Rational Mathematics Education

Won't Get Khanned Again: How Can Education Help Democracy Trump Capitalism?

2011-11-21T17:04:00.000-05:00

The other evening, I saw an article about Salman Khan's latest plans to expand his "education" empire into the world of brick and mortar schooling (how unrevolutionary of him, I must say), and it set me to thinking. One sentence caught my attention in particular:

"They played a 'paranoia' version of the game Risk to understand the theory of probabilities using Monopoly money, where kids trade securities based on the outcome of the game."

There's nothing obviously new about this: there have been teachers offering stock market simulation "games" in various grade bands for decades. So what's the big deal? Maybe nothing, maybe something significant. Here we have Sal Khan, former Wall St. hedge fund analyst (not a job for which I hold a great deal of respect, for some reason, particularly in connection with the education deform movement), giving summer campers, some of whom by his own words, "couldn't see the board" (which I assume means that they were quite young), the opportunity to find out how our capitalist system works. Not that the word "capitalist" or its variants ever gets mentioned of course. But it's certainly Sal's prerogative to ~~indoctrinate~~ ~~proselytize~~ ~~rationalize~~ um, expose students to The Market.

What popped into my head was, "How do educators (in mathematics or any other subject) who are concerned not only about social justice and the obvious inequities (and iniquities) of the current American system give students the opportunity to critically examine the assumptions about what it means to be human that are inherent in our so-called "free-market" capitalist system and how that system impacts our alleged belief in "core democratic values."

For those of you not playing along at home, let me remind you that the reason we've fought wars over the last 60 years in places like Korea, Vietnam, Panama, Nicaragua, Somalia, Kosovo, Iraq, Afghanistan, and Libya (to name just some of the low-lights; for a more thorough list of our military interventions, both foreign and domestic, over the last 120 years or so, go here), is to spread our "democratic core values" and bring freedom to oppressed people throughout the world (or so I've heard it said). Yet, some people lately have been making a lot of noise that includes the notion that we don't really have democracy or anything vaguely like it here at home. And some folks are linking capitalism and the activities of Wall Street, banking, multinational corporations, globalization, and much else that I suspect Mr. Khan finds perfectly fine, to the absence of meaningful democracy, social justice and equity in these United States.

So while I can't stop Sal Khan from expanding his influence into the hearts and minds of our children, particularly since Bill Gates isn't backing my on-line screeds financially or otherwise, nor has the O'Sullivan Foundation offered me $5 million in grant money to spread my vision to physical classrooms, I guess I'm still free (until the Internet comes under complete control of the government and its corporate and oligarchical masters) to try to get others interested in offering kids a different viewpoint. In particular, I invite people to offer ideas (information on and/or links to already-existing projects, speculations on projects that might be, or inklings of possibilities) on non-didactic education (how's that for an oxymoron?) on the human and humane implications and costs of unchecked capitalism.

My initial thought last week was that I wanted a game that allowed students to explore various economic, social, and political arrangements and systems in ways that made it likely (perhaps unavoidable) that players would need to think hard about what it costs people to live and work in our system, not just Americans, of course, but people all over the world, and not just monetarily, but in terms of physical, intellectual, emotional, ethical, spiritual, and other aspects of what might be called human health and well-being. Naturally, the environment in which we live, the planet we inhabit, would likely need to be considered carefully as well.

Lest I appear more completely ignorant than I actually am (which is, of course, quite ignorant of a host of things), I should mention that I'm aware that Bucky Fuller was up to something at least in part like what I'm raising above with his World Game. I'm most certainly not in his league, but I'm thrilled to have discovered in reading about his game ideas that during the 1960s, Fuller several times proposed them as the core curriculum for Southern Illinois University. I'd like to see his college curricular idea and raise him a K-16 and beyond curriculum, one that begin as close to the start of formal education as possible and finds ways to lead students into conversations about the what happens to us when we operate in a capitalist mindset. I think there are fundamental ethical questions and assumptions that kids of school age care about and are quite capable of discussing intelligently.

I do put in that oxymoron about non-didactic education advisedly. There's no question that it's possible to readily create lessons the entire point of which is to propagandize an anti-capitalist moral. That's not what I'm interested in, however. I think that if such a curriculum is to be useful, it needs to be much more open-ended than what being the flip side of Mr. Khan's little market game seems grounded in. But maybe I'm chasing a chimera here. What do you think?

Huh? The ETS Discourages Thinking Again

2011-10-28T14:25:00.000-04:00

Every day, the ETS/College Board kindly e-mails to subscribers an "SAT Problem of the Day": about one in every four is a math problem. Twisted soul that I am, I frequently start my day by answering the problem of the day, and while I'm generally able to nail better than 98% of them, I particularly look forward to the math problems, and even more so to ones that relatively few people who've attempted them have gotten right (yes, I'll admit to still having a sad little place in my soul that is stroked by such silliness).

This morning's poser was definitely challenging for a majority of respondents. As of this writing, with nearly 80,000 folks attempting the problem, only 40% had answered it correctly. Before we continue, here it is for your smoking enjoyment:

The stopping distance of a car is the number of feet that the car travels after the driver starts applying the brakes. The stopping distance of a certain car is directly proportional to the square of the speed of the car, in miles per hour, at the time the brakes are first applied. If the car’s stopping distance for an initial speed of miles per hour is feet, what is its stopping distance for an initial speed of miles per hour?

(A) feet (B) 51 feet (C) 60 feet (D) 68 feet (E) 85 feet

Okey-doke. I'll give you some time to think about it, preferably before you look further down the screen for the official answer and explanation. And it's that explanation I want to address. No peeking now! After all, your self-esteem is riding on how well you do here (not to mention your entire future).

All done? Great. Now before we see the official explanation/answer, let's take this opportunity to do what the folks in Princeton and Berkeley (East and West Coast headquarters of the ETS) apparently don't want us to do: think!

First of all, were this problem to appear in an actual test (despite my enormous collection of past actual SAT tests, I don't recall having seen this one before), my best guess is that it would be one of the last ten problems or so in a section of mathematics problems, based on the 40% correct response rate (I've seen math problems where fewer than 10% of students attempting it got the right answer; as I have no way of knowing what percent of the folks who answered this one today are actually high school students, this might actually be a bit harder for the intended audience than the numbers indicate). It seems fair to suggest that this isn't something that is ridiculously difficult, but that many high school students would nonetheless get wrong for various reasons.

One reason, of course, is literacy: if students read this carelessly or are unable to comprehend the point, they may overlook or simply fail to understand the significance of the words "directly proportional to the square of the speed of the car." If so, one obvious trap answer, a distractor, if you will, that will draw a disproportionately high number of incorrect replies, is (A). It's relatively easy to miss/ignore the word "square" and figure that since 40 mph is twice the speed 20 mph, the stopping distance will be twice that of the slower car and hence 2 x 17 ft = 34 ft.

But if you're on your toes, even if you're not a math whiz you should be suspicious of this answer. It's just too damned easy to come up with. A late-appearing SAT or ACT math problem isn't going to be quite so simple. And in fact, any simple equation you set up here that ONLY involves the given numbers and an unknown will be wrong. That "square" thing isn't put in for show, and it's going to come into play somehow or other.

Sometimes it's possible on problems at this level of difficulty to do more reasoning and eliminate additional choices after avoiding the "big trap" answer. I've not thought this one through further other than to solve it, but perhaps interested readers can offer ideas as to why the other three wrong answers were chosen by the test-makers as possible replies. For me, the problem itself spoke to such a simple solution that I just cut to the chase and answered it. And then, I eagerly awaited learning what the experts had to say as to why my answer was (of course) correct. ;^) So with no further ado, here is the official explanation from the ETS/College Board:

Explanation

The stopping distance is directly proportional to the square of the initial speed of the car. If represents the initial speed of the car, in miles per hour, and represents the stopping distance, you have that the stopping distance is a function of and that , where is a constant. Since the car’s stopping distance is feet for an initial speed of miles per hour, you know that . Therefore, , and the car's stopping distance for an initial speed of miles per hour is feet.

Say what? Gosh, but that seems like a hell of a lot of work to me, and quite frankly, 0.0425 never came into play in my calculations. Having done (way too) many SAT/ACT math problems, when I see something that involves ratios (usually geometry problems, but the underlying issue here is the same), I think about dimensions: am I looking at a linear/linear ratio (if it's a challenging problem, probably not)? If not, is it linear/square? Linear/cubic? Some other ratio where the dimensions change? In this case we're looking at a proportion in which we have a linear ratio (speed to speed) that is being set equal to another linear ratio (braking distance to braking distance), but we're told that the second varies as the SQUARE of the first. So we've got (20/40)^2 which simplifies to (1/2)^2 or 1/4. Thus, the other side of this proportion needs a number on the bottom (conveniently, our unknown) that is 4 times the number on top (17) - of course, you might set things up another way. As 4 x 17 = 68, that's the right answer. Doable, I might add, in your head if you have a bit of facility with mental arithmetic. On the other hand, I challenge you to do the number-crunching in the official explanation in your head. Not impossible, of course, but not exactly as simple as squaring (1/2) and then multiplying 17 by 4 (one way is to take (15 + 2) x 4 = 60 + 8 to get a total of 68.

My point is, of course, that the best place to go for explanations on SAT/ACT math problems is not the test-makers. Considering that these are timed tests, they will NEVER do a "process of elimination" approach, which is necessary when you don't know how to proceed (and when you have to deal with certain question structures, both in math and elsewhere, where it's much easier to eliminate wrong answers than arrive at the right one). They'll never talk about the errors they presume you'll make or the whole concept of distractors. Nope, that would be a bit too helpful. And it would undermine the nice illusion that want us all to harbor that these tests are simply logical extensions of the sorts of things we teach/learn in school.

But of course, a word to the wise guy should suffice: these aren't school tests, never have been, never will be. They're part of a game that's been set up to reward those most adept at cutting through the baloney. It's not absolutely necessary to be able to do that in order to do well, but it certainly helps, particularly on a timed test, where the longer you take to do any given problem, the less you can spend on others, some of which may take more than the average allotted time per problem (about 60 seconds on math). The better you are at cutting to the chase, the more time you buy for the ones that you really need a little more time to figure out, and the more likely you are to actually be able to see and think about all the problems. That's an edge every student needs, but few actually get. And if everyone follows the "official" explanations, few ever will.

Duncan, USDOE Stonewall on His Ties to SALF

2011-10-11T12:52:00.000-04:00

Arne Duncan and Carol Spizzirri

Regular readers of this blog recall that in June of this year I sent an open letter to US Secretary of Education Arne Duncan in an attempt to get his input on open questions surrounding his involvement with the Save-A-Life Foundation and its founder/president, Carol Spizzirri. What ensued was a four-month runaround from various functionaries working for Mr. Duncan which concluded with a classic non-answer answer that advised me to seek the information I requested about Mr. Duncan's views from . . . wait for it! . . . officials in the Chicago Public Schools. How they were supposed to be able to read Mr. Duncan's mind was not specified. But I didn't need to be a mind reader to recognize that I wasn't going to get a meaningful reply from Arne Duncan. Was I disappointed? You bet. Was I surprised that someone who appears to have skeletons in his closet he'd prefer the American public would either forget or never hear rattling in the first place was not forthcoming about them? Not in the least.

Yesterday, Peter Heimlich, who has been vigorously pursuing this story, posted his findings on his blog, The Sidebar:

Haunted by his ties to tainted nonprofit, Education Secretary Arne Duncan ignores questions about $174,000 "phantom" program he arranged for the group

If you've not been following this story here, Peter's piece is probably the best summary of the entire scandal, and even if you have, you would do well to read his expert take on Duncan and SALF.

A Partial Bridge Over Troubled Mathematical Waters: Mumford and Garfunkel Try To Fix US Math Education

2011-08-28T22:50:00.003-04:00

William R. Robinson

William R. Robinson, my former mentor at University of Florida's Department of English, used to say that the further someone was from getting it right, the more useful it is to find something in what they have to say that is “heuristic” (by which he meant ‘thought-provoking’) and that the closer someone is to ‘getting it right,’ the more significant are the ways in which they ‘get it wrong.’

That useful binary construct came to mind again last week as I read NEW YORK TIMES opinion piece by Sol Garfunkel and David Mumford, "How To Fix Our Math Education."

Sol Garfunkel

David Mumford

Much of what Garfunkel and Mumford have to say is praiseworthy. Certainly, their main point - that a 'one-size-fits-all' approach to mathematics curricula is a bad idea - is mostly well-made and sensible. It is also true that many students would benefit by a more practical, applied approach to teaching and learning mathematics (though I would suggest that all students should have the chance to see the beauty of mathematics in itself, to see it as a living, growing, very human body of knowledge that not only has practical power, but also profound and stunning beauty. Neither the traditional American approach to mathematics for the vast majority of its students nor a strictly applied or modeling approach addresses the aesthetics of mathematics).

Nonetheless, my fundamental complaints regarding Garfunkel and Mumford's piece do not revolve around pure mathematics and mathematical aesthetics. Rather, my shock and disappointed came from their failure to debunk some of the assumptions of the current educational deform movement that they raise and then pass over without any examination at the beginning of their proposed remedies.

First, the authors state unquestioningly that

widespread alarm in the United States about the state of our math education. . . can be traced to the poor performance of American students on various international tests, and it is now embodied in George W. Bush’s No Child Left Behind law, which requires public school students to pass standardized math tests by the year 2014 and punishes their schools or their teachers if they do not.

So much nonsense left unexamined in one paragraph. First, how widespread is this "alarm," exactly, and who accepts that the sky is falling when it comes to US mathematics education? While it is true that we are not doing justice when it comes to educating most of our students meaningfully about mathematics, this is not a recent problem and likely can be claimed about any era of American education one cares to examine closely. Readers of Gerald Bracey's work are familiar with the fact that our "Sputnik moments" in mathematics, science, and other areas are not limited to now or the late 1950s. Education punditry going back at least as far as 18th century America has found fault with public schooling, though the claimed consequences of the alleged failures have varied from the decay of the moral fabric of the nation to the current received wisdom that our economy and future ability to "compete in the global marketplace" are at risk because public schools are so poor and our youngsters so uneducated, ignorant, and intellectually lazy.

These notions have been debunked repeatedly by less credulous scholars, from Bracey to Richard Rothstein to Diane Ravitch; many others have or are starting to look more critically at such claims, including the ludicrous notion that economic success or failure depends directly and primarily on the quality of public education. Many have noted, too, that none of our successes are ever attributed by the critics to something that public education may have done right. Teachers, schools, colleges of education, and professors make increasingly-convenient scapegoats for sins that are more obviously attributable to people and institutions far removed from the world of public education and teacher preparation.

But even if there were truth to the causal linking of K-16 education to our economy (and even if one accepts the very arguable notion that the main or sole purpose of public education is to provide career preparation to and the winnowing of young people for the cost-free benefit of businesses - like many other historical facts the punditry and deform machine conveniently ignore, the fact that it used to be the responsibility of companies to train workers for their jobs and to figure out who was best fit for what kinds of work has be neatly pushed under the rug), the notion that comparative performance on international tests of mathematics and science gives sufficient and meaningful information about how well or poorly the schools in a country are doing is simply ridiculous. This is another bit of received wisdom that has repeatedly been debunked by a host of experts and scholars, going back at least as far as the criticism the mathematician Banesh Hoffman leveled in THE TYRANNY OF TESTING, his 1962 critique of standardized tests. For Mumford and Garfunkel to take at face value that these international tests are telling us anything of importance seems like another important opportunity missed.

Finally in the quoted paragraph, we have an accurate description of what NCLB are about: punishing public schools, teachers, administrators, students, their parents, and the communities that are their homes. But not a word of criticism from the authors to this shocking and cynical policy. Nothing about the deeply-flawed mathematics through which the inevitable branding of all schools as "failing," sooner or later, is the consequence, if not the outright goal of all of its authors and supporters. This oversight by two mathematicians is difficult to understand.

As stated earlier, Garfunkel and Mumford do a commendable job of outlining reasons to rethink and reorganize what mathematics should be available to students and how mathematics can be made more relevant and appealing to many young Americans. But then they offer near the end this disastrous assertion:

It is true that our students’ proficiency, measured by traditional standards, has fallen behind that of other countries’ students, but we believe that the best way for the United States to compete globally is to strive for universal quantitative literacy: teaching topics that make sense to all students and can be used by them throughout their lives.

The authors make two egregious errors here: first, they accept that we've fallen behind base on traditional standards. But in fact, many experts have shown that when apples are compared to apples, American students more than hold their own on such tests. The problem has been that the education deform movement has made much of results where we have a broad sample of American students, including those from the neediest, most impoverished urban and rural schools being compared with a far less representative sample of students and schools from many of the countries against which we are being judged and, ostensibly, found wanting. It simply makes no sense to draw conclusions about our schools, teachers, or students based on such invalid comparisons.

Second, Garfunkel and Mumford, sadly continue to accept the notion that we should make educational policy on a national level (a questionable notion in itself that I will not go into here) based on the idea that we educate students to compete (and, of course, conquer) in a global competition. This sort of social Darwinist, survival of the fittest nonsense is a huge key to how the education deformers get their agenda accepted by so many politicians and uncritical members of the public. Not only does it go against many of the basic principles of child-rearing and education, but it also undermines the collegiality among educators that is crucial to how many of the countries (Japan and Finland immediately come to mind) that are held up as "beating" us construct their schools.

Finally, let me make clear again that while I support the authors' suggestions about bringing applied mathematics and modeling much more deeply into what is commonly available to students, I don't think they suffice to give every student a fair chance at numeracy or at a variety of pursuits (not merely jobs) that can emerge from mathematics. I believe that they have properly tried to get more leverage for aspects of mathematics that have been mostly pushed aside over the last century or more in our schools, but they've left out some things of vital importance for enriching students lives. And of course, they've either overlooked or willfully ignored some key assumptions and myths that, left unchallenged, guarantee that the very changes Garfunkel and Mumford advocate will never be seriously considered by the real educational policy makers, let alone actually implemented.

Wrong Again, Jonathan: Mr. Alter Doesn't Get Public Education (and neither does Obama)

2011-08-13T17:21:00.000-04:00

Once upon a time, back in the Ronald Wilson Reagan era, I used to look forward to reading Jonathan Alter's column in NEWSWEEK. He seemed to be one of the guys who got it. I particularly remember what he wrote about Gary Hart during his rise and fall in the 1984 campaign.

Sad to say, Mr. Alter seems to have completely jumped the shark. Every post he makes on education is so blindly wrong, so clearly ignorant of what's going on with the current education deform movement in which Mr. Duncan and, at least passively, Mr. Obama, are willing partners, that as an educator who has worked most of his career with students, parents, teachers, and administrators in districts, schools, and communities devastated by extreme poverty, I cannot help but be appalled.

Looking at his latest education commentary ("Obama Shows Spunk Pushing Brave Education Plan"), there is simply no way that Mr. Alter could be truly in touch with what's going on in places like Chicago, New Orleans, Detroit, Los Angeles, Atlanta, Philadelphia, Boston, Miami, or New York City, given how shallow and erroneous is his analysis of education and what Obama's administration is up to. The recent waiver offer by Mr. Duncan is a classic case of figuring out long after everyone else has done so that a horrid piece of legislation like NCLB is about to crash and potentially bring loads of hard-working people down with it. This impending disaster was completely predictable for anyone who had a grasp of the nonsensical mathematics informing "annual yearly progress" and the absurd notion of 100% proficiency in math and literacy AT ANY POINT IN TIME. But I believe the authors of NCLB for the most part knew exactly what they were doing: creating a way to destroy public education and turn even the most outstanding public schools into "failing schools" through its impossible demands.

When Obama put Duncan in charge, we saw another strategic move, cynically called "Race To The Top," that was nothing more than openly bribing states to comply with anti-union, anti-teacher, anti-student, pro-testing, pro-charter, pro-privatization policies in exchange for the federal dollars, or otherwise be denied the funding to which ALL kids should have equal access, regardless of where they live. It doesn't get any more undemocratic, any less fair, any more patently wrong.

In light of NCLB and RttT, many people have correctly counseled states to refuse to take the waivers being "boldly" offered by Duncan/Obama. In fact, the absolutely sanest strategy for every state, district, administrator, school, teacher, parent, and student would be to say, "No!" to high-stakes testing abuse and insanity, to refuse to allow, administer, or participate in these heinous tests. Please look at the Bartleby Project and join those of us who will not bend to the forces of billionaires and politicians who care nothing about learning, kids, education, or democracy, but only about lining their pockets and those of their friends, colleagues, and masters. The folks pulling Mr. Obama and Mr. Duncan's strings (and one merely need look at how they operated in Chicago with public schools there to know that what's happened at the federal level since they took power was all too predictable) have to be rubbing their hands with glee at the "spunk" being shown in the US Dept. of Education. If this is the best Mr. Obama and his minions can come up with, we need a different nominee in 2012.

The Three Most Important Words in Education: Assessment, Assessment, Assessment.

2011-08-10T21:58:00.002-04:00

Today (August 10, 2011), Alfie Kohn posted a piece entitled, "Teaching Strategies That Work! (Just Don't Ask 'Work to Do What?')"

As I read it (with the usual enjoyment and anger Alfie Kohn's posts elicit from me), I found myself thinking about this paragraph in particular: "Thus, 'evidence' may demonstrate beyond a doubt that a certain teaching strategy is effective, but it isn't until you remember to press for the working definition of effectiveness -- which can take quite a bit of pressing when the answer isn't clearly specified -- that you realize the teaching strategy (and all the impressive sounding data that support it) are worthless because there's no evidence that it improves learning. Just test scores."

In countless arguments I've had on-line with people about education and assessment in general, and mathematics education and testing in particular, invariably my antagonists (and I use that word advisedly) would reject any curricular materials, pedagogical strategy, tool, task, theory, activity, etc., by stating, "Where is your gold standard research that shows that X is effective?" And as night follows day, when pressed, they would make clear that inside that "gold standard" was what for them comprised a platinum standard: only 'objective' (and hence machine-scored, multiple choice tests if given on a wide-scale, or, if the assessment was small and local (e.g., in one classroom), only tests that were scored with no partial credit, no discretionary judgment or rubrics for the scorer, but rather those that had single answers that were either 100% right or 100% wrong, so that the results couldn't be shaped by the scorer (who might, of course, be inclined to be subjective or, even worse, fuzzy!)

Now, I think we all want reliable and valid tests, but I find it intriguing that these folks were SO suspicious of any test that allowed for a "human" factor in the scoring (let alone one that had human factors in the tasks themselves, of course!), and so absolutely convinced that given practicality, costs, and the fuzziness factor they so abhorred, only those nationally-normed, multiple-choice standardized tests would count. They were the true measurement of anything one might wish to measure in education.

Alfie Kohn raises the opposite question: what good are your 'results' if all they are is improving test scores, not learning? And these ideologues I finally gave up arguing with about 15 months or so ago only want to talk about just that: test scores, and a very particular type of test at that. With no wiggle room at all. As an advocate for increased intelligent use of meaningful formative assessment (see the work of Paul Black, Dylan Wiliam, et al.), I find myself realizing with increasing dismay that everything I value about education is precisely what is dismissed by the folks I'm trying to either convince or, yes, defeat in the court of not only reason, public opinion, and school policy, but in the halls of government and the meeting rooms of the monied and powerful. If I and others who agree with my viewpoint are not able to get people to see that improving scores on lousy tests is an utter waste of time by ANY reasonable criteria one might choose to use, then US public education is doomed.

And we cannot afford to let that happen, to allow control to be ceded to self-interested greedy profiteers and people with various political, social, or especially religious agendas that are at odds with our democratic core values. Assessment, what it means, and how we do it isn't the ONLY issue we need to struggle with, but it tends to be the one that touches upon where the rubber meets the road for a lot of folks.
Naturally, I agree with people like Alfie Kohn, Marian Brady, and others who want to focus clearly on WHAT we're teaching and why we are teaching it. There's no getting away from content and the curriculum that frames it. But I think assessment remains key because it is still the one thing that people are guaranteed to pay attention to: kids, parents, teachers, administrators, politicians, media, and the general public. If we can't win the assessment fight, we're in very deep trouble.

Trust Us, We're The ETS (And we have a nice bridge for sale in Brooklyn, too!)

2011-07-24T16:04:00.000-04:00

This was today's (July 24, 2011) SAT question of the day from our good friends at the College Board and Educational Testing Service:

Read the following SAT test question and then click on a button to select your answer.

If a, b, and c are numbers such that a / b = 3 and b / c = 7, then (a+b) / (b+c) is equal to which of the following?

A. 7 / 2

B. 7 / 8

C. 3 / 7

D. 1 / 7

E. 21

It's an old problem. I remember it from the '80s (some of the problems that appear in the Problem of the Day go back as far as the '70s). Nothing particularly wrong with it, nothing particularly great about it. But here's what bothers me, though it doesn't surprise me based on experience. Here's the explanation they offer:

Explanation

From a / b = 3 is implied that (a+b) / b = 4 (1)

And from b / c = 7 is implied that b / (b+c) = 7 / 8 (2)

If we multiply (1) and (2) together, we have that (a+b) / (b+c) = 4 times (7 / 8) = 7 / 2 ( b is canceled out).

Hmm. Well, maybe I don't know high school kids all that well after tutoring SAT math for 30-odd years, but I don't think very many of them would get that explanation or think to do the problem that way, assuming they tried to solve it. And that's one of my gripes with many of the ETS solutions to their own problems: they're generally not the quickest or most intuitive way to get at the answer.

My approach was to solve for a and c in terms of b, the variable that's common to both of the equations. It follows that a = 3b. Substituting for a in the expression (a + b) yields 3b + b = 4b. The solving for b in the second statement gives c = b/7 and substituting for c in the expression (b + c) yields b + b/7 or 8b/7. So the entire expression (a + b) / (b + c) in terms of b is 4b/ (8b/7). Indeed, the b's simplify to 1 (it's implied in the original problem that none of the variables are zero), and the rest simplifies to 4 * 7/8 or 7/2.

Maybe that's not "better," that what ETS offers, but I think it's more clear as to what's going on and why the approach works. The ETS explanation feels a bit like magic to me, looking at it from a student's perspective. That kind of thing in textbooks always frustrates me, and I know it loses a host of kids at the starting line. While this sort of answer from them may not be their most egregious sin, it is one example of why I warn students to not assume that when ETS explains something in math (or anything else) that it's always going to be coin of the realm. It's not wrong, but how helpful is it, really, for most kids?

Open Letter to Arne Duncan Follow-Up

2011-06-02T15:00:00.001-04:00

Ronald McDonald, Carol Spizzirri, Arne Duncan

Yesterday, I posted here an open letter to US Secretary of Education Arne Duncan about a burgeoning scandal involving the Save-A-Life Foundation, its founder, Carol Spizzirri, and Chicago Public Schools under the leadership of Mr. Duncan.

Here are follow up links to a blog post the late Gerald Bracey was working on when he died in 2009, examining this same troubling situation.

First, two posts by Susan Ohanian: AN OPEN LETTER TO US SECRETARY OF EDUCATION ARNE DUNCAN.

Second, her original post of Bracey's column on Duncan and Save-A-Life:

News delayed is news denied.... The Skeleton in Arne Duncan's Closet

Ohanian's post of Bracey's column was picked up at the same time at two other sites, SUBSTANCE and SCHOOLS MATTER.

Finally, here are two videos from Chicago's Channel 7 I-Team investigation of the Save-A-Life Foundation, "The Maneuver, Part I" and "The Maneuver, Part II."

An Open Letter To US Secretary of Education Arne Duncan

2011-06-01T06:00:00.020-04:00

Click here for supporting documents regarding the following.

Michael Paul Goldenberg
6655 Jackson Rd Lot #136
Ann Arbor, MI 48103
(734)644-0975
mikegold@umich.edu
http://rationalmathed.blogspot.com

June 1, 2011

The Honorable Arne Duncan
US Secretary of Education
Department of Education Building
400 Maryland Ave, SW
Washington, DC 20202

Dear Mr. Duncan:

I'm a mathematics educator working in an urban public school district. On my blog, I'm reporting about the Save-A-Life Foundation (SALF), an Illinois nonprofit with which you were associated when you served as CEO of the Chicago Public Schools (CPS). I'd greatly appreciate your answers to two brief, but serious questions.

You'll recall that SALF's charter was to provide in-class first aid training to students. According to an October 11, 2009 Chicago Tribune article, SALF founder/president Carol J. Spizzirri claimed “2 million children took the classes, many of them from the Chicago Public Schools.”

According to news reports, press releases, and other records, from 2003 through late 2006 you lent your support to Spizzirri's organization in various ways, including appearing as an animated cartoon character on SALF's website.

Subsequently, a November 2006 ABC7 I-Team story reported that SALF and Spizzirri engaged in a variety of serious misrepresentations. In that broadcast, you yourself raise doubts about SALF's claims. Since then, the organization has been the subject of dozens more media exposes including an October 11, 2010 article in The Hill reporting that SALF was under investigation by the Illinois Attorney General's Charitable Trust Bureau. An investigation by the US Centers for Disease Control and Prevention also appears to be underway.

Here's why I'm writing. In response to a federal court subpoena and FOIA requests, the only records produced by CPS indicate that at best a few dozen students ever participated in SALF training classes. As result, Chicago Schools Inspector General James M. Sullivan has been asked to investigate what happened to approximately $62,000 CPS awarded to SALF, most of which was arranged by you. Records show that you contracted with Spizzirri to provide first-aid training for approximately 18,000 students from 2004-2006. You signed off on $49,000 in CPS funds and Ronald McDonald House Charities provided an additional $125,000, making a total of $174,000 paid to SALF for what appears to be a program that never happened.

Given the facts, do you think Inspector General Sullivan should proceed with an investigation? And would you co-operate with such an investigation?

Thank you for your consideration and I look forward to receiving your answers.

Sincerely,
Michael Paul Goldenberg

cc:
Justin Hamilton, Press Secretary
US Department of Education
Peter M. Heimlich, MedFraud.info

You Say You Want A Revolution? Try Some "Inconvenient Truth" About Deformers

2011-05-22T09:11:00.002-04:00

Brian Jones and Julie Cavenaugh:
Two Courageous Teachers

I had the pleasure and privilege of attending the premiere of THE INCONVENIENT TRUTH BEHIND Waiting For Superman on Thursday night and hearing the excellent panel discussion afterwards. This is a movie that, as Diane Ravitch said, needs to go viral. Go here to request a copy: http://www.waitingforsupermantruth.org/ Consider showing it wherever you can. And think about this: we're individuals, working collectively, to fight a small number of billionaires and their pawns and puppets. There are vastly more of us than there are of them. Our futures and those of our children and grandchildren are at stake. The very essence of democracy and the role of free public education therein are facing serious threats, not from foreign forces or terrorists, but from corporate interests and the super-rich. Every major US city, particularly where there are a lot of poor and minority people, faces a direct threat from privatization, but ultimately, if the deformers are allowed to win, they will spread their poison everywhere.

Consider making versions of ITBWFS that focus on your city or region. LAUSD, Chicago, Washington, DC, Baltimore, Detroit, New Orleans, Philadelphia, St. Louis, and Miami are some of the most immediate, obvious targets, but no doubt there are many others.

Diane Ravitch said on Thursday that no billionaires are coming to save us or do the hard work for us. We have to do it ourselves. And it's about time we did.

Fear and Loathing in Calcville: Who Makes Kids Anxious About Math?

2011-05-19T11:57:00.006-04:00

Recently, another study (Researchers Probe Causes of Math Anxiety: It's more than just disliking math, according to scholars) has appeared proposing to explain the causes of mathematics anxiety. It shows up as part of a book called CHOKE: What the Secrets of the Brain Reveal About Getting It Right When You Have To, by Sian Beilock. If what's in the article is an accurate depiction of what the study has to tell us, there's not much new to see.

On my view, math anxiety is obviously not something many people, if any, are born with: for the most part, we catch it from others. However, it is worth noting that there are many carriers who are not themselves suffering from the disease. Contemptuous, arrogant mathematics teachers can readily drive someone into math anxiety, and frequently do, I strongly suspect. So can rigidity about what doing and being "good at" mathematics entails. Given how most US teachers present the subject in K-12, math is only or primarily the following: calculation, arithmetic, and speed (with accuracy, of course).

Yet none of those things are particularly what mathematicians deal with. No mathematician is judged by speed of calculations - arithmetic or otherwise. Calculation may not even be a particular strength of a professional mathematician. Mathematicians by and large deal with abstractions, patterns, connections. Of course, some deal with applications of mathematics to sciences and engineering and other "real world" problems and situations, some straddle the territory between "pure" and "applied mathematics," and most couldn't care less whether what they work on has applications beyond mathematics itself. Calculation isn't their interest and they know that when it comes to pure calculation, it's hard to beat a computer for speed and accuracy. They also know that the computer won't offer insight, leaps of heuristic thinking that connects seemingly unrelated ideas in two or more areas of mathematics, or the recognition of underlying structural similarities, etc. While by definition computers excel at computation, the fact remains that they don't think or "do" mathematics.

Unfortunately, neither do most American schoolchildren after a few years of exposure to what accurately should only be called "school math." Is it any wonder that, confronted in early elementary school with high-pressure tests that demand the calculation of 100 arithmetic problems (mixed or not) in 3 to 5 minutes depending on the teacher or school, many students just bail out of mathematics for the rest of their lives? The "stand and deliver" approach may work for those kids who happen to be quick at the given task demanded of them (I was one such kid) and enjoy the concomitant competition, but for many that's the fast track to tuning out mathematics permanently.

Of course, I was no more doing mathematics when I crunched all those numbers quickly and accurately than is a computer today when it does in a nanosecond what it took me a few minutes to complete. It took me close to another thirty years to find out what mathematics actually is about. And I'm one of the lucky ones: I stumbled into more useful viewpoints about the subject, along with learning a reasonable amount of mathematics. Most Americans don't: not because they were born deficient in the ability to do and appreciate "higher" mathematics, but because they were denied the opportunity to get anywhere near higher mathematics due to an approach to the subject that is demeaning, alienating, and clearly grounded in some sort of bizarre notion of competition and "winnowing wheat from chaff." Who knows how much mathematical talent is wasted every day in our country due to such absurd notions of the subject and its teaching? For how much longer can we afford to tolerate such an anemic view of this vital, powerful, and - dare I say it - beautiful discipline?

Education Notes: I'm Boycotting change.org due to deceptive Rhee petition attacking teacher protections

2011-05-01T07:16:00.000-04:00

Education Notes: I'm Boycotting change.org due to deceptive Rhee petition attacking teacher protections

GFBrandenburg's Blog

2011-04-28T12:20:00.000-04:00

GFBrandenburg's Blog

Regrets & clarification…after signing one of Michelle Rhee’s petitions « Parents Across America

2011-04-28T12:18:00.000-04:00

Regrets & clarification…after signing one of Michelle Rhee’s petitions « Parents Across America

Another "Load" From A Familiar Source: Sweller and Friends Try To Deceive Again

2011-04-10T14:07:00.000-04:00

John Sweller

Paul Kirschner

Richard E. "Dick" Clark

The trio pictured above are Australian educational psychologist John Sweller (known particularly for "cognitive load theory," Paul Kirschner, a professor of educational psychology based in Holland, and Richard E. Clark, an educational psychologist and clinical research professor of surgery at USC. These gentlemen have written several articles that intend to show that progressive, discovery-oriented, student-centered approaches to mathematics education are not viable. The first such article that caught my attention is their 2006 EDUCATIONAL PSYCHOLOGIST piece, "Why minimal guidance during instruction does not work: An analysis of the failure of constructivist discovery, problem-based, experiential, and inquiry-based teaching."

Note, please, the subtle, intellectually modest language of that title. It isn't that such instruction may be in some ways flawed, in need of refinement, or in any way worth employing. No, in the view of these good professors, it DOES NOT WORK and is a FAILURE. And what do they propose we should use in place of this list of approaches? Not to keep you on tenterhooks, it is, of course, direct instruction. In the words of Hamlet to Polonius, "My lord, I have news to tell you. When Roscius was an actor in Rome. . . "

Apparently, however, the 2006 article did not suffice to remove the scales from everyone's eyes. So in 2010, this stalwart band of clear-eyed thinkers saw fit to address their ideas directly to the American Mathematical Society, one of the two organizations of professional mathematicians in the United States. While their previous piece was twelve pages long, "Teaching General Problem-Solving Skills Is Not a Substitute for, or a Viable Addition to,Teaching Mathematics," last year's opus took but two.

Again, the authors make no pretense of intellectual modesty: not only is teaching problem-solving methods not a replacement for teaching mathematics (who would disagree?), but there's no benefit from teaching problem-solving methods at all! Polya be damned, it turns out that these ed psych guys know WAY more about how mathematicians do mathematics than did the world-class mathematician known for having dedicated a sizable amount of work to how to solve mathematical problems and develop heuristic methods for improving students' problem solving.

In this recent article, we are treated to the following:

Recent “reform” curricula both ignore the absence
of supporting data and completely misunderstand
the role of problem solving in cognition.
If, the argument goes, we are not really teaching
people mathematics but rather are teaching them
some form of general problem solving, then mathematical
content can be reduced in importance.
According to this argument, we can teach students
how to solve problems in general, and that will
make them good mathematicians able to discover
novel solutions irrespective of the content.

That's some heady, alarming stuff indeed. The only problem is that it has no foundation in reality. And that is likely why our heroes are able to offer not a single citation to tell us who, exactly, is making the argument with which they wish to further bash any approach to mathematics teaching that isn't business as usual.

Now, if I were not a veteran of the history of the Math Wars and someone told me that there were folks who believed that we could improve mathematics education by reducing the importance of mathematical content, I'd be alarmed. And I suspect that a majority of readers of AMS Notes are generally ignorant of the specifics of the two-decade-long fight between a small number of educationally conservative mathematicians and the leadership of the mathematics education research community (though, of course, there are more than two sides and there are many more players on the two main sides than those subgroups I've mentioned). Like previous conservative efforts in various mathematics publications to scare the bejeezus out of the community of working mathematicians, this one is intended to convince people that there are some really crazy folks who want to take the mathematics out of mathematics education. It's got all the appeal of the usual efforts to win a political, ideological war through sound-bites rather than facts. And like the vast majority of such efforts, it just ain't so.

Given that fully 1/4th of the 12 pages offered in 2006 were references, it's noteworthy that there isn't a single citation to let us know who it is that's making the "argument" we're supposed to be so contemptuous of: downplay mathematical content because we can teach problem-solving methods absent actual mathematics! Well, not to put too fine a point on it, but were anyone making that argument, I'd be in the forefront of those pointing out its absurdity. The reason I'm not leading the parade, however, is because in more than twenty years of work in mathematics education, I've never seen evidence that anyone believes anything of the kind. To suggest that our ed. psych. friends are using the straw man technique is to underrate the outrageousness chutzpah that goes into writing something founded completely on myth (no little irony in the fact that one of the authors, Professor Kirschner, puts himself forth as a debunker of "intellectual urban legends." Apparently, that gives him license to sign on to promulgating an egregious whopper of his own).

It's not that these academics are supporters of direct instruction uber alles that makes them so dangerous. It's that they appear willing to simply make things up in order to try to rid the world of all competition to their favorite pedagogy. While repeatedly claiming that educators and learning theorists who take issue with direct instruction or, to the dismay of our heroes, dare to advocate for other sorts of instruction have NO evidence to support their views, Sweller, Kirschner, and Clark are hardly above making unsubstantiated claims about ghostly demons who believe things one only reads in the writings of. . . well, people like Sweller, Kirschner, and Clark. Further, they ground their own work in the usual "gold standard" sorts of laboratory research that generally seem to have nothing to do with what goes on in actual classrooms, while complaining that their adversaries aren't doing the same thing.

Without wanting to once again go over all the reasons that educational research in the field differs dramatically from educational and psychological research in the laboratory, I'll simply point out that the kinds of things these fellows tend to base their arguments upon are either disconnected from what is feasible in schools and classrooms, or are at least as questionable as the ideas and practices of which they are so contemptuous (see, for example, the chess analogy in their 2010 piece).

Why is it that critics of progressive ideas in education (particularly those grounded in respect for students' interests, their need for ownership of their own learning, and their desire to be listened to and taken seriously, as well as those designed to promote democratic values and build skills necessary for actively participating in democratic societies) are so quick to load the dice when they "critique" those ideas? Why, too, do so many of them seem to operate with the same sorts of rhetorical tricks and propensity for utterly dismissing everything connected with educational methods at odds with their own? Is it really necessary, I wonder, to claim that progressive ideas in education are UTTERLY without merit or application in order to question and view them skeptically?

I am increasingly convinced that absolutism is a common thread amongst anti-progressives whether regarding education or just about anything. The lack of intellectual modesty and humility is to be expected from poltical pundits these days, but academics are supposed to show a little more restraint than Glenn Beck or Rush Limbaugh. The more I read from Messrs Sweller, Kirschner, and Clark, however, the more I expect to see them getting a weekly show on Fox News.

No matter how cynical you become, it's never enough to keep up:

2011-03-10T15:19:00.000-05:00

Ordinarily, I try to keep the focus here directly on mathematics education issues, but it's impossible to view what's going on in this country right now without seeing how a host of threads come together through almost any lens one chooses to look.

The situation in Wisconsin and the rhetoric being deployed by the anti-worker governor and his supporters would be funny if the stakes weren't so high and the incredible hypocrisy and cynicism so deep and real. However, there are times when it seems that the only source that completely captures just how unprincipled and transparently dishonest such influential sources as Faux Snooze are is THE DAILY SHOW. After viewing one of Jon Stewart's most recent broadcasts, I feel that there's really nothing more that I need say than to urge you to watch. If you have any doubts as to which side is lying through its teeth every time it says anything, just watch this Thursday, March 3, 2011 excerpt:

Daily Show: Crisis in Dairyland - For Richer and Poorer - Teachers and Wall Street

A Must-Read Book on mathematics education

2011-02-13T15:23:00.000-05:00

Derek Stolp

I am nearly finished reading a truly remarkable book about mathematics education in the US, what we can do about it to make it more effective and meaningful, and a call for a return to democratic core values in our entire approach to schooling (okay, I'm not sure the author goes quite that far explicitly, but let me do so for him if he doesn't). The author is Derek Stolp, pictured above, and the book is MATHEMATICS MISEDUCATION: The Case Against A Tired Tradition.

It's a definite must-read if you are concerned about just how off-base both traditional and many reform efforts are in US mathematics education. For example, Stolp makes a very telling point about NCTM's commitment to real-world mathematics in his examination of the article content in THE MATHEMATICS TEACHER from 2002, the year he was writing his book. Unsurprisingly, to me at least, he found a rather dramatic mismatch between NCTM's public commitment in, say, PSSM to real-world connections and the focus of the vast majority of articles that appeared two years later in MT. I suspect that a similar study regarding the productive, creative use of technology in mathematics teaching would show even greater disconnections between NCTM's talk and walk. But they are hardly alone in this regard.

Stolp has a lovely website with a lot of curricular ideas that jibe with his philosophy of mathematics education. It looks like it could prove to be an excellent resource for teachers who want to create a similar sort of mathematics teaching and opportunity for students in their classrooms.

I have only one quibble thus far with Mr. Stolp: at several points in his book, he mentions as an example of the difference between conceptual and procedural understanding the importance that students "get" that multiplication is repeated addition. Regular readers of my blog will recognize that those are (nearly) fighting words to my ears. But, hey, I don't expect perfection from anyone, not even myself. ;^) So even if Stolp is at least temporarily in the MIRA camp, I hope he will come to see the limitations of that viewpoint, and even if he doesn't, he's got way too much of value in his writing for me not to promote it here.

Finally, I'd love to see what Derek Stolp and Dan Meyer would make of each other's work and viewpoints. I see the potential for a very productive exchange and potential collaboration between them. But then, I always tend to think synergistically when it comes to math and math teaching: probably comes from knowing how little I'd know in this domain if it weren't for the brilliance and inspired work of a host of other people.

It Ain't Just Claptrap: Can We Meaningfully Critique Our Schools?

2011-01-31T16:24:00.000-05:00

On an education discussion list I follow, Jonathan Groves made that I will not reproduce here, in which he was critical of many aspects of US math and literacy education, based on what he sees in his own teaching at the college level. One reader of the list replied to Jonathan's post, "This is just a repetition of all the claptrap, generalizing, and stereotyping [this list] is supposed to combat."

I think the situation, particularly from the perspective of mathematics educators, is a bit more complex than would allow the dismissal of Jonathan's post as "claptrap."

The fact is that if you spend time in classrooms where things are not going well (for a host of reasons, not merely the short list that the deformers cite, if those apply at all), you can't help but be frustrated at how poorly we are presenting mathematics to a vast majority of students there. In that context, I speak of places like Detroit, but that list includes many other poverty-stricken places, large and small, in this country.

Next, looking at many classrooms in less economically depressed districts, one sees teachers who don't know the mathematics they're expected to teach (particularly, but not exclusively, at the K-5 level); those who 'know' the math, but don't know nearly enough about how to teach it well and effectively to any but a small number of so-called "mathy" kids - generally those who already like math, particularly when math is reduced to quick, accurate calculation; and then the lovely but all-too-rare cases of inspired teachers allowed to do their jobs without absurd shackles placed on them by national, state, or district tests and other idiocy that has little or nothing to do with mathematics or education.

We also know that this overall inadequacy doesn't result in an under-supply of mathematically competent folks. There's no shortage of mathematicians, economists, engineers, physicists, etc. That was made crystal-clear in THE MANUFACTURED CRISIS, but those numbers are ignored by the deformers, the anti-progressives in the Math Wars, by Obama and his "Sputnik moment," and many others.

So the question becomes much more an ethical one about whether to teach mathematics better to more kids for its own sake (the pleasure, power, and beauty of mathematics), for the sake of equity (everyone has a right to become mathematically competent and rise to whatever heights she desires and is capable of reaching), and for reasons of core democratic values (an innumerate citizenry is one easily deceived by politicians, demagogues, advertisers, and other scam artists). It is NOT a question of "saving" the economy with a new wave of math and science folks. So many jobs in the predictable future will NOT require all that much math or, for that matter, what's typically viewed as a college education, though competition and raising the bar may make folks have to have college degrees to get jobs that don't really require much of what colleges generally teach and students generally study there (in other words, even if employers choose to make a college degree a basic requirement for a service job, that doesn't mean the course of study will better prepare anyone to DO that job).

What this all boils down to, for me, is the issue of criticism from the right versus criticism from the left - to use a crappy metaphor - when it comes to public education. Not being of the educational deform mindset, my back goes up every time I read or hear an attack on our public schools that comes from Duncan, Klein, Rhee, the usual think tanks and foundations, the typical education reporter, ad nauseum. At the same time, I don't think our current public educational model is sound, I don't think it's been sound for a long time, and I think we can and must do better.

The problem becomes how to level correct, meaningful, constructive criticism at the system that leads to real change without throwing in with the deformers or inadvertently winding up supporting their causes (it's rather unlikely that they will help ours, no matter what their rhetoric about choice, accountability, raising the bar, and a host of other catch phrases and buzz words).

It becomes exhausting to have to keep repeating that I don't want to see our public schools dismantled or privatized, that I don't want to see teachers sacrificed on the altar of real or phony economic shortages (and misplaced priorities), that I "get" why we have teachers' unions, that teaching is a very difficult job, etc., etc., and that I STILL know that there are many crappy schools, crappy teachers, crappy administrators, crappy tests, crappy politicians with their fingers in the education mess making it worse, and a bunch of greedy asshats trying to suck away billions from kids into their own pockets, all the while singing psalms about global competition, accountability, 21st century skills, and so on.

Your mileage, of course, may vary.

A MUST read: "PISA: It's Poverty Not Stupid"

2010-12-24T11:54:00.000-05:00

"PISA: It's Poverty Not Stupid" by Mel Riddile

http://tinyurl.com/povertyandPISA

My comments:

The sky isn't falling in US public schools, folks. Some readers likely already knew that, particularly those who are followers of the late Gerald Bracey's work, amongst other debunkers of educational disinformation.

The results of the recent PISA exams highlight this fact, but you wouldn't know it from US Sec. of Education, Arne Duncan or any other doom-sayer, bloviator, pundit, or deformer. And with good reason. Because what the numbers say is disturbing, but not because our schools are failing us.

The facts are that our schools are amongst the best in the world, but to see it you have to compare apples to apples. And what's being done with the PISA scores for the most part is what's been done with so much other data: the wrong things are compared and the apparent results make our public schools look like they're at best doing a mediocre job.

What is being missed or hidden by the 'experts' who want to convince us to fire teachers, bust unions, turn public education over to Wall Street, promote charters, hand out vouchers to parents (particularly rich and upper-middle class ones), and generally dismantle our public schools in order to turn them over to drooling private market entrepreneurs and (many) charlatans? The fact that the US has an enormous disparity between rich and poor compared with other industrialized nations, and the impact of poverty on the average scores. But disambiguate scores so that we compare similar economic strata across nations and suddenly we're just where one might guess: Number One.

But please, don't take my word for it. Read a detailed analysis. Then consider why Duncan, Rhee, Klein, and so many others are SO invested in convincing you that it's our ENTIRE nation's schools that are in crisis, at risk, failing, collapsing, corrupt, incompetent, bleeding money, and all the rest of it. Why they look at the public schools alone as having failed to solve the effects of severe poverty in rural and urban settings alike. Or in other words, why they don't want you to realize that poverty and greed at the top of our economic system, not teachers' unions, is what's keeping the folks at the bottom from lifting themselves by their non-existent bootstraps.

http://tinyurl.com/povertyandPISA

Another Cartoon Worth Tens of Thousands of Words

2010-10-28T11:58:00.000-04:00

Does Arne Duncan Have A Soul?

2010-09-20T10:19:00.001-04:00

"Not even this much of one."

You'd think that after the recent infamnia the LA TIMES perpetrated against teachers, Arne Duncan, the US Secretary of Education with no background or credentials as an educator, would have had the good sense to either repudiate this clueless act or at least keep his mouth shut about it. Instead, he outdoes himself with the following:

U.S. Secretary of Education Arne Duncan last week urged school districts across the country to disclose more data on student achievement and teacher effectiveness, saying too much information that would help teachers and parents is being kept out of public view.

The education secretary told an audience in Little Rock, Ark., that schools too often aren't disclosing data on student achievement that could not only help parents measure teachers' effectiveness, but also help teachers get better feedback.

Mr. Duncan said his remarks were prompted by a Los Angeles Times series analyzing teacher performance through value added scoring to show which elementary teachers were helping students make the most gains. The secretary said he was not advocating posting the results online, as the Times plans to do, but he urged transparency.

From Education Week [American Education's Newspaper of Record], Wednesday, September 1, 2010, Volume 30, Issue 2, p. 4. See http://www.edweek.org/ew/articles/2010/09/01/02brief-3.h30.html?r=1277642202

So what, exactly, is the difference between posting the results of misleading or meaningless data on-line and Mr. Duncan's vision of "transparency"? As he apparently offered no specifics, it's impossible to know how he distinguishes what the LA TIMES did and plans to do from something else. But of course, the problem isn't transparency versus secrecy. It's between meaningful data and utter bullshit. And using student scores on questionable tests and a phony formula with no credibility to rank teachers in terms of "effectiveness" is simply the latter: complete, utter, political bullshit.

How many times are we going to be asked to swallow the patent lie (speaking of 'transparency') that any of these pols and pundits are interested in learning about what kids actually know and can do, or in seeing that TEACHERS (as well as parents, kids, and other stake holders in education) get useful data that will potentially lead to more effective instruction and great student learning and achievement? For in fact, were that the goal of the folks at the LA TIMES, at various right-wing foundations,at the US Dept. of Education, or anywhere else where self-righteous 'experts' wring their hands over the results of fraudulent excursions into experimental statistics and psychometrics, then they would be calling for a very different sort of assessment and holding their peace until such assessments were being give and data collected from them. Further, they would all demand that the data be collected in ways that gave teachers, students, and parents specific feedback on each and every relevant data point: kids and parents would know how the student did on each item, what his/her answer was, what are the likely weak points or areas of confusion in the subject based on the answers, and what is recommended for that student to improve; teachers would receive similar data, but for both individual students and classes as a whole, as well as expert recommendations on how to address the weak areas. Something might actually happen to make things better. But such is not the case, nor is it likely that it will be until at LEAST 2014 when the folks being paid to make better tests to fit the Common Core Standards roll out their first products.

It remains to be seen what those "better tests" will look like, given the much higher cost in time and money that creating, administering, and particularly grading non-multiple choice, non-short answer, non-true/false items of a performance-task nature entails. Creating ANY good test item is challenging, but creating test items that actually tell us what we need to know to improve teaching, learning, and parenting when it comes to academic subjects is a major challenge. If the deform crowd is seriously committed to these goals (as opposed to merely paying them lip-service and instead focusing upon destroying teachers' unions and public schools in order to promote profit-based, private takeovers of public education - quite frankly precisely their real goals, on my view - then they must publicly and privately commit to paying the price to create excellent assessment and seeing that only such instruments that pass reasonable professional and public scrutiny are used for "high-stakes" purposes.

I'd prefer, of course, to see the whole notion of high-stakes testing interrogated with as much care and brutality as the pundits and deformers have been using on kids and teachers. I've said on multiple occasions that as things stand, the only fair way to go if we're going to stick with the multiple-choice nonsense and weak 'student-generated' and 'free-response' questions that dominate the current crop of high-stakes tests is to demand that the pundits and pols who advocate and vote for these tests be made to take the things themselves and allow their scores to be circulated on the internet and published in newspapers. Until then, I doubt that many of them, even those with truly good intentions, will start to look closely at what's being tested, how it's being tested, and what practical uses the results of such tests can be put to. If they really pay attention, they might start to see how antithetical to the alleged purposes of improving education - i.e., teaching, learning, parenting - these instruments are in practice.

Meanwhile, Mr. Duncan will no doubt continue to alienate the vast majority of educators with his ham-handed, anti-teacher proclamations. It would be lovely to see him placed under the same sort of microscope and held to the same sorts of standards he advocates for teachers. It would be more lovely still if President Obama would get his head out of his behind regarding education. For all his own experiences, none of which had a bloody thing to do with the sorts of garbage he and Duncan have been pushing on our nation's public schools, Obama seems purblind about education. While I didn't grouse when the Obamas chose to send their kids to Sidwell-Friends School, I'm now starting to wonder if a dose of ordinary reality isn't just what the doctor ordered for our "socialist" president.

Clearly, he's not a stupid man. Can he really believe that the more we test kids, the more we scapegoat teachers, the more we put weapons into the hands of privatizers and right-wing education deformers, the more we bully and bribe states, the better things are going to be for kids? For the country as a whole? For the future of American democracy? Or has that never really been the point of public schools?

The LA TIMES Cracks Out of Turn When It Doesn't Know The Shot

2010-08-16T01:13:00.000-04:00

David Mamet*

I'm not sure what I was doing or where I was the day the LOS ANGELES TIMES went from being a newspaper to being a national leader in evaluating teacher quality. Perhaps it was supposed to be kept secret, like what was discussed and said at the meeting Dick Cheney had with Big Oil executives a decade ago. If so, several TIMES reporters have blown it with a recent article, WHO'S TEACHING LA'S KIDS?

In it, three reporters, Jason Felch, Jason Song and Doug Smith, present ratings of "teacher effectiveness." In particular, they single out one particular fifth grade teacher, John Smith, and claim he is the least effective teacher for his grade level in his school. Mr. Smith's photo is at the beginning of the article that purports to know, based on kids' scores on administrations of a single standardized test, which teachers are helping their students, teaching effectively, and making a positive difference, and, of course, which, like Mr. Smith, are allegedly failing to move their students ahead.

I wrote the following to these reporters today and will be fascinated to see if any of them respond. I know that were I John Smith, I'd be speaking to my attorney and considering lawsuits against several parties, not the least of whom are Jason Felch, Jason Song, and Doug Smith. As I am not, the best I can do is try to point out how wrongheaded, how irresponsible, and how ultimately counterproductive is both their article and the methods they employ to smear the professional integrity of many fine teachers who for any number of reasons may not "measure up." The professional integrity I call into question, however, is that of these reporters, their editors, and others who profit from the publication of this sort of cheap-shot, ignorant journalism.

If you start with the absurd assumption that multiple-guess
standardized test scores tell us anything (let alone EVERYTHING) we
need to know about teacher effectiveness or student learning of
subject matter or all the other things that teachers and schools are
about (not all of which are good, but that's another debate entirely),
then it follows that the LA TIMES is as qualified as anyone else with
no expertise whatsoever in psychometrics to determine which teachers are "most
effective" and which are "least effective." Further, with the same
starting assumption, there's nothing unconscionable about reporters
and editors from that noble publication choosing to print a photo of
a so-called "ineffective" 5th grade teacher and include the following
in the article:

Yet year after year, one fifth-grade class learns far more than the
other down the hall. The difference has almost nothing to do with the
size of the class, the students or their parents.

It's their teachers.

With Miguel Aguilar, students consistently have made striking gains on
state standardized tests, many of them vaulting from the bottom third
of students in Los Angeles schools to well above average, according to
a Times analysis. John Smith's pupils next door have started out
slightly ahead of Aguilar's but by the end of the year have been far
behind."

But if the assumption is false, then what the TIMES and its reporters have done is to pillory one 5th grade teacher on the wheel of meaningless test scores. They have, in fact, violated two fundamental principles of psychometrics: never use a test designed to measure one thing (e.g., student achievement) to measure something it was not designed to measure (e.g., teacher effectiveness), and never use a single test score or measurement type to draw definitive conclusions (particularly not in the social sciences). Further, they have made the fundamental error of assuming that correlation (Teacher A's kids scores are higher than Teacher B's scores) equates with causation (Scores rose primarily BECAUSE of the superior teaching skills and methods of Teacher A).

In fact, the above-cited article is so fraught with error and leaps of logic (and bad faith) as to be utterly, irredeemably worthless, not unlike the test scores upon which its false (and probably libelous) conclusions are based. But then, the article's authors began with a patently incorrect assumption, and they very likely had its conclusions well in mind to begin with.

So I am moved to ask: may we expect in the near future an article by the same reporters on which LA TIMES journalists are "most effective" and "least effective" based on how sales of the paper are impacted by their articles and reportage?

May we expect that the reporters will be taking, say, the tests given to high school kids in LAUSD (I assume all these journalists graduated from college) or perhaps the SAT or ACT (or, Darwin forbid! the GRE) and publishing the results in the paper? How about the politicians who pushed and voted for using these tests as fair measures of a host of things they were never designed to assess? (And I include in that list not only state and local officials, but every US senator, congressperson, US Department of Education secretary, and every US president from William Jefferson Clinton to George W. Bush to Barack Hussein Obama who has supported these tests as the measure of all things.)

Let's shed more sunshine on the test-based competence of our reporters and politicians. Publish and publicize their scores. Threaten, meaningfully, to hoist these folks by their own petard and we'll see some critical examination of the assumption that the tests are valid and reliable, as well as adequate measures alone of effectiveness or lack thereof. My suggestion is no worse than what politicians and reporters are doing now with kids, administrators, schools, districts, and, of course, everyone's favorite scapegoat, public school teachers.

Until such time, shame on Messrs. Felch, Song, Smith; shame on the LA TIMES. Shame on anyone and everyone who buys into the ridiculous test-mad nonsense that has this country by the throat.

Michael Paul Goldenberg

p.s.: Lest someone suggest otherwise, my last official GRE scores, taken in October 1991, are Verbal 800; Mathematics 780; and Logical Reasoning 720. I feel safe in putting those scores up against those on any comparable test of the three reporters, any member of the LA TIMES staff, any current legislator in California or the United States Congress. I've spent over 30 years preparing students for various standardized tests and debunking many of the myths surrounding them. I'll happily meet anyone on the standardized test battle ground, No. 2 pencils aready at dawn or high noon.

*For those wondering what David Mamet's photo is doing at the beginning of this blog entry, it has to do with the title of my post. Mamet is very fond of the language of con artists. Apparently, our intrepid LA TIMES reporters are not unfamiliar with both the short and long cons. Or perhaps it's just their editors, the publisher, and others with vested interests in destroying US public education.

A Cartoon Is Worth 5,000 Words

2010-08-11T13:19:00.000-04:00

My Favorite Week: Math Circle Summer Teacher Training Institute

2010-07-13T19:19:00.000-04:00

Ellen and Bob Kaplan, Jordan Hall, University of Notre Dame, 7/7/10

I just got back from Notre Dame, and boy, is my brain tired! Well, actually not. Despite a lot of walking for this out-of-shape math educator and a lot of strenuous mathematical thinking, I'm exhilarated after my first Math Circle Summer Teacher Training Institute with the wonderful Bob and Ellen Kaplan.

This was the third such institute held at the University of Notre Dame, a campus that seems as isolated from time as a medieval European monastery (though the food and accommodations are vastly better). I had wanted to go two years ago but couldn't get released from a relatively new teaching position to go. Last summer, I simply didn't have adequate funds to do it. But the third time was indeed the charm, and I can say without hesitation that this was the best-spent $800 I've invested in a very long time.

The Institute ran from July 4th through July 9th, with five days of sessions in the morning and afternoon from the 5th until the 9th, plus evening informal sessions at the dorm in which Leo Goldmakher, a number theorist from University of Toronto, and Amanda Serenevy, a mathematician from South Bend who organized the Institute and runs local Math Circles groups at the non-profit Riverbend Community Math Center, led interested participants in the pursuit of various math problems (some held over from our daily sessions).

I can't begin to express my pleasure at spending this much concentrated time with people who are really passionate about quality mathematics content, teaching, and learning. Some were K-12 teachers across the spectrum of grade bands. Some taught community college mathematics. Some were interested home-schooling parents. We had non-teachers who have a love for math and who have started or plan to start local math circles in their communities. There were high school students from the Riverbend Community Math Center. All of us participated in morning sessions led by Bob, Ellen, Amanda, and Leo on various mathematical topics, as well as regarding how to set up and run a Math Circle in keeping with how the Kaplans have been doing it. In the afternoons, we planned teaching sessions and then worked with students whose attendance Amanda arranged: some were in elementary school, while the eldest were in high school or getting ready to start college in the fall. I worked with three groups, doing a problem from Computer Science Unplugged on finding minimal spanning trees with a group of 2nd and 3rd graders on Tuesday, then trying a modified version and the original problem with 4th and 5th graders on Friday. The first group struggled a lot with both the language of the problem and with the complexity of the diagram. However, I finally came up with a very simple version on the spot that they were able to work with. I'd say that with proper modifications, they could have gotten the original problem, and indeed several students from this group came by later in the week to show me their correct solutions. The second group was somewhat "bi-modal" in that four students blew through both the modified and original problems: I left them to consider how many unique minimal spanning trees could be found for the given graph. One student was able, with help, to get through the modified problem. One student, who may have been cognitively impaired, seemed thoroughly out of his depth and likely would need to learn some requisite skills before tackling these problems, along with simplified examples and language.

On Thursday, I worked with a mixed-age group of students on several questions I'd been interested in going back to November 2009 surrounding something called "number bracelets." Specifically, after the students worked through some of the basic questions that arise when working the problem modulo 10, I posed the following to them: a) for a given base b, how many disjoint orbits will there be? and b) for a given base b, what will the lengths of those disjoint orbits be?

This proved to be a deeply intrigued problem for all six students, regardless of age. One of the older students, who was perhaps 14 or 15, really sank his teeth into it. On Friday, he returned with a partially developed solution that could potentially answer part or all of each of my questions. He has promised to follow up with me as he continues to work on them.

We also had the opportunity to observe Bob, Ellen, Leo, and Amanda work with these students on several lovely problems on Monday, and to observe some of our peers teaching when we ourselves were not so engaged. I had originally planned to teach only the Friday group, so I lost out on opportunities to do more observing of others' styles and interests, to my regret. But since we debriefed collectively for about 90 minutes every day after the one-hour classes, there was ample opportunity to hear feedback from those who did observe and comments from my fellow student-teachers.

I cannot overly praise Bob and Ellen Kaplan's work in every aspect of this institute, as well as that of Leo and Amanda. And my "classmates" were all bright, dedicated, and highly-motivated to talk seriously about and work on mathematics, mathematics teaching, their experiences with the students we worked with, and much else. The week was enormously rich in terms of math content, thinking about proof (we spent one morning with Ellen leading us through some of the early section of Lakatos' PROOFS AND REFUTATIONS, not directly, but through exploring the Euler Formula for polyhedra, Cauchy's well-known attempt at proving it, and problems with his method that are raised by Lakatos), pedagogy, and various notions about what it means to teach mathematics effectively.

2010 has been one of my favorite years, and the week of July 4th has certainly been the most memorable thus far.

Who Was George Polya's Intended Audience? (Or More Mathematically Correct Lies)

2010-06-02T14:46:00.000-04:00

George Polya, c. 1973

One of the more difficult aspects of wars, even ones where the main ammunition is words, is separating lies from facts. Every side in a war has a proclivity for propaganda. Inconvenient facts are brushed aside. Inaccuracies, petty or gross, become the coin of the realm. The Big Lie rules.

Of course, sometimes, it is possible to sort through the fog of war to arrive at what appears to be incontrovertible truth. It may take years, even decades, to find the facts, even when they are readily available to anyone who bothers to look in the right place for them. Sometimes, they've been staring everyone in the face for a very long time.

Thus, it is with no small embarrassment that I present a long-overdue and clearly definitive retort to one of the lies frequently promulgated a decade or so ago by Professor Wayne Bishop and some of his Mathematically Correct and HOLD anti-progressive allies, namely that George Polya's work on heuristic methods (from the Greek "Εὑρίσκω" for "find" or "discover": an adjective for experience-based techniques that help in problem solving, learning and discovery) was intended only for graduate students or perhaps undergraduate mathematics majors, not for the general student of mathematics, and certainly not for high school students or younger children.

Of course, in the Math Wars, it is of the utmost importance to the counter-revolutionaries and anti-progressives that nothing that broadens access to mathematics be allowed to stand unchallenged or unsullied. Any curriculum, pedagogy, tool, etc., that is brought forward by reformers as "worth trying" must be smashed. That has been the tireless task of members of groups like Mathematically Correct and HOLD: to undermine any and all efforts to change what they view as immutable approaches to the teaching and learning of mathematics.

It's almost as if they were the American Medical Association, fearful that if too many people get into medical school - indeed, if there are no arbitrary, meaningless gates, such as requiring a full college calculus sequence, put up to block the pathway to the profession - some of their members who managed to get through the gauntlet but who in fact are not all that good at being actual doctors might suddenly be threatened by "others" who happen to have all the requisite skills, including some that these doctors lack. Such folks are inclined to argue that the established path is absolutely correct, the ONLY reasonable one that could possibly be allowed. Anything else would clearly be "fuzzy," "unscientific," "watered-down," etc. Even when, in fact, the main difference might be an emphasis on people skills, psychology, and perhaps looking critically but with open-mindedness at various sorts of holistic, non-Western, and other alternative medical approaches. If the goal is to help as many patients as effectively as possible, what would be the harm in looking scientifically at alternatives? It has been known to happen that methods once dismissed by mainstream science turned out to be highly effective (for one such example, look at the work on treating infantile paralysis by Sister Kenny).

Instances of complete dismissal of a wide variety of innovations or, as in the case of lattice multiplication, the return to an older, mathematically valid algorithm (see "Looking Further At Multiplication" and "Who Invented Lattice Multiplication?" , are legion in the 'work' of these educational reactionaries and conservatives. But one particularly amazing instance surrounds the work on heuristics by Polya, author of several books on the subject, most famously HOW TO SOLVE IT: A NEW ASPECT OF MATHEMATICAL METHOD, first published in 1945, followed in 1954 by Mathematics and Plausible Reasoning Volume I: Induction and Analogy in Mathematics, and Mathematics and Plausible Reasoning Volume II: Patterns of Plausible Reasoning, and in 1965 by the two-volume Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving.

I am hardly alone in suggesting that the above work, while many of the examples geared to undergraduate and graduate students of mathematics, has many deep implications for earlier mathematics education. I cited both the books of Polya and the more famous of the videos of his teaching, LET US TEACH GUESSING in support of the notion that K-12 teachers and there students would gain much from considering and making use of Polya's approach to problem-solving, and that grounding K-12 curricula in this approach would be an improvement over business as usual. (See David Bressoud's 2007 MAA column "Polya's Art of Guessing" for more details on part of what Polya is up to in that video).

My notions were fiercely rejected by Wayne Bishop and others. They denied that Polya was thinking in any way about seeing his methods used in K-12 education and that it would be disastrous to introduce such methods into the public school curriculum, particularly in lieu of teaching traditional algorithms (it's remarkable how everything in the Math Wars comes down to 'either/or' in the hands of the MC/HOLD crowd. The notion of "as well" seems unknown to them).

Well, let me cut to the chase. Here is what Polya himself says at the end of the introduction to HOW TO SOLVE IT:

We have mentioned repeatedly the "student" and the "teacher" and we shall refer to them again and again. It may be good to observe that the "student" may be a high school student, or a college student, or anyone else who is studying mathematics. Also the "teacher" may be a high school teacher, or a college instructor, or anyone interested in the technique of teaching mathematics. The author looks at the situation sometimes from the point of view of the student and sometimes from that of the teacher (the latter case in proponderant in the first part). Yet most of the time (especially in the third part) the point of view is that of a person who is neither teacher nor student but anxious to solve the problem before him. pp. xx - xxi [emphasis added]

Well, slap my face and call me "Susan," but it surely appears that Polya is at minimum open to seeing his method used in high school, though it's not outlandish to suggest that he is suggesting that it's appropriate in some form for any student and any teacher of mathematics, as well as those who are neither.

But perhaps I'm just overstating the case. Maybe heuristic methods are just too daring, too advanced, too non-traditional, surely TOO SOMETHING! to be risked with our K-8 students and faculty. Skies may fall, dams may break, heads may explode, should we try making problem-solving methods a major foundation of our mathematics curriculum, rather than calculation, as has far too long been the case in this country.

Well, here's an interesting bit of evidence from a wonderful article by Tibor Frank, "George Pólya and the Heuristic Tradition Fascination with Genius in Central Europe" in which he explores the general intellectual traditions of Hungary in the period from which Polya, Von Neumann, and many other brilliant mathematicians and physicists emerged:

[He]uristic thinking was also a common tradition that many other Hungarian mathematicians and scientists shared. John Von Neumann‘s brother remembered the mathematician‘s „heuristic insights” as a specific feature that evolved during his Hungarian childhood and appeared explicity in the work of the mature scientist.

Von Neumann‘s famous high school director, physics professor Sándor Mikola [note: another of Mikola's students was the Nobel physicist Eugene Paul (Jeno) Wigner], made a special effort to introduce heuristic thinking in the elementary school curriculum in Hungary already in the 1900s

So it may not in fact be stretching anything at all to say that the intellectual and educational tradition out of which came thinkers like Polya favored the heuristic approach to mathematics education for ALL students, at any grade level or stage of growth.

Generally, of course, the rebuttal that is leveled by educational conservatives to mention of such things is that the students that Polya and others had in mind were "math people." That is to say, they were students who came prepared to do serious mathematics, already had mastered the basics, and were already showing the necessary mathematical interest and 'talent' for doing higher-level thinking and mathematical problem solving.

The problems with such a claim are two-fold: first, there is no evidence that Polya or Mikola or anyone in Hungary or anywhere else who promoted these notions was looking at a narrow, highly-gifted group. But second, and perhaps more importantly, even if such were the case, that does not prevent American mathematics teachers, teach-educators, and researchers from considering how to implement heuristic approaches into their teaching. It does not mean that such approaches are "verbotten" other than in the conservative minds of anti-progressives. And given that such people seem utterly closed to ANY innovation, any change, any departure from what they assert produced a "Golden Age" where all American kids learned WAY more of EVERYTHING than do today's kids ("See?" the self-serving story goes, "Back in MY day, teachers really taught rigorous content and EVERYONE learned it; today's teachers are slackers and today's kids even more so.")

Can we afford to buy such arrant nonsense without close examination of the facts and of the motives of those who are purveying this bizarre version of history? I, for one, think we cannot. I believe that George Polya, John Von Neumann, and many of their contemporaries and instructors would agree. But do read the source material and decide for yourself. Then check out Dan Meyer and other teachers who aren't waiting for the approval of Mathematically Correct, HOLD, or any other dinosaur or nay-sayer.

I Saw Mathematics Education Future and Its Name Is Dan Meyer

2010-06-01T17:05:00.000-04:00

Dan Meyer

Some thinkers/innovators and their ideas are simply too good for even the media to miss. Dan Meyer is one such thinker. He happens to be, amongst other things, a high school mathematics teacher in Felton, CA, barely a stone's throw from Santa Cruz. He teaches geometry to kids he describes in ways that makes them sound like non-enthusiasts in mathematics, at least when they enter his classes in the fall.

Dan blogs at dy/dan on a variety of issues related to teaching mathematics, but his most innovative contribution thus far, in my view, has been a series of problems, lessons, and pedagogical experiments collected under the label "What Can You Do With This? (WCYDWT?)

Rather than offer my lame description of what Dan is up to with this ideal, I think readers would do far better to see for themselves. Towards that end, I've done you the favor of collecting links to all of the WCYDWT? blog entries thus far, along with the dates on which they appeared and the (sometimes cryptic) titles. Because it is currently only possible to go backwards chronologically, I took the trouble to arrange what follows in ascending chronological order so that interested readers can see the evolution of the WCYDWT? approach.

New readers are encouraged to look at the comments as well as Dan's entries. He is a polite and frequent respondent to those of his readers who offer useful alternatives, refinements, criticisms, and departure-points for further entries. So, much may be lost if the comments are skipped.

WCYDWT? The Story Thus Far

2007

6/8/07 http://blog.mrmeyer.com/?p=255 "I Need Another Blog

7/9/07 http://blog.mrmeyer.com/?p=284 "Bizarro Blog: [title redacted]"

8/10/07 http://blog.mrmeyer.com/?p=311 "Science Owes Me A Beer"

2008

10/4/08 http://blog.mrmeyer.com/?p=1220 "Pilot"

10/4/08 http://blog.mrmeyer.com/?p=1526 "License Plates"

10/6/08 http://blog.mrmeyer.com/?p=1565 "Out of Control"

10/8/08 http://blog.mrmeyer.com/?p=1510 "The Bone Collector"

10/21/08 http://blog.mrmeyer.com/?p=1680 "Schrute Bucks"

11/23/08 http://blog.mrmeyer.com/?p=1896 "EXIF"

2009

1/14/09 http://blog.mrmeyer.com/?p=2783 "How Can We Break This?"

1/16/09 http://blog.mrmeyer.com/?p=2796 "ELA Edition"

2/16/09 http://blog.mrmeyer.com/?p=3001 "Becky Blessing"

2/17/09 http://blog.mrmeyer.com/?p=2992 "The Woman Who Didn't Swim Across the Atlantic"

3/11/09 http://blog.mrmeyer.com/?p=3159 "YouTube URLs"

3/23/09 http://blog.mrmeyer.com/?p=3243 "2008 World Series of Poker"

3/23/09 http://blog.mrmeyer.com/?p=3251 "Projectile Motion"

4/7/09 http://blog.mrmeyer.com/?p=3467 "Global Math Geeks"

4/19/09 http://blog.mrmeyer.com/?p=3622 "What You Can?t Do With This: NLOS Cannon Challenge"

4/20/09 http://blog.mrmeyer.com/?p=3667 "Yes We Can, Etc"

4/21/09 http://blog.mrmeyer.com/?p=3624 "The Door Lock" part 1

4/23/09 http://blog.mrmeyer.com/?p=3675 "The Door Lock" part 2

4/29/09 http://blog.mrmeyer.com/?p=3732 "Flight Control"

5/14/09 http://blog.mrmeyer.com/?p=3826 "Other People"

6/4/09 http://blog.mrmeyer.com/?p=3905 "Glassware" part 1

6/10/09 http://blog.mrmeyer.com/?p=4018 "Glassware" part 2

7/22/09 http://blog.mrmeyer.com/?p=4255 "Club Soda"

7/24/09 http://blog.mrmeyer.com/?p=4276 "Don't Forget Answers, Iteration"

9/6/09 http://blog.mrmeyer.com/?p=4636 "Groceries" part 1

9/8/09 http://blog.mrmeyer.com/?p=4646 "Groceries" part 2

9/14/09 http://blog.mrmeyer.com/?p=4472 "Excellent Math Blogging"

9/19/09 http://blog.mrmeyer.com/?p=4778 "Check For Understanding"

10/5/09 http://blog.mrmeyer.com/?p=4838 "What I?m Trying To Say:"

10/7/09 http://blog.mrmeyer.com/?p=4893 "Pocket Change" part 1

10/8/09 http://blog.mrmeyer.com/?p=4905 "Pocket Change" part 2

10/29/09 http://blog.mrmeyer.com/?p=5046 "Redesigned: Kyle Webb"

11/19/09 http://blog.mrmeyer.com/?p=5260 "Dan & Chris

12/28/09 http://blog.mrmeyer.com/?p=5394 "The $6400 Question"

2010

1/6/10 http://blog.mrmeyer.com/?p=5633 "How Do You Turn Something Interesting Into Something Challenging?"

1/16/10 http://blog.mrmeyer.com/?p=5674 "This Blog Is Counterproductive"

2/7/10 http://blog.mrmeyer.com/?p=5903 "The Weak WCYDWT Brand"

2/9/10 http://blog.mrmeyer.com/?p=5945 "Two Excellent Entries For The WCYDWT Course Catalog"

2/9/10 http://blog.mrmeyer.com/?p=5789 "Will It Hit The Corner?"

2/11/10 http://blog.mrmeyer.com/?p=5954 "Follow Up: Will It Hit The Corner?"

2/13/10 http://blog.mrmeyer.com/?p=6007 "Nick Hershman?s Follow Up: Will It Hit The Corner?"

2/22/10 http://blog.mrmeyer.com/?p=5990 "Water Tank"

2/23/10 http://blog.mrmeyer.com/?p=6110 "Check For Understanding"

2/25/10 http://blog.mrmeyer.com/?p=5983 "Math curriculum makeover"

2/25/10 http://blog.mrmeyer.com/?p=6132 "Who cares?"

3/23/10 http://blog.mrmeyer.com/?p=6339 "The Italian Job"

3/23/10 http://blog.mrmeyer.com/?p=6383 "The WCYDWT Workflow"

3/24/10 http://blog.mrmeyer.com/?p=6404 "Dy/Dan Teacher Prep Academy ? Certification Exam Question #42"

4/5/10 http://blog.mrmeyer.com/?p=255 "Gimme Friction / Taberinos"

What Time Is It?

2010-05-27T12:46:00.001-04:00

Lillian R. Lieber

Always fascinating to find out that it's 1959.

POSTULATES are, as you know,
the RULES for playing some "game".
Surely anyone in his right mind
would not even try to
play a game without knowing its
BASIC RULES!
And yet,
some people, young and old,
try to play the games of
arithmetic and algebra
WITHOUT EVER realizing
that these HAVE basic rules!
Now, mind you
it is NOT BECAUSE these rules
are difficult,
NOT BECAUSE there are too many of them!
On the contrary, they are very simple
and very few,
as you will soon see.
Why is it then that
youngsters, in their study of
these subjects,
are usually NOT given
the POSTULATES?
But, instead, are given
various directions for
doing various little things ------
thousands of them! -----
just as if a beginner in football
were told
"now grab the ball",
"now run this way",
"now run the other way",
etc., etc.,
without ever telling him
about the "goal",
or what he is really supposed to accomplish
or what he is allowed to do
or not allowed,
in short just pushing him around
in ways that may be clear enough to
the "pusher" or "teacher"
but which
the learner does not understand at all,
for he does not know what it is that
he is trying to do,
and gets quite bewildered by
the enormous number of details
with which he is overwhelmed!
Surely no one would ever think of
teaching football this way,
and yet this is the way
mathematics is often taught!
No wonder so many people
think they "hate math."

Now why is this so?

Is it because a certain psychologist
once emphasized the idea that
there are millions of
"S - R" bonds"
(Stimulus-Response bonds),
in arithmetic, for instance,
each of which is
something separate and distinct
and must be individually learned -----
thus the Stimulus "1 + 1"
must bring the Response "2",
"2 + 1" must bring "3",
etc., etc.,
ad infinitum.

Now I do not presume to
criticize this,
but surely it must be good psychology
to get a BROAD view of a game
and be aware of
the set of rules which govern it!

Another objection someone raised to
this "postulational" approach
is that words like
"commutative",
"associative",
"distributive",
etc.,
(which describe some of the postulates,
as you will see)
are just too hard for teenagers!
To which I can only say -----
let us not underrate teenagers!
If we do not believe in them
and in their great drive to achieve,
we may turn them aside altogether
from good, hard pursuits,
and they may then use their strength
in other ways -----
not necessarily good ways -----
for strength they HAVE and
MUST use it.
And to think that
the above-mentioned words are
"too hard" for them
is just arrant nonsense!

Lillian Rosanoff Lieber
(LATTICE THEORY, pp. 34 - 36)

Going by the book? Why math texts are resources, not bibles.

2010-04-24T11:48:00.002-04:00

I have addressed the issue of how to view mathematics textbooks in K-12 several times before ("Changing order of topics: an example from practice", and "The Book Gods" amongst others, but it seems to rear its unlovely head periodically amongst real teachers and mathematics educators. A recent post to a math teacher list I read posed the following:

I know this question might sound silly, but I need to know what you think. Do you believe it is counterproductive to literally teach math by the book? Today's text books are very complete and I have seen teacher developing strict routines on the book activities. Do you fully rely on your textbooks to teach Math?

Given the relatively progressive orientation of the list in question, I was a bit surprised to see this particular notion raised there, but I figured that when I posted what follows, I was merely reflecting pretty much universally-held sentiments there:

I think textbooks provide resources for teachers and students. Treating them as bibles is a huge error, one that far too many teachers fall into making. The unstated assumption behind teaching a textbook as given is that the author(s) know(s) more about your students than does the teacher. This is an absurd notion, one that any honest author (or publisher) would have to reject.
I have been amazed and appalled by teachers who cannot believe it's not only possible to depart from the textbook but in fact necessary for effective mathematics teaching. The extreme opposite position from mine is expressed by the late John Saxon, who insisted in the introduction to his books that every problem must be covered, and in the order given. If one analyzes Saxon Math books, this claim on his part becomes even more glaringly ridiculous than I hope it sounds on
first hearing. His Algebra 1 text, for example, looks like he took the table of contents, cut it up, threw the topics in the air, and then inserted them back in whatever random order they fell.
But even the most logically and carefully constructed textbook cannot possibly meet the needs of ANY class or student. It is the job of the teacher to change things around, supplement, omit, re-order, pare, edit, and otherwise perform thoughtful experimentation upon textbooks, even those books which the teacher selects herself. I recommend strongly looking at a twelve minute TED talk by Dan Meyer in this regard, though there is so much he packs into that talk that the issue of dealing with textbooks is only one important idea in it.

Thus, I was caught off-guard when Tad Watanabe, a respected colleague wrote:

My answer is "it depends." If a textbook series is carefully and thoughtfully developed, I think it is a good idea to follow it very closely at least a few years. The changes we make should be based on the actual "data" of how students respond to the instruction recommended by the series. I think too many teachers make too many changes prematurely. It is particularly troublesome when teachers change lessons from an NSF curriculum because what it suggests just doesn't fit their conceptualization of mathematics teaching. Now, if the textbooks aren't well written, then that will be a different issue. But, even in that case, there must be some convincing evidence that the quality of the book just isn't there.

Having coached mathematics at every grade level from 3rd through 12th, I was well-aware of the propensity of teachers being asked to implement a progressive reform program such as EVERYDAY MATHEMATICS (EM) in K-5, CONNECTED MATHEMATICS (CMP) in 6-8, or CORE-PLUS/CONTEMPORARY MATHEMATICS IN CONTEXT (CMIC) in 9-12 to undermine the textbook authors' philosophy in a host of ways. In the case of EM, I saw teachers who taught the book as written but because they never understood or did not buy into the underlying pedagogy managed to weaken the potential effectiveness of the program: leaving out the games that were included to help students build and reinforce basic arithmetic facts; refusing to allow discussion of student errors and instead immediately correcting mistakes so that, as one third grade teacher explained to me, they wouldn't have a chance to take root in the minds of the rest of the students (why correct ideas and information never seemed to do quite so good a job of spreading themselves she did not say); and generally making a very traditional, teacher-centered class out of a non-traditional, student-centered textbook.

In a near-by district, I saw the department chair at the middle school simply refuse to teach from CMP at all except when she knew I was coming to observe, and even then using the texts in such a perfunctory and disconnected manner that it couldn't have been lost on the students that their teacher had little or no respect for the material she was presenting.

In the case of CMIC, while I supervised secondary math student teachers for the University of Michigan at two enormous comprehensive high schools in Ann Arbor, I discovered that the books were only being used for students deemed 'difficult,' in need of remediation, and likely to be non-college-bound. At the same time, the department chair at one of these schools misled parents by citing the PSAT scores of these same low-level students as 'proof' that CMIC was an inferior program that CAUSED standardized test scores to decline. Later, she bragged to me that she had "killed Core Plus Mathematics" in Ann Arbor.

Thus, I realize quite clearly that there are serious reasons to worry about how 'teacher choice' might impact the use of any textbook, no matter how good, and particularly those which challenge assumptions and habits of new and veteran teachers alike.

Nonetheless, I offer the following in response to the above-quoted suggestion that "it depends":

I think this needs to be looked at from the perspective of pedagogical content knowledge (PCK).

If we assume that the textbook author(s) as viewed through the lens of the textbook have consistently superior PCK and that the teacher in a given classroom has consistently superior PCK, there's no problem. The teacher will use her PCK to make appropriate adjustments to the textbook that arise, as do many such decisions, in the heat of the moment: see the quintessential example of Deborah Ball's reaction to a third grade student asserting that "I think some numbers can be both odd AND even," one that no textbook author on the planet could reasonably be expected to have anticipated, nor any classroom teacher, for that matter. But only one person, the classroom teacher, actually is in a position to make a decision based on PCK at that moment, and only that person MUST make that decision.

If she decides to bow to the wisdom of the textbook/author(s), then since this incredibly powerful moment was not anticipated, she must pay it little or no heed and move on with the lesson as written. Of course, such moments have the potential to bifurcate into a host of possibilities depending on more variables than anyone can conceivably enumerate, let alone take into account. Many such moments get lost, the less fruitful paths pursued (for reasons that may have to do with: 1) following the text as written; 2) weak PCK on the part of the teacher; 3) reasonable decisions that just don't pan out; and 4) 'merely' the vicissitudes of complex human interactions that occur in activities like classroom mathematics teaching).

But to immediately ensure that more will be lost than need be the case because reason #1 MUST be adhered to (in order to satisfy the needs of textbook authors, publishers, project developers, researchers, etc.) simply denies the fundamental importance of teacher PCK in the making of every teaching decision in the moment that it arises.

Now, of course, rarely, if ever, do textbook author(s) and/or classroom teachers possess superior PCK. In the lower grades, particularly in the post-Liping Ma era in which we all know that most (American) elementary teachers don't know enough mathematics content and hence almost assuredly lack superior PCK, we tend to assume that a randomly selected teacher will have inferior PCK, and will come up short in PCK when weighed against that of the textbook author(s). And so it SEEMS like a no-brainer to agree with Tad here (and generally, I agree with Tad on most things). But I think we would be wrong to do so. Not because it isn't probably true that the PCK (or at least the CK) of the author(s) is deeper and broader and grounded in more experience, thought, research, etc., but because even with all that, mistakes are made. Mistakes that for any given set of kids might not be so bad but for others will make a lesson sink like a stone and create more difficulties than are probably good for anyone.

This isn't to say that 'problematic' situations are to be avoided in math. On the contrary, I think certain kinds of problematizing mathematics are essential for effective teaching and learning. But knowing in advance that doing a given problem at given point in a sequence of problems is going to result in good rather than in destructive problematizing is a very difficult if not impossible task, and certainly knowing this in a way that will apply nearly universally to class upon class, year after year, regardless of circumstances, would require god-like omniscience. Indeed, I doubt it's possible even for a deity. There are simply too many variables and likelihood of combinations of them that will make a given problem in a given sequence a loser FOR A SPECIFIC KID OR GROUP.

And that's why ultimately it must be the combination of the PCK of the author tempered by that of the teacher that rules.

Now, of course, we can all agree that there are teachers who will undermine a well-constructed lesson or unit simply because they "don't get it." I've been in PD sessions where the refrain is, "My kids won't do this," or "My kids CAN'T do this," when of course what is meant is "I don't get this" and/or "I don't like this" and/or "I can't teach like this."

So what does one do as a supervisor, department chair, content coach, author, researcher, principal, publisher, project director, etc.?

First, accept that teachers will tend to follow the path of least resistance unless they are confident, flexible, curious, and secure. When things go wrong, most teachers return to their comfort zone, which is generally teacher-centered, direct instruction with WAY TOO MUCH EXPLAINING, over-scaffolding, etc. While some folks can resist that temptation, they are few and far between. Those listed above had best do everything possible to raise the teacher's believe that she is safe: that heads won't roll, jobs won't be lost, people won't die, salaries will not drop, etc., simply because something doesn't quite go as planned or predicted. Try doing that in today's education deform, "teacher accountability" atmosphere, with everyone now worrying about "racing to the top" for the $$ (Oh, you thought that meant that we're all speeding to get kids to reach some ideal peak of learning? My ass, it does.)

Second, to be crude, bad teachers can fuck up a wet dream. You can hand them the most perfectly constructed, well-thought out lesson and they can and will make it fail if it's simply not in them to make it succeed. They'll dilute everything, they'll change the balance of the lesson to center on them, they'll micro-manage all the empty space in the lesson that should be left for the students to think in ways that turn an investigation into mere practice, thinking into watching, inquiry into imitation, a problem into an exercise.

But then, good teachers can still fail with the same wonderful lessons if, in fact, those wonderful lessons are for the most part simply text-book centered rather than teacher-centered but STILL NOT STUDENT-CENTERED. If the burden of the work and thinking is taken over by the text rather than by the teacher, it's not an improvement. Only if the STUDENTS have to do the thinking and the real work is a book a truly helpful resource. All the little subsets of questions so thoughtfully provided by authors because they assume that the kids must have such things laid out for them in bite-sized pieces with everything resolving neatly at the end, a simplistic moral lesson learned by everyone (e.g., "follow order of operations or bad things will happen!")

Which of course brings me back to that 12 minute TEDxNYCE talk by the brilliant Dan Meyer. Didn't watch it yet? Shame on you! Stop what you're doing (reading my silly diatribe) and watch it:

At the risk of infantilizing my readers and trivializing the power of Dan's talk by reducing it to my notes, here is something I crafted after multiple viewings of what he presented in New York. And since I fear some folks just won't click on that link or take the time to watch the talk, I present my notes here:

Notes from Dan Meyer TEDxNYED talk 3/6/2010

Five symptoms that you're doing mathematical reasoning wrong in classes:

1. Lack of initiative (students don't self-start)
2. Lack of perseverance
3. Lack of retention
4. Aversion to word problems
5. Eagerness for formulas

An impatience with irresolution:

“[Contemporary drama] creates an impatience, for example, with irresolution. And I'm doing what I can to tell stories which engage those issues in ways that can engage the imagination so that people don't feel threatened by it.” - David Milch, creator of NYPD BLUE, DEADWOOD

Sit-com sized problems that wrap up in 22 minutes, three commercial breaks, and a laugh track. No problem worth solving is that simple.

Our textbooks create impatient problem solving attitudes.

What problems worth solving give you exactly the information you need?

Real problems either have a surplus of information that must be sorted through or a lack of adequate information, some of which must be searched for.

Ski Life/Slope problem with four separate layers of information: 1) Visual; 2) mathematical structure (grid, labels, measurements, axes, points); 3) subsets (of questions) all leading to the question we really want to talk about: 4) which section is the steepest?

The layers are presented at once, breeding impatience with real problems because everything is handed to the kids (#3, the subsets of questions, over-scaffolds the question so that #4, the real question of interest, becomes trivialized)

Instead, Meyer starts with the (stripped-down) visual and immediately asks the final question, starting conversations, argument, disagreement; the situation is problematized in part for students because there is no vocabulary, terminology, labeling, etc. to frame the conversation. It's the difficulty of talking meaningfully about the skiers and what steepness means that leads to the labeling, the mathematical structure, etc.: the math serves the conversation, the conversation doesn't serve the math.

“The formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill.” - Albert Einstein

But we just give problems to students; we don't involve them in the formulation of problems.

Water tank: Eliminate all the sub-sets, all the specific, non-distracting information, and reduce this to an under-represented math question where the students have to decide what matters (or not):

A water tank. . . . how long will it take to fill it? Plus a real photograph (not line art or clip art) or, even better, a video of someone filling a tank, slowly, tediously, driving kids to look at their watches and asking, “How long is it going to take to fill up?”

We don't get our answers from an answer key at the back of the book: we just watch the end of the movie.

1. Use multimedia

2. Encourage student intuition

3. Ask the shortest question you can

4. Let students build the problem

5. Be less helpful

The textbook is helpful, but in all the wrong ways.

We have all these cheap or free tools available to do these things. This makes it a most exciting time to be a mathematics teacher.

You Just Can't Make This Stuff Up (or can you, if you're James Milgram?)

2010-03-25T14:26:00.006-04:00

If someone sent this to me without the accompanying link, I'd never believe it was real. Sadly, it's very real.

R. James Milgram, a renowned mathematician who has long been in the forefront of opposition to progressive mathematics education reform, was interviewed for something called the Baltimore Curriculum Project in 2006 and the entire interview appears here.

However, the following excerpt is perhaps the single most bizarre-sounding thing I've ever encountered in the history of the Math Wars (and that's saying something).

Where Dr. Milgram got these notions from I have no clue. Who the mathematics educators are that he claims hold the views he mentions below would be amazing to discover, if such people actually exist. My guess, however, is that if they do, they're almost certainly not in the mainstream of the mathematics education research community today.

Perhaps he found someone from a more conservative era of NCTM and the research community who actually believes what Milgram states. If so, that is hardly grounds to make the assertions about a large number of mathematics educators.

Dr. Milgram cites NO specifics. He seems to be in fact making this up as he goes. And on my view, it has absolutely nothing to do with the real world. I've been involved in mathematics education officially since I started graduate work at University of Michigan in 1992. I've attended many conferences of mathematics education researchers and teachers. I participate in many on-line discussion about mathematics education and have corresponded individually and in small private lists with dozens of university mathematics educators and researchers, both in the US and abroad.

I have NEVER heard or read anyone else say anything even remotely like what Milgram asserts below. And I believe he has no facts to support his claims. Even the interviewer appears completely caught off guard by the assertion that there's a conflict between mathematics educators and other professors of education, and that the former feel constrained by the latter and disempowered by them.

Further, his off-the-cuff claim that there is "a lot more [agreement] than you would expect" amongst mathematics educators with the anti-reform traditionalists Milgram represents appears to be another howler he invents for the occasion.

If this is actually what Milgram believes, and the general community of mathematics educators and other school of education researchers and professors has gotten wind of it yet has remained to a person silent about it, I'd be amazed. I think this interview has sat unread by anyone in a position to expose it for what it is. Or else people are being far too polite to reply.

Not being quite so polite, I challenge Dr. Milgram to back up his claims about what professors of education see as the purpose of education, what mathematics educators generally believe about their colleagues in schools of education, and what mathematics educators believe regarding the views that Milgram espouses about mathematics teaching and learning. I think he'd be very hard-pressed indeed to find more than the tiniest trickle of support, if he can find any current working mathematics educator or other ed school professor who backs any of his points.

Here is the excerpt to which I am referring:

Q: Why do the philosophies of mathematicians and educators seem to vary so widely?

A: The people holding the power in the education community today hold the belief that the major function of the public schools is to keep children out of the workforce.

The recollection is the horrors of child labor from the 19th century. The objective was to keep them out of the workforce as children, but that was it. They also believe that kids should have a good time in school because implicit in their belief is the conviction that kids will not have a good time as adults.

Q: That's shocking to me. I've read about the Math Wars, but I've never hear that viewpoint expressed.

A: The debate in the Math Wars was between math educators and mathematicians. Somehow the people in the education schools proper stayed out of it. But when you come right down to it, you have to deal with the people in the education schools.

Ultimately and what really was remarkable to me when I got to know a number of these math educators is they were consistently telling me of their feelings of powerlessness. We were assuming they were the ones that are responsible. They don't necessarily agree with us 100 percent, but they agree with us a lot more than you would expect.

Sure, Dr. Milgram. And maybe I'm a Chinese jet pilot.

Where Are The Explanations? Davydov, Vygotsky, Measurement, and Scientific Knowledge

2010-03-14T22:00:00.000-04:00

Someone asked today on math-teach@mathforum.org why it is that students fail at math. His answer?

The short answer: They don't get it.

The long answer...

What is it that they don't get? They don't get the chain of reasoning involved in mathematics.

This struck me as a rather useless tautology, a non-explanation purporting to reveal all. I could not see what he subsequently offered as any sort of realistic answer. I thought instead about Davydov's curriculum.

I first became aware of the mathematical curriculum of V. V. Davydov and his colleagues early in 2009 when several people independently wrote to ask me my opinion of his "measurement-based" approach to elementary mathematics. I had no idea who Davydov was or what his approach entailed.

After reading some papers that these folks sent me, I knew that Davydov followed and grounded his work in some key ideas of Vygotsky, and that at least one attempt had been made in the US to develop and implement a measurement-based elementary curriculum: MEASURE UP! at the University of Hawaii's Curriculum Research and Development Group. One of those developers was someone I knew, Barbara Dougherty, now at Iowa State University.

Further, I learned that Jean Schmittau at SUNY@Binghamton had written several papers about the Davydov curriculum and had, in fact, translated Davydov's elementary books (which comprise an early-grade three-year program) into English in the late 1990s and used these translations to work with elementary students in a local school district.

Perhaps coincidentally, at the time I became aware of and began looking into Davydov's work, I became interested and embroiled in the subject of my previous post (and many others): the controversy Keith Devlin generated beginning with his June 2008 column on multiplication, "It Ain't No Repeated Addition." By the time I jumped in, first to oppose Devlin's views and later to agree with them, the arguments about whether there was any reason NOT to teach young students that multiplication is repeated addition (MIRA) and whether in fact it was ever true to claim that MIRA is correct was raging hot and heavy in the blogosphere. That didn't stop me from putting both feet in my mouth (I took umbrage at Devlin, a mathematician I'd once met briefly and whom I respected as a solid thinker, trying to tell elementary teachers how to teach basic mathematics to kids.) And then I actually started to think, both about the nature of multiplicative reasoning and problems I saw in typical elementary school mathematics education (and our way of "schooling" so many students into intellectual passivity) that Davydov's approach seemed well-positioned to address.

As I began to post about both Davydov and the multiplication/MIRA issues both on my blog and on a couple of on-line discussion lists, I noticed how quickly the educational conservatives lined up against Devlin, against Davydov, against anything that wasn't precisely how they were taught or taught others themselves in the cases where they instructed on elementary mathematics (usually in remedial situations).

I wondered why they would not find Vygotsky's theories and Davydov's implementation of them in early mathematics curricula appealing or at least interesting. After all, that viewpoint is all about the idea that kids, left to their own devices or to typical educational approaches to teaching them mathematics, will have little reason to grapple with abstraction. Schmittau and Morris make this clear when they write regarding Vygotsky and Luria that they

"found in their studies of the development of primates, 'primitive' peoples, and children, that cognitive development occurs when one is confronted with a problem for which previous methods of solution are inadequate [Comment: not unlike how much mathematics has developed, by the way]. Hence, Davydov's curriculum is a series of very deliberately sequenced problems that require children to go beyond prior methods, or challenge them to look at prior methods in altogether new ways, in order to attain a complete theoretical understanding of concepts. More importantly, their consistent engagement with this process develops the ability to analyze problem situations at a theoretical rather than an empirical level, and thus to form THEORETICAL rather than EMPIRICAL generalizations, which is the distinguishing feature of Davydov's work." ("The Development of Algebra in the Elementary Mathematics Curriculum of Davydov," THE MATHEMATICS EDUCATOR, 2004, p. 62)

One might expect that this approach would have great appeal to anyone concerned with how little is asked of US students for the most part in elementary mathematics teaching and curricula. And given the fact that for many disadvantaged students, reading is a huge impediment to using many of the often-maligned progressive reform elementary books such as EVERYDAY MATHEMATICS and INVESTIGATIONS IN NUMBER, DATA & SPACE, the following should be enormously heuristic:

"The curriculum itself. . . consists of nothing but a carefully developed sequence of problems, which children are expected to solve. The problems are not broken down into steps for the children, they are not given hints, and there is no didactic presentation of the material. There is nothing to read but one problem after another. The third grade curriculum, for example, consists of more than 900 problems. Teachers, in turn, present the children with these problems, and they do not affirm the correctness of solutions; rather, the children must come to these conclusions FROM THE MATHEMATICS ITSELF [emphasis added]. The children learn to argue their points of view without, however, becoming argumentative." (loc. cit.)

I find it interesting, however, that in speaking with several US mathematicians and mathematics educators who either have developed or are in the process of developing curricular materials (or both) based on Davydov's ideas and work state that they don't feel that Davydov's original materials would fly in the US (in translation, of course).

I strongly suspect that the concern is not so much for the ability of kids to succeed with Davydov's materials (after all, if they work for kids in Russia, and they are still successful there (at least up to about 10 years ago; I have no more updated information) as evidenced by the 1999 testing at the end of third grade of 2300 Rusian students who used the El'Konin-Davydov curriculum. According to Catherine Sophian's latest book, Vorontsov concluded based on the results of the testing that

"'pupils learning in [this] educational system completely fulfill the requirements of the existing state standard.' The percentages of students who succeeded on 'tasks corresponding to the "standard" level of elementary school' ranged from 86 to 96% with the exception of a task that involved dividing multi-digit numbers, on which 76% of the pupils were successful. In a comment reminiscent of the debate surrounding reform mathematics in the United States, Vorontsov goes on to state, 'The results obtained dispel the myth, current in pedagogical circles, of the poor results of mastery of subject matter among children in [these] programs.'" (Sophian, Catherine: THE ORIGINS OF MATHEMATICAL KNOWLEDGE IN CHILDREN, 2007, p. 168).

In fact, I suspect the problem, such as it is, has little to do with kids, but rather with the low ability and willingness of so many US elementary teachers (and beyond) to teach mathematics without step-by-step guidebooks and their relatively weak knowledge of the relevant mathematics and content-based pedagogy. Considering that any non-traditional mathematics program introduced in the US that does not have the imprimatur of such educationally conservative groups as Mathematically Correct, NYC-HOLD, and a host of think tanks that condemn each and every "progressive math" program being used in this country, but praise uncritically particular programs from high-scoring Asian countries (particularly Singapore) for reasons that may not be completely non-political or grounded in reality for American kids and teachers, are subject to criticism for being TOO verbal, TOO hard, TOO sparse AND TOO watered-down (I am not kidding; the willingness of these self-proclaimed curriculum experts to offer utterly contradictory criticism in order to destroy programs they don't like is remarkable) it takes a daring teacher, school or district indeed to risk using something that is so starkly non-verbal as are the original Davydov materials.

Nonetheless, I think that teachers in inner-city, poverty-stricken, and other challenging environments would, if properly helped to understand how to teach and implement a Davydov-like program, be interested in mathematics books that don't have the serious disadvantage of being inaccessible to non-native and native English speakers alike who struggle with reading English.

Obviously, to fully evaluate the potential of a program like that described by Schmittau and Morris, it would be necessary both to examine the problems and see the instructional guides that come with them, and/or to view classrooms in which children are taught by competent teachers who have been trained effectively in its use. Thus, it is impossible to draw definitive conclusions about exactly what students are doing and how they are led to do it. While the reported results are naturally intriguing, it remains an open question, to my mind at least, as to whether American teachers could be brought to use willingly (and ultimately successfully for their students) a translation, with or without adjustments, of the original Davydov problems and guides. Absent DIRECT access to the problems or teacher materials, and knowing that replication of Schmittau's and her colleagues' research has not been attempted elsewhere. However, neither is it fair to draw any definitive conclusions from that. One would also want to look at the research emerging from various Americanized programs that purport to take an approach similar to Davydov's (though clearly they do not follow the entire Vygotskian theoretical approach as Davydov and his colleagues implemented it).

All the above said, I find it curious that some critics feel well-positioned to draw any conclusions either about Davydov's work or what is actually behind much of American student failure in mathematics. No expansion upon the tautology that students fail because they "don't get it" is likely to be adequate to the task unless one has looked adequately at approaches that purport to be able to take students to much higher degrees of success in mathematics, on average, by instead of giving students pre-algebraic experiences that are numerical, giving them pre-numerical experiences that are algebraic. I'd say that 2300 Russian students succeeding with such an approach is worthy of serious consideration as at least one piece of the puzzle as to how to remedy student failure. Offering up tautologies and then speculating about the causes based on one's own limited imagination? Probably not so much.

Terezinha Nunes and Peter Bryant Dole Out The Multiplicative Harshness

2010-03-13T23:00:00.002-05:00

From CHILDREN DOING MATHEMATICS by Terezinha Nunes and Peter Bryant:

"[I]t would be wrong to treat multiplication as just another, rather complicated, form of addition or division as just another form of subtraction.

The reason for this is that there is much more to understanding multiplication and division than computing sums. The child must learn about and understand an entirely new set of number meanings and a new set of invariants, all of which are related to multiplication and division but not to addition and subtraction. [p. 144]

One is tempted to stop there. The above comes from the opening of the authors' chapter on multiplication and division, in a section called "Multiplication, Division, and New Number Meanings. For the debate that has been raging all over the internet since Keith Devlin published his first column about multiplication in 2008 called "It Ain't No Repeated Addition," it seems that for many people, the ideas raised in the above quotation are absurd. After all, one can compute products by doing repeated addition and quotients by doing repeated subtraction. If you can get the right answer to a computation by two different operations (which begs the question as to whether repeated addition is actually a well-defined operation), aren't those operations the same? Isn't multiplication, when all is said and done, repeated addition? Certainly this is true for the whole numbers, right? And the analogy used there can readily be made to fit integers, rational numbers, maybe even the real numbers in their entirety. It's merely common sense, and this is, after all, the way CHILDREN think, isn't it?

Let's look further at what Nunes and Bryant have to say before considering the above question again. Summarizing the number meanings and situations for additive reasoning they write:

Additive reasoning is about situations in which objects (or sets of objects) are put together or separated. All the number meanings in additive situations are directly related to set size and to the actions of joining or separating objects and sets. Number as a measure of sets involves putting objects into a set where the starting-point is zero; number as a measure of transformations relates to the set that is joined to/separated from another set; number as a measure of a static relation (in comparison problems) relates to the set that would have to be joined to/separated from another in order to make two sets equal in number. [p.144]

Anything too radical there for the folks who seems to cling so passionately to the Multiplication Is Repeated Addition (MIRA) point of view? I would think not. For all that Nunes and Barnes are doing is reflecting the essential nature of addition itself. And there is no real debate about that raging underneath the battle over multiplication. The trouble comes when Devlin and others suggest that multiplication not only need not be defined as "repeated addition" (even though it often is in many places, including reputable ones), but also that it simply should not be so defined. It is a separate operation that stands on its own, with no need to be defined as something else.

To be sure, as previously stated and as Devlin and others on the other side of the aisle in this particular debate have stated many times, we can readily arrive at the result of multiplying two whole numbers, say, 3 x 2, by adding 2 + 2 + 2. No one will argue that the resulting product and sum are not the same whole number, 6. But is that coincidence (in the literal sense of two things "falling together") sufficient to determine that we are looking at the same operations (indeed, there is good reason to question whether repeated is a well-defined binary mathematical operation (Devlin says it is not)?

Well, if not, why not? What IS different about multiplication and multiplicative reasoning? Nunes and Barnes state:

Situations which give rise to multiplicative reasoning are different because they do not involve the actions of joining and separating. We will distinguish three main kinds of multiplicative situations: (1) one-to-many correspondence situations; (2) situations which involve relationships between variables; and (3) situations which involve sharing, division, and splitting. [pp.144-145]

Those of us familiar with some of the arguments of those in the MIRA camp know that one typical response from such people is to suggest that their antagonists are presenting "distinctions without difference." Is that in fact the case with the analysis of Nunes and Bryant? Let's examine each of the situations they mention above in this light.

One-to-many correspondence

Most of us are familiar from ordinary experience with the notion of one-to-many correspondence. Some obvious examples are: a car has four wheels (1-to-4); a hand has five digits (1-to-5); a triangle has three sides (1-to-3); a piano has eighty-eight keys (1-to-88), and so on. As Nunes and Bryant point out:

There are some continuities between these multiplicative situations and additive situations. The most salient is that some of the number meanings here are also connected to sets.

In the examples above, "four wheels," "five digits," etc., do refer to set size. But Nunes and Barnes claim that there are four greatly significant differences:

First, the multiplicative situations involve a constant relation of one-to-many correspondence between two sets. This constant one-to-many correspondence is the invariant in the situation, a type of invariant which is not present in additive reasoning. The one-to-many correspondence is the basis for a new mathematical concept, the concept of ratio. In order to keep, for example, the correspondence '1 car to 4 wheels' constant, each time we add one car to the set of cars we must add 4 wheels to the set of wheels - that is, we add different numbers of objects to each set. This contrasts with the additive situation where, in order to keep the difference between two sets constant, we add the same number of objects to each set. [p.145]

In raising the issue of ratios (and thereby proportional reasoning) quickly, Nunes and Bryant cut to one of the core issues that mathematics educators must take seriously: if multiplication is fundamentally not about joining/separating sets, but about things like ratios, is it reasonable to suspect that one major cause for the well-known difficulties students have with rational numbers and proportional reasoning comes from the propensity of so many teachers to introduce multiplication as if it were simply repeated addition, a viewpoint that is, of course, naively reinforced by those teachers and other adults who were taught to think about multiplication the same way?

It is hard to believe that many young children who run into difficulty grasping multiplication will NOT be told by most adults, "Look, honey: it's just repeated addition. See it now?" Amazingly, I have read some MIRA supporters claim that many kids don't know this "fact" (which of course these same teachers presume is correct), and go on at length about how they must remediate such students by showing them this "obviously true" model. One of them has gone so far as to argue that Devlin and his supporters wouldn't be making claims against MIRA if only he and they had studied with her as a teacher. I'm not confident that this was intended as humor.

Getting back to Nunes and Bryant: they state that this new kind of reasoning

leads us to the second difference: the actions carried out to maintain a ratio invariant are not joining/separating but replicating (to use Kieren's expression) and its inverse. Replicating is not like joining, where any amount can be added to one set. Replicating involves adding to each set the corresponding unit for the set so that the invariant one-to-many correspondence is maintained. For example, in the relation 'one car has four wheels', the unit to be considered in the set of cars is one, whereas the unit in the set of wheels is a composite unit of four wheels. The inverse of replicating is removing corresponding units from each set. If we remove one car we must remove four wheels, in order to maintain the 1 : 4 ratio between cars and wheels. [p. 145]

Here we see one of the most problematic points in trying to convince MIRA supporters that multiplication is not repeated addition. They do not see that in a question like, "If each car has four wheels, how many wheels are there on six cars?" that there are two sets being operated upon differently from the way sets are in additive situations. I suspect strongly that this has something with the difficulty they have in seeing (or simply the refusal to grant) that there are implies or explicitly stated units associated with each set in the problem as stated and they are not the same for each set, whereas in addition, we would be talking about joining or separating sets of the same kind of object (with the same unit, say, 'tires') in each set. With the one-to-many concept grounding this view of multiplication, we have a set of cars with a unit "one car" and a set of wheels with a composite unit, "four wheels per car." Thus, if there are six cars, this problem can be thought of as six cars * four wheels/car. Multiplying the numbers yields

24 cars * wheels/car which equals 24 wheels. In my view, a final non-composite unit, in this case, wheels, can emerge by itself only from multiplication or its inverse. Were this truly repeated addition, we would start with a set containing zero wheels and add four wheels at a time. Joining sets of wheels can only result in wheels. Cars have nothing to do with the case. We could, in fact, add 1 wheel, then 3 wheels and then 2 wheels and then 5 wheels and then 4 wheels and then 6 wheels and then 0 wheels and then 3 wheels and get the same set of 24 wheels. But it would not have the same meaning as 6 cars times 4 wheels/car = 24 wheels. This is the point the authors make when they state that in joining "any amount can be added to one set." And I am convinced that MIRA supporters simply do not see this crucial difference. (There is something here worth looking into regarding how the units behave in rational number arithmetic that I shall delay until another post.)

Turning to the third crucial difference between addition and the one-to-many correspondence situation, Nunes and Bryant state:

[A] ratio remains constant when replication is carried out even if the number of cars and the number of wheels change. In a set where there are 3 cars and 12 wheels, the ratio is still 1 : 4. This is the case because the ratio does not represent the number of objects in either set but is an expression of the relation between the two sets. [pp. 145-156]

This distinction, too, seems utterly absent when reading what MIRA defenders speak about. They harp repeatedly on the fact that the calculations are the same, and thus the operations must be the same or at least that multiplication is reasonably viewed as definable in terms of addition (or the "functions" must be the same, terminology that on my view does absolutely nothing to clarify the underlying meanings or fundamental mathematical ideas under consideration, though perhaps it does help obfuscate those ideas for adults, including some teachers, who aren't sure what a function is or whether calling something a function will help focus, rather than obscure, the issues.)

The last distinction Nunes and Bryant raise for the one-to-many correspondence situation is that:

a new number meaning can be identified in the number of times that a replication is carried out. For example, if we start with the simple situation where we have 1 car and 4 wheels and replicate this starting situation six times, "'6' refers to the number of replications - called the scalar factor. A scalar factor is neither about cars nor about wheels; it does not refer to the number of objects in the sets but to the number of replications relating the two set sizes of the same type. 'Six' expresses the relation between 1 and 6 cars and between 4 and 24 wheels. For the ratio to remain constant, the same scalar factor must be applied to each set.

It is worth pointing out that ratios do not need to involve a unit: for example a recipe may involve a 2 : 3 ration between the number of eggs and cups of flour. When you increase the number of cups, you also need to increase the number of eggs so that the ratio remains constant.

The number meanings in one-to-many correspondence situations are schematically represented in figure 7.1 [Note: the figure consists of two drawings, each with a truck on the left and four wheels on the right. The first drawing has the words "1 truck, 4 wheels: each replication keeps the same ratio." The second has the words "2nd replication of 1 truck, 4 wheels"] In short, one-to-many correspondence situations involve the development of two new number meanings: ratio, which is expressed by a pair of numbers that remains invariant in a situation even if the set size varies, and the scalar factor, that refers to the number of replications applied to both sets maintaining the ratio constant. It should be clear that neither of these meanings relates to set size: the ratio and the scalar factor remain constant even when the set sizes vary. [p. 146]

I will remind the reader that the above four differences are only raised about the first of THREE situations Nunes and Bryant examine. Still to come are those that involve relationships between variables - co-variation, and those that involve sharing and successive splits. We will examine these in subsequent posts. However, I believe that there is enough in just this first situation to seriously damage the idea that multiplication is just repeated addition, even looking strictly at whole numbers. The supporters of MIRA as a reasonable way to introduce children to multiplication and to help children who are struggling with multiplication and multiplicative reasoning have a lot of 'splainin' to do. Or, more likely, explaining away.

An Open Letter to Wayne Bishop (and the MC/HOLD posse)

2010-03-10T07:58:00.011-05:00

Responding to some positive remarks about progressive mathematics education, Wayne Bishop (seen above), a founding member of the anti-progressive reform group Mathematically Correct wrote:

Such speculation sounds beautiful, of course, but I have yet to meet
any mathematician who was taught in a full-blown "discovery"
environment.

This prompted me to write the following open letter:

Dear Wayne and Posse:

Reading your comment about what sort of mathematicians you've never met, I must point out that I've yet to meet one who was taught in a post-racial American classroom, either. That's because neither environment exists, or perhaps like post-racial American classrooms, a fully-realized (is it just a coincidence that you used 'full-blown,' a term I only hear used in connection with cases of AIDS?) K-12 discovery environment exists in some tiny little isolated pockets of the country, so tiny and so few that it's merely a drop in the ocean of indifferent, mostly traditional teaching, materials, and curriculum. Of course, there are districts that use some progressive reform curricula in K-5 or K-8, a very few who use them in K-12. But then, books aren't "discovery learning" or student-centered teaching. They're books. Last I checked, teachers and pedagogy are a major component of what goes on in mathematics classrooms. Classroom culture. School culture. Loads of other factors. Which textbook was purchased may not even reflect which resources are used. I've been in many classrooms where teachers have two or more sets of books and pull from all of them, none of them, or some other variant. At any rate, finding those discovery classrooms that might or might not produce mathematicians, doctors, lawyers, or members of Mathematically Correct is a challenge. Finding ones that fit MY view of high-quality, inquiry-based, student-centered discovery learning with good mathematical content and good problem tasks is not a trivial matter.

This is especially so if we're discussing, ahem, full-blown discovery from K to 12 and then through college and graduate school. And mathematicians would necessarily have to have completed graduate school, right? So you're complaining about not meeting someone who under current conditions cannot likely exist. Let's not miss the fact that the curricula and methods you decry weren't even in the wind for the most part until the early to mid 1990s. Doing the math. . . gee, whiz: it would be pretty surprising to find a professional mathematician who was in a discovery learning oriented mathematics classroom from K - 20 (Kindergarten through Ph.D) Because the environment required doesn't exist in all likelihood and certainly not in one district in enough classrooms to cover K-12 over the requisite years, let alone a university where one would find it from freshman year through the end of doctoral classes.

So you've given us a tautology. No doubt you've not found a lot of dead people who are alive, either. Great job, Brownie!

Then again, have you met many mathematicians at all lately, Wayne? Particularly recently-minted ones? I didn't think so.

You went on:

Many of us correctly believe that we should have been
taught more, and more quickly, but the idea of not learning as much
as possible (ostensibly, from knowledgeable teachers and/or
well-written mathematics books) before embarking on discovering new
and exciting mathematics is purely the stuff of ed school insight,

You made that up. Where is it written in "ed school insight" that we should not want students to learn as much as possible, and from excellent teachers, texts (and other sources you seem to always forget about).

But then we have your usual intentional distortion of what "discovery" entails.

Let's keep this simple for the slower readers in the audience: when is 15 - 9 = ? a problem, and when is it an exercise?

Not to keep you in suspense, it's a problem when you haven't been explicitly shown how to do it or make sense of it. It's an exercise when it's something you already know and are asked to merely repeat what you know to demonstrate that you know it.

If I give that question to the vast majority of first grade students, for them it's a problem. If I give it to the vast majority of high school students, it's MOSTLY an exercise, though for some it's STILL a problem, sad to say. Left to figure out what this could mean, students will figure out one or more ways to have it make sense to them. Given the chance to share ideas under the guidance of a wise and knowledgeable teacher, they will decide what makes mathematical sense and choose the method(s) that work well for them. And given the chance to think, they'll likely keep right on thinking. For 13 years of K-12 and well beyond that. Of course, your own genius children and yourself aside, you don't trust most kids. You really think most kids need Saxon Math or something equally dull. And that most teachers (whom you trust even less than you do children) can't possibly learn to do anything better for their students than take them through a million years of a billion exercises culled from the teacher-proof materials of the late Saint Saxon of the Increments. So utterly pessimistic. So utterly mind-killing. But I suspect that for the majority of kids, that's just what you would love to see.

What IS discovery learning? Is it requiring that students "re-invent" all of the K-12 mathematics curriculum as they go? Of course not. No one has ever suggested anything of the kind except for you and your buddies when you try to scare the pants of the ignorant and gullible. And you do SUCH a good job of it. Kind of like the mathematics educational equivalent of WMD and yellow-cake uranium from Niger, etc. Tell your ugly Big Lies often enough, of course, and no one who didn't already know what you're up to might have a hard time distinguishing reality from your fantasy spook stories.

Is discovery learning creating "new and original" mathematics in K-12? Yes and no. It's new and original to each student as she constructs her own understanding of mathematical ideas. (And once in a great while, K-12 kids actually DO come up with something original. But that's not really the point and you know that fact fully well (or at least that we over here in the real world know it), despite your willingness to feign otherwise). But kids are generally not going to go to many places that they haven't been put in a position to go. If the questions asked and the manner in which they are asked and the classroom and school cultures in which they are asked are suitable, the sky may well be the limit as to what is possible or even likely that kids will do in mathematics or any subject. And when the opposite is the case, then kids will almost never go anywhere worthwhile or meaningful when it comes to thinking about mathematics as much more than a set of rules, facts, and procedures to be memorized at least long enough to pass today's test.

It's interesting that someone as utterly lacking in intellectual joy or curiosity as you managed to become a professional mathematician, though not a particularly prolific one from what I can gather of your output. Seems like you settled into a very mundane position at a less-than-demanding school and decided that to puff up your own importance you'd declare yourself an expert on K-12 mathematics education. And you did a nice job of blustering your way into some level of national prominence (or notoriety, depending upon one's perspective). And so you got to show up at some school board meetings and some state or local hearings, maybe a national one here and there, and declare that the sky is falling because people want to provide kids with a richer, more exciting environment in which to learn mathematics than you ever had. You must be VERY proud of yourself, indeed. Seriously.

But boy, does that progressive education stuff threaten and disturb your little world. And so you latched onto the post-Reagan rhetorical ploy of how to undermine progressive thinking and work: preemptive strikes! It's brilliant. If you're politically, socially, personally, educationally,or philosophically regressive, accuse the other guys of being what you clearly are before they get a chance to point out the obvious about you.

MC, HOLD and similar groups are just a small part of the national manifestation of this sort of tactic. You get to call black people, native Americans, and anyone else you choose "racists" if they advocate an approach to math education that goes against your grain. What could be sweeter? You get to call yourself a reformer, when your idea of reform is "Back to the one-room Iowa school house of my youth" or just back to SOMETHING, even a something that for the vast majority of us never existed and never will. Or back to Saxon Math. Gevalt, it's enough to evoke tears from a gargoyle.

It's hard not to laugh when you cite the seminal group who created Mathematically Correct. Even though you managed to attract a couple of self-proclaimed "socialists," to the last member MC comprises people with conservative souls when it comes to education. You tried to pass yourself off for years on the math-teach list-serve as a "life-long liberal Democrat." That may be the single most absurd and transparent lie ever told.

While no one would suggest that a few of the MC/HOLD cabal are highly-regarded mathematicians, you don't quite get to declare yourself (or David Klein or Jerry Rosen) to be in the elite just by rubbing elbows or what-have-you with Jim Milgram. And having Jim Milgram in your fold doesn't make any of the rest of you (or him, for that matter) knowledgeable about ANYTHING that goes into effective K-12 mathematics TEACHING (I have to suspend my disbelief about college and graduate school teaching).

This really does come back to Lou Talman's recent question to Robert Hansen about how arrogant Lou would be were he to declare himself an expert in engineering because he's a knowledgeable professional mathematician. Robert didn't get it, or played at not getting it, but no one else can miss the point. It's arrogant to step way outside one's area of expertise (alleged or real) and then bash the knowledge and professionalism of the actual experts in that field, merely because there is some area of overlap (yes, the word 'mathematics' does overlap). But knowing math well and teaching math well or understanding what it takes to do so are not the same thing. The second two clearly require the first but don't necessarily follow from having it. And that's where you and your MC/HOLD buddies just go utterly off the rails and never come close to jumping back on again in the two decades or so that you've been trying to call yourselves everything you're not.

Deborah Ball and Magdalene Lampert, to name two non-mathematicians you no doubt would deride (behind their backs if not to their faces, or at least I recommend you not try the latter tack) as 'educationists' whose schools should be blown up (you're lucky: they work at the same one here in Ann Arbor), know more in their little fingers about teaching K-12 mathematics than you'll ever know if you live to be 1,000 years old. You are simply not ever going to figure out the things they figured out without all your advanced knowledge of abstract algebra (even when you try to pull the wool over the eyes of some readers here by throwing out a bunch of jargon to hide the fact that multiplication isn't repeated addition and never is going to be repeated addition. If it were, you'd long ago have addressed my inquiry about why those real mathematicians amongst whom you fancy yourself to belong all seem to think we need two fundamental operations for rings and fields and all the structures in between. You know it's not because they think that the latter is just some version of the former. But you can't bring yourself to say it because you'd be agreeing that there's something wrong about the traditional American curriculum. Horrors!!!)

Were you lucky enough to see Ball or Lampert teach kids, I think your head would explode. Well, not really, because your ability to shut out what you don't want to see, to come up with a thousand reasons why what you see can't be what's really going on is truly remarkable. It no doubt keeps you sane in the face of tons of facts that would produce an overload of cognitive dissonance in most people.

And if all else fails, you'll bring up test scores. Or religion. Or one of your dozens of other dishonest ploys.

I wonder if it ever occurred to you that we can tell more about what goes on in a classroom by actually observing what goes on in a classroom than by all the multiple guess tests in the universe?

Probably not.

But please, Wayne: no more red herrings and lies about what discovery learning is or why YOU'VE never met a mathematician who was trained in such an educational environment in K-12. Instead, talk about the millions who never had a chance in hell of becoming mathematically educated even minimally because they were never shown actual mathematics or any way to think mathematically. And hang your head in shame for continuing to try to prevent that from happening merely because it threatens you in a host of ways.

Well, not to be unfair, let me give you most of the last words:

not professional mathematicians let alone (and statistically
speaking, more important) those who need a strong mathematics
background to pursue their areas of interest. For example, none of
the seminal group who created Mathematically Correct word is
mathematics per se (although some of us became involved very early).
Two were PhD's from Stanford, one of statistics and another in
genetics (later recruited as full professor with tenure to Brown),
another was a PhD in geophysics from USC, another was an independent
contractor electrical engineer, etc., united serendipitously one
evening with a single common thread; all were teaching their children
(and sometimes small groups of their children's friends as well)
mathematics to compensate for their school's use of one of the better
of the math reform curricula, CPM under the misnomer College
Preparatory Mathematics about which I have had some experience:
http://mathematicallycorrect.com/cpmwb.htm

Groovy. Just remember: your Ph.D wasn't from Stanford or anywhere close to it in quality. Nor do you teach at Brown, USC or anywhere near that caliber of institution. You don't get credit because some people who do happen to agree with you to sit in the same room sometimes and share your narrow and elitist views. But if you bet me one Jim Milgram, I'll see you with a Hyman Bass, and raise you a Deborah Ball and a Magdalene Lampert. You've got nothing on your side to match the likes of them, or the many outstanding K-12 mathematics teachers who get what all this is really about. You know: kids learning and doing mathematics and thinking mathematically. Not being little Saxon robots. Or robots of any kind.

Independent, democratic, student-generated thinkers and inquirers: they're not just in English class any more.

Keith Devlin - Extended

2010-02-19T18:12:00.008-05:00

Yes, the debate about multiplication-as-repeated-addition debate is still raging. It popped up again recently on math-teach thanks to a post by the always-thoughtful Jonathan Groves on February 16th, 2010, and has already engendered some fifty-five responses (whether the growth of the thread is additive, multiplicative, or exponential is left as an exercise for the student).

Interested readers are urged to check both the discussion on math-teach (see the link above) and to go to the columns Mr. Groves links to in that post, along with Devlin's most recent column on this issue (from January 2010) and two related columns he cites there from December 2008 and January 2009.

With those as background (or at least Devlin's first column on this issue, "It Ain't No Repeated Addition," from June 2008), let me try to extend the discussion.

Devlin's first column closes with the plea, "In the meantime, teachers, please stop telling your pupils that multiplication is repeated addition."

That's not exactly something you'd think would be devastating. But apparently it really shook up a lot of folks, so much so that Devlin realized that many of those responding actually DO believe that multiplication IS repeated addition. Period. And the most troublesome part is that many of those who do are K-5 teachers (that some of them are 6-12 mathematics teachers is beyond troublesome. That there are a few mathematicians who seem to want to defend such folks is nearly beyond belief (though not if you're familiar with the Math Wars).

Further, even some teachers who know better seem to think there's no problem with telling kids a "white lie." They say that in due time they (or, more likely, some teachers down the line) will say to the students, "Oh, by the way. Not quite" and that this will be completely harmless. (I wonder if some of the thinking that informs this has to do with Tooth Fairies, Santa Claus, and Compassionate Conservatives).

Devlin mentions in several of the columns that he views this sort of thing as far less than harmless, and cites explicitly in the third column, "Multiplication and Those Pesky British Spellings," a body of research that supports his views. I won't attempt to recapitulate that column or the research here, though I have been reading extensively in it, as well as in related work, particularly that of Catherine Sophian of the University of Hawaii in her outstanding book, THE ORIGINS OF MATHEMATICAL KNOWLEDGE IN CHILDHOOD.

I would suggest that this sort of explicit lie or as Devlin calls it "brittle metaphor" - though in the examples below perhaps "brittle rule" would be more on-point - is not exceptional, sad to say. Let me introduce a few more:

1) "You can't take a bigger number from a smaller number," and

2) "You can't divide a smaller number by a bigger number."

Both of these are, of course, false, yet teachers say this every day school is in session and have done so for likely several centuries in this country and elsewhere. I very much doubt there is any American reading this blog who was never told those two things (and probably the third one I will introduce below). When challenged, teachers who know these things are false are likely to offer up the same sorts of arguments that Devlin cites regarding the "multiplication-is-repeated-addition" issue: it really does no harm to tell kids this. They'll "unlearn it" soon enough. My experience with students of a wide variety of ages, including adults, suggests otherwise. I doubt very much that experience is an isolated case.

So what might teachers do instead? Am I about to recommend that they teach students about number systems, modern algebraic notions like complete ordered fields, ordered fields, and integral domains? Or perhaps the Peano axioms (which as Devlin points out are a way to define whole-number arithmetic to first-order logic, not, as are the number systems already mentioned, a descriptive axiom system that tells us how to work within it)? Not likely, though Devlin has been repeatedly accused of wanting to do something along these lines by a number of critics, including at least one mathematician, who know perfectly well that Devlin has explicitly said that is NOT his recommendation).

No, I simply suggest that teachers say to students something like, "We haven't yet learned how to do that, but you will learn [and here the teacher should say something appropriate to the age/grade of the student, though as I will mention, even that may require sensitivity and insight on the part of the teacher, along the lines of "next year" or "in a couple of years"], keeping in mind that there will already be students who know perfectly well that you CAN subtract bigger numbers from smaller ones, or may have at least speculated that this is possible. Some will have already learned outside of school about negative numbers. And so teachers need also be prepared to point students to good resources (on-line, in the library, or perhaps even on the math shelf in class), and to offer to talk with students individually or in a small group if they would like to do so). And obviously, all of the above holds for dividing smaller numbers by bigger ones, given that there will be students who know something about division. (I'm tempted to say that it's not a terribly good idea even to introduce division strictly with problems that have no remainders, but that's one of those things that isn't left open very long at all, and I don't know that teachers actually ever SAY that "You can't divide a number by another number that doesn't "go into" it exactly.)

I think a third commonly-told school mathematics lie brings us to the most general point thus far:

3) "You can't take the square root of a negative number."

But of course, we can. And most high school students have to come to grips with this before they graduate, though generally at a very shallow level.

So what should teachers who aren't teaching about complex solutions to quadratic equations and "imaginary" numbers tell their students instead? By the time students are learning about square roots, it hardly seems like it would be mind-blowing to tell them, "You have not yet learned about the kinds of numbers that we need to solve these kinds of problems [e.g., x^2 + 1 = 0, or just sqr( -16)], but you will in high school" [or whenever is in fact the case].

Thus, what I'm suggesting (nay, calling for) is that we demand that our mathematics teachers in K-12 stop lying to students unnecessarily and instead offer sensible, yet simple responses to questions that require mathematical knowledge that students don't yet have (and indeed, their teachers may not really have very solidly, if at all). Such replies are honest, do not require being "untaught" or "unlearned" at any point down the line. They DO require teachers who have a clue as to how what they teach in mathematics fits into a bigger picture, of course, but it doesn't necessarily require that they have mastered that which they are not actually going to teach. Not that it would be in any way undesirable if they DID have mastery, but I'm not looking to try operating schools with a handful of teachers until we develop enough of them who have such mastery.

In agreement with Professor Devlin, I'm also saying we need teachers to understand the difference between interesting connections between the operations and what those operations actually do. That they know why we really do need multiplication and multiplicative reasoning, not just addition and additive reasoning. And that beyond arithmetic we need exponentiation and exponential reasoning. That some of our models and metaphors don't tell us as much as we may think they do (or have been led to believe they do by others). That every metaphor and model in mathematics breaks down at some point. And that, to quote Korzybski, "the map is not the territory."

Elitism and mathematics education: Why tracking is still wrong

2009-11-17T20:38:00.013-05:00

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?

2009-10-29T16:09:00.005-04:00

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!

2009-06-13T13:17:00.006-04:00

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.

2009-06-03T17:07:00.006-04:00

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!"

2009-05-25T13:15:00.000-04:00

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

2009-05-22T23:15:00.000-04:00

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

2009-05-13T09:18:00.011-04:00

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

2009-05-11T11:06:00.007-04:00

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

2009-04-11T18:43:00.003-04:00

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

LEARNING MATH BY THINKING - Hassler Whitney, Louis P. Benezet, and how many more wasted lives and decades will it take?

2009-04-03T09:47:00.014-04:00

It was 1986, folks (or perhaps 1929), for those keeping score at home. Twenty-three (or eighty) years later and the same arguments are going on, the same mistakes are being made, as if nothing at all has been said like what Louis P. Benezet or Hassler Whitney offered. As if Constance Kamii's work has never been done or published.

My thanks to G. S. Chandy for pointing me to this article. It was published while I was in the process of taking undergraduate mathematics courses in NYC and slowly gravitating towards changing fields from literature to mathematics education. I'd never heard of NCTM or any of the other organizations involved in mathematics teaching and research, the Math Wars hadn't officially started yet, and had I read this piece at the time, I would have naively wondered how anyone could be on the other side from people like Benezet and Whitney. Having suffered through a K-12 mathematics education that was about as inspiring as a dead fish in the gutter, it is remarkable to me even today that I took it upon myself in my thirties to go back to school just to prove to myself that I could indeed learn more mathematics, from calculus to where I was led. Didn't plan to go into the field, and it was fortune, more than anything, the attention of one of my instructors, that led me to start teaching remedial mathematics and, eventually, to do graduate work in math education at the University of Michigan in the 1990s.

Yes, there has been some progress since then, but the entrenchment of traditionalists is fiercer than ever. The lies, distortions, selective quotations, meaningless and carefully culled data, shifting criteria for what "counts" when it comes to evaluating programs, teachers, schools, kids, materials, etc., and many other shady tactics continue unabated, fueled by a hatred for innovation and purveyed by politically-motivated, educationally conservative and reactionary pundits, think-tanks, and foundations, all fiercely determined to see to it that mathematics teaching and learning in this country remain in the hands of a smug, patronizing elite. As long as they are successful in reducing us to a standardized-test crazy culture, as Whitney so accurately describes below, the country as a whole and millions of children will suffer unnecessary torment and boredom when it comes to mathematics. And a populace that is mathematically ignorant is a populace that is far easier to lead by the nose.

Are Benezet's and Whitney's ideas really just those of a couple of isolated cranks, as the anti-progressives from groups like Mathematically Correct would have us believe? Consider for a moment the following anecdote about the great mathematician, Augustin Louis Cauchy:

A mathematical friend of Cauchy's father, Lagrange, recognized the young boy's precocious talent and commented to a contemporary, 'You see that little young man? Well! He will supplant all of us in so far as we are mathematicians.' But he had interesting advice for Cauchy's father. 'Do not let him touch a mathematical book till he is seventeen.' Instead, he suggested stimulating the boy's literary skills so that when eventually he returned to mathematics he would be able to write with his own mathematical voice and not one he had picked up from the books of the day.

It proved to be sound advice. Cauchy developed a new voice that was irrepressible once the floodgates protecting Cauchy from the outside world had been reopened." (Marcus du Sautoy's THE MUSIC OF THE PRIMES pp. 65-66)

No, the analogy to Benezet's experiment is not perfect by any means. But it does suggest that there has long been an awareness that some aspects of formal instruction as they become institutionalized can be stifling to creativity and originality.

Does this mean I am advocating for the destruction of public (or any) formal education and schooling? Not quite. What I am advocating for is a coming to sanity on the part of educators in this country when it comes to mathematics teaching (if nothing else). We are destroying our children, en masse, with the most stultifying approaches imaginable to learning and doing real mathematics, substituting instead a phony "school mathematics" that serves no one truly well, and from which only a small minority emerges able to actually do mathematics, in spite of, rather than because of, the way the subject is taught in most instances.

Few American K-12 teachers have the smallest idea what mathematics is, what it means to do mathematics, or what it means to be a professional mathematician. And what these teachers wind up doing, consciously or not, is to guarantee that very few students will ever find out.

Every time I bump into a piece like the one about Whitney and Benezet below, I am both amazed and sickened: amazed that I hadn't seen this before (though I've been aware of Benezet for over a decade now); sickened that the same lies rise like a foul smokescreen every time Benezet's name or any idea that sounds even vaguely like his, is presented. How much longer do our children have to be tormented by meaningless mathematics education? When will it be time for real mathematics to be taught and learned, in a manner suitable to how children are? If not now, when? If not us, who?

ABOUT EDUCATION; LEARNING MATH BY THINKING

By FRED M. HECHINGER
Published: June 10, 1986

SCHOOL reformers, business executives and politicians are demanding more mathematics for American children. Schools are responding, at least in terms of the hours given to math. Not all mathematicians are cheering. They worry that pressures for more hours of mathematics may hurt rather than help, unless mathematics is taught differently.

Dr. Hassler Whitney, a distinguished mathematician at the Institute for Advanced Study in Princeton, says that for several decades mathematics teaching has largely failed. He predicts that the current round of tougher standards and longer hours threatens to ''throw great numbers, already with great math anxiety, into severe crisis.''

Dr. Whitney has spent many years in classrooms, both teaching mathematics and observing how it is taught, and he calls for an end to what he considers wrongheaded ways.

Long before school, he says, very young children ''learn in manifold ways, at a rate that will never be equaled in later life, and with no formal teaching.'' For example, they learn to speak and communicate, and to deal with their environment. Yet the same children find much simpler things far more difficult as soon as they are formally taught in school.

Learning mathematics, Dr. Whitney says, should mean ''finding one's way through problems of new sorts, and taking responsiblity for the results.''

''This has been completely forgotten'' in most schools, he finds. ''The pressure is now to pass standardized tests. This means simply to remember the rules for a certain number of standard exercises at the moment of the test and thus 'show achievement.' This is the lowest form of learning, of no use in the outside world.''

Dr. Whitney, in a recent report in The Journal of Mathematical Behaviour, recalled an experiment begun in 1929 by L. P. Benezet, then superintendent of schools in Manchester, N.H. Mr. Benezet was distressed over eighth graders' poor command of English and their inability to communicate ideas.

''In the fall of 1929,'' he wrote in 1935, ''I made up my mind to try an experiment of abandoning all formal instruction in arithmetic below the seventh grade and concentrate instead on teaching the children to read, to reason and to recite'' by reporting on books they had read and on incidents they had seen. The children were no longer made to struggle with long-division. ''For some years,'' Mr. Benezet went on, ''I had noticed that the effect of the early introduction of arithmetic had been to dull and almost chloroform the child's reasoning faculties.''

Over the years numbers crept into the children's experience, Mr. Benezet said. They learned to deal with ''halves'' and ''doubles,'' with estimates of size, with a natural development of multiplication tables and slowly, with formal arithmetic.

Mr. Benezet concluded that children who had not been dragged into early but only dimly understood mathematics eventually outdistanced those who had. Literacy in English and a capacity to think independently and to speak and write clearly helped many to do well in mathematics, too.

Dr. Whitney points to that experiment as he looks at today's mathematics teaching. He cites the responses to a problem on a recent test given by the National Assessment of Educational Progress: John and Lewis are planning a rectangular garden 10 feet long and 6 feet wide, and they want to put a fence around it. Ignoring such real matters as the need for a gate, the question was simply how many feet of fencing was needed.

Of the 9-year-olds who took the test, 9 percent chose 32 feet; 59 percent, 16 feet; 14 percent, 60 feet, and the remaining 18 percent gave other answers. Of the 13-year-olds taking the test, 31 percent said 32 feet; 38 percent, 16 feet; and 21 per cent, 60 feet, with 10 percent giving other answers that apparently did not use any arithmetical formulas.

''Why did not all the children get the correct answer?'' Dr. Whitney asks. ''If they were involved in it as a real problem they could have drawn a picture or made it real in some way, and looked to find the answer.'' Instead, he said, they did it ''the school way,'' guessing at what kind of ''operation'' to use - multiplying or adding the numbers.

Numbers, Dr. Whitney says, become a tool when you use them for a purpose. In a class of 6-year-olds, he recalls, the teacher explained how to find the sum of 3 plus 5 by drawing ducks on the board, not noticing a boy in the back of the room saying to another, ''Yesterday I gave you 10 cards; now you gave me 7, so you still owe me 3.''

In the traditional school climate, Dr. Whitney writes, children's natural thinking ''becomes gradually replaced by attempts at rote learning, with disaster as a result.'' In high school, students increasingly say, ''Just tell me which formula to use,'' a way of saying, ''Don't ask me to think.''

Because teachers must ''cover the material,'' Dr. Whitney adds, there is less time to think. When students are called on, they must answer instantly. Wrong answers are not discussed.

''Students and teachers are all victims'' as national commissions clamor for more mathematics without realizing, Dr. Whitney warns, that they may create less knowledge and more anxiety. He says it is crucial to stop just learning the rules.

Dr. Whitney's views are controversial, as were Mr. Benezet's in 1935. Some mathematics teachers and other experts may denounce them as soft on mathematics, but others may welcome relief from demands that turn youngsters off mathematics. Of course, some teachers, ignoring the demands of the moment, actually do teach in the Benezet-Whitney fashion.

However controversial his views, Dr. Whitney deserves a hearing. Present attitudes, he writes, ''lead to the lowest of goals, passing standardized tests,'' instead of encouraging the kind of thinking ''essential for true progress in science, techology and elsewhere.''

The mathematics teaching Dr. Whitney talks about makes children want to know the answers in situations that are real to them. It makes mathematics come alive for them as they do their own thinking and take control over their work, not for tests but for themselves.

Stoller’s Rules: Thoughts on Psychoanalysis and Education Research

2009-03-19T23:49:00.001-04:00

In his 1985 book, OBSERVING THE EROTIC IMAGINATION, the late psychoanalyst, Robert J. Stoller, offers a number of rules for psychoanalytic research that could well be of usefulness in considering what exists and what is possible in conducting (and consuming) education research (as well as the research in other social sciences. I first give Stoller’s list of rules, without putting them in the precise context in which they were originally presented, for fear of prejudicing some readers about taking these ideas seriously. Note that words in brackets are changes I have inserted to make the rules more obviously relevant in the context of educational research. Original language being replaced is specifically psychoanalytic.

Rule 1: anyone can assert anything.
Rule 2: no one can show anyone is wrong, since no one can check anyone’s observations (including his or her own).
Rule 3: ignorance can be wisdom (“The way toward better understanding, then begins with our understanding how little we understand.”)
Rule 4: use [description of motives] warily
Rule 5: ease up, forswear rhetoric, love clarity, relax.
Rule 6: describe people as we see, hear, or otherwise sense them, carefully and in detail. Do not use [educational jargon] in the midst of . . . descriptive sentences.
Rule 7: When it comes to [using educational jargon], less is more
Rule 8: stop picking on students or teachers
Rule 9: let us then, regarding [education], start afresh.
Final note: this is not really a report about [psychoanalysis], but, rather, one that uses psychoanalysis] as an example of the failure of [educational research], so far, as science.

So, what is this? More bashing of education research but from an unexpected source? Hardly. In fact, it is a call for honesty on the part of educators (at various levels), administrators (from buildings, to districts, to states, to Washington, DC), researchers, policy-makers, politicians, journalists, talking heads, business leaders, think-tank pundits, foundation funders, ideologues, and various other stake-holders when it comes to what education research is, can be, or should be.

Like Stoller in his remarks on psychoanalytic research literature and its jargon, I offer this commentary out of concern that both practitioners and critics are destroying education research by trying to make of it something it is not: a scientific literature akin to that of physics. I will resist at this juncture the temptation to go further by looking at the work of Paul Feyerabend and other post-modern philosophers of science who are critics of the notion that even the hardest science is not devoid of subjectivity, judgment, and, for want of a better term, humanity. It suffices to state that it is an error to try to make a "pure" science out of an area of inquiry that cannot afford to lose contact with its essential humanness and humanity.

Thus, allowing those who think that "the answer" to problems in education and education research are the same, and that this alleged single answer is foundational, I suggest that we turn to the instructional precedent of the history of the philosophy of mathematics itself, where logicism (as exemplified by the efforts of Russell and Whitehead in PRINCIPIA MATHEMATICA) failed to establish purely logical grounds for all of mathematics, due to underlying problems with completeness and consistency revealed by the work of Kurt Godel. The failure of the Russell and Whitehead project did not lead to the death of mathematics, of course, but rather to the demise of foundationalist attempts at creating some "ultimate" underpinnings for mathematics. Among the implications of Godel's Incompleteness Theorem are the realization that there will always be unprovable but true theorems as well as false hypotheses that forever resist refutation. Is this a tragedy, or in fact, as I believe, a signal to mathematicians that new ideas and inventions in their discipline will never be exhausted?

Returning to education research, if mathematics, the "queen of sciences" cannot be ultimately grounded, why should it be necessary to ground education or its research literature strictly in quantifiable, statistical/mathematical terms? The fact is that no such grounding in some sort of absolute and objective reality is possible. Neither is such a chimerical pursuit necessarily desirable if it could be attained.

One of the peculiar things I noticed as a new (but hardly young) graduate student in mathematics education at the University of Michigan in July, 1992, was the on-going controversy and conflict in my newly-chosen field between those who advocated for purely quantitative research and those who supported primarily qualitative methods. Given that my adviser and the principal investigator on the project that funded my graduate work at the time was engaged in fundamentally qualitative research, it might seem predictable that I would be unduly prejudiced against quantitative methods. It bears noting, however, that I had already taken two graduate-level courses in statistics and quantitative methods and experimental design while doing graduate work at the University of Florida in psychological foundations of education. I was not ignorant of or intimidated by statistics. I did, however, find myself somewhat skeptical of the notion that educational issues would readily be settled through the kinds of experiments that could be well-analyzed by the quantitative methods I studied (somewhat ironically, the other students in the courses I took were all doctoral candidates in clinical psychology, many of whom planned to be psychotherapists, not research psychologists. And the text we used, the classic STATISTIC FOR EXPERIMENTERS by Box, Hunter, & Hunter, seemed to draw all its examples from industry: nothing could have been more quantitative and, seemingly, objective. I could make sense of it, but I didn't see it applying readily to educational research and still do not.

Neither, I suspect, would Robert J. Stoller. He makes clear in book after book that he feels obligated to remove a false sense of objective truth (conveyed in no small part through the use of psychoanalytic jargon, which he both decries and eschews) from his psychoanalytic studies of particular patients (several of his books deal primarily or exclusively with but a single case history) and issues of gender identity, sexual attitudes, practices, and feelings, etc. In the latter part of his career, he worked directly with an ethnographer/anthropologist, Gilbert Herdt, and even traveled to meet with and study the Sambia tribe in New Guinea that Herdt had been investigating (they co-authored a book on this collaboration). Stoller began doing a form of what he termed clinical ethnography that extended to studying several marginalized segments of American society.

Throughout this work, he makes crystal clear in the books he authored that in the sort of science he engaged in, he, the analyst, interviewer, ethnographer, IS the instrument of (not under) investigation, and hence cannot be kept out of the awareness of readers. The idea is not to guarantee a sort of false reliability by, in effect, telling readers: "Look, I'm drawing your attention to the fact that I'm the lens through which all of this is being filtered (it's important to note that Stoller made a major point of reviewing analytic notes and manuscripts with patients and those he interviewed for other studies before publishing and always got their explicit approval), and so you can trust me because I'm telling you this." Rather, he repeatedly states that the best he can do is to periodically remind readers of his own choices of what to report and how, and to do his best to get his viewpoint, biases, and methods "out there on the table" for readers to consider and examine. This practice doesn't make him right, or reliable, or objective, or anything of the kind. But it does make him much more honest than many other writers. And it does go a long distance towards demystifying and "descientizing" what he is up to.

On my view, this approach of Stoller's is precisely what educational researchers should be doing. While there are places where purely quantitative research may be possible, I believe those places are far fewer than many would have us believe, especially those during the last eight years who have pushed for so-called "data-based research" as the alleged gold standard. My sense of this move is that it has been a smokescreen for marginalizing research that looks closely at individual and small cases, the very sort of work that brings to life what happens in real classrooms with real teachers and kids. It may be possible that a blend of qualitative and quantitative research methods will emerge during this century that will allow research teams to include close studies of individuals with "bigger picture" statistical studies, the former fleshing out and giving real life to the latter. My fear, however, is that the biases both outside and inside the research community will continue to make such work difficult. The misguided desire to appear scientific will allow pressures from politically-motivated forces to keep qualitative researchers on the defensive, when in fact it may well be that it is purely quantitative research that we need to be most suspicious of. The idea that "data speak," that there is some objectivity about deciding what data to collect, how to collect it, what experiments to conduct, what statistical methods to employ, and how to interpret the results of statistical analysis, are all highly doubtful and dangerous. This sort of think is similar to that of people who believe that documentary films are somehow objective, when in fact every single aspect of them entail subjective choices by the filmmaker(s) that shape what the viewer gets to see and attempts to craft a particular sort of response. The fact remains that such films can be, if anything, less true, less honest than "fictional" movies.

If it were possible to do so, I would strive to convince all educational researchers to cast off the guise of objective science and to turn to methods similar to those employed by Stoller. I would urge them to put themselves and their prejudices and viewpoints more explicitly into their research reports, while preserving the sound tradition within social science research of examining alternative interpretations and hypotheses, something Stoller does with great frequency. And I would counsel that they abandon as much as possible jargon-filled writing that alienates teachers in the field, parents, and other stakeholders from being able to make sense or use of much educational research, jargon that primarily is aimed, I believe, in making the research appear to be weightier and more objectively scientific than in fact it is or could possibly be.

Like Stoller, I suspect that I am here trying to swim upstream. But if his work is as valuable to psychoanalysis as I believe it to be, and if he, a physician who could readily have hidden behind both psychoanalytic and medical jargon, refused to do so in the belief that his work would be far more meaningful and beneficial if written in more accessible language, then how much easier should it be for educational researchers to abandon much of the pseudo-scientific trappings of their work? And how much more effective would that work be, in the long run, if through honesty and simplicity they stopped trying to pretend to be physicists and stood up to those who demand that educational research produce theorems and laws to which the enormous complexity that comprises both teaching and learning can be supposed to have been reduced.