Professor Frank Quinn Says: "Calculators? Whoa!"

Frank Quinn doesn't like calculators:
(what about his dog?)

The current (May 2008) NOTICES OF THE AMS contains the following opinion piece from Virginia Tech mathematician, Frank Quinn. It bears noting that the mathematics education folks at his university are members of the math department, which must make for some fun faculty meetings.

K–12 Calculator Woes

In the third grade my daughter complained that she wasn’t
learning to read. She switched schools, was classified as
Learning Disabled, and with special instruction quickly
caught up. The problem was that her first teacher used
a visual word recognition approach to reading, but my
daughter has a strong verbal orientation. The method
did not connect with her strongest learning channel and
her visual channel could not compensate. The LD teacher
recognized this and changed to a phonics approach.

My daughter was not alone. So many children had
trouble that verbal methods are now widely used and
companies make money offering phonics instruction to
students in visual programs.

It appears that Professor Quinn has been perusing the Mathematically Correct play book carefully. The attack on so-called "fuzzy" math grew at least in part out of the attack on whole-language. I earned certification as a secondary English teacher in the early 1970s, and taught English at the University of Florida for 3 1/2 years while doing graduate work in the mid-'70s, but it would never occur to me to claim to be an expert on early literacy instruction. Yet folks with mathematics, engineering, and science backgrounds, none of whom are vaguely involved with teaching reading or writing to kids, have emerged as self-proclaimed experts on the best way (and of course, there's only ONE such way) in K-5. And that way just happens to be - (Drum roll, please!) - phonics-only instruction.

Oddly, every K-5 teacher and literacy education professor I've spoken with in the past two decades who is a fan of whole-language states unequivocally that phonics is part of what they teach and/or advocate. It's just not the entirety of that instruction. Does this sound at all familiar? Like the debate about "fuzzy" math supposedly being devoid of facts, not caring about right answers, etc.? If so, don't be surprised. Ken Goodman, one of the pioneers of whole language instruction, identified many of the foundations and think-tanks and the experts they fund that are opposed to whole language and who promote phonics-only literacy teaching. I was not completely shocked to realize that these were often the same groups and individuals who were attacking progressive reform in mathematics education. And the tactics and rhetoric employed in the Math Wars had long ago been developed in the Reading Wars. So it is no coincidence, or at least not much of one, that Professor Quinn opens a piece about calculators with an anecdote about the alleged horrors of and fallout from whole language teaching.

The concern here is with serious learning deficits associated
with calculator use in K–12 math. Calculators may
not be making contact with important learning channels.
Are they the latest analog of visual reading?

See? It's SCIENCE!!! Guilt by association. Tinker (with every progressive effort) to (Bill) Evers to (Leave Nothing To) Chance!

For brevity, connections are presented as “deductions”
(this about calculators causes that in learning). However
the deficits described are direct observations from many
hundreds of hours of one-on-one work with students in
elementary university courses.(1) The connections are after-the-
fact speculations. If the explanations are off-base, the
problems remain and need some other explanation.

Stop right there, please, Professor Quinn. What you're saying seems to be that you've worked a lot with students in lower-division courses (math, I presume) and found many of them to be wanting? Such courses are typically calculus and below, and depending upon the college/university in question may include precalculus, college algebra (high school algebra 2, for the most part, though the level of the class might start slightly lower and still result in college credit), or even lower-level classes that carry no college credit (at the community college level, such courses are, of course, a major part of what mathematics departments teach). So it's not exactly shocking that a lot of the students are not all where we might hope them to be mathematically. Indeed, some are no doubt deeply deficient. And of course that is a matter for concern.

But what, exactly, is new about this situation? On what basis other than ideology do you imply that you are seeing something that is news and that can be attributed directly or significantly to calculator use in K-12 mathematics teaching? Reading the fine print above, it seems even YOU realize you don't have anything that you would accept as proof if someone were to assert just the opposite: that calculators (or other calculation and number-crunching tools, like computers), were improving the quality of mathematics students emerging from our high schools and entering our post-secondary institutions. Somehow, I don't think you'd buy for one second "personal observations" and "deductions" of that sort without a great deal of supporting data with careful statistical and methodological analysis and detail. I give you credit for the honest admission above, even if you rather quickly gloss over it and don't hint at any alternative explanations to what you think you've seen.

Disconnect from mathematical structure. Calculators
lead students to think in terms of algorithms rather
than expressions. Adding a bunch of numbers is “enter
12, press +, enter 24, press +,…”, and they do not see
this either figuratively or literally as a single expression
“12+24+…”. Algorithms are less flexible than expressions:
harder to manipulate, generalize, or abstract; and
structural commonalities are hidden by implementation
differences.(2) The algorithmic mindset has to be overcome
before students can progress much beyond primitive numerical
calculation.

Intriguing, in that one of the oft-repeated complaints from the folks on the anti-progressive side of the Math Wars is that "fuzzy" math doesn't teach the current "standard" algorithms of arithmetic, or presents or encourages kids to develop their own alsternative algorithms. Now we hear from Professor Quinn that the very mindset of mathematics as calculation comes NOT from teaching kids to think that way (which has long been the contention of progressive reformers) but rather from letting them use calculators. Who knew? This will no doubt shock the heck out of the leading spokespeople for Mathematically Correct and NYC-HOLD, should they actually notice what you've said. (Of course, they'll also be able to spin it to mean something else. If by some miracle they cannot, you've seriously risked being drummed out of the club!)

Disconnect from visual and symbolic thinking.
Calculator keystroke sequences are strongly kinetic. But
this sort of kinetic learning is disconnected from other
channels: touch typists, for instance, often have trouble
visually locating keys. Many students can do impressive
multi-step numerical calculations but are unable to either
write or verbally describe the expressions they are evaluating.
Their expertise is not transferred to domains where
it can be generalized.

That's a really interesting assertion, and if it is supported by research data, I'd be truly fascinated to look at it. Absent such studies, however, you appear to be indulging in some convenient speculation that ALMOST sounds like it's grounded in the work of the multiple-intelligences and differentiated instruction folks that is so thoroughly dismissed by those who hate progressive reform in K12.

I also like the nifty use of the word "transferred to domains where it can be generalized." That sounds really scientific, too. Except that it begs a lot of questions. Does WRITING mathematical expressions and equations with pencil-and-paper that one doesn't understand for purposes of calculations using algorithms that one doesn't understand either transfer in the way you mention above? Indeed, the whole issue of transference of learning is a thorny one with a history that suggests that its VERY difficult to pin down. Not all that long ago, I blogged about a study that purported to show that kids couldn't transfer from using concrete objects to model ideas, with the claim being that this study called the use of "manipulatives" in math instruction into question. The problem was that the study seemed rather rigged to produce the desired conclusions, and the tasks appeared to have little, if anything to do with mathematics or justify the desired beliefs.

Even among high achievers calculators leave an imprint
in things like parenthesis errors. The expression for
an average such as (a + b + c )/3 requires parentheses.
The keystroke sequence does not: the sum is encapsulated
by being evaluated before the division is done.
Traditional programs also encourage parenthesis problems(3),
but they seem more common among calculator-oriented
students.

I must be more English-language impaired than I thought. In my experience with calculators, (a + b + c)/3 gives the average of three numbers when entered into a calculator and evaluated; a + b + c/3 gives the sum of a, b and one third of c. The calculator forces the student to think about order of operations very consciously if the correct answer is going to result.

Is Prof. Quinn saying that calculators "know" what the student intends? If so, he's wrong. Is he talking about a statistics function on a calculator where the three numbers can be entered into a list and then one-variable statistics can be run on that list, giving, among other things, the mean? In that case, obviously no parentheses are required, but then, neither is the formula for the mean (or the median, or the variance, the standard deviation, or a lot of other statistics such devices or computer programs can spew out simply by entering all the data points and a few relevant commands).

I fail to see how ignorance of or incompetence with order of operations and proper use of grouping symbols can be ascribed to use of calculator or other computing tools. Nor is failure to use mental arithmetic and estimation excusable in students whether they use computational aids or simply figure a simple average on paper or even in their heads. If a student doesn't ballpark results, silly errors are more likely to be taken as correct. But it's still perfectly possible to MAKE the silly or careless errors. The difference is that regardless of HOW those errors are introduced (and I don't think Quinn really knows), students who think are more likely to CATCH and CORRECT such errors than students who do not. Checking one's work is another way to employ intelligence and care that is something that should be mandatory for all students, but which was never popular for many students before the advent of wide-spread use of calculators. Is that now also to be blamed on them? It wouldn't surprise me in the least to hear that argument made by those who oppose these devices and alternative tools.

Lack of kinetic reinforcement. It is ironic that calculators
might be too kinetic in one way and not enough in another,
but this seems to be the case with graphing. In some
K–12 curricula, graphing is now almost entirely visual:
students push keys to see a picture on their graphing calculators
and are tested by hand math actually connected
with ways our brains learn, and the way calculators are
used to bypass drudgery has weakened these connections
and undercut learning.

This is more absurdity. Any competent teacher has students learn how to sketch graphs by hand. the calculator is used first as a check of one's work and as a way to explore a lot of graphs in a short time to see the relationship between changing parameters and the resultant graphs (when that is what is being focused upon). As new sorts of functions are graphed, hand techniques are still introduced. This is true all the way into calculus, and most students can readily appreciate the improved power of the methods taught in first semester calculus for sketching graphs.

However, over the long haul, and especially as graphs become increasingly complex, having calculator and computer tools available is enormously useful for most students (in know they have been and continue to be for me). But as good teachers are quick to note, it's vital to THINK about the graphs produced with these tools. They can be misleading. So just as with number crunching, students have to use their brains, and good teachers make this fact clear and push students to do heed it) (often by giving problems that highlight the dangers of being overly-credulous).

If the explanations offered are correct, then there are
several further conclusions. First, the learning connections
in traditional courses are largely accidental, and a
more conscious approach should significantly improve
learning. Second, calculators are not actually evil, but we
must be much more sophisticated in how such things are
designed and used.(4) But most of all, learning must now be
the focus in education. Not technology, not teaching, not
learning in traditional classrooms, but unfamiliar interactions
between odd and variable features of human brains
and a complex new environment.


1 At the Math Emporium at Virginia Tech, http://www.emporium.
vt.edu.
2 For further analysis see “Beneficial high-stakes math tests: An example” at
http://www.math.vt.edu/people/quinn/education.
3 See the Teaching Note on Parentheses at http://amstechnicalcareers.
wikidot.org.
4 See “Student computing in math: Interface design” at the site in
footnote 2 for an attempt.

The last paragraph above conveys a considerably different tone from most of the rest of Quinn's piece, particularly the title. They aren't exactly all in keeping with my own views (Calculators aren't "actually evil"? Gee, thanks for damning with faint praise!) but by the end, Professor Quinn actually seems to be advocating that we study the impact of new technology in classrooms on people and attend to how learning is enhanced, hindered, or simply done differently.

It would have done much more good for everyone if the piece had been written with a more open spirit of inquiry in mind from the beginning, with a good deal less of the usual anti-calculator, anti-technology, and anti-progressive tone. The problem is, in no small part, I think, that Professor Quinn starts with a very doubtful assumption about whole language, draws an analogy to math, and then jumps to a host of conclusions, none of which necessarily comprise anything that couldn't be said of mathematics teaching and learning prior to the invention of affordable hand-held devices and personal computers. Further, the idea that technology needs to be used intelligently is true but hardly new.

Perhaps I've erred in my suspicion that Professor Quinn is just shilling for the MC/HOLD camp. He may genuinely believe on his own that he's found some serious problems with calculator use and is merely calling for careful studies of their use and of other technology in mathematics teaching and learning. While the first part seems doubtful (at least from what he gets specific about) the second notion is always reasonable. It simply would have been better if he'd made clear from the beginning (and through a less inflammatory title) that he was calling for a bit less panic than would seem to be the case. I hope that in the future, when mathematicians decide to offer up calls for caution, they themselves exercise their own care in how they do so. Because the idea that technology and texts and much else that has been promoted by progressive reformers in mathematics education are just a "bunch of fuzzy crap" is all-too-easy to find on the Internet and elsewhere, and such hysterical claims that the sky is falling merely produce a great deal of heat without shedding much light, if any, on the real issues of improving teaching, learning, and achievement in US mathematics classrooms.

Do I Repeat Myself? - Getting Rote Right

Joseph Mazur, dangerous guy

If you want to get educational traditionalists all aflutter, say something the implies that rote learning may not be all it's cracked up to be. It's about as effective in stirring up ire as burning a US flag in front of the local branch of the American Legion (though I think impugning rote learning isn't likely to get one arrested, jailed, or fined. Yet.)

Because I'm basically a bad person, I like to post without comment quotations I consider interesting and potentially provocative on lists inhabited by knee-jerk anti-progressives and educational conservatives. The resulting furor is remarkable, more so because I don't say a word about what I think is noteworthy, supportable, brilliant, or absurd in the passages. Naturally, on lists like math-teach@mathforum.org, where I've participated in various ways since about 1994, my reputation precedes me, and it's a safe bet that those who aren't fond of me or my ideas are sure they always know what I'm saying, even when I haven't said anything at all.

So last week I posted a quotation from Joseph Mazur's THE MOTION PARADOX at math-teach, under the subject line "More on rote learning":

Teaching was dictatorial, and rote memorization of Aristotle's works played a central part in the curriculum. The seven liberal arts -- grammar, logic, rhetoric, arithmetic, geometry, music, and astronomy -- were required, though how much of each was under local control. This rote learning numbed the intellect so severely that nobody thought to criticize the classic works of science, especially the unshakable doctrines of Aristotle. Moreover, except for rote learning of arithmetic and computation, mathematics was completely neglected. (Joseph Mazur, THE MOTION PARADOX, pp. 60-61)

Of course, it was a bit unfair of me not to offer a little bit more of this passage to help give it meaningful context, so I later added the following sentence which immediately followed the quotation above:

The names of Euclid and Archimedes were empty sounds to the mass of students who daily thronged the academic halls of Bologna, the ancient and the free, of Pisa, and even the learned Padua.

One final addition from a few paragraphs down the page, sent later that day, completes the context, I think:

Young Galileo was studying the usual courses of philosophy and medicine, but under stiflingly rigid training, rather than through the kind of education he was used to at home with his father, who taught him to weigh, examine, and reason the truth of each assertion before accepting it. He despised university training, which professed truth by authority and regarded any contradiction to Aristotle as blasphemy.

Now, silly me, I would have thought this would be enough to balm the seething souls who saw what Mazur had written as "bashing" rote learning. But then, I try to resist accepting the depths to which some folks are entrenched when it comes to such things and the deep-seated fear they seem to have (or at least the compulsion to claim) that progressive math educators want to banish facts, formulas, algorithms, proofs, etc., from classrooms. The result is that no amount of context for Mazur's negative comments on the stultifying atmosphere of university education in Galileo's day can mollify the staunch traditionalists, at least not on math-teach.

To their protests, I offered the following reply:

I think a key question here is: given two possible approaches to learning something not consisting strictly of unrelated facts (random items on a list, dates and names from history being taught strictly or primarily for the purpose of having them regurgitated on an exam with little or no concern for their actual significance or meaning, etc., or in other words, learning as a "bunch o' facts"), one grounded strictly or primarily in rote, the other in grounding the important facts in context with a focus on what the important ideas involved with those facts are (and this could readily apply to mathematics, science, history, literature, philosophy, or many other subjects), which approach would you prefer as a student? As a teacher? Which do you believe would be more likely to produce successful mathematicians, scientists, historians, philosophers, literary critics, etc.?

Note that I do not suggest that the latter approach be devoid of facts, or that some degree of being able to recall those facts "on demand" as you are fond of saying, be kept strictly off the table. But I think Mazur's point about education was, and I know mine is, that: 1) rote learning as the sole or even predominant focus of education is deadening and tends to be used in ways that discourage active thinking and questioning of what is being memorized; 2) that to no small extent, historically this deadening of the mind and suppression of skeptical and critical thought has been a major GOAL of such educational methodology; and 3) that we would do well to employ the requirement that material be learned strictly or primarily by rote sparingly and always with other alternative approaches.

That said, I would certainly offer alternatives to students about how to master important facts. I've mentioned this before, written about it extensively here at times, seemingly with little impact on some others' viewpoints about rote. I continue to hold that in cases where relatively arbitrary facts are involved (e.g., the names and order of the cranial nerves, the names and order of the presidents of the United States, or the names and atomic numbers of the chemical elements - where the usefulness of 'at one's fingertips mastery' can be debated as more or less important in each case, as in the case of many other such examples, especially given how readily one can access such information these days), there are effective mnemonic methods available that should be taught or at least made mention of by any instructor who insists that it is necessary for students to memorize a great deal of such material. To not do so is, on my view, irresponsible. To not even know of and have explored such methods suggests a certain self-centeredness on the part of some teachers who may be particularly adept at rapid memorization without regard to either special techniques or a great deal of rote repetition, or who simply enjoy such engagement in "mastering" facts and really don't care whether their students like doing so, are successful at it, or actually are better off for having done it: their viewpoint seems to be that such tasks are necessary rituals that must be respected and gauntlets that must be run by each member of every generation. In other words, if long bouts of rote were bad enough for me, they're certainly bad enough for my kids or students.

I wouldn't forbid students who feel that they wish to indulge in rote learning from so doing, though likely it would be something I'd suggest they do on their own time. If I were spending "precious" classroom time on memorizing, it would be in ways I believe are more efficient and effective: through teaching or helping students develop their own mnemonics, and through games and other activities that help students do drill and practice in ways that reinforce memorization but not without elements of thinking and enjoyment. I'm afraid I'm much to anti-Puritanical to swallow the notion promoted endlessly in the Math Wars and Education Wars that it is necessary to torment students in order for them to learn.

On the other hand, to return to another favorite sore point, I am all for challenging students to stretch their thinking, and believe strongly that asking mathematics students to take what they've learned and use that to make attainable leaps beyond what has been directly instructed or analyzed in class is a reasonable and useful expenditure of that precious time. The degree to which such tasks need to be scaffolded is a useful and open pedagogical question that those of us who actually do such teaching continue to explore. I'm sure that it's something those Japanese teachers who use these sorts of problem tasks think about and discuss on a local and national basis. This remains yet another one of those delicate instructional questions for which I doubt there is a simple answer that would apply to everyone all the time (either teacher or student). But I know the mention of these sorts of problems, like the criticism of rote learning, upsets traditionalists. It seems ironic to me, however, that many of these same people have no compunctions, it seems, about promoting one kind of instruction that many students find boring and painful, while disdaining another kind which many find frustrating and painful. Or perhaps I only THINK this is a contradiction and source of irony.

Calavitta and Rote

The response of one defender of rote learning was to cite a recent article about a Los Angeles-area private school mathematics teacher and a video of a small snippet of his classroom work. I read the article when it was first mentioned on the list, but somehow didn't see the video (perhaps I didn't read the article on the LA TIMES web site) and decided when his name kept being thrown out as proof of the wonderful usefulness of rote learning that I needed to view him in action. Having done so, I posted this follow-up to my previous comments:

I just watch the very short video. Calavitta's a very exciting guy. I have too little context and content available to me to judge the scope of his methods. What we see is kids having a lot of fun. We don't see any math, of course, or have any way to judge what the kids can do mathematically, or how they learn to do those things. We do know that they seem to know and be able to recognize definitions, theorems, etc. and repeat them, one student doing it so absurdly quickly that Calavitta points out that he can't understand what the student said. Do you think that's a good thing? Why not get that guy from the old commercial who specialized in rapid-fire talking? Wouldn't he be an even BETTER instructor for these kids, if that's the real goal?

But of course, it isn't. Calavitta's teaching on that video is grounded in kids playing a game, and it looks like it's a positive, pretty much student-centered activity. I suspect there's more to his classes and instruction than that, or these kids wouldn't be doing well on calculus tests. Rapid fire repetition of theorems and definitions won't get you far on the tests with which I'm familiar, in high school or college.

You seem hung up on the surface of what he's up to. That's really too bad, as it would sell him badly short as an effective instructor. The guy actually cares about his students, according to the article. They clearly feel and reflect that. Do you think that's contained in his recitation exercises? Could you or most teachers learn to care about kids more from having them do those exercises? Would students believe you cared about them because you had them do such exercises?

You're mistaking the gift wrapping for the present. But kids won't make that mistake.

The Torture Never Stops?

Of course, to quote from BEN HUR, "It goes on." No matter that I thought I'd done a fair job of suggesting that Calavitta must be up to something other than (and more than) rote, the same idea that his teaching was a matter of "recitation-based memorization work" persisted (and continues to persist amongst the faithful. And so today I offered what I hope will be my last words on this matter:

As we still don't seem to have any definition of "recitation-based memorization work that Sam Calavitta advocates," it's a bit hard to talk intelligently about what he's doing. We see a video of kids responding quickly to flash-cards of formula and theorem names. So we know that some students have memorized the requisite information.

This begs several questions: 1) have all the kids done this, or only those who are quickest on the draw in the contest? 2) HOW have they memorized this information? We are being asked to believe without the smallest support that the contests ARE the learning. This seems highly doubtful. How could a student who hadn't already memorized these facts and words possibly learn them from hearing another kid rattle them off at breakneck speed? WHERE, WHEN, and HOW is the actual "learning" going on? 3) What can those who have memorized this information do with that information? Does being able to recite consistently translate into understanding of the meaning of the words and concepts? Into being able to use the information? How so? How do we know this? The video certainly provides virtually no information along those lines, either; and 4) Where do we see Calavitta advocating "recitation-based memorization"? We see him advocating caring about students. We see him letting kids have fun. We see him doing something along the lines of Math Jeopardy or Math Bee, but we don't see anyone doing "recitation-based memorization," unless I'm really blind to what's on that tape.

Lou Talman (a Denver-based mathematician) has pointed out that there are more powerful methods of memorization. So have I. I've also spoken towards the importance of review. None of this is a matter of "mere rote." Rote is repetition, as far as I understand the term. Repeating things over and over guarantees nothing for most people.

Here's one very simple example: my best friend had a business in lower Manhattan in the 1970s and 80s, until he moved it to Brooklyn. The phone number was 212 925-6095. I haven't called that number in about 20 years, probably more, since he's been based in Brooklyn for at least that long, has moved the business twice, and has had different numbers each time. Yet I can still recall that phone number. I didn't learn it or ingrain it in my mind by rote. I used a simple mnemonic system that translated the numbers into consonant sounds, and then I constructed a phrase that I also connected a couple of simple images to. The images, the phrase, and his business were somehow interconnected in ways that allowed me to easily call up "on demand" that number, and decades later, it's still there, probably as permanently as anything in the way of completely useless (now) and arbitrary information can be ingrained in one's mind intentionally.

Am I saying that rote wouldn't have worked? Of course not. But it wasn't necessary and would have comprised increased time and effort with less effective results (for me, in my experience with that approach). I doubt highly that rote alone would have resulted in my knowing that number today. I can cite more complex and less trivial examples from personal experience.

Perhaps there is some study out there that some list member can point to that shows that rote is an effective and efficient method for making students good at mathematics. I'd love to see it. I'm sure everyone on this list would love to. I simply am skeptical that any such study exists. It would be interesting to see what top researchers in the field of memorizing (especially of MEANINGFUL, rather than random or arbitrary material) have to say about the effectiveness and efficiency of rote or drill in achieving long-term retention and understanding (of mathematics, or of other subjects).

Please note that quizzes, recitation, and other assessments that require quick recall do not TEACH anyone how to attain recall. They only "demand" that students be able to do so. And of course, there are many types of quizzes that aren't about recitation. My calculus II teacher in NYC and my calculus teachers at University of Michigan gave quizzes at the beginning of every class. They simply comprised short problems that were based on what was studied in class and the previous homework, to encourage students to actually do work outside of class. But one needed to actually be able to do some problem-solving, calculating, etc., making use of what was studied, not spew easily-regurgitated facts or definitions that could readily be forgotten at least until a chapter test, when they could again be crammed, spewed, and forgotten.

Of course, maybe that's what mathematics is really all about. Maybe my high school teachers had it right, and my university instructors were confused.

See How They Run, Like Pigs From A Gun. . .: Vern Williams and the NMP Report

Last week, there was a public conversation/Q & A session on the National Math Panel Report between two members of the body that produced it: Vern Williams and Francis "Skip" Fennell. Unfortunately, work commitments prevented me from hearing it live and attempting to participate by submitting questions. I had to satisfy my curiosity as to how anti-progressives through the voice of Mr. Williams, would try to spin matters by reading the transcript this morning.

Or, I should say, as much of it as I could tolerate given that I had made the error of eating my breakfast before assaying the task. Your mileage may vary, and it is to be hoped that stronger stomachs than mine can slog through it all. I gave up after Mr. Williams' third answer, but then, I'm dangerously close to overdosing on the rhetoric and nonsense of the educational right wing.

I will simply look below at the first few comments from Mr. Williams:

[Comment From Matt]
How do we use the recommendations to improve daily instruction for our children?

Francis (Skip) Fennell: I figure I will just jump in here too - I think the recommendations re: conceptual understanding, fluency and problem solving have immediate impact in classrooms - every day, as do those involve formative assessment, as starters here.

Vern Williams: I have been concentrating more on the basics of arithmetic even with my algebra classes because as the report stated, there is a problem with the arithmetic backgrounds of even our brightest students.

Yepper, the lines couldn't be much more clearly drawn between a progressive mathematics educator like NCTM's recent past president, Skip Fennell, and an apologist for business as usual in mathematics education like Vern Williams. Not that I have any objection to seeing that kids know arithmetic well. Or doubt that everyone would do well to have a deeper understanding of the many subtleties arithmetic contains, present company included. Maybe even Mr. Williams has a few things to learn about arithmetic, and, perish the thought, he might learn some of them from listening to kids. But rest assured, that's not part of his educational philosophy. Direct instruction and sage-on-the-stage uber alles.

Let me at this point insert some words from Mr. Williams own web site that I find incredibly telling:

My main goal is to support and mathematically challenge intellectually gifted middle school students and to help them survive the educational establishment's war on intellectual excellence.

During the past thirty years, I have experienced educational fads from brain growth plateaus to professional learning communities. Some of the more destructive fads involve those that have taken the math out of mathematics and replaced it with calculators, watered down content and picture books. Many outstanding traditional mathematics teachers have left the field because they were forced to lower their standards and replace them with fuzzy standards championed by the NCTM. I'm one of the lucky ones. I have somehow managed to teach Real math for over thirty years and have no intention of changing my methods.

This sort of thing is disheartening coming as it does from a fellow who is supposed to be a professional mathematics teacher. He spouts the usual reactionary hyperbole, attempting with a flourish of rhetoric to dismiss all innovations as "fads," all those who disagree with him as supporting idiotic, watered-down "fake" math, while he is one of the bastions of "Real math." Of course, there's nothing fake about the mathematics NCTM has supported. What Williams and his ilk object to is never grounded in a substantive charge that the mathematics being promoted isn't math. As some folks on their side of the debate like to say, somewhat ironically quoting Bill Clinton, "Algebra is algebra." And of course, "math is math." There's nothing fuzzy about the content of progressive mathematics books. It's the pedagogical approaches, the infusion of issues in applied problem situations and subjects perceived as politically liberal by the anti-progressives, and the emphasis on student-centered teaching and democratic core values that seems to be at the root of many self-named "parents-with-pitchforks" groups. (On that point, I always remind people that there was ALWAYS politics in K-12 textbooks: the exclusion of non-white faces, non-Anglo names, and indeed many social issues from those books fit the political agenda of one facet of American society, but that facet is on the wane. As they perceive that they are losing control of the more subtle sorts of propagandizing that tends to slip into various aspects of public education, they simultaneously have sought to undermine public schools through various means, not the least of which is test-mania via No Child Left Behind legislation, and to increase the privatizing of public education through vouchers, charter schools, and other programs and policies. Of course, none of THOSE tactics are political. And I'm the Emperor of China).

Here's a bit more from Mr. Williams' web site:

I had junior high school teachers who were intellectual and had a passion for their subject. Those great teachers helped me to develop a passion for learning and a respect for hard work that remains with me to this day. At Paul Junior High School in Washington DC, intellectual excellence was the norm and it was celebrated. There was no cooperative learning, fake self esteem, differentiated instruction or ten pound textbooks loaded with pictures and useless content. I decided to become a math teacher during my last year of junior high school because I knew that I wanted to return some day to continue the fun, the learning, and the celebration of excellence that I experienced in seventh, eighth, and ninth grades.

I am of course thrilled for Mr. Williams' experiences and the high quality of the teachers he had. However, he continues to imply that teachers with philosophies and approaches he decries do not have passion, are not intellectual, and in general are not fit to teach their subject. This is simply nonsense bordering on libel against colleagues who don't follow his beliefs, yet are highly-knowledgeable, deeply-dedicated professionals who dare to teach mathematics in ways different from him. Were I to suggest that Mr. Williams was unsuitable for his job because he decries cooperative learning or differentiated instruction (the other two comments about "fake self-esteem" and "ten-pound textbooks" are red herrings; I have written here before about the rhetorical use of "fake self-esteem" as a blanket criticism of student-centered education, and I know of no teacher who is happy about the size of textbooks that the mainstream publishers sell. However, the textbook culture we live in did not arise as a one way street in which publishers forced their will upon America. It took complicity by a lot of teachers, the vast majority of whom never heard of cooperative learning or differentiated instruction when they were selecting big fat books with lots of expensive colored pictures, none of which had or have a damned thing to do with progressive reform. In fact, Core Plus (published as CONTEMPORARY MATHEMATICS IN CONTEXT) is a very plainly presented textbook series with no color photos), yet it is always attacked by folks like Mr. Williams. Since part of the strategy of such people is to throw as much crap at reform as possible, they never state publicly that what might be true of one given book isn't true of another. Rather, all progressive books are lumped together and any flaw that is found with one is ascribed to all. It's clever, it's effective, and it's thoroughly dishonest. And despite having this called to their attention many times for well over a decade, nothing ever changes about the tactics and rhetoric that comes out of web sites that Mr. Williams proudly links to on his own site, like NYC-HOLD and Mathematically Correct. For folks who claim to be interested in "Honest, Open, Logical Debate, they likely can only accurately lay claim to the "debate" part, and that only to the extent that they sometimes have to speak with progressive reformers in a forum that resembles debate.

Returning to the conversation about the NMAP Report:

Let me follow up on Matt's question with one that may be relevant to some members of our audience. Are there recommendations in the panel's report that a relevant not only to teachers -- but to parents, for helping children? This one's for both of you.

Francis (Skip) Fennell: Well, sure - the fact that effort makes a difference. Teachers need support. They can't do it all. Parents and other caregivers can provide support at home, at rec centers, wherever. We needs kids to care about this subject - everyday.

Vern Williams:

I would hope that parents become more proactive in the math education of their children by researching the materials/textbooks used at their children's school. They would be wise to make sure that their children are receiving an excellent grounding in the basics.

Dr. Fennell addresses ways that parents can help their kids. Mr. Williams, however, is all politics. Parents, he says, need to be vigilant. And indeed, the spirit of vigilantism is exactly what informs so much of the atmosphere in the Math Wars. From New Jersey to California and back again, the radical right has enlisted and stirred up through fear and disinformation groups of parents who are sure that their kids are being led down the road to mathematical ignorance and some sort of socialist hell through the influence of progressive mathematics education.

I wonder if Mr. Williams, were he a science teacher, would welcome the incursion of religious parents into what he got to teach in his classroom. If he would be thrilled at being told that he needed to teach Intelligent Design with at least as much attention as he gave to legitimate science. If he would accept input from parents as to which science books he was allowed to use so that they could ensure that natural selection and the theory of evolution were taught in ways that made them appear to be mere speculation by secularists who want to undermine the moral and religious fiber of our country. I believe it is a safe bet that he would not. And yet he suggests that the way for parents to help their kids learn math more effectively is to monitor those textbooks. Yeah. Great idea, Vern. That's REALLY what it's all about.

One last excerpt from the discussion, the one that killed my willingness to read further:

Sean Cavanagh: Thank you. I will direct this next question to both of you, about teaching in Japan, and collaborations between teachers.

[Comment From Guest]
As seen in the TIMSS study, Japan does something where teachers are required to get together so many hours a week to work on perfecting one lesson plan. They focus on major drawback areas such as fractions, and work and rework the lessons together. Throughout the process the lesson is taught, with other teachers present to evaluate how effective the lesson is. Do you think a process like this would help math education in the lower grades.

Francis (Skip) Fennell: The Japenese Lesson Study model has become quite popular in this country and seems to be making a real difference as a Professional Development model. That said, just the opportunity to think carefully (as a group of teachers) has potential in improving teaching.

Sean Cavanagh: Just a bit of background about the panel:

The National Mathematics Advisory Panel was created through an executive order signed by President George W. Bush on April 18, 2006. The panel was given two years to produce recommendations, based on the “best available scientific evidence,” on the “critical skills and skill progressions for students to acquire competence in algebra and readiness for higher levels of mathematics.” The panel had 19 voting and five non-voting members, who included cognitive psychologists, mathematicians, representatives of think tanks and professional organizations and other math experts.

Vern Williams: Yes, however we need to first make sure that our elementary and middle school teachers understand the math content enough to actually plan and conduct excellent lessons as they do in Japan.

While I'd be the last person on the planet to suggest that we don't need to do everything in our power to increase the knowledge base of mathematical content in all teachers, and not just in grades K-8, it is astounding that each time something interesting is raised, Mr. Williams move is to turn the subject elsewhere. Again, rather than indicate that he has the slightest clue what lesson study entails or why it would be useful for teachers to engage in it, he goes to a standard right wing complaint about public school math teachers: they don't know their subject well enough.

No one denies that this is true, of course. What is ironic, in fact, is how much effort progressive mathematics researchers and educators have been making towards improving the content knowledge of teachers in the lower grade bands, while also trying to determine what that knowledge needs to be for effective teaching in K-5 and 6-8 (something the right wing doesn't ever do, because they turn up their noses at such research unless it carries the proper political messages with it). But are we seriously to believe that we should wait until some critical mass of K-8 teachers emerges with "enough" content knowledge to satisfy Mr. Williams and his friends before ALSO exploring increased cooperation (oops! that dirty word again) among professionals? Is not the point of lesson study, in fact, to deepen professional knowledge of math content, pedagogy, and lesson planning, amongst other relevant things? I pointed yesterday to a talk Jim Stigler gave at Harvard in which he says precisely that. There is nothing radical about Japanese-style lesson study. Unless, of course, it's radical to think that teachers can and need to learn from each other. But of course, that doesn't suit the mavericks like Mr. Williams who in part appear secretly happy to portray and believe themselves to be the smartest, best teachers in the room, the ones who are serious, the ones who are dedicated, the ones who teach "Real math," as he puts it. Everyone else? Incompetents.

Given three opportunities to speak about how to do something constructive, Vern Williams runs the other way. It's remarkable how the educational right wing is vastly better at telling us what not to do than it is at showing us how to do better than we've done using precisely the approaches they advocate for, approaches that have never served a sizable percentage, and possible the majority, of American kids in mathematics classrooms. Such failures and shortcomings are rationalized and glossed over time and again by Mr. Williams and his conservative colleagues. I do not in any way impugn his own teaching ability or professionalism, but I resent like hell his baseless attacks on those of me and my colleagues through his absurd blanket attacks on anything and everything progressive. It's just politics, not an interest in the maximum possible success of all kids, that fuels such attacks. And they cannot be allowed to stand unchallenged.

Don't Assume, Teach: Why Good Educators Must Model and Scaffold More Than Just Academics

Yesterday I posted to several lists something about a recent presentation by Jim Stigler entitled, "Reflections on Mathematics Teaching and How to Improve It." Quotations from Prof. Stigler's presentation engendered one puzzled reaction from an anonymous skeptic who opined:

The way I understand the word is used in the U.S., diversity is to be celebrated, and the schools are to accommodate the students rather than the students being made to conform to the schools.

Japan, on the other hand, is famously one of the least diverse places on earth. And yet, even in Japan, according to the article, individual Japanese students do not know exactly how to be students so they are explicitly instructed. This sounds to me like the student is made to conform to the expectations of the school, not the school "accommodating diversity" in the American sense. This does not seem to support the "every country has diversity" assertion and therefore "different strategies" are required.

One of the most interesting things I picked up from reading LEARNING TO TRUST by Watson and Ecken, a book that looks at Ecken's experiences teaching a combined grade 1-2 classroom over a two-year period in inner-city Louisville, is the necessity to teach a host of skills to kids that those of us raised in middle-class communities and homes take for granted as a given that everyone brings with them to school. These include such "obvious" things as listening to and following directions, having a one-to-one conversation with a peer without turning it into a brawl (physical, verbal, or both), taking turns, and so forth.

I don't find it surprising that effective teachers in any country realize the necessity of "schooling" kids in some of these expectations and skills. The hard part is being the first teacher to try to do this for kids who are in grades 6-12. By that time, the horse has long left the barn.

It is puzzling that anyone would be surprised or confused by this: kids come to schools from a wide range of cultures and sets of attitudes about school and learning. One merely needs to set foot in a classroom with a female teacher and a male student of, say, Middle Eastern Muslim descent to note that it is a cultural norm for boys to assume that females are neither qualified to teach nor to administer discipline to them. (While this may not be universally true, I've seen it so often in SE Michigan, an area with a very sizable population from that background, that it is a lesson that can and needs to be learned quickly for most teachers here.) Clearly, such kids are going to be very problematic in typical public school settings, where a majority of teachers are female, if they are not "schooled" and to some extent enculturated. Obviously, there is no single approach taken around here to this or similar issues, and no doubt some people would argue that schools have no business treading on anyone's cultural beliefs and values. But from a practical perspective, it's likely to be imperative that these sort of things be addressed in the best interests of everyone concerned.

It would not be difficult to multiply the above example greatly. But even without something so obvious, teachers are going to have their own classroom rules and expectations, and it is foolish of them to assume that all or even most kids will come to class with the requisite skill set to adapt. Similarly, teachers are likely to meet with great difficulties when trying to implement pedagogical approaches with which students are not familiar. Something as simple as the commonly-used "Think-Pair-Share" method is going to go down in flames if kids are not used to being asked to work in pairs or simply cannot conduct themselves effectively in such situations, as described well in Watson/Ecken.

Cultural diversity and individual differences are huge factors for most American teachers, and certainly the latter play a significant role globally. To the extent that no country is without some sort of diversity (ethnic, , economic, etc.), the notion of cultural diversity is also relevant. Of course, I'm not speaking of paying lip-service to "celebrating cultures and diversity," but rather to actually knowing enough about the sorts of issues that may arise as a result of cultural differences that one can allow for and deal effectively with them as they arise in challenging or problematic ways in one's classroom or school. The culture of bullying that has been a serious concern in Japan for several decades is but one example that continues to challenge educators there.