Friday, December 24, 2010

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

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

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.

Monday, September 20, 2010

Does Arne Duncan Have A Soul?

"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

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?

Sunday, August 15, 2010

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

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

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.

Tuesday, July 13, 2010

My Favorite Week: Math Circle Summer Teacher Training Institute

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. 

Wednesday, June 2, 2010

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

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.

Tuesday, June 1, 2010

I Saw Mathematics Education Future and Its Name Is Dan Meyer

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


6/8/07 "I Need Another Blog

7/9/07 "Bizarro Blog: [title redacted]"

8/10/07 "Science Owes Me A Beer"


10/4/08 "Pilot"

10/4/08 "License Plates"

10/6/08 "Out of Control"

10/8/08 "The Bone Collector"

10/21/08 "Schrute Bucks"

11/23/08 "EXIF"


1/14/09 "How Can We Break This?"

1/16/09 "ELA Edition"

2/16/09 "Becky Blessing"

2/17/09 "The Woman Who Didn't Swim Across the Atlantic"

3/11/09 "YouTube URLs"

3/23/09 "2008 World Series of Poker"

3/23/09 "Projectile Motion"

4/7/09 "Global Math Geeks"

4/19/09 "What You Can?t Do With This: NLOS Cannon Challenge"

4/20/09 "Yes We Can, Etc"

4/21/09 "The Door Lock" part 1

4/23/09 "The Door Lock" part 2

4/29/09 "Flight Control"

5/14/09 "Other People"

6/4/09 "Glassware" part 1

6/10/09 "Glassware" part 2

7/22/09 "Club Soda"

7/24/09 "Don't Forget Answers, Iteration"

9/6/09 "Groceries" part 1

9/8/09 "Groceries" part 2

9/14/09 "Excellent Math Blogging"

9/19/09 "Check For Understanding"

10/5/09 "What I?m Trying To Say:"

10/7/09 "Pocket Change" part 1

10/8/09 "Pocket Change" part 2

10/29/09 "Redesigned: Kyle Webb"

11/19/09 "Dan & Chris

12/28/09 "The $6400 Question"


1/6/10 "How Do You Turn Something Interesting Into Something Challenging?"

1/16/10 "This Blog Is Counterproductive"

2/7/10 "The Weak WCYDWT Brand"

2/9/10 "Two Excellent Entries For The WCYDWT Course Catalog"

2/9/10 "Will It Hit The Corner?"

2/11/10 "Follow Up: Will It Hit The Corner?"

2/13/10 "Nick Hershman?s Follow Up: Will It Hit The Corner?"

2/22/10 "Water Tank"

2/23/10 "Check For Understanding"

2/25/10 "Math curriculum makeover"

2/25/10 "Who cares?"

3/23/10 "The Italian Job"

3/23/10 "The WCYDWT Workflow"

3/24/10 "Dy/Dan Teacher Prep Academy ? Certification Exam Question #42"

4/5/10 "Gimme Friction / Taberinos"

Thursday, May 27, 2010

What Time Is It?

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
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
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
(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)

Saturday, April 24, 2010

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

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.