Monday, May 26, 2008

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.

Monday, May 12, 2008

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.

Thursday, May 1, 2008

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 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.