Elitism and mathematics education: Why tracking is still wrong

As I survey the many pressing issues facing US education in general and mathematics education in particular, one that never goes away is tracking (or "streaming" or whatever euphemism is being served up by its advocates these days). To me, it is anathema. Not that I see no reasons to mix things up within classrooms frequently so that there are occasionally homogeneous as well as heterogeneous groupings along various constructs, including ability, gender, ethnicity, and others). But I will never tolerate or support segregating kids all or most of the time along such lines. And therein lies an endless debate.

What is disturbing is how easily people lose sight of the deeper implications of tracking, in no small part because they have long ago lost sight of the role public education plays in forming a healthy democratic society. Without wanting to go off-track into the many social and political forces that keep that notion of school far from public consciousness and conversation, I'll merely suggest that there are a host of reasons that it's easy to forget or ignore the fundamental principles upon which both our nation and public education were founded. Or the vital importance of not seeing public schools as the tools of industry. Heretical as it might seem, it's possible to imagine a nation in which companies pay to train their workers when they hire them, and public schools do quite a number of other things, not to spite these companies, but to pursue more important goals that private companies cannot be expected to worry about very much. The problem comes when we forget that just as companies must pay for raw materials, machinery, buildings or factories, and a host of other production and operating costs, it was never a given that they should NOT have to pay for specialized training of their employees in particular skills they want them to have for doing their jobs. Once we started allowing business to dictate what the purpose of public education was, things began to go very badly indeed, in a host of ways, for the core democratic principles and values upon which the nation was based (no matter how much jingoistic, sloganeering lip-service is paid to these issues by those looking to bend our public schools to their personal profit).

In the context of the above, I wish to present my most recent response on math-teach@mathforum.org in response to a post alleging to represent my ideas about tracking and related issues in mathematics. My antagonist here, a Florida-based fellow named Robert Hansen, manages to get virtually nothing right about my views, while promulgating many he holds with which I strongly disagree.

Quoting Robert Hansen :

> I have no problem with experimentation on how to get all those kids
> that are disinterested in math, interested. I just think it should be
> labeled properly and should not affect the curriculum for the kids
> that are interested and that do get it.

Oh, please, Robert. This is such an enormous red herring that the smell of it defies description. Why is it necessary or desirable to pitch everything in such terms so that there is some sort of scarcity mentality: limited amount of mathematical knowledge available to the quicker kids, is there? Are they competing with the less quick to control what mathematics is available to them?

The fact is that what limits the mathematics ANY kid can study in school is . . . duh, all these moronic state (and soon national) "standards," the narrow vision of most schools, the narrow knowledge of most K-12 mathematics teachers, and so forth. But putting the needs of the quickest and the needs of everyone else at odds is, as usual, the wrong way to go about things. The illusion that somehow the fast ones would be best off if they were kept out of contact with everyone else so that they can get to calculus by the time they hit, what, seventh grade? is just a huge pile of baloney. It's elitist, of course, and it helps promote snooty attitudes and idiotic class notions - mathematical 'social strata,' if you will - so that everyone knows his place: gloating at the top or mucking in the trenches, as the case may be.

As opposed to, of course, a democratic approach to mathematics education that fosters notions of community. We MUST NOT have that by any means. Imagine raising a generation of kids who don't sneer at those who aren't as fast at mathematics as some of them may be. Who go on to become mathematicians, mathematics teachers, high end users who aren't automatically inclined to see anyone who doesn't as beneath contempt. Boy, would that screw things up, wouldn't it?

Imagine what sorts of people those would be. And how they might benefit by having to rub shoulders and share ideas, strategies, etc., with the hoi polloi. Must make your skin crawl. Imagine benefiting from having to explain things to peers, or occasionally learn from their "inferior" ideas. How awful.
>
> If this is what MG said ...
>
> "There is no argument whatsoever about the fact that (as Mike
> Goldenberg has observed) to learn standard formal mathematics, one
> has to DO standard formal mathematics."

[The above is quoted from a post http://mathforum.org/kb/thread.jspa?threadid=2008638&messageid=6900648#6900648 by list member G.S. Chandy in response to an earlier one of my own that may be viewed by going to http://mathforum.org/kb/thread.jspa?threadID=2006296&messageID=6899654#6899654 ]
>
> Then I would say we agree quite a bit.

Of course I said that, Robert, but I doubt that we agree past the surface level of my remarks.

> But I don't remember MG ever saying anything like that.

I can't be held accountable for your flawed memory or having failed to read the post to which Mr. Chandy refers, Robert.


> If I remember correctly, the last thing he said was that parents should provide an
> education for their star students outside of school, after hours or
> something like that.

Did I really? Or did I suggest that there's no limit to how far parents can choose to supplement their child's mathematics education through a host of free resources? Quite a different matter entirely, of course. Your spin is typical and of course very misleading as to my thinking. What a non-surprise.


> But I took that as just his frustration with the
> star students getting the say as to what math gets taught. A policy I
> never advocated.

I have no such frustration. You really don't understand my thinking on this at all, Robert. It's embarrassing to see you butcher it, but again, predictable. Do you really think my fondest hope is to keep an upper bound on what students get to explore? If so, you couldn't be more wrong. But I'm not going to see the majority crippled in order to ensure that a relatively small minority gets all the good resources, the best teachers, the maximum opportunities. I don't believe in meritocracy because it sooner or later comes down to the meritocracy of race, gender, social standing, and money, not necessarily in that order, of course. In the typical tracking malarkey, the whole ugly game is obvious.

That does NOT mean we need to deny brighter kids all sorts of opportunities to rise to whatever levels they wish. But always in a broader social context that considers more than simply bending to the will of their (mostly) well-to-do, more educated, more influential parents (and folks whose social agenda is, always has been, and always will be to get as much for theirs and those they see as like themselves as possible at any cost, on the backs of the poor, the socially disadvantaged, the less "in." Of course it's a balancing act, but the false dichotomy you insist upon is all about making this a contest for limited resources, the answer to which is to always separate "wheat" from "chaff" (or is it goats from sheep?)

> Although, if we only get one choice then it had better be authentic
> since it would be quite difficult to convince those that pay for the
> damn schools that it should be something less than that.

Whose "authentic," Robert? Yours? Not on my watch. Yours is way too narrow. Kirby's <http://www.4dsolutions.net/ocn/> would be far more to my liking because it's not just one choice and it isn't predicated on one group pissing on the backs of another. Everyone gets to play. Everyone gets to do WAY more than is dreamed of in your philosophy.

> I don't think that it should be just one choice, but then you have that
> tracking thing. One group of teachers turn off tracking and then
> another group want to try new things for the masses that don't get
> it. Seems to me that the two groups should work out their differences
> first on this less tracking / more filling dilemma.

Right. Accept the fact that most kids are just too bloody stupid to learn mathematics, right, Robert? Isn't that your essential truth? Isn't that the implicit truth in EVERY tracking system (regardless of the subject in question)? Isn't that the message that some of your like-minded fellow here are delivering all the time?

Come out to Detroit some time, Robert. I could show you a few things that might change your thinking a bit, but you'd need to scrape the scales off your eyes first.

Good Tests, Bad Tests: Can The Testing Fanatics Tell The Difference?

And do they really care?

Consider the following problem:

=====================================

3, 4, 6, 7, 10, 12

The number n is to be added to the list above. If n is an integer, which of the following could be the median of the new list of seven numbers?

I. 6

II. 6 1/2

III. 7

(A) I only (B) II only (C) III only (D) I and III only (E) I, II, and III

==============================================

In considering your evaluation of the question, you may wish to know the following facts: this is the 13th of 16 questions for which the total time allotted is 20 minutes. The question appeared on the last of three mathematics sections and on the 8th of 10 sections overall in an exam that takes a total of 225 minutes. The test is one commonly used as part of college admissions in the United States.

What factors enter into your determination of how good or bad a question this is? What assumptions are you making, if any, about what comprises a good mathematics assessment in general?

Some interesting responses

I posted the above on a couple of my usual math-education lists, primarily to elicit reactions from the usual suspects who seem to mindlessly support multiple-choice standardized testing in mathematics as the only valid way to assess learning, students, teachers, schools, districts, books, pedagogy, etc. I had a particular issue in mind with this problem and didn't expect to see several people become concerned about the use of the word "added" in the problem's exposition. Here is part of one such comment:

This seems more of an aptitude question than a content
question, and it's probably in the upper 3rd to upper
4th portion of all math questions in difficulty (but
the actual difficulty level is obtained from the item
statistics after pre-testing and not by people sitting
in a room rating how hard the question is). This is not
the kind of question that should be on a college math
placement test, but from what you said, it's not. In
isolation, I don't really like it all that much (the
test taker may have forgotten what 'median' means),
but as one of many other questions, I don't see any
major flaws with it right now, except I don't like
the use of "added" in the statement of the problem,
since a possibly valid mathematical interpretation
(at least before the appearance of "seven" at the end)
is that the new list is the old list translated by n,
giving 3+n, 4+n, 6+n, 7+n, 10+n, 12+n.

A similar response appeared on another list to which I posted the problem, followed by one post that questioned whether there was really any real ambiguity. I then made the following reply:

I think it's a stretch to say this is ambiguous, and if we allow that it is, what word(s) should be substituted for "added"? "Appended"? You immediately lose a host of kids who've never seen the word and have no clue what that means. Further, it's too restrictive given the intent of the problem.

"Concatenated"? Even more problematic. "Inserted"? Fails to make the point that the number could come at the beginning or end of the list (though of course given how n is restricted, it can't be placed between two pairs of values in particular, which winds up being rather significant).

That said, I wasn't thinking about that aspect of the problem when I posted it. Indeed, my "real" target was people who seem to have enormous faith in standardized tests except for when they don't. When don't they? When they don't like the results or the implications of the results for their educational politics.

I don't think the problem is horrid. But I did note at least one point I found a little annoying, and it's yet another example of the difference between SAT/ACT-type math problems and actual assessment. (For the record, this one is from the SAT; that might have been obvious to some readers from the additional information I gave, since the ACT only has ONE section of mathematics).

The annoying thing is the restriction that n is an integer. Because of this fact, there is no number that can come between 6 and 7, and hence it's impossible for 6 1/2 to be the new median. Either the new number is 6 or less, in which case 6 is the median, or it is 7 or more, in which case 7 is the new median. No other possibilities exist.

What's wrong with that? Nothing, per se. Except that I believe it's really easy for a student who is trying to do problems with an average of 1 minute 15 seconds per problem available to miss that restriction. Of course, in HINDSIGHT, the restriction is pretty much the point, and if the purpose of the problem was to illustrate that point and teach students something, I'd have no complaints.

But of course TEACHING students is the last thing the SAT is used for and certainly is NOT the reason the test-writers construct it. The next conversation or article you encounter in which those who are responsible for the SAT or ACT address what students learn from these tests will be the first. They do NOT provide formative feedback, at least not in the vast majority of cases. Students take the tests and await the numbers. End of story.

So trying to trip kids up, which is so often the strategy test-makers use to help produce the holy bell curve they seek, is increasingly the rule as the problems "increase in difficulty." There's an upper bound on what topics the test makers have decided to address, and that means they need to get trickier rather than more thought-provoking. And why do the latter when no one is going to be reflecting on the problems afterwards?

So what does a teacher, a student, a parent, an administrator, a politician, or anyone else learn about what any given student, class, school, district, state, or nation knows about the median from this problem? Can anyone state with a straight face what it MEANS about a student's mathematical knowledge if s/he gets this one wrong (or right)? What's being tested here, really? What does the teacher tell the student about where s/he went amiss? What does the teacher need to teach "better" for the next time?

On my view, while of course there is mathematical content here, it's impossible for anyone to claim to know whether any student gets this wrong because of a misunderstanding about what the median is or how to find it, because of misreading, because of not knowing what an integer is, because of forgetting or missing the restriction of n to integers, because of not correctly figuring the implications of that restriction, or anything else, EVEN if we know what answer choice the student selected.

And yet huge claims are made and consequences suffered based on the results of student performance on such problems. Imagine if you can a state deciding to make its high school graduation test the SAT. Impossible, you say? What state would abuse a test like that? What state would use a test not based closely on its curriculum framework? Well, Michigan, for one, which has been using the ACT to that end for the last few years, despite the fact that its curriculum framework and grade-level content expectations (GLCEs) do not in any way concern or inform the production of the ACT.

Amazing? Not really. We continue to be in the grips of testing insanity. All one needs remember to understand this is one simple thing: it has NOTHING to do with learning or education.

You Want Proof? I'll GIVE You Proof!

Once again, the fires of discord are raging on math-teach@mathforum.org. One of the threads I've been embroiled in revolves on several axes: one is about teaching pure mathematics in K-12. Another is about visual proofs. The one I wish to specifically deal with here is the one that links the two: what comprises the nature of proof in elementary school mathematics classrooms and how do we get students in those grades to develop their notions of what a mathematical proof actually is?

The problems in having this sort of conversation in a hostile forum like math-teach are legion. One problem is that only a few of the participants are K-12 teachers, and fewer still are K-5 teachers or spend time working with elementary students and/or their teachers. Another is the long history of enmity from the Math Wars that tends to inform most conversations there. Fortunately, the one I'm going to pull from is not being polluted by the more egregiously nasty sorts of name-calling (due in no small part to the absence of two of the more troll-like participants from that list when it comes to progressive math education, as well as a sort of detente' that I was able to reach off-list with the person to whose ideas and arguments you'll see me replying below. It isn't quite all milk and honey, but it at least doesn't reek with epithets. I take what I can get these days, as I try to moderate my own propensities for vitriol. It's a long, hard slog, and I am fallible.

For background, things had reached a point where I was arguing for letting kids develop their own ideas of proof, and Paul A. Tanner, III made the following post:

I mean it could be made available as part of the mix in the same way that things you like could be made available, like the lattice method. (You want the lattice method made available, right? You don't want it left to chance as to whether they have the opportunity to see it, right?) It could be taught to them as part of the main course or
as a supplement. It could be in some sort of enrichment context, as something to be learned at the end of a guided method or as something to be learned in a direct teaching context. It could also be in written form, in which case they could study it if and when they wished, consulting with others outside the class or with their teacher.

Here is my reply, which touches, I believe, on several really central themes for the direction effective, student-centered mathematics education must go:

Paul, you have a unfortunate propensity for mixing up issues in ways that seem to come from your drive to win arguments rather than understand what others are saying. It's not a helpful habit and really makes discourse more difficult.

We were talking about notions of proof and whether formal proofs need be introduced earlier in K-12 and if so in what manner.

I raised the suggestion that for younger children, it would benefit them more to promote conversations amongst them about what constitutes proof and to help that notion grow OVER TIME (as in, over the course of years, not weeks or months). And I expressed concern that it would be difficult for some adults to resist pushing their more sophisticated notions on kids rather than trust that they will, if nurtured intellectually, move towards more mature and precise notions of what comprises proof, and eventually will start to see a need to know and understand what the mathematics community at large considers to be proof (and as you see, there is not exactly universal agreement about what that is: note the on-going debate here about visual proofs, for instance).

You then switch, as you all-too-often do, to algorithms, specifically so you can "trap" me with the example of lattice multiplication. But your move is flawed. Here's why. First, if you recall, I am inclined to agree with the work of Kamii, among others, who suggest that one of the big errors we make is to PREMATURELY introduce formal algorithms for arithmetic to kids. Please note: that's ANY formal algorithm, which would include the standard multiplication algorithm, the Russian Peasant algorithm, lattice multiplication, partial products, area models, et al.

And before you jump on the word "prematurely," which you have done erroneously in the past, this is NOT an issue of developmental "readiness." Rather it's a concern about crushing students' confidence in and reliance upon their OWN ideas before they've had a chance to test them and compare them with those of their peers. Adults need not be so bloody worried that the kids will get everything wrong and be irreparably harmed. Using traditional teaching methods and algorithms, teachers are ALREADY eliciting student misconceptions and errors.

So what's the rush? Can't wait to see if the kids can, by interacting with peers in guided conversations by a knowledgeable teacher, find their own way and correct misconceptions? If not, why not? Where is the evidence that this method would be WORSE than what we've done for decade upon decade? Lots of kids get mired in buggy algorithms as it is. I see the evidence all the time. A much smaller percentage of kids seem to blow it when they use lattice multiplication than the traditional algorithm because they don't wind up with misaligned partial products and they don't forget to carry out all the necessary multiplications. They can, of course, still do one digit multiplications wrong, and they can still add wrong, and they can still forget to carry. So this isn't foolproof. But the incredible number of the other two kinds of errors I mention simply don't occur with this method. And yet, the forces of anti-reform refuse to look at this, refuse to consider that it not only might this true, but that it makes perfect sense that it should be true. Given that sort of rigid thinking, can we seriously expect that such people are going to trust kids to use their own algorithms, test those algorithms against problems and against those of their peers, and choose reasonably which one(s) they prefer to use? Clearly, that isn't a possibility for educational conservatives.

Getting back to that word 'prematurely" again, the issue isn't development, as I said. It's letting kids develop their sense of what works well for them, and that they can figure this out, and that they can develop, test, and refine their own ideas about mathematics. There's lots of time to then introduce, if necessary, any algorithms that we might feel kids need to see that haven't arisen. Later.

But as I wrote in my previous post, what tends to happen is that either nothing but the traditional algorithm is taught, or when it is taught it's given explicit or implicit pride of place. And this occurs so early, in lower elementary, that what I discuss about about kids' creativity, self-confidence, and judgment about what works is crushed or suppressed. And we wind up with what? Passive kids who wait for Teacher to tell them everything. Who don't take risks. Who don't think except along the narrowest possible lines. All this well entrenched, from what I've seen, by third grade. It's a tragedy.

So, getting back to PROOF: the same rush to formalism and tradition is equally ill-founded. Let kids develop their own standards and ideas about what comprises proof in general and mathematical proof in particular over the elementary grades. Guide them towards more sophisticated notions, and by the time they've gained enough mathematical maturity, they'll want to know what professional mathematicians (about whom they might actually be allowed to learn something) consider to be proof. It may not happen in K-5. Or maybe it will. But it will happen. If we trust kids' curiosity and the ability of wise teachers not to shove things down their throats at the first sign that the students aren't doing things "by the book."

Do I trust YOU, Paul, and those who think like you, to show this wisdom and patience? I do not. I have seen ample evidence of how the majority of even elementary school teachers think and work when it comes to traditional vs. other algorithms to believe that it would be any different with notions of proof. The drive that is grounded in mistrust of kids is so strong that I had a third grade teacher tell me in 2005 that she doesn't explore student errors with students in class because "the other kids will fixate on the errors and then I can't extinguish them." I wondered why they ostensibly don't fixate on the correct solutions and methods: only the wrong ones they hear. Why would it not be useful to discuss the errors, where they come from, and try to reveal why they are grounded in things that don't correspond to what kids already know to be true from experience inside AND outside of school when it comes to math. But apparently it's just too dangerous to talk about those errors. Just say, "No, dear. You do it like this."

It's the adults, not the kids, who have problems. The kids will be fine if we let them. They are not morons, but we make them into empty-headed robots in short order with our insistence upon spoon-feeding them when they're perfectly capable of a great deal of self-nurturance. All we need do is provide a safe, rich environment and keep our eyes and ears open and, much of the time, our mouths SHUT. But few of us can do any of that: as Bob Kaplan told me in Chicago in 2003, it's easy to find teachers for the Math Circle who know the mathematics. The problem is finding teachers who know how and when to keep their mouths shut.

I know you will continue to try to lawyer this into something that allows you to shove things down kids' throats that you don't trust them to want to know later. And here's the part you miss: I don't insist that any particular algorithm MUST be taught. What I want is going to happen if we let kids breathe and think. When they want to see more methods, they'll let you know. When they become discontented with their ideas of proof, they'll let you know. And it WILL happen. Because there will always be kids who ask themselves and their peers: "Why does that work? Why does that make sense? How do you know?" And that's all we need to nurture in them: their own natural curiosity, rather than suppress that and replace it with curiosity about only the following: What does the teacher think? What does the teacher want me to say or do? What do I need to do to get an A?

Math Test - a poem.

I occasionally get requests to post guest pieces at RME, but invariably they prove to be from people looking to promote some sort of commercial web site or service. As those so requesting fail to offer to adequately grease my palm, I always refuse, sometimes in less than friendly fashion.

However, I received a work today of such outstanding literary merit, so fraught with relevance to many of the concerns of this blog, that I could not refuse despite the lack of financial incentives from the author.

For now, I am going to publish this without attribution to the author, though in due course his or her name will be appended in an updated version. I will state unequivocally that I am NOT the author, though I did, with permission, make one substantive change to improve a rhyme. It did not change the intended meaning, however.

The Math Test

by Zane Goldenberg Dietz

Oh no.
I'm screwed.
Why didn't I study?
I blankly stare at Question Three.
I feel it staring back at me.

I feel my heart pounding in my chest.
I despise this dumb algebra test.
I wonder how long it'll take
Before my hands begin to shake.

What could the answer be?
Is it "line of symmetry"?

Y is ᴨ so complex?

Polynomial? More like "Idunnomial."

Who ever cared about 2x^2?

Calculus is ridiculus.

Geometry makes me vomitry.

Why is tangent so abhorrent?

And I'm not fanatical about that radical.

Then, it hits me.
The answer to Three.
I let out a big sigh.
The answer all along was blueberry ᴨ.

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.

Memorizing vs. Rote Learning & Drilling: More Ways In Which Anti-Reformers Get It Wrong

In an on-going discussion about Benezet on the math-teach@mathforum.org list that unfortunately is shedding limited light on the subject, the suggestion was made that "promoters" of Benezet and other reformers (which in this context means "progressive education, student-centered learning, humanistic, and constructivist types") are opposed to memorization. This sort of false accusation is typical of the muddying of the waters by certain fanatic critics of anything outside the neo-Prussian school of how to teach mathematics to children.

On Apr 10, 2009, at 1:53 AM, Paul A. Tanner III wrote:

[T]here is much written by reformers against this necessary part. This includes all that anti-rote rhetoric which serves as backdoor anti-memorization rhetoric.

My Response

Wrong again, Paul. I know of no one who opposes memorization (which by the way is NOT the same as rote learning. More on that in a moment). What many people oppose or question (again, not the same thing; I feel it is necessary to bore the people who read fluently when addressing ANYTHING to you, because you are SO adept at misreading, misconstruing, misinterpreting, garbling, or otherwise distorting or changing what people actually say and believe into nifty little straw ideas you can "decimate" with your famous logic "traps"), is MINDLESS rote learning of things that can be learned effectively, possibly MUCH more effectively, in other ways.

I don't claim to be a mnemonist, but I am quite experienced with and knowledgeable of memory training methods. I know quite well the methods that were used by Bruno H. Furst in his courses (published in a number of forms, one of which was a printed course called STOP FORGETTING) and later Harry Lorrayne (with Jerry Lucas) in THE MEMORY BOOK, and on his own in REMEMBERING PEOPLE. None of the methods Furst or Lorrayne taught were original to them, as far as I could determine from researching this field around 1980 when I was asked to do so for a company in New York City called The College Skills Center. The methods are, however, extremely easy to understand and put into practical use for a wide range of applications.

One thing that became quite clear to me at the time was that these methods pay off in inverse proportion to the arbitrariness of the material being memorized: what that means is that when faced with , say, a random or arbitrary list of items, dates, facts, etc., the more random the list, the less conceptual links or "common knowledge" might be involved, the more a person using mnemonics would gain from using these techniques, because otherwise the main option would be. . . {wait for it} some variation on pure rote.

However, less time was needed for memorizing information with more structure, because the "inherent logic" or interconnectedness of the information helped one memorize. Arbitrary or random information (and that's a subjective thing in many cases: what might seem unstructured to me might be very clearly structured to someone more knowledgeable about the things in question) is damnably difficult to hold onto. Getting more that 7 +/- 2 bits of information into short-term memory is challenging for most people, as researchers in the field have known for a very long time. Getting such bits into long-term memory usually requires apply structure where no such structure may have been obvious (or even exists until an individual human mind creates it).

Mathematics already is based on logical and conceptual links. Hence, it is often the case that what needs to be "memorized" in the sense mentioned above is minimal and arbitrary. What sorts of things would be arbitrary in mathematics? Well, things like Order of Operations, which consists of conventions, not something that simply HAS to be. Terminology. Notation. Axioms. Things that do not follow from first precepts.

Even going beyond that, it is undoubtedly true that we need to "memorize" certain fundamental relationships and identities in specific areas of mathematics in order to not have to tediously look them up for every single instance in which they arise. In trigonometry, for example, understanding the definition of sine, cosine, and tangent in right triangle trigonometry is a key "fact" that one does much better to have at one's mental fingertips than not. Hence, Big Chief SOHCAHTOA. However, knowing that tangent = sin/cos reduces three things to two. Knowing about the reciprocal functions and cofunctions reduces a lot of other facts to far fewer. Knowing the angle sum and difference formulas for sine and cosine allows one to figure out sine and cosine and other trig functions for a bunch of "non-standard" angles if one knows and gets the derivation of the quadrant angles and the angles of the 30-60-90 and isosceles right triangles, and how to rotate them through the unit circle. One can also derive the double angle formulas for sine and cosine (etc.) with the angle sum formulas.

My point in the above is that the amount that "must" be memorized is often far smaller than one originally believes (or is falsely led to believe by dull or incompetent instructors), because of underlying relationships and concepts that create natural connections among a smaller set of facts. Anyone who is led to believe that it all needs to be memorized by drill or rote is being mistaught.

Unfortunately, Paul, you never seem to see such distinctions, or if you do, you seem to steamroller over them in your fervor to turn yet another issue that could be reasonably debated into something absolute. In this case, it is, for you, either one hates, loathes and refuses to allow students to engage in memorization of any sort, or one very reasonably indulges in and encourages it.

But such a dichotomy is, quite frankly, utter nonsense. It's impossible to get through mathematics without certain facts. It's also impossible to actually DO mathematics without a lot of concepts and habits of mind. The two areas are clearly interrelated, and only a fool would utterly proscribe learning facts. The key questions in this regard when it comes to mathematics however, are not quite as simple as it might seem. They are: 1) What exactly MUST be known, and 2) How exactly MUST what must be know be learned?

I don't know if you honestly think that everything in the first category must be learned through rote. But I do know that there is a great deal that need not be learned that way, and I think that what a lot of us who find your views off-putting think is that you don't seem to grasp the power for students of NOT memorizing, but rather understanding the linking ideas so that anything not used frequently (for practice is absolutely essential for recalling "raw, arbitrary facts," as anyone who know about memory training will tell you; and that means not only initial practice during the learning, but periodic revisiting/refreshing of previously learned material if one isn't using it regularly through the natural course of one's life) will deteriorate.

On the other hand, that which is understood conceptually has a better chance of lasting, and can be more readily recreated through the concepts even if the "at one's fingertips" recall has been weakened or extinguished. Most people are well aware of this through personal experience. Perhaps you are unique or one of the few who has not figured this out or had it happen (in which case, I can more readily understand your viewpoint, even if I think it's not one that applies to very many people, and in fact need not apply to most people unless they have some actual organic damage that would prevent them from using their memories the way most folks can do naturally.

So while it's lovely of you to talk about "all those anti-memorization reformers," I think they are essentially a myth, more straw people for you to demolish. But the argument isn't "Pro-memorization vs. anti-memorization." It is pro-rote & drill vs. those who wish to minimize the time spent UNNECESSARILY on such an approach where it is not appropriate and where other methods may have far larger payoffs in the long run, especially for the time invested.

Very little in life worth knowing gets memorized with zero effort as we get older (though kids manage to absorb inordinate amounts of information, STAGGERING amounts, in fact, with little or no conscious effort at all). This fact is not unrelated to the Benezet-Kamii arguments, of course. On their view (and mine) traditionalists prematurely try to treat little kids as if they're older, more rigid learners. Stuffing one "absolutely right and best" way of doing arithmetic down little kids' throats is, according to Kamii's research, both unnecessary and for many of them a practice with a negative expected value. I know you, Wayne, and a few others can't accept this, in part at least because that likely SEEMS to run counter to your own personal childhood experiences. The problem is, of course, that first, you don't know how things might have gone had you been taught less traditionally, and you certainly don't know that what worked for you automatically is best for everyone else. You seem absolutely wedded to the notion that everyone SHOULD learn identically, and that that way they should identically learn happens to be the way you did or believe you did, or still do.

How absolutely egocentric of you (collective you here and above, for the most part). How selfish. And how utterly wrong-headed. Ditto when it comes to the issue of memorization through primarily or exclusively rote methods. That just is NOT a sufficient repertoire and it is clearly a huge turn-off, road block, and ultimate mistake for a lot of kids. I realize that it offends you deeply to call rote learning what it is: drill. But it offends a lot more educators to be told that rote is a better way to teach subjects that are fraught with already existing structures that students can use to build understanding (and hence have a framework upon which to hang facts as they acquire them). I know, too, that you will move not one single angstrom away from your entrenched views, no matter how many studies you might be shown, no matter how clear or eloquent my arguments and examples might be, no matter how many experts in cognition and memory might be called forth to explain to you that you just have it wrong for the most part. But some other people here may benefit from reading what I've written, and so I have taken the trouble to discuss this as if I thought you might indeed learn something