, a horrifically expensive and useless reading program that has recently been shown to have no impact on improving reading, at a cost of billions of taxpayer dollars. Don't fret, of course: as we saw in the Hurricane Katrina crisis, GWB & friends are doing a heck of a job. And let's not mention the $3 trillion dollars it is expected that Iraq will have cost us, according to a Nobel Prize-winning economist. So you can be sure that they packed the NMP with hacks who by and large had the required viewpoint on mathematics teaching.

Could there already be some math textbook projects in the pipeline that are aligned with the Panel's say-nothing recommendations, backed by the Bushies and created by those with hands deep in government pockets? Nah! Now THAT'S a bunch of progressive paranoia. Just look at the track-record of READING FIRST: everything strictly above-board. No child's parent's wallet left behind. It's for the KIDS!

Suffice it to say that there is reason to question the idea that the NMP represents the best available thinking and research about mathematics teaching and learning, this despite the presence on the panel of Deborah L. Ball, current dean of the University of Michigan School of Education, a widely respected researcher, former elementary school mathematics teacher, and mathematics teacher educator extraordinaire. Also on the panel was Liping Ma, she of KNOWING AND TEACHING ELEMENTARY MATHEMATICS fame (and deservedly so). And the panel also had Francis M. "Skip" Fennell, then president of the National Council of Teachers of Mathematics (NCTM). With three such respected mathematics educators on board, what could go wrong with the NMP? The sad answer is: everything. But that has been discussed and documented elsewhere.

So much for Mr. Quirk's quirky and deceptive title. In Part 2, I will examine some of his "substantive" complaints about INVESTIGATION's alleged failures to anticipate and meet a document that was published only a few months ago. It goes without saying that no currently published curriculum at any grade level could honestly claim to have anticipated, adjusted to, or met the demands of the NMP report, even if the authors and publishers believed that the report and panel had the legitimacy and authority to demand ANYTHING. But of course, the panel, like NCTM in its various standards volumes published since 1989, does not demand. It recommends. Apparently, having been among those who was unable to make that distinction when viciously criticizing those NCTM publications, Mr. Quirk has lost any sense of what "recommends" means. Were that the worst of the flaws in his "analysis," there would be little reason to analyze his piece. But of course, he has much, much worse to offer. And whether he believes the lies and inaccuracies he serves up is far less important than whether others do.

Shaved Decks, Loaded Dice, Cognitive Psych, and "Concrete Instantiations"

After my previous post about a recent article in the NEW YORK TIMES about some "earth-shaking" cognitive science research, I was reprimanded on math-teach@mathforum.org by Mark Roberts for not having the decency to read any of the original research. While I don't feel too guilty about my critique of Kenneth Chang's credulous reportage (I would think a science writer would by nature want to be skeptical, but my take on Chang's coverage is that he was not only anything but skeptical, but also inane in his including a ridiculous and irrelevant problem about trains, as if they had ANYTHING to do with the research. It's one thing to swallow the conclusions without questioning in the slightest the experiment itself or the implications being made (well beyond the parameters of the data), but quite another to try to sex up the story by including a tired old example from grandpa's algebra nightmares and pretending it's relevant to the issues. I wonder if Mr. Chang would permit anyone to draw a diagram?

Okay, so I don't nominate the coverage for a Pulitzer. But was the research itself really so bad? Mark Roberts, despite insinuating I'd not been fair and in fact had lacked decency in what I wrote, did provide a useful link to a follow-up article by the same research team, available free on-line, entitled "Do Children Need Concrete Instantiations to Learn an Abstract Concept?"

I have to say, the title seemed ominous. NEED? Who'd ever said anything about concrete instantiations (and what a clunky phrase, if ever there were one, likely intended like so much social science prose to obscure more than to enlighten) being NEEDED? The question is whether they're useful. 

Meanwhile, Mark commented:

This research is not about learning the rules of a
 silly game as you seem to think it is. Both the sixth
 graders and the college students managed to learn the
 rules of the silly game in either the `concrete
 situation' or the `generic situation' (the very few
 who did not were excluded from further research). The
 research is about TRANSFER. The question is whether
 the students can use what they learned in a novel but
 very similar situation. In this piece of research the
 students who learned in the `generic situation' did
 this much better than the ones who learned in the
 `concrete situation'.

I'm skeptical about the concept of transfer of learning between domains, but less so about transfer within the same subject. So the idea that learning a problem solving method would pay dividends when attacking similar problems AND unrelated problems that might still yield to the method (I'm think in broad, Polya-like categories of strategies, rather than very specific techniques like how to factor a quadratic by grouping).

One key question I have about this research is whether there's good reason to suspect that the students who learned with what the authors call "generic" methods, i.e., purely symbolic ones, were advantaged given the tasks. On my view, they were because the so-called "concrete" instantiation quite likely adds both an unnecessary layer of complexity to the learning task, and takes time to apply to the new situation. There's a lot of processing that has to go on, and I am not surprised that the group with the supposedly concrete examples which, if I'm reading the research correctly is not at all concrete, but rather consists of some pretty rudimentary symbols meant to look like cups that are either 1/3, 2/3 or 3/3 full, performed indifferently. What ever happened to empty cups? Wouldn't that be an important question kids would be thinking about? Is an empty cup the same as a full one? How can that be possible? We're talking about youngsters with little exposure, if any, to abstract algebra. If I wanted to confuse young kids, I can't think of much better ways to do it than with this particular approach.

On the other hand, learning rules about a small additive structure, even if you don't recognize it for what it is (in this case, addition modulo 3), is pretty simple, and well within the capacity of many sixth graders, especially if they've been exposed to "clock arithmetic" or any similar concepts and/or problems. I wonder in particular why the researchers chose pictures rather than concrete objects to make their point about, well, concrete objects (and I really believe they had it in mind to MAKE the point, not to find something out that led them, inevitably, to the only possible conclusion, in spite of their utter objectivity prior to the experiment) that such objects may interfere with the transfer of learning math concepts. And further, I wonder why they chose to look at a bit of mathematics that can be concretely represented in ways that seem more related than their cup illustrations communicate (to me, at least). I said it before I read this follow-up and I say it now that I have: the game seems rigged to get the desired result.

That aside, they just about completely give their intentions and biases away with the following:

"Is it possible that concreteness is helpful but only for younger participants who CANNOT acquire an abstract concept otherwise? In particular, children may need a concrete instantiation to begin to grasp an abstract concept. This argument finds support in constructivist theories of development (e.g., Inhelder & Piaget, 1958) that posit that development proceeds from the concrete to the abstract and therefore learning SHOULD do the same." [my emphasis]

There are several problems here. First, I don't hold that we can know in advance that any given child CANNOT acquire any concept without a particular method being applied. I'm not a mind-reader or a fortune teller. I can't make such predictions about my own learning, let alone about that of someone else. The purpose of using concrete models - pictorial, palpable, narrative, electronic, or otherwise - is to help ground understanding of mathematical ideas, rules, and procedures. It's a fait accompli that kids can learn division without a clue as to what they're doing, why it works, or what the results mean. But if one's goal is to impart more than rote knowledge and procedural facility, simply teaching the rules and having students practice them MAY not suffice for any deeper understanding to develop. And so we use models and tools of varying kinds to help kids (and older students) who may be stuck in grasping what's going on and why.

The researchers seem fixated on the notion that these concrete methods will always be where we begin when teaching math to kids or anyone else. And that sort of rigid teaching is foolish. But the goal of the study appears to be to "prove" that since we only have one choice - to either start with the "generic" abstraction or to start with a concrete instantiation. And then we should CLEARLY pick the former, given the results of this research. My sense is that having created a false dichotomy, these researchers are committed to saving us from the evils and shortcomings they believe are perhaps inherent in "concrete" pedagogy. Have they thus provided real evidence, or merely the results of a rigged game?

In fairness, those who think this research is pristine and conclusive need to consider if they'd accept unquestioningly the contrary results from a study which appeared to be slanting matters in advance to give a desired outcome. I agree fully with Mark's comments below. We really do need to see if this study leads us somewhere useful when we look at results from K-5 kids and if the research results are replicable, but I wonder if the research as constructed is actually worth redoing. I think more thought needs to go into the selection of the mathematical task and the "concrete instantiation." Perhaps results will emerge that show that, not unlike light, which exhibits different properties of particle or wave depending on how one tests it, the suitability of particular methods for grounding or illuminating abstract mathematical ideas will vary depending both upon the ideas and the particular methods. I would suspect that further, the individual learner (not to mention teacher) will prove to be an important variable. If I'm in any way representative of people trying to learn something new in mathematics, different approaches may work better depending on a host of issues.

But there's more to consider. One huge question this study fails to address is that of the INSTRUCTION. Who, exactly, constructed the teaching in this little game? Were learners getting excellent, interactive teaching about how to make connections between the concrete objects and the underlying mathematical structure? Or were the manipulatives (okay, the PICTURES of the weird little symbols) supposed to MAGICALLY convey understanding to students? Having apparently not done a bang up job, the concrete objects are blamed for latter confusion. How pat. It's fascinating that the authors reference one of the most important articles on this whole issue, Deborah Ball's "Magical hopes: manipulatives and the reform of mathematics education," and yet manage to COMPLETELY miss her point: the mathematics isn't in the objects, it isn't in the model, and it isn't in the metaphors. Nor is the teaching or the learning. That exists only in the interplay amongst the mathematical concepts, the teacher, and the learner, with the objects or model merely providing tools. No one, least of all Deborah Ball, who really understands teaching mathematics, would ever believe otherwise.

Perhaps most importantly, why is the goal of the researchers to show that we should either always or never use concrete methods (or the "generic," for that matter)? Who is putting guns to our heads and insisting upon all concrete methods, all the time? Of course, it's not hard to find teachers who use all "generic" abstraction, all of the time in math classrooms at various grade levels, or who lean very heavily in that direction. I'm sure that such teachers would be outraged if forced to abandon symbolic methods entirely, or only to be allowed to introduce them after working through concrete ones. And the reverse would be true for those who are inclined to use the concrete or other "not generic" approaches first and/or foremost.

Reasonable people take a less dogmatic, more pragmatic approach, knowing that there are no panaceas, but that one size doesn't fit all, nor should it have to. Mathematics and its teaching and learning are not Procrustean beds, even if some folks prefer that they be so.
 

Mark Roberts said,

 There are things about this study that can be
 questioned and certainly replication of this study
 with a different underlying mathematical concept,
different teaching materials etcetera is needed
 before any definitive conclusions should be drawn.

Yes, Mark. Lots to question here. Or to swallow whole if one is so inclined. I, for one, am not. I hope the NEW YORK TIMES hires science and education staff writers with a little more skepticism. Maybe it's time to bring back Richard Rothstein.

Much Ado About Nothing: Cog Sci Gets It Wrong Again.

A recent article in the NEW YORK TIMES, "Study Suggests Math Teachers Scrap Balls and Slices," by the all-too-credulous reporter Kenny Chang, uncritically touts the results of a research study done by Jennifer A. Kaminski, a research scientist at the Center for Cognitive Science at Ohio State, and her colleagues, Vladimir M. Sloutsky and Andrew F. Heckler. Why is this of relevance to mathematics educators? Well, the study employed the so-called gold standard of research: a randomized, controlled experiment. And the educational conservatives, always hot for "data-based" evidence (as long as it's evidence that supports their politics), were quick to notice.

Well, now we know how many holes it takes to fill the Albert Hall, but I doubt we know anything useful about teaching and learning math from this study. It has only slightly more holes than the above-mentioned concert venue.

First, as mentioned in the article, the researchers studied only college students and extrapolated boldly to the entire rest of the population, particularly K-12 students. But aside from age/developmental issues, they are comparing K-12 kids with students at Ohio State University, a group that is not representative of a cross-section of K-12 students, many of whom would never gain entrance to that institution. Are these inferences really valid? Any mildly skeptical person should wonder at the enthusiasm coming from the anti-reformers.

Second, compare what the OSU students were asked to do with what K-12 students are asked to do in math class. Do you agree that K-12 mathematics is a decontextualized set of arbitrary rules that can be taught PURELY through abstract symbols? Try that in K-5, for starters. Just introduce the notation of arithmetic of whole numbers, then move on to integers, then to fractions. No context AT ALL. I'm not talking about "real-world applications" so much as any sort of correlation whatsoever with prior experience and knowledge. Have fun.

This sort of experiment, which of course is impressive to those who are looking for confirmatory evidence of what they already believe, is always problematic because it creates a nonsensical, decontextualized task, often one of the few things where it's possible to control for a lot of variables and noise, but unfortunately not necessarily like anything outside of the lab, and uses the results to judge things that are at best VAGUELY like the controlled situation.

I would bet a year's salary that if someone were to do a similar study that "proved" the opposite thesis, reform critics would bury it and dismiss the results if someone brought it to their attention.

What seems to be the conclusion of this study is that K-14 math should be taught like upper-division college and graduate school math. What an idiotic idea. If anything, we need a bit more of K-12 math informing some of how college and graduate school math below the doctoral level is taught (I omit doctoral level courses on the assumption that anyone who makes it that far has likely bought pretty deeply into the status quo and is in any event capable of surviving bad math teaching regardless of whether s/he accepts it as good math teaching).

I don't doubt, of course, that concrete examples of meaningless, arbitrary rules are not any guarantee that one will do well at learning those meaningless, arbitrary rules. But if our goal were to have students learn such things, we wouldn't be teaching them mathematics. Most of mathematics is neither meaningless nor arbitrary. Aside from trappings like notation, terminology, and things like order of operations, or what to make an axiom, what terms to accept as undefined, etc., which are more a matter of judgment, taste, or style, I have operated under the assumption that mathematical ideas make sense. I don't think I'd evaluate the usefulness of manipulatives, applications, or the like based primarily on this experiment.

As for the anti-reformers who praise of this study, well, they'll believe anything. As long as it supports their biases, of course.