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.

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.'

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).

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.

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