A riveting piece by Elizabeth Green in Sunday’s Times magazine on the coming revolution in teacher preparation. Rarely for a Sunday magazine piece it is well worth reading the whole thing. She takes as her starting point the movements for deselection (firing teachers — my rule of thumb is that the more someone talks about firing teachers the less likely it is that they will actually do it) and merit pay, and points out:

So far, both merit-pay efforts and programs that recruit a more-elite teaching corps, like Teach for America, have thin records of reliably improving student learning. Even if competition could coax better performance, would it be enough? Consider a bar graph presented at a recent talk on teaching, displaying the number of Americans in different professions. The shortest bar, all the way on the right, represented architects: 180,000. Farther over, slightly higher, came psychologists (185,000) and then lawyers (952,000), followed by engineers (1.3 million) and waiters (1.8 million). On the left side of the graph, the top three: janitors, maids and household cleaners (3.3 million); secretaries (3.6 million); and, finally, teachers (3.7 million). Moreover, a coming swell of baby-boomer retirements is expected to force school systems to hire up to a million new teachers between now and 2014. Expanding the pool of potential teachers is clearly important, but in a profession as large as teaching, can financial incentives alone make an impact?

There is no alternative but to prepare teachers better for the task.

Figuring out what makes teachers effective instructors, and what we can do to prepare a prospective teacher to be more effective, is not easy, and is not something that has commanded a great deal of attention till relatively recently. The leaders of the effort are in Michigan, where the Dean of the UofM School of Education, Deborah Lowenberg Ball, has been leading a reform of the teacher education program [1]. She describes how Ball’s experiences as an elementary teacher shaped her view that “Teaching, even teaching third-grade math, is extraordinarily specialized, requiring both intricate skills and complex knowledge about math”. Describing a 3rd grade math lesson that Ball was teaching while still a classroom teacher she says:

Ball had a goal for that day’s lesson, and it was not to investigate the special properties of the number six. Yet by entertaining Sean’s odd idea [that 6 could be both an odd number and an even number, because it is composed of two odd numbers, and an odd number of even numbers], Ball was able to teach the class far more than if she had stuck to her lesson plan. By the end of the day, a girl from Nigeria had led the class in deriving precise definitions of even and odd; everyone — even Sean — had agreed that a number could not be both odd and even; and the class had coined a new, special type of number, one that happens to be the product of an odd number and two. They called them Sean numbers. Other memorable moments from the year include a day when they derived the concept of infinity (“You would die before you counted all the numbers!” one girl said) and another when an 8-year-old girl proved that an odd number plus an odd number will always equal an even number.

Dropping a lesson plan and fruitfully improvising requires a certain kind of knowledge — knowledge that Ball, a college French major, did not always have. In fact, she told me that math was the subject she felt least confident teaching at the beginning of her career. Frustrated, she decided to sign up for math classes at a local community college and then at Michigan State. She worked her way from calculus to number theory. “Pretty much right away,” she told me, “I saw that studying math was helping.” Suddenly, she could explain why one isn’t a prime number and why you can’t divide by zero. Most important, she finally understood math’s secret language: the kinds of questions it involves and the way ideas become proofs. But still, the effect on her teaching was fairly random. Much of the math she never used at all, while other parts of teaching still challenged her.

Working with Hyman Bass, a mathematician at the University of Michigan, Ball began to theorize that while teaching math obviously required subject knowledge, the knowledge seemed to be something distinct from what she had learned in math class. It’s one thing to know that 307 minus 168 equals 139; it is another thing to be able understand why a third grader might think that 261 is the right answer. Mathematicians need to understand a problem only for themselves; math teachers need both to know the math and to know how 30 different minds might understand (or misunderstand) it. Then they need to take each mind from not getting it to mastery. And they need to do this in 45 minutes or less. This was neither pure content knowledge nor what educators call pedagogical knowledge, a set of facts independent of subject matter, like Lemov’s techniques. It was a different animal altogether. Ball named it Mathematical Knowledge for Teaching, or M.K.T. She theorized that it included everything from the “common” math understood by most adults to math that only teachers need to know, like which visual tools to use to represent fractions (sticks? blocks? a picture of a pizza?) or a sense of the everyday errors students tend to make when they start learning about negative numbers. At the heart of M.K.T., she thought, was an ability to step outside of your own head. “Teaching depends on what other people think,” Ball told me, “not what you think.”

The idea that just knowing math was not enough to teach it seemed legitimate, but Ball wanted to test her theory. Working with Hill, the Harvard professor, and another colleague, she developed a multiple-choice test for teachers. The test included questions about common math, like whether zero is odd or even (it’s even), as well as questions evaluating the part of M.K.T. that is special to teachers. Hill then cross-referenced teachers’ results with their students’ test scores. The results were impressive: students whose teacher got an above-average M.K.T. score learned about three more weeks of material over the course of a year than those whose teacher had an average score, a boost equivalent to that of coming from a middle-class family rather than a working-class one. The finding is especially powerful given how few properties of teachers can be shown to directly affect student learning. Looking at data from New York City teachers in 2006 and 2007, a team of economists found many factors that did not predict whether their students learned successfully. One of two that were more promising: the teacher’s score on the M.K.T. test, which they took as part of a survey compiled for the study. (Another, slightly less powerful factor was the selectivity of the college a teacher attended as an undergraduate.)

Ball also administered a similar test to a group of mathematicians, 60 percent of whom bombed on the same few key questions.

[1] Full disclosure: I am a Senior Project Adviser at the Spencer Foundation, which funds Green’s fellowship, and of which Ball is a Director. I have also been involved in one aspect of the reform of UofM’s Teacher Education Program, and count Ball as a friend.

{ 88 comments }

Barry 03.05.10 at 3:24 pm

“Teaching, even teaching third-grade math, is extraordinarily specialized, requiring both intricate skills and complex knowledge about math”.

I think that the difference between people who know *anything* about math and/or teaching and those who don’t, is that the latter would be surprised by that statement.

Harry 03.05.10 at 3:29 pm

Well, I don’t disagree with that. But the way we train and attract teachers doesn’t reflect the truth in the statement, and quite how specialized it is is neglected by the reformers who focus solely on reconfiguring the incentives, as if the actors all along had the knowledge, capacity and skills to do what needed to be done and just didn’t want to or couldn’t be bothered. (That includes the “improve the markets” side and the “we need to spend more money” side).

Matt L 03.05.10 at 3:47 pm

Wow. Well, that explains a lot. Then there is a lot to change about how we train all sorts of teachers at all levels: K-12, undergraduate, and graduate school.

First, I have been teaching college history for 5 years (15 if you could count the time I spent as a TA in grad school). I have Ph.D. in history, enough content knowledge to choke a horse, but I never had a single pedagogy class in grad school. All I ever learned about teaching college was through experience on the job, by watching my instructors, and talking with my colleagues. I was conscientious about trying to learn how to teach, but I always knew I was re-inventing the wheel. I guess I am going to have to get into the literature on this even more, but I do not think there will be much support for this at an institutional level. Its always easier to monkey around with things at the edges (end tenure, merit pay, fire the incompetent, etc) than it is to rethink training and pedagogy.

Second, we train a lot of students to teach high-school and middle school social sciences. They universally hate most of their ed school classes and deride them as a waste of time. I have had former teachers tell me the same thing. I always dismissed this as carping about requirements, but this article, or at least the experience of Ms. Ball and others, suggests that they were right. Maybe the same is true of some of our history content classes too. What does history look like to a third grader?

Thanks, this is a great post.

ajay 03.05.10 at 4:02 pm

There are a lot of good tips in the article, some of which I’ve come across before – like asking the question before selecting someone to answer it (aka “pose, pause, pounce”.)

Barry 03.05.10 at 4:34 pm

I would add that the phenomenon of coming to work on day 1, and fiding large gaps in one’s education, is not confined to ed school; I had a crapload to learn as a working statistician, and spent years not doing a great job. I think that the big differences are (a) a new teacher is not physically practicing with their colleagues (they have less interaction than many other professions, I’d guess), and (b) the foundation skill is in dealing with people, which is really hard to teach.

Harry 03.05.10 at 4:44 pm

Barry’s (a) is a huge problem — Ball and her colleagues are working on that one too – -how to structure the practices within the school so that teachers really can interact and learn from one another over time. It takes a radical volte face in the way that administrators think of what their job is (they have to become leaders, focussed on instruction, rather than managers acting as a buffer between school and district).

Using their ideas I’m trying to think about what could be done at the college level for teachers of undergraduates (as Matt L says we need to). More on that if I make some progress…

Matt L 03.05.10 at 5:08 pm

Harry, do you have a complete cite for the Times article and one of the studies by Ball? Is there something (book or article) that would serve as a good entry point for the literature? I’ve been delving into the ‘teaching history’ scholarship for a while, but the things your post talks about seems like it would make a productive point of comparison. Thanks.

Vance Maverick 03.05.10 at 5:20 pm

To grasp this by the wrongest possible end — is it even true that “one isn’t a prime number”? It’s a special case, and as far as I can tell it’s completely arbitrary whether we rule it in and out. (See MathWorld on this — the classification has changed over time.)

Vance Maverick 03.05.10 at 5:20 pm

(should be “in

orout”, obvs.)Sumana Harihareswara 03.05.10 at 5:23 pm

The full article, to be published in the NYT Sunday Magazine on 7 March 2010. “Building a Better Teacher” by Elizabeth Green.

Salient 03.05.10 at 5:44 pm

This is a good place to mention that anyone interested in comprehensive school reform might like to read the SII findings. Little old right-wing-reactionary me found much to love in the success of America’s Choice for content-specializing teachers, though it’s also a delicate reform plan which is really sensitive to being blown up by protesting teachers or insufficiently invested administration.

(My own little theoretical contribution is to argue that AC’s central focus on literacy actually helps teachers reorient their internal understanding of what mathematics is / what mathematics can be, focusing more on problem solving, pattern recognition, the logical reasoning behind the formulas we have, etc, because they have to find more activities in which students are asked to write and contemplate. Drill-kill is out implicitly by design, so then what? That this internal rethinking and revision carries over to other core topics as well. As for students, writing/communication skills and critical thinking skills track together, but that’s already been covered all over the place.)

Since this thread’s about building better teachers, let me suggest that (a) the America’s Choice comprehensive program should be on the table, because it provides infrastructure which would facilitate that building process among teachers already in/entering the field, and (b) it would be interesting and profitable to investigate how we could apply the ideas motivating America’s Choice to design of an education-major curriculum, at least for future middle-school and high-school teachers who content-specialize.

I will note that AC helps contribute to the solution of Barry’s paradox above, somewhat, by creating support staff positions whose job it is to facilitate that kind of practice, and by instituting a macro-structure in which teachers are required to collaborate.

Salient 03.05.10 at 5:45 pm

Here’s a scary thing, only tangentially related: as it stands, having a deep understanding of mathematics, or little such understanding, doesn’t currently have much substantive impact on success in an education program. So we might all agree, “teaching third-grade math, is extraordinarily specialized, requiring both intricate skills and complex knowledge about math,” but we’re not necessarily ensuring that the people we send into that job are well-prepared for it.

LizardBreath 03.05.10 at 6:12 pm

8: I was looking at that as well — I didn’t think there was any non-arbitrary explanation of

whyone isn’t a prime number, and was vaguely thinking that I should look it up in case it turned out to be interesting.On the article in general, I spent a couple of years teaching high school math in Samoa, very badly, and training of the kind described in the article sounds very valuable. It took an immense amount of stupid trial and error to figure out how to run a classroom functionally, and in two years I didn’t get very far along that path. (I’d like to apologize to anyone who took math from me in 1993 or 1994 at Vaipouli College.)

Matt 03.05.10 at 6:15 pm

When I lived in Russia for a while I worked at a “Pedagogical University”, a university nominally dedicated to producing teachers. (It was set up for producing teachers, but there was no illusion that most graduates would become teachers- most Russian universities are still officially specialized in function though people don’t necessarily go to them for that function.) I was really very impressed with the emphasis that was placed on 1) mastering the material that people were learning to teach, and 2) learning pedagogy. I don’t love everything about the Russian schools (they are especially bad in many humanities and some of history) but their general methods were vastly better, and better teacher training, focused on specialized knowledge and pedagogy, was essential. People there were also shocked that in the US “general purpose” teachers are regularly used through the 5th or 6th grade (usually ones who know very little math.) In Russia there’s specialization after the 2nd grade, and it’s a big improvement.

North 03.05.10 at 6:18 pm

I loved that article, because it’s one of the very few I’ve ever read about school reform that actually took seriously the truly enormous number of teachers that need to be replaced or retrained. I find it kind of laughable that people think KIPP (or the other schools whose strategy is basically “hire good people, make them work 16-hour days, don’t pay them more”) is replicable at 3.5 million scale.

Once people figure out how to reliably improve training, though, they’ll have to figure out how to reliably deliver that training. Giving how crap most teacher ed is – and professional development run by my former school district was much worse – it’s going to be interesting to figure out how to use this information at scale.

b9n10nt 03.05.10 at 6:40 pm

I’ll be intersted to read Letmov’s book. At any rate, a lot of this teacher training can be done by teachers themselves. There’s not necessarily institutional support required for teachers to learn and practice Knowledge For “X” Teachers.

What absolutely does require institutional support, however, is a progressive redesign of curriculum towards multidisciplinary, project-based learning.

At any rate, this article gives me hope for my profession.

Stuart 03.05.10 at 6:57 pm

#15: What I could see working would be for a couple of weeks each summer (or whenever the main break is in different countries) you could get a bunch of teachers of a given subject together to work with each other – doing things like covering the curriculum, recalling questions that students asked and discussing the best way to respond maybe including role playing sessions along those lines, meeting with the curriculum developers to discuss the contents/changes each year and any issues with the exam, and various other aspects of pedagogy. Don’t know if anywhere does/has tried anything along these lines though, I wouldn’t be surprised.

Jim Harrison 03.05.10 at 7:03 pm

The sorrow and the pity of the thing is that effective teaching require high levels of training and skill, but we insist that our educational system do everything on the cheap. How do you get around that contradiction?

Maurice Meilleur 03.05.10 at 7:38 pm

Green’s piece is fascinating. It’s worth, on a related note, also taking a look at Diane Ravitch’s recent about-face on the best way to improve public schools (h/t Mark Kleiman). I’m very pleased by these turns in the conversation about public education. But I’m worried that all these discussions–about the relative benefits of merit pay versus firing unproductive teachers, about different pedagogical strategies, about how to recruit the best students for teaching programs, and so on–will remain meaningless until we examine and reform the actual experience of being trained as a teacher in college.

At my institution, the University of Illinois, for example, if you want to teach social studies in secondary ed, you cannot major in political science. Why not? Because

your entire four-year curriculumis so tightly prescribed–not a single elective–that because of major restrictions on taking certain courses, you would not be able to complete your courses to certify. And, apparently, there are no mulligans for any specialty in teaching–no changing your mind about one major and deciding to become a teacher. If you don’t start from your very first term down the proper path toward certification, no cert for you. You’ll need to get grad certification, perhaps through an MAT.Illinois may be particularly fussy about this, but I’ve taught in five states now, and students in ed programs all report the same sorts of problems: to be certified as a teacher, states require such specific and (thanks to the vagaries of state legislatures and ed boards) often-shifting courses that the people who are most likely to succeed in them are the ones who best know how to learn and follow rules, and who appreciate the bean-counter approach to outcome assessment. Students who are intellectually imaginative and expansive, who are interested in pursuing ideas like Deborah Ball’s in Green’s piece, are not rewarded for their ingenuity–they’re more likely to be punished for it. And it’s not just them: even students who want to develop a more traditional form of expertise in a field, like math or sociology or history, as a basis for their teaching career will be unable to do so, because they just will never be able to develop that depth and also finish their degree. We in higher ed talk about the demise of a liberal arts education a lot, but for teaching majors, it seems like even focusing on pure content knowledge in one discipline is ruled out.

No surprise, then, that ed majors bring that predilection for following rules and counting beans with them to their K-12 ed jobs, and tend to filter the challenges they face as teachers through it. And no surprise that their students in K-12 are most likely to succeed, all other things being equal, if

theyalso learn and follow rules well, and count their beans proficiently. It’s this systematic selection for successful rule-following and number-crunching that will make a joke of any implementation ofanysort of ‘knowledge for teaching’. The anecdote about Ball’s math class and the ‘Sean numbers’ made me smile, but it was a bittersweet read, because the way things are now, these stories will remain simply anecdotes–rare exceptions to the rule across the US.Matt Warren 03.05.10 at 7:40 pm

Good pointer. That article is worth a thorough read.

My wife is wrapping up student teaching right now. The process has been a nightmarish ordeal. Between being rigorously scrutinized, dealing with kids that treat you like a part timer, and having to constantly revise lesson plans, it’s beat her ego up quite a bit.

All this is in spite of having a decade’s worth of special ed, learning center, and general subbing experience. There’s so much that needs changing that it’s tempting to think you can just roll up your sleeves and tackle it. But the problem proves to be bigger than it seems at first glance.

The largest problem in my field of view is that there is still more emphasis on abstract factual data (names, dates, places) than there is on critical thinking skills. The former will be almost entirely forgotten while the latter can serve a student throughout their educational (and broader) life and is applicable to many types of inquiries.

Thanks for the pointer, Harry, AND to all you commenters that have added a nice stable of reference materials for my wife and I to explore.

Ahistoricality 03.05.10 at 7:52 pm

The article’s probably fine — I’ll read it over the weekend — but the introduction annoyed me. “Teachers” is a moderately useful category, but even a modifier like “K-12” covers multiple varieties of age-appropriate pedagogy (aside from subject-matter, of course) which are not really transferable. Someone trained in elementary education is going to be about as lost in a HS Social Studies classroom as a Ph.D. historian used to teaching college students. The idea that there’s such a thing as “teaching” which is fungible….

Yes, it’s hard to get through an undergraduate teaching program in four years. Because it

shouldtake longer. It’s complex professional work and should be compensated and evaluated appropriately.Christine Cheng 03.05.10 at 7:57 pm

I enjoyed the article as well and have passed it on to friends who work on education policy.

However, there was one thing that jumped out at me: there was not enough evidence that Lemov’s techniques worked, except on an anecdotal basis.

“While study after study shows that teachers who once boosted student test scores are very likely to do so in the future, no research he can think of has shown a teacher-training program to boost student achievement. So why invest in training when, as he told me recently, “you could be throwing your money away”?

There is an easy solution to this problem: if you want to know whether the Lemov techniques work, all that needs to be done is to conduct a field experiment. This would be a perfect application for it.

Take a group of teachers-in-training and randomly divide them into two. (The group needs to be large enough to get statistical significance in the results. Use a matching technique if necessary.) Put one of these groups through the Lemov course, but not the other (to be used as a control). Track their students’ test scores (or some other way of measuring student outcomes) for some number of years, say 3-5. Compare the results.

If Lemov’s techniques work as well as we’d hope, then the results will speak for themselves. This will provide the proof that is needed.

jcs 03.05.10 at 8:11 pm

@20 “The largest problem in my field of view is that there is still more emphasis on abstract factual data (names, dates, places) than there is on critical thinking skills.”

I have been teaching for 15 years and this seems to me to be exactly correct. For the last two years I have been teaching at an international high school with students attending from acr0ss North America as well as all NATO and EU countries and it is my observation that failure to teach critical thinking skills and promote creative thinking is not unique to American schools at all.

Matt 03.05.10 at 8:17 pm

Yes, it’s hard to get through an undergraduate teaching program in four years. Because it should take longer.That’s another thing about the Russian system- it’s 5 years long and has fewer general ed requirements so the graduates of the program are specialists in their area as well as trained as teachers.

piglet 03.05.10 at 8:19 pm

I am not a high school teacher (I now do teach though, in college, with exactly zero formal preparation for the task) nor particularly knowledgeable in these matters but nothing I read in the excerpted article was new or surprising to me. It should be pretty obvious that knowing a subject matter is a necessary but not sufficient condition for being able to teach it.

I also share some of the nitpicking here. That 1 is not a prime is in fact arbitrary and precisely not a good example of what it means to understand maths, as opposed to merely follow instructions to derive solutions to (usually unrealistic) maths exercises.

Another statement that strikes me as besides the point:

“It’s one thing to know that 307 minus 168 equals 139; it is another thing to be able understand why a third grader might think that 261 is the right answer.”Knowing the result of 307 minus 168 is not, in my book, the goal of maths education. The kids know how to use a calculator.On the broader question of how to improve education, the argument that we “only” need to find better teachers has always mystified me. Anybody read the article in the New Yorker in which Malcolm Gladwell suggested teacher selection was like identifying the best athletes? (http://www.newyorker.com/reporting/2008/12/15/081215fa_fact_gladwell) Such a plain silly idea and this stuff is being taken seriously. Never mind that the whole point of athletics is that a small number of elite athletes suffice to entertain whole nations. A small number of elite teachers aren’t going to make a dent in educational outcomes. Of course, like in any other profession, some teachers are better than others. Among a force of millions of teachers, there are bound to be some really horrible and some really excellent individuals. But what matters is the average teacher, the bulge of the bell curve, not the tail. those who fail to understand this can’t have paid much attention in school.

Substance McGravitas 03.05.10 at 8:20 pm

The Ravitch piece linked by Maurice above is all in favour of general ed requirements.

vaipouli_grad 03.05.10 at 8:43 pm

“(I’d like to apologize to anyone who took math from me in 1993 or 1994 at Vaipouli college.)”

Don’t worry you did fine. Anyway I doubt anyone is still affected as it’s 2010 now so it was more than 25 years ago…

Omega Centauri 03.05.10 at 11:04 pm

This sort of reminds me of my ex weekend fun job -Ski Instructor. There a huge amount of attention was paid to diagnosing what students were doing wrong, and how to correct student mistakes. It was never about, just do X Y and Z, and turn in your homework. Obviously teaching a simple physical skill differs from teaching important intellectual skills, but I think the emphasis on diagnosing common -and not-so common learning impediments amoung the students is relevant.

Kind of an example of the opposite, was my eleventh grade geometry class. The teacher was highly credentialed, PhD and all. But she basically taught as if she had a lesson tape recording. Any questions were always responding to by “rewinding the tape and replaying that part of the lecture”. There seemed to be no give and take whose purpose should be to figure out what the student wasn’t getting.

Seeds 03.05.10 at 11:18 pm

@piglet, 25:

Does anyone take Gladwell seriously? It’s a terrifying thought.

@vaipouli_grad, 27:

Ha!

And, as a general point:

Pity the poor students of English. I teach English as a Second Language with the best qualification available, and qualifying took me 4 weeks. I had no previous teaching experience. One year on I still don’t feel as if I provide particularly good value to my students, many of whom urgently need English to get into university, find a job, etc.

ScentOfViolets 03.06.10 at 2:35 am

Well no, not really. Something completely arbitrary would be having the natural numbers start at one rather than zero, or vice versa (and in fact, a lot of people do define the natural numbers as including zero – the important thing about them is that there be a

smallestnatural number so that induction holds.)As mathworld notes, you can have a different definition, but if you do, you get different theorems, more complicated ones at that, and it’s just not worth it. Also, when you want to generalize, moving from the naturals to the integers to the rationals, etc, the definitions and theorems have to be successively modified as well. For example, if you don’t exclude units in the integers your “nonuniquess” is even more nonunique. I suspect the problem here is that pedagogically, the way to treat the naturals at the lower levels has historically been to classify them as either prime or not and leave it at that with no discussion of what a unit is, or the difference between numbers that are irreducible and numbers that are prime.

ScentOfViolets 03.06.10 at 2:57 am

Blink. This distinction came up rather quickly. One of the problems is that the definition you learned in school was probably wrong. What you learned was the definition of irreducibility, that a number p is irreducible(prime) if the only factorizations of p result in at least one of the factors being a unit. The definition for primality is different: a number p is prime if p divides ab implies p divides a or p divides b.

So for example, in clock arithmetic 3×9=3 and neither 3 nor 9 are units so 3 is reducible. Otoh, 3 is clearly seen to be prime by the second definition. Neither definition contains the other, btw. An element can be nonprime and irreducible as well as prime and reducible.

ScentOfViolets 03.06.10 at 3:14 am

As someone who teaches math, this touches on my personal bugbear for all these schemes for “improving” teachers: while I emphatically agree with the bolded part, it means – wait for it – homework. I need my students to do homework because that way I have the information to run a diagnostic on. If I don’t have the homework, or the homework is arbitrarily done in a very poor way so that I can’t get a handle on what’s wrong (it’s not what you don’t know, it’s what you do know that ain’t so. Usually), I can’t effectively teach.

So my first, my very first action when looking at a problem student is to look at their homework. And if it’s not done, or done incompletely, I say problem solved: somebody needs to have a talk with their parents. If this doesn’t happen, and if the student continues to not turn in their homework, well, there’s just not a whole lot I can do. And no amount of “coaching” or “training” is going to make me into a better teacher for dealing with those instances. Sadly, this seems to be a very common problem, perhaps the predominating problem, as opposed to a comparatively minor one

Grim 03.06.10 at 8:21 am

Harry (and Salient),

Well, a whole lot more responses to your post this time. Did it touch a few nerves, perhaps ?

For a tiny bit of perspective, I once spent some time absorbing a book titled ‘Planning Theory’ written by Andreas Faludi (published in 1973). Faludi, who was a town and urban planner by profession, spent some time distinguishing between what he termed:

Procedural theory – the theory of how to do planning; and,

Substantive theory – the theory of what it is you are trying to plan.

Seems to apply just a little to ‘educating’ doesn’t it. Of course, the point is that unless you are proficient in both, you get the kind of planning (educating) that we have all suffered for many decades. And, surpringly, the acquisition of both kinds of ‘theory’ needs theory and a lot of practice – in short, applying the ‘theory’ and learning from actual results that you have personally experienced. So it goes with homo sapiens, yes ?

However, to me it seems that the really key point in Ball’s account has been all but completely overlooked:

“Teaching depends on what other people think,” Ball told me, “not what you think.”

In short, it isn’t so much a matter of ‘Can you teach ?’ as of ‘Can you work out how to get your students to learn ?’ And that is very, very much a matter of “what [and how] other people think…”.

But I look forward to another post wherein Deborah Ball can gleefully recount that her 3rd graders (oh, ok, maybe the 4th graders), have spontaneously generated Goldbach’s Conjecture, all on their own.

Justin 03.06.10 at 12:17 pm

@25 Responding to comments like this is a waste of time, but this morning, I’m just amazed that you’ve gotten to your present age you are without learning what’s wrong with your reasoning. If getting the results of three digit subtraction was the end goal, the calculator would do just fine. But mathematics is cumulative, and having procedural knowledge ends up being useful in an indefinitely large number of ways. Compare:

You don’t need to know dates to do history, because you can just check wikipedia.You don’t need to know the names of functions to program, because you can just look them up. You don’t need to have a strong vocabulary to write well, because you can just check your thesaurus.

Now, we can definitely argue that history courses spend too much time emphasizing dates, and not enough time developing critical thinking, for instance, but that’s not to say that you can teach history without making sure the students know quite a few dates.

lal 03.06.10 at 3:18 pm

thinking about my many years of teaching (at the undergraduate and graduate level) i realize that it took me a while to realize that one key to effective teaching is understanding what is “hard” about the material being taught—sometimes new professors have an advantage because they remember their relatively recent travails finding the path to what seems “obvious” now.

teaching young children magnifies the complexity of pedagogy because they don’t

understand the world as adults do and it simply won’t work to “try to remember what

was hard about manipulating numbers.”

it doesn’t seem remotely practical to require all elementary school teachers to retrain

in the developmental psychology of cognition but it might be practical to

develop teacher teams led by someone who does have such training to

regularly discuss lesson plans and inform pedagogy based on greater

insight into children’s needs and propensities.

i very much like the example of labelling the parity of zero as even—it is a

priceless lesson on having the courage of your convictions–once finding a

definition of parity asking and answering : where does it lead?

lgm 03.06.10 at 6:03 pm

This post was a blast from the past. I thought Ball and what we call “fuzzy math” had been discredited along with whole language reading. Fuzzy math (called “math reform” by its advocates, including Ball). It is ironic that Ball refers to subject knowledge as important, given that the fuzzy math developers excluded actual mathematicians and felt free to decide for themselves what mathematics was. Check out

this old site for more on the “math wars”.

piglet 03.06.10 at 6:25 pm

Justin, you are not paying attention. The quote is “Knowing the result of 307 minus 168”, and that as I say is irrelevant. You are talking about “understanding how to do a subtraction, and understanding why some students would do it wrong”, and that is something entirely different. Maybe that’s what the author meant to say but it’s not what she did say. And I am not nitpicking here on semantic nuances. The point is that a whole lot of people, perhaps the author of this article included, don’t understand that mathematics is not what calculators do.

piglet 03.06.10 at 7:36 pm

ScentOfViolets: I may have to refresh my number theory but in any case, the reasons you give for excluding unity from the primes cannot be taught and understood in the context of 3d grade mathematics. So the objection voiced by several commenters is still valid. The way I remember is that the primes are usually (in high school I mean) defined as those numbers divisible only by 1 and themselves, and then 1 has to be arbitrarily excluded.

Walt 03.06.10 at 8:24 pm

You could certain explain it at the high school level. They cover unique factorization of integers in high school. And since unique factorization is the point, it’s not really arbitrary.

engels 03.06.10 at 9:02 pm

There’s no need to treat 1 as a special case in your high school definition as you can equally well define it as a number that has _exactly_ two factors – 1 and itself.

piglet 03.06.10 at 10:57 pm

engels, you can use that definition but you could as well use a different definition which would include 1. Again the issue is that it is hard to explain at high school level why this definition is better than the other, and why it isn’t simply arbitrary.

Walt: you are right. But the argument for excluding 1 is again one of convenience and not of principle. One could rephrase the unique factorization theorem to exclude unit factors instead of choosing a definition of prime number that excludes 1. I doubt whether these intricacies are really relevant at high school level. I do think that the teacher should have been exposed to these intricacies and there I actually agree with the article.

tbrober 03.07.10 at 12:15 am

ScentOfViolets, I couldn’t have said it better

I am also a public high school math teacher, and the fact that I can’t get most of my students to do homework means that there is no retention. Many of my students (some days, almost all of them) understand most of what I teach on a given day, but forget it the next day. This is because they have had no opportunity to reflect on it, or struggle with a problem on their own (they do problems on their own in class, but the teaching is too fresh in their minds. It doesn’t really show that they will retain it).

And no retention means I can’t build on prior knowledge. In our district (one of the largest in the country, with a 95% minority population) we have a new reform every year or two, and all the cringe-inducing ‘professional development’ that goes along with it. But until I can persuade students to do homework, nothing is going to change.

ScentOfViolets 03.07.10 at 12:34 am

Piglet, it really doesn’t matter, because the definition of primality at the elementary school level (and probably the high school level as well) is simply wrong. And no, you can’t simply rephrase the theorems this easily. Part of the problem is that whether or not an element is irreducible(or “prime”) is not an intrinsic property, rather it is in the context of the underlying number system. 4 is not prime in the integers, it is prime in the integers mod 9, and then in the rationals it makes no sense to say that 4 is prime at all since every nonzero element is a unit.

You can of course with a sufficient amount of effort rework all of the relevant theorems. But that’s rather more work than is put into the effort of asking why zero is not a natural number, and why is there a separate class of numbers called whole numbers made up just to include one extra element. If you’re going to allow the one, it seems to me that you better allow the other, and yet, this never seems to be the case.

ScentOfViolets 03.07.10 at 1:07 am

I have acquaintances who report the same thing happening in chemistry and physics as well. So my thought is that until “reform” advocates start making noises about holding the children and the parents of the children accountable for actually

doing the homework– and that’s a minimal as well as minimum requirement I would think – all of this is just so much scapegoating the teachers.MikeM 03.07.10 at 3:00 am

C’mon now. Every single one of us has experienced dozens of teachers (with their unique mannerisms and idiosyncrasies), and we must have picked up

somepedagogical skills this way. My wife, a sociolinguist, calls this “learning lyrically,” which is how she learned Spanish — from the families she worked with in her ethnographic research. [In fact, when she goes to academic conferences in Mexico and Spain they laugh at her “ranchero” accent.] I taught research methods and statistics for 30 years. I was pretty tense the first two years, but afterwards was able to relax and integrate some of the qualities I had picked up from my teachers.North 03.07.10 at 5:53 am

@45 (MikeM) that doesn’t mean those techniques were *good* – I’ve always been amazed at how much of teacher education involves prospective teachers talking about their own experiences, which of course varied widely in quality. Also some of the effectiveness of any particular technique has to do with the student. I know that I’m a very atypical student, and thus what worked for me is a particularly unreliable guide to becoming a good teacher myself.

Justin and piglet, I think you are having different conversations. There are two things that I thought were important for my math students to know about arithmetic operations before they could learn algebra (which is what I taught, and they largely didn’t know these things when they got to my class). First, they should know the principles of the operations – what subtraction is and why and how you do it. piglet, I read you as thinking this is important, which it is. BUT. I also found that my students learned algebra much much faster if they had a supply of arithmetic operations they could do quickly in their heads, so they could solve math problems on the board without calculators, check for plausibility easily, and – perhaps most importantly – have a sufficient stock of problems which were dead simple for them. When you’re teaching new notation, like solving equations, it’s important to have problems where the answer is intuitively obvious (3x=6, what is x?) and problems where it requires a little calculation but is still very simple (7x=63, what is x?) and problems that are simple to work out in your head or on paper but make it clear that you might actually want the procedure rather than being able to guess accurately (200x=80000, what is x?). Most of my students weren’t comfortable enough with multiplication to do problems two and three without a calculator, or to verify their answers without a calculator. That’s a problem for teaching. It slows you way down and makes it harder for your students to build from simple to complex examples.

None of this is an argument for drilling long division or putting a huge amount of emphasis on the mechanics of borrowing in subtraction or thinking that not being able to do long arithmetic problems means you’re dumb. But if 217 doesn’t look wrong to a student as the answer to that subtraction problem, I’d bet on that kid to have more trouble in algebra.

magistra 03.07.10 at 7:46 am

On the broader question of re-training teachers, how easy is for someone to abandon bad habits/ineffective teaching practices that they have already and adopt new and better practices? A lot of the discussion in the original article seemed to assume that teachers just didn’t know any better and once they had been told the correct techniques they would automatically change. But I found in the very limited training I had as a lecturer that even when I ‘knew’ that I should be doing something different (like speaking more slowly or allowing more interaction), it was hard to put that into practice because my instincts tended to take over when I was tense.

Does any plan about retraining teachers need to think rather harder about how these teachers might be helped to improve sustainably, rather than just saying ‘this is what you should be doing, now go and apply it’?

piglet 03.07.10 at 4:55 pm

ScentOfViolets, I don’t disagree with the definition of primes.

North, I agree completely.

And I agree with what has been said about homework. There is nothing that can be learned without practice. Nobody expects to learn a foreign language without practicing it. The same is true for maths and any other subject. A good teacher may design more effective exercises and be a better motivator but without the student doing the actual work, even the most brilliant teacher will have only limited success.

Harry 03.07.10 at 6:09 pm

I didn’t know you were a KCL graduate, magistra. Me too.

The question you raise is hard. Here’s a composite view drawn from various sources. Basically, if you really organise initial training around the skills and knowledge base that teachers really need to teach well, you’re not going to get much push back from the trainees. The problem is changing practices of existing teachers, and creating an environment in schools in which your trainees maintain and build on the skills they have. Doing that requires reforming the whole school around a focus on instruction. In other words, you need data about who succeeds with which students, embedded time for colleagues to observe another and to pick up the skills that the successful teachers have, and an administration and department leadership that sees itself as leading instruction and facilitating learning about instruction. This is all extremely difficult when you lack common assessment instruments; and even more difficult because it requires going completely against the well established and entrenched norms of the profession. Principal preparation, and massive work to reconfigure the internal life of schools is needed. (“QWERTY times 100”, as one friend described a similar problem at the college level).

piglet 03.08.10 at 3:58 am

“But I found in the very limited training I had as a lecturer that even when I ‘knew’ that I should be doing something different (like speaking more slowly or allowing more interaction), it was hard to put that into practice because my instincts tended to take over when I was tense.”

magistra and Harry, Is there any reason to assume that your discussion is more than anecdotal? I’m genuinely curious.

Anarch 03.08.10 at 4:08 am

Oh please. If you’re teaching a course on ring theory that distinction might be acceptable, but unless you’re prepared to offer examples of non-PIDs to a third-grader it has no place in this conversation.

This particular example is a pet peeve of mine, as it happens, since it’s one of those cases where the “advanced” definition is inconsistent with the standard one for no compelling reason. Yes, when you hit ring theory you realize that the concepts now termed “irreducibility” and “primality” are distinct, but there was no excuse in overturning several hundred years of mathematical terminology in recognizing the distinction. It smacks of the kind of mathematical elitism that drives me crazy.

As an aside: LizardBreath, ScentOfViolets is entirely correct that the exclusion of 1 as prime is somewhat arbitrary, chosen to make theorems work out more neatly. If I were pressed to justify it, it’s that primes fundamentally relate to divisibility or, more accurately, to

non-divisibility. In a mathematical “universe” where you can divide by anything — the real numbers, for example, or fields in general — there are neither primes nor irreducibles. There’s only a multiplicative structure of interest when you can’t divide, where there are things which can be considered “building blocks” by virtue of their non-divisibility. So 1 and -1 are excluded from being primes precisely they can divide anything, and themselves are divisible (by 1 and -1 respectively), hence are uninteresting from that point of view.Luther Blissett 03.08.10 at 7:15 am

Sure, practice is key to retention. However, homework is not the only way to practice. In a 45 minute period, there’s no reason why the lesson cannot be taught in 25-30 minutes, leaving 15 minutes for practice every day. That might not seem like a lot, but research suggests that there’s no increase in learning when K-12 students do 50 math problems each night rather than five well-chosen problems.

As a high school English teacher, I’ve stripped homework down to the essentials: reading the primary texts, preparing notes for the next day’s activities (which might mean asking three open-ended questions and noting three quotations that might help answer them), and working on essays outside of class (‘tho I usually schedule computer lab time as well). For vocabulary and grammar, I find that practice works better when I’m around to answer questions as they come up.

Doug 03.08.10 at 8:12 am

49: “even more difficult because it requires going completely against the well established and entrenched norms of the profession.”

It’s even worse than that, because chances are good that a given teaching corps has already experienced one or more rounds of attempts to do just that. Whether they were politically motivated or research-oriented (if, indeed, that’s a meaningful distinction from the point of view of people being asked/told to change significant parts of their working habits); whether a new regime arrived with a new principal, a new superintendent or someone else; some serious effort at reform has probably come through a given school within three to six years.

6: “It takes a radical volte face in the way that administrators think of what their job is (they have to become leaders, focussed on instruction, rather than managers acting as a buffer between school and district).”

Even a buffer would be an improvement on several of the situations my mom has seen over nearly 30 years of teaching: principals who used their schools as a power base in local politics, principals keen to move up in the hierarchy and acting as a transmission belt for the powers-that-be, principals promoted according to the Peter principle. Principals who are more about the school than something else are already ahead of the game.

Salient 03.08.10 at 3:02 pm

Oh let’s please not have another “if they’d only do their homework” thread; sure, let’s for the sake of not arguing all agree at the outset that if the kids all understood how to do all their homework and all did all of it each day, they’d all understand everything perfectly, retain it, and the world would be wonderful. But we’re working within a system in which that will not and cannot happen, and there’s nothing to be done intra-school to change that.

Yes, in theory we could somehow make sure everyone’s home life is wonderful (or at least conducive to study and schoolwork) and that 11-year-olds aren’t working full time after school on their parents’ tobacco fields and so on. We could mandate that nobody under the age of 18 is allowed to have a full-time job, and that any family which can’t make ends meet unless their kid is working get enough social welfare to get by okay, and we could hire sufficiently many social workers to check in on kids whose home life is terribly painful and offer whatever amelioration is necessary. But that’s not the case here, and that’s not the case elsewhere, and just because we don’t have a sufficiently good system to ensure that our students are unoccupied outside of school doesn’t mean school becomes automatically worthless, right? Given unfortunate situation X, namely that enough students will have no time outside of school for schoolwork to ensure that assigning out-of-classroom work will be broadly ineffective, how do we maximize Y, student success? Wishing for not-X doesn’t solve that problem.

JB 03.08.10 at 3:15 pm

Salient, most of the students who don’t do their homework are not experiencing the level of challenge that you’re talking about. Most of them could do it, but decide not to. To be honest, two of my children fell into that category. Part of why they didn’t do the homework had to do with the fact that their interests and talents lay in other directions, yet the high school insisted on aiming all kids at college.

ScentOfViolets 03.08.10 at 3:46 pm

Blink. I’d hardly call it “elitism” (I happen to be an algebraist, or rather, an algebraic geometer, so ymmv). If you read the entire subthread, this is about a pernicious question that arises at the lower levels when the concept is introduced: “Why is one not a prime?” I’m merely pointing out that it would be quite easy to change the definition so that this question is either trivially easy to answer or never even comes up. Particularly since it isn’t technically correct anyway. If you’re going to have inconsistent definitions, you might as well use ones that are easier on the student.

I’m also one of those people who happen to think that you don’t particularly need a whole lot of algebra to succeed in life and would be quite happy if an alternate probability/statistics sequence were offered to satisfy the math requirements.

Harry 03.08.10 at 3:52 pm

The population that I’m interested in doesn’t have the problem that JB’s children had.

Its not exactly true that there is nothing intra-school that can be done about homework, salient. The school day can be lengthened, and after-school tutoring which is closely aligned to the daily instruction could be provided, so that high need children would be provided with an environment very loosely approximating that of the middle/upper middle class home (in which the parents get the kids to do their homework and have more than passing acquaintance with what is needed to get it done).

I think that “homework” call is basically right, but it neglects the fact that across classes the homework experience is radically different — being one room away from someone who is invested in your success and can help when needed is different to being in the same room with someone who, however invested they are in your success, has other things on their mind and did not find the academic side of school enjoyable themselves.

ScentOfViolets 03.08.10 at 4:14 pm

Sigh. No one is trying to imply this that I can see. Further, if you don’t do the homework, you’re not going to learn much of anything. Period. Suggesting otherwise is like saying let’s do health care reform that doesn’t antagonize the insurance companies, hospitals, or doctors . . . and it’s got to be “bipartisan”. Or let’s do financial reform that doesn’t lay “burdensome restrictions and regulations” on banking institutions, and it’s got to be “bipartisan”. Or let’s balance the local/state/federal budgets . . . but you can’t raise taxes, and you can’t cut military spending, ag subsidies, etc. The “Let’s do W, but you can’t resort to X,Y, or Z” approach is what people who don’t want real reform advocate so they have a fig leaf – a very tiny one – of deniability.

Anyway, doing the homework doesn’t guarantee anyone an ‘A’ or even a ‘C’. It is just that, a minimum requirement. And this noncompliance is by no means something that can be laid at the feet of extreme poverty and privation: Every year in the fall semester it is my privilege to teach one of the algebra courses that one should have passed to meet the high school graduation requirement. Given that a significant fraction of my students drive nicer cars than I do, and given they’re not paying for college themselves, Mom and Dad are, I don’t think you can plead lack of opportunity or time as an extenuating circumstance.

In fact, a rather substantial fraction are indignant that homework that’s not turned in, quizzes that are missed (and even exams!) will be counted against their final scores. They have this strange idea that unsubmitted work should either be allowed to be made up, or that alternate make-up work can be assigned . . . often something along the lines of “an extra credit book report on a famous mathematician.” And – of course – there’s always the parents who will come in and apply pressure, saying “that’s what they did in high school.”

Now, I’m not denying that those inevitable hard cases exist. But I don’t see a lot of evidence that they make up a large fraction of all students. What I do see is parents communicating to their kids in not very subtle ways that school and scholarship is a waste of time – even amongst the so-called privileged classes[1]. Chalk it up to American anti-intellectualism if you like.

[1]I suspect that this is yet another instance where you see the erosion of social capital painfully accumulated over the decades. Time was, riding herd on the kids, making sure they did their homework and helping out with their various projects was the province of the stay-at-home Mom. And Dad? Traditionally, his children struggling with their homework was one of the very few things that would get him out of the barcalounger and seated at the kitchen table. Yeah, that’s romanticizing, and with a broad brush at that. But it’s not terribly far off the mark either.

ScentOfViolets 03.08.10 at 4:20 pm

Oh, I’m not trying to imply that’s all that needs to be done. That’s an absolute minimum floor, that’s all. Anybody who’s trying to lay deficient academic performance at the feet of the teachers and who also says that it’s their responsibility that the kids learn the material, but they aren’t allowed to assign any out-of-class homework or studying or projects is simply not being serious.

Justin 03.08.10 at 4:32 pm

@37 The next sentence is “mathematicians only need to understand a problem for themselves.” Understanding a problem is obviously more than knowing that the answer is 139. Your interpretation is excessively literal, and your own comment exhibits it. After all, if you just needed to know that particular answer, you could write it down in lieu of a calculator.

JSE 03.08.10 at 5:48 pm

I know this is somewhat off-point, but ScentOfViolet’s definition of primality

certainly calls 1 a prime, for vacuous reasons: 1

alwaysdivides both a and b, whatever a and b may be. This is why, in the definition of a prime ideal, you require that the ideal be proper. I’m also confused by the statementsince 4 is a unit in the integers mod 9 just as it is in the rationals (its inverse is 7.)

As for myself, I don’t see the usual definition of “prime” as arbitrary — or rather, I don’t see it as any more or less arbitrary than other customary mathematical definitions. The right way to “carve nature at the joints” here is to divide the positive integers into three classes: 1, the primes, and the composites. Which is indeed what we do. But I tend to agree that it would be a serious challenge to convince a third-grader of the naturality of this definition. Lots of smart people with advanced degrees in areas other than math are confused about why it’s unnatural and perverse to say “zero is not an even number,” and that’s much easier.

bianca steele 03.08.10 at 7:25 pm

@53 is right. (I am also the child of a teacher.) There is likely a 50-year gap in the graduation dates of the oldest and the youngest teachers in a given school: “going against established practices” would seem meaningless after even a couple of rounds of reform.

Salient 03.08.10 at 7:33 pm

Its not exactly true that there is nothing intra-school that can be done about homework, salient.Well, ok, this is an unfortunate consequence of me being awfully pedantic.

I would call what you described an (appropriate)

alternativeto homework, and one that we’re pursuing with vigor around here… If “homework” = “independent practice in a semi-structured environment” then all I would insist is that schools ought to provide that semi-structured environment, as well as a free^1^ dinner meal and free^1^ transportation home, to every student, and school administrators ought to be given the legal ability to mandate attendance for students who do not complete their assignments (basically so that, e.g., farmer parents can’t demand their kids come home to work the fields instead of doing schoolwork, something we’re currently seeing much too much of in rural Appalachia, and so that students themselves can’t decide that participating in extra-curriculars is more important than the schoolwork, etc.)I suppose what I meant to say is: given for the sake of not arguing, say, that the last two paragraphs of comment #58 are correct, and further given that that won’t ever change,^2^ what should the public schools do in order to improve?

^1^As in potentially free for low-income students, like school lunch.

^2^If it changes, so much the better, but it’s out of our hands.

bianca steele 03.08.10 at 7:39 pm

And my experience with consultants (as in the article) is that they sometimes make that problem worse, especially in conjunction with external standards. If you have a manager/principal who doesn’t really understand how things are supposed to be done and why, they have no alternative but to memorize what the consultant says, basically, and to impose that on their subordinates. The whole process relies on perfection at every step; if the consultant conveyed the wrong information, at “best” the organization doesn’t get the intended benefit, and at “worst” the organization fails the external assessment.

marcel 03.09.10 at 3:05 am

My daughter just started teaching, 7th grade math this year. My son is in a math PhD program, and long, long ago, I had a BA in math. Because of my daughter, my wife and I read this article with particular interest. My wife asked me about this (1 not being prime), and I told her that it was essentially arbitrary, based on what I recalled from long, long ago. We ended up having a family discussion, and my son explained something along the lines of, in ring theory, it makes no sense for the multiplicative identity to be considered prime. I understood the reason yesterday during the discussion, but now, about 36 hours later, on Monday night, I can no longer recall the details. It may be that in rings, there are not multiplicative inverses, uhm, nevermind.

Walt 03.09.10 at 8:33 am

You all are killing me here with your talk of whether 1 being prime is arbitrary. Killing me! I can believe that it’s not an important point to get across to an 8-year old, but 1 behaves very differently than primes do. And it doesn’t require an excursion into ring theory to see why.

Start with 12. How many times can you divide by 2 without resorting to fractions? Well, you can divide once, and then you get 6. You divide again, and then you get 3. And then you have to stop.

But how many times can you divide 12 by 1 without resorting to fractions? You can divide once, and then you get 12. You divide again, and then you get 12. You divide again, and then you get 12. Hmmm, this looks like it’s going to take a while. Apparently, you can divide by 1 as many times as you want.

You can decompose any natural number as a product of primes. 12 = 2 times 2 times 3, for example. If you throw 1 in there, what’s the decomposition? 12 = 2 times 2 times 3 times 1 times 1 times 1 times 1 times 1 times …?

engels 03.09.10 at 12:25 pm

I agree with Walt. I think an interested 3rd former can understand this. The thing about 1 is that it doesn’t do anything (under multiplication). Primes are the basic building blocks from which you can make all the other numbers. But 1 isn’t a building block; you can’t make anything (else) using 1. Having 1 as a prime would be a bit like having ‘transparent’ as a primary colour.

harry b 03.09.10 at 1:58 pm

Yes, I think 53 is right, too, depressing as that is. You have very good reasons to think some reform will improve things; but teachers cannot hear those reasons because they have experienced endless cycles of reform generated by district officials who have a thin grasp of what should be done and very little connection with what is going on in the schools (see my previous, related, post).

My only experience with consultants was pretty good, but that was because the CEO of the organisation was completely on the ball, and recruited onto the strategic review committee a consultant whom he knew would be critical of what the consultants he employed would come up with.

Margaret Atherton 03.09.10 at 3:25 pm

Ball’s work on Mathematical Knowledge for Teaching in the original article and highlighted in the post here is especially interesting in its suggestion of a content-based way to improve teaching. I do know, however, that work has been done along these lines for decades by Herbert Ginsberg, at TC, Columbia working with clinical interviews to establish children’s reasoning flaws. I believe his work has had impact on curricula, but one can’t help thinking that there is never any progress being made in pedagogy, just wheel reinvention.

Harold 03.09.10 at 9:01 pm

The Times article had two components. The first involved simple steps to achieve discipline and order in the classroom without resorting to yelling, but through using very detailed, explicit and positive instructions and drawing attention to those who are following the instructions correctly so that others can imitate them.

The second part of the article was about how helpful it is to have a somewhat deeper knowledge than average of the subject being taught so that errors and digressions can be turned into teaching moments. Still, I imagine that effective teaching of the subject also must involve the ability to give detailed positive instructions in the same way as bringing order to the classroom. The Soviets, as someone above pointed out, devoted a lot of research to this; and their method, sometimes euphemistically known in the USA as the “Singapore” method, appears to be very effective. The problem is in encouraging teachers to be motivated to follow these proven methods — it is not only children who do not want to do their homework!

Jim Harrison 03.09.10 at 11:40 pm

A grad school friend of mine use to tutor kids in D.C. Like many economists, he was an excellent mathematician and apparently got wonderful results because he was able to understand what the kids were trying to do when they made errors. Thing is, though, we’re not going to stock the public schools with brilliant teachers. According to the ed school profs I’ve met–and I’ve met a great many of ’em in my former job–we’re lucky if potential elementary school teachers can do long division themselves and can be counted on not to communicate their panic terror of all things mathematical to their students. I’m certainly sympathetic to efforts to improve math teaching, but I wish that the various committees that come up with recommendations would do a better job of taking into account the realities of American education. A teaching method that works magnificently when administered by talented, motivated people may just make things worse in practice. You have to ask yourself, what will the run of teachers actually do with these recommendations?

Salient 03.10.10 at 12:00 am

According to the ed school profs I’ve met—and I’ve met a great many of ‘em in my former job—we’re lucky if potential elementary school teachers can do long division themselves and can be counted on not to communicate their panic terror of all things mathematical to their students.…Which ought to lead us to the question of why their programs are graduating a large number of such folks on a regular basis.

Grim 03.10.10 at 5:43 am

Hi Salient @72

That’s a really good question you’ve asked. Do you have a really good answer to go with it ?

Since this is all about mathematics – and absolutely nothing to do with M.K.T. – does the number 3.7 million have anything to do with it ?

Jim Harrison 03.10.10 at 6:32 am

Dear Grim,

The ed profs I mentioned weren’t cynics but they were realists. They spent a lot of the time in their methods classes on math content because so many of their students simply couldn’t do basic math. They couldn’t just flunk out unprepared students without destroying their programs–back in the 70s, or so they told me, the quality of ed majors declined. The usual explanation was that the bright women who formerly went into elementary education had better options while students in general were less idealistic than earlier generations.

Things may or may not be better now than a few years ago. My evidence is certainly dated, and I’d be interested if anybody could make a more recent report on the view from education departments, especially departments in non-elite institutions.

Vance Maverick 03.10.10 at 6:39 am

You can decompose any natural number as a product of primes.Any natural number, that is, except 1. (Or 0, if you count that as a natural number.) I don’t disagree with any point you make, Walt, except your implied claim to have banished arbitrariness.

Grim 03.10.10 at 9:21 am

Jim Harrison @74

Yes, we all get mugged by reality from time to time, and it’s been a long time (if ever), since the ‘best and brightest’ beat a path to the Ed. school.

The main question as I see it is whether any of the approaches, techniques, methods etc of the very talented – such as Deborah Ball or your grad school economist friend – can somehow be imparted to the very large numbers of not so talented who do find their way into education and without whom there simply would be no public education.

If it requires the trainee educators to be “excellent mathematician[s]”, then the attempt is doomed from the start. But Deborah Ball did not start out as an “excellent mathematician”, she merely had the motivation and talent required to become good enough to be able to focus on her students, rather than on herself.

Can any of that actually be taught to the majority of Ed. students ?

Matt 03.10.10 at 11:55 am

If it requires the trainee educators to be “excellent mathematician[s]”, then the attempt is doomed from the start.Again, something that seems to be important in Russian (and before that Soviet) schools doing quite well in teaching math is that they have specialized teachers for different classes much sooner than in the US. I’m pretty sure that having only one teacher for all (or nearly all) subjects stops at 2nd grade, if that late. Math is taught be people who studied math at the pedagogical university, even to pretty young kids. Some teachers are better than others, of course, but teaching kids in general isn’t a refuge for those who can’t do even low-level math. This would be a pretty big change from the US system, where I think having one teacher is still common through at least 5th, often 6th grade, but would be a clear improvement, I think. (Not just for math, either, but also other subjects as well.)

Salient 03.10.10 at 3:32 pm

That’s a really good question you’ve asked. Do you have a really good answer to go with it?Not really, but here are some disconnected-yet-related observations.

It occurs to me that my class is (probably) the last math class any elementary education or special education major at this university ever takes, in their life. Now bear in mind, I’m not some Ph.D. in instructive awesomeness. I don’t have decades of experience to bring to bear. I’m a

graduate student.And most of the mathematics-education reading/research I do is entirely of my own volition.We’re accepting students into the program who struggle with core skills — not just in mathematics, but also spelling, writing, etc — and there’s only so much one year of mathematics instruction can do for them (elementary ed majors are only required to take two math courses, which begin with core-skill adding and multiplying and estimating techniques, and end with three-dimensional geometry — nothing that hasn’t already been taught in the high schools, and most of which was taught at the grade levels they’ll be teaching).

Many students go into the program, by their own admission, because they “like working with kids” and making kids happy (the most common reason we receive, regarding why students entered the program, is students reporting positive babysitting and/or day-care and/or after-school-YMCA-program experiences, and second in frequency, students reporting that one of their relatives, who they loved, was a teacher).

Of course, these kids got straight As in high school and are somehow managing nearly straight As here at university. I was surprised to learn just how high the proportion of GPA-based scholarships going to education majors is. It seems the average GPA of an education major is substantially higher than the average GPA overall.

Madeleine Conway 03.10.10 at 5:33 pm

The UK and US are in the same boat – schools are not serving their students, by most measures poorer students are failed by the system and consequently are unable to build any kind of future for themselves. While the Green article was interesting, I felt that resorting to Lemov’s taxonomy is like sticking a bandaid on the still-bleeding stump where the victim’s hand once was. Just as the UK is preparing to embrace Charter Schools, Swedish and US evidence is suggesting that they do not necessarily contribute a long-term rise in achievement.

Following Finland’s continuing success in the PISA tests, the World Bank published a report (Aho Pitkanen and Sahlberg 2006) on Finland’s educational system, and it gives clear guidance on what works – high social status and professional respect for teachers, a holistic approach to schools as part of their community – healthcare and law enforcement representatives visiting frequently, interaction between the community and the school, and a system which allows students to switch streams between academic and vocational and across disciplines until they are 18/19. There is far less focus on standardised testing than in the US and the UK.

There is plenty of research out there to tell us what is necessary to improve education – investing in teachers is critical, encouraging teachers to believe in themselves and to develop both their skills and their subject knowledge is critical – but ultimately, in market-driven economies like the UK and the US, taxpayers are simply not willing to foot the bill, even when it comes to helping their children to be the best they can be.

engels 03.10.10 at 10:38 pm

Madeleine’s points are spot on of course but in the British case one also has to consider the ingrained anti-intellectual stupidity and crass materialism of its rulers together with the fact that they mostly wouldn’t let their own children anywhere near a state school.

Harry 03.10.10 at 10:58 pm

I’m always a bit sceptical about the relevance of the Finland studies (a society in which there is almost no child poverty is going to be somewhat unlike the US and UK). BUT, and I’m not going to quote the figure because I can’t find it and I’ll get it wrong, the relationship of teacher’s income to median income is massively different than in the UK and US — as I say, I can’t give the exact figure, but I was really taken aback by it. High pay, a career structure, AND insignificant child poverty.

Harold 03.11.10 at 1:41 am

Insignificant child poverty may have something to do with social welfare policies!

Grim 03.11.10 at 6:57 am

Once we got past the number/ring theory tutorial, and whether kids doing their homework at home is the secret of life, the universe and everything, I for one have been very appreciative of the information conveyed.

Though I have to say that the discussion is something of a reminder of a paper that was influential in my former profession titled “Archipelagos of Information”. It strikes me that there is much valuable information held in some quite distinct islands, or, if you prefer a traditional description “the right hand doesn’t know what the left hand is doing” (and vice versa, of course).

What we seem to have is the age old problem (to quote myself): “if you don’t know what to do, do what you know”, which, as usual, doesn’t work very well in complex situations. So some, like Lemov and Ball, due to both personal qualities (mainly intelligence, motivation and persistence) and external cicumstances (ie funded opportunities), have worked long and hard to come up with approaches, techniques etc that, at least in part (though I share Madeleine’s concerns about Lemov), could make for better teaching which in turn could make for more accomplished students.

I tend to share Matt’s view of emphasising subject-matter expertise and early concentration thereupon, but then again, as I said in a previous thread: ” for most students, maths education is too much too soon, but for those who will go on to higher maths, it’s much too little too late”. Maybe, if all students are taught as though they were going on to ‘higher maths’, it would greatly support those who do without permanently injuring those who don’t. But how many subject areas could be given this kind of treatment without seriously overloading the kids (and the teachers, I suspect) ?

Perhaps the secret to Finland is that it is an extraordinarily unified country because of its history of external threat. Maybe the Finns just think that “no Finn should be left behind” or something.

Otherwise, it appears to me, if there are problems with how well teachers teach kids, then maybe there’s even worse problems with how well educators teach teachers. Precisely what Lemov and Ball (et al) are trying to address, each in their ‘separate island’ way: if you can’t rely on the inventiveness and ‘native talent’ of proto-teachers, then train ’em like you would a Navy Seal … or something like that, anyway.

Just one thing though: does anybody here teach music or sport ? Doesn’t seem to be any shortage of candidates for the Julliard School, or for football, baseball or basketball teams ? Do the music and sport teachers know something that the ‘academic’ teachers don’t ?

harold 03.11.10 at 3:17 pm

Well, a lot of the candidates for Juilliard come from China, which has a big advantage as far as population goes.

But yes, music teachers have figured out how to get high levels of performance from students who seem to have little discernible “talent” to begin with. One reason the Suzuki method works well is that it entails three or more lessons a week, for example. And I understand that Asian (non-Suzuki) music classes have daily practice

in schoolon desktop keyboards, or so I have heard. Where there’s a will, there’s a way. It is understood that music instruction from the earliest years develops the brain and helps in other subjects. In the Waldorf Schools early math is combined with movement — throwing a bean bag while practicing the times tables — dance-like moving in groups, and the like. These methods use mastery learning, which means that everyone (not just the “talented” is supposed to master the subject before moving on — rather than “failing” and being eliminated.Salient 03.11.10 at 5:59 pm

My wife teaches music. There’s a strong self-selection (and parent-selection) process at work there, of course, so there’s an assumed incentive structure. If everyone were forced to take music and sport as classes, the teachers would end up in the same boat as maths teachers. (See: mandatory PhyEd class.)

Even so, I guarantee you that every single sport instructor on the face of the earth past or present has one word in their vocabulary that they deploy ten times more than any other, and that word is “hustle!” These instructors still need to encourage and motivate their charges, but because most of their charges in some sense already want to be there badly enough to be there, the instructors

have cruder tools at their disposal.That’s a blessing.Otherwise, it appears to me, if there are problems with how well teachers teach kids, then maybe there’s even worse problems with how well educators teach teachers.In a way, yep, though another problem is recruiting. As an administrator very patiently explained to me on this end not long ago — not long after my CT comment on the subject :) — each state needs x many new educators, and most teachers go to university in-state, so the state schools need to graduate x many new educators. There must be sufficient supply to meet demand each year, give or take a few percent, and it’s basically up to the universities to ensure supply, and to ensure that that supply is as well-prepared as we can get them. So we can’t require all entrants to the major to be superstars. But when only x + 31 many people are even interested in the major across the state, recruitment becomes a huge concern. So let’s trace this back to roots;

Why don’t more college students want to become teachers?(I don’t know of any other major which tracks its number of graduates across-state and the number of open positions across-state so meticulously. I don’t even know of any other major which

coulddo this. I really do think it’s neat that we are able to observe what open positions tend to go chronically unfilled, and then guide students toward those career opportunities. Unlike most majors, I think, the education major really can be geared toward career preparation for a very narrow range of public service jobs. This is potentially a big thing in our favor.)Maybe, if all students are taught as though they were going on to ‘higher maths’, it would greatly support those who do without permanently injuring those who don’t.Like you, I doubt this, but perhaps unlike you, I think it comes down to a question of what the definition of “prepare for higher maths” is. Put a bit paradoxically, because I’m up against the limits of grammar here, I don’t think that preparation for higher maths helps prepare students for higher maths.

A lot of ‘higher maths’ can be cruised through — e.g. blue Rudin — by students with (a) powerful reasoning skills, like the ability to make comparisons and notice particular parts of a diagram, and (b) a big tinker-toolbox, problem solving tricks that commonly work in a given subfield.

And given solidity in (a), the size of (b) that a student is less important. Or at least, it’s easy to plunk more stuff down in (b) in college. Heck, I’m still adding to my tinker-toolbox as a master’s student in maths. But of course the huge difficulty is both in defining what the heck (a) means exactly (thankfully this job is handled admirably well by the NCTM), and then, in determining how best to get students to develop (a), which is hard to even measure/assess.

Plenty of time for open-ended exploration, with conjecture and follow-up to see if the conjectures are true, with the teacher walking around and guiding students to interesting results, helps students develop (a) no matter what their level of facility. This benefit’s fairly well-established, with certain constraints, i.e. assuming a given baseline of teacher preparation and comfort, and overall classroom management facility, etc. Worth acknowledging that lots of teachers are

amazingat providing this environment. And lots of teachers who are too scared of maths to feel secure providing this environment.Oops, this is becoming a long low-substance comment. I think I’m going to start collecting and collating good research/information links to share for future CT posts, and maybe be useful for once, promise!

Grim 03.12.10 at 1:57 am

harold @84

Thanks. Maybe, yes, that’s the real secret: requiring that every student achieve ‘mastery’ instead of the tried (and tried and tried) and failed (and failed and failed) approach of “Let’s run it up the flapole to see who can salute.” And those who, for whatever reason, can’t salute are declared “underachievers” and henceforth ignored.

Salient @85

Yes, some ‘salient’ research (ha ha) from you could be enlightening. Perhaps in terms of Matt @77’s point about the earlier you have subject-specialist teachers, the better is your final result, at least for some (though I doubt if the majority of Russkis are any better at understanding the difference between the arithmetic mean and the median than the majority of Americans, Britains or Aussies are. I wonder if the majority of Finns are any better).

Perhaps in terms of your own experience, generalised. Were you part of a ‘whole of class mastery’ approach, or just the usual ‘flagpole’ approach ? Are you going to go on to do original mathematics research, or just ‘teach’ ? Or will you become the ‘new Tom Lehrer’ ? :-)

If we (ie homo sapiens) need original mathematics researchers, is it ok to concentrate on that for the few in each year that will go on to accomplish this, even if it means ‘failing’ the majority of students ? Or would a more determined approach of ‘whole of class mastery’, no matter how much it holds back the potential ‘overachievers’, be justified ?

Or can we somehow achieve both ? Would a determined ‘whole of class mastery’ approach overcome the ‘lack of talent’ problem with those who turn up at the Ed. schools’ doors wanting to become teachers ?

bianca steele 03.12.10 at 2:11 am

The Boston Globe had an article a few weeks ago about how surprisingly few New Englanders there were on the Olympic ice hockey team. The BU coach complained about a lack of local players as well IIRC. Seems there was

too muchhustle being asked of pretty small kids, and they were burning out and deciding that competitive hockey wasn’t for them, long before they got to college. This was largely physical burnout, though, which wouldn’t apply to math or to music.Grim 03.12.10 at 4:25 am

bianca steele @87.

That’s interesting. Thanks. [Aside: I wonder if that’s what happened to Australia’s generation long superiority at tennis ? And if so, why it hasn’t happened like that to swimming at which Australia is still world competitive ? Maybe because many more Aussies swim than play tennis nowadays, so there’s a bigger ‘pool’ (so to speak) to choose from ?]

But personally, that’s exactly what I would expect with maths and music (two of the core things the entire human race has in common). But then, many people can enjoy music for itself, even if they aren’t at the professional forefront.

How many people nowadays do maths at home as a preferred leisure activity simply because they enjoy doing it ? How many ever did, other than perhaps Fermat ?

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