Category Archives: engagement

Realizing Radians: Teaching as Stagecraft

Teaching Objective: Introduce radian as a unit of angle measure that corresponds to the number of radians in the length of the arc that the angle “subtends” (cuts off? intersects?).  Put another way: One radian is the measure of an angle that subtends an arc the length of the circle’s radius.  Put still another way, with pictures:

How do you  engage understanding and interest, given this rather dry fact?  There’s no one answer. But in this particular case, I use stagecraft and misdirection.

I start by walking around a small circle.

“How far did I walk?”

“360 degrees.”

“Yeah, that won’t work.” I walk around a group of desks. “How far did I walk?”

“360 degrees.”

“Really? I walked the same distance both times?”

“No!” from the class.

“So what’s the difference?”

It takes a minute or so for someone to mention radius.

“Hey, there you go. Why does the radius matter?”

That’s always an interesting pause as the kids take into account something they’ve known forever, but never genuinely thought about before–the distance around a circle is determined by the radius.

“Yeah. Of course, we knew that, right? What’s that word for the distance around a circle?”


“Yes. And how do you find the circumference of a circle?” There’s always a pause, here. “OK, let me tell you for the fiftieth time: know the difference between area and circumference formulas!”

“2Πr” someone offers tentatively.  I put it up:


“So the circumference is the difference between this small circle” and I walk it again “and this biiiiigg circle around these desks here.” Nods. “And the difference in circumference comes down to radius.”


“Look at the equation. 2 Π is 2 Π. So the only difference is radius. The difference in these two circles I walked is that one has a bigger radius.”

“So the real question is, how does the radius play into the circumference?”

“Well,” it’s always one of the better math students, here: “The bigger the radius is, the farther away from the center, right?”

“So then…you have to walk more around…more to walk around,” some other student will finish, or I’ll ask someone to explain what that means.

“Right. But how does that actually work? Can we know exactly how much bigger a circle is if it has a bigger radius?”

“A circle with a radius of 2 has a circumference of  4Π. A circle with a radius of 4 has a radius of 8 Π. So it’s bigger.” again, I can prompt if needed, but my class is such that the stronger students will speak their thoughts aloud. I allow it here, because they can never see where I’m going. See below for what happens if they start with spoiler alerts.

“Sure. But what’s that mean?”


I pass out pairs of circles, cut from simple construction paper, of varying sizes, although each pair has the same radius.

“You’re going to find out exactly how many radius lengths are in a circle’s circumference using the two circles. Don’t mix and match. Don’t write annoyingly obscene things on the circles.”

“How about obscene things that aren’t annoying?”

“If you can think of charmingly obscene comments, imagine yourself repeating them to the principal or your parents, and refrain from writing them, too. Now. You will use one of these circles as a ruler. All you have to do is create a radius ruler. Then you’ll use that ruler to tell me how many times the radius goes around the circumference.”

“Use one of the circles as a ruler?”

“You figure it out.”

And they do. Most of them figure it out independently; a few covertly imitate a nearby group that got it. Folding up one of the circles into fourths (or 8ths) exposes the radius.


Folding up one circle exposes the radius.

It takes most of them a bit more time to figure out how to use the radius as a ruler, and sometimes I noodge them. It’s so low-tech!


Curl the folded circle around the edge of the measured circle. 

But within ten to fifteen minutes everyone has painstakingly used the “radius ruler” to mark off the number of radius lengths around the circumference, and then I go back up front.


“Okay. So how many times did the radius fit into the circumference?”

Various choruses of “Over six” come back, but invariably, someone says something like “Six with and a little bit left over.”

“Hey, I like that. Six and a little bit. Everyone agreed?” Yesses come back. “So did everyone get something that looks like this?”


“Huh. And did it matter what size the circle was? Jody, you had the big two, right? Samir, the tiny ones? Same difference? Six and a little bit?”

“So no matter the circle size, it appears, the radius goes into the circumference six times, with a little bit left over.”

No one has any clue where I’m going, usually, but they’re interested.

“‘Goes into’ is a familiar term, isn’t it? I mean, if I say I wonder how many times 2 goes into 6, what am I actually asking?”

Pause, as the import registers, then “Six divided by two.”

“Yeah, it’s a division question! So when I ask how many times the radius goes into the circumference, I’m actually asking…..” The pause is a fun thing. Most beginning teachers dream of using it, but then get fearful when no one answers. No. Be fearless. Wait longer. And, if you need it:

“Oh, come on. You all just said it. How many times does 2 go into 6 is 6 divided by 2. So how many times the radius goes into the circumference is…”

and this time you’ll get it: “Circumference divided by the radius.”

“Yeah–and that’s interesting, isn’t it? It applies to the original formula, too.”


“Cancel  out the radius.” the class is still mystified, usually, but they see the math.

“Right. The radius is a factor in both the numerator and denominator, so they can be eliminated. This leaves an equation that looks like this.”


“The circumference divided by the radius is 2Π. Well. That’s good to know. Does everyone follow the math? Everyone get what we did? You all manually measured the circumference in terms of radius length–which is the same as division–and learned that the radius goes into the circumference a little bit over six times. Meanwhile, we’re looking at the algebra, where it appears that the circumference divided by the radius is 2Π.”

(Note: I have never had the experience where a bright kid figures it out at this point. If I did, I would kill him daid, visually speaking, with a look of daggers. YOU DO NOT SPOIL MY APPLAUSE LINE. It’s important. Then go to him or her later and say, “thanks for keeping it secret.” Or give kudos after the fact, “Aman figured it out early, just two seconds before figuring out I’d kill him if he spoke up.” Bright kids learn early, in my class, to speak to me personally about their great observations and not interrupt my stagecraft.)

And then, almost as an aside: “What is Π, again?” I always ask it that way, never “what’s the value of Π” because the stronger kids, again, will answer reflexively with the correct value and they aren’t the main audience yet. So the stronger kids will start talking yap about circles, and I will always call then on a weaker kid, up front.

“So, Alberto, you know those insane posters going around all the math teachers’ walls? With all the numbers?”

“Oh, yeah. That’s Π, right? 3.14.”

“Right. So Π is 3.14 blah blah blah. And we multiply it by two.”


That’s when I start to get the gasps and “Oh, MAN!” “You’re kidding!”

“….so 3.14 blah blah times 2 is 6.28 or…..”

“SIX AND A LITTLE BIT!” the class always shouts with joy and comprehension. And on good days, I get applause, too, from the stronger kids who realized I misdirected them long enough to get a deeper appreciation of the math, not just “the answer”.


So a traditionalist would just explain it, maybe with power point. I don’t want to fault that, but I have a bunch of students who would simply not pay any attention. They’ll take the F. I either have to figure out a way to feed them the math in a way they’ll remember, or fail more kids than I’m comfortable failing.

A discovery-oriented teacher would probably turn it into a crafts project, complete with pipe cleaners and magic markers. I don’t want to fault that, but you always get the obsessive artists who focus on making a beautiful picture and don’t care about the math. Besides, it takes forever. This little activity has to be 15-20 minutes, tops. Remember, there’s still a lot to explain. Radians are the unit measure that allow us to talk about circles in terms akin to similarity in polygons–and that’s just the start, of course. We have to talk about conversion, about the power that radians gives us in terms of thinking of percentage of the entire circle–and then actual practice. I don’t have time for a damn pipe-cleaning activity.

As I’ve written before somewhere between open-ended, squishy discovery and straight discussion lecture lies a lot of ground for productive, memorable teaching. In my  opinion, good teachers don’t just transmit information, but create learning events, moments that all students remember and can use as hooks for further memories of learning. In this case, I want them to sneak around the back end to realize that  Π is a concrete reality, something that can actually be counted, if not exactly.


Teaching as stagecraft. All the best teachers use it–even pure lecture artists who do it with the power of their words (and an appropriate audience).  Many idealistic teachers begin with fond delusions of an enthralled class listening as they explain math in terms that their other soulless, uncaring teachers just listlessly put up on the board. When those fantasies are ruthlessly dashed, they often have no plan B. My god, it turns out that the kids really don’t find math interesting! Who do I blame, myself or them?

I never had the delusions. I always ask my kids one simple question: is your life better off if you pass math, or if you fail?  Stick with me, and you’ll pass. For many, that’s a soulless promise. To me, that’s where the fun starts. How do you get them interested? How do you create those moments? How do you engage kids who don’t care?

It’s not enough. It’s never enough.

But it’s a good way to start.

Understanding Math, and the Zombie Problem

I have been mulling this piece on the evils of explanations for a while. There’s many ways to approach this issue, and I highly recommend the extended discussion at Dan Meyer’s blog, as it captures experience-based teachers (mostly reform biased) with the traditionalists, who are primarily not teachers.

What struck me suddenly, as I was engaged in commenting, was the Atlantic’s clever juxtaposition.

All the buzz, all the sturm und drang about Common Core and overprocessed math has involved elementary school. The cute show your thinking pictures are from 8 year olds and first graders. Louis CK breaks our hearts with his third grader’s pain. The image in the Atlantic article has cute little pudgy second grade arms—with just the suggestion of race, maybe black, maybe Hispanic, probably male—writing a whole paragraph on math. The evocative image evokes protective feelings, outrage over the iniquities of modern math instruction, as a probably male student desperately struggles to obey meaningless demands from a probably female teacher who probably doesn’t understand math beyond an elementary level anyway. Hence another underprivileged child’s potential crushed, early and permanently, by the white matriarchal power structure unwilling to acknowledge its limitations.

And who could disagree? Arithmetic has, as John Derbyshire notes, “the peculiar characteristic that it easy to state problems in it that are ferociously difficult to solve.” Why force children to explain place value or the division algorithm? Let them get fluency first. Garelick and Beals (henceforth referred to as G&B) cite various studies finding that elementary school students gain competence by focusing on procedure first, conceptual understanding at some later point.

There’s just one problem. While the Atlantic’s framing targets elementary school, and the essay’s evidence base is entirely from elementary school, G&B’s focus is on middle school.

Percentages. Proportions. Historically, the bane of middle school math. Exhibit C on high school math teachers list of “things our students should know but don’t” (after negatives and fractions), and an oft-tested topic, both conceptually and procedurally, in college placement.

G&B make no bones about their focus. They aren’t the ones who chose the image. They start off with a middle school example, and speak of middle school students who “just want to do the math”.

But again, there’s that authoritatively cited research (linked in blue here):


Again, all cites to research on elementary school math. The researched students are at most fifth graders; the topics never move above arithmetic facts. G&B even make it clear that the claim of “procedure without understanding is rare” is limited to elementary school math, and in the comments, Garelick discusses the limitations of a child’s brain, acknowledging that explanations become more important in adolescence—aka, middle school, algebra, and beyond.

G&B aren’t arguing for 8 year olds to multiply integers in happy, ignorant fluency, but for 14 year olds to calculate percentages and simply “show their work”. And in the event, which they deem unlikely, that students are just going through the motions, that’s okay because “doing a procedure devoid of any understanding of what is being done is actually hard to accomplish with elementary math.” Oh. Wait.

Once you get past the Atlantic bait and switch and discuss the issue at the appropriate age level, everything about the article seems odd.

First, Beals and Garelick would–or should, at least–be delighted with math instruction in 8th grade and beyond. Reform math doesn’t get very far in high school. Not only do most high school teachers reject reform math, most research shows that the bulk of advanced math teachers have proven impervious to all efforts to move beyond “lecture and assign a problem set”. Most math teachers at the high school level accept a worked problem as evidence of understanding, even when it’s not. I’m not as familiar with middle school algebra and geometry teachers, but since NCLB required middle school teachers to be subject-certified, it’s more likely they profile like high school teachers.

G&B don’t even begin to make the case that “explaining math” dominates at the middle school level. They gave an anecdote suggesting that 10% of the week’s math instruction was spent on 2-3 problems, “explaining thinking”.

This is the basis for an interesting discussion. Is it worth spending 10% of the time that would, presumably, otherwise be spent on procedural fluency on making kids jump through hoops to add meaningless detail to correctly worked problems? And then some people would say well, hang on, how about meaningful detail? Or how about other methods of assessing for understanding? For example, how about asking students why they can’t just increase $160 by 20% to get the original coat price? And if 10% is too much time, how about 5%? How about just a few test questions?

But G&B present the case as utterly beyond question, because research and besides, Aspergers. And you know, ELL. We shouldn’t make sure they understand what’s going on, provided they they know the procedures! Isn’t that enough?

Except, as noted, the research they use is for younger kids. None of their research supports their assertion that procedural fluency leads to conceptual understanding for algebra and beyond. We don’t really know.

However, to the extent we do know, most of the research available in algebra suggests exactly the opposite–that students benefit from “sense-making”, conceptual approaches (which is not the same as discovery) as opposed to entirely procedural based instruction. But researching algebra instruction is far more difficult than evaluating the pedagogy of arithmetic operations—and forget about any research done beyond the algebra level. So G&B didn’t provide adequate basis for making their claims about the relative value of procedural vs conceptual fluency, and it’s doubtful the basis exists.

I’ll get to the rest in a minute, but let’s take a pause there. Imagine how different the article would be if G&B had acknowledged that, while elementary school research supports fact fluency over sense-making (and fact fluency seems to be helpful in advanced math), the research and practice at algebra and beyond is less well established. What if they’d argued for their preferences, as opposed to research-based practices, and made an effort to build a case for procedural fluency over comprehension in advanced math? It would have led to a much richer conversation, with everyone acknowledging the strengths and weaknesses of different strategies and choices.

Someday, I’d like to see that conversation take place. Not with G&B, though, since I’m not even sure they understand the big hole in their case. They aren’t experienced enough.

Then there’s the zombie quote, where Garelick and Beals most tellingly display their inexperience:

Yes, Virginia, there are “math zombies”.

In high school, math zombies are very common, particularly in schools with a diverse range of students and thus abilities. Experienced teachers commenting at Dan Meyer’s blog or the Atlantic article all confirm their existence. This piece is long enough without going into anecdotal proof of zombies. One can infer zombie existence by the ever-growing complaints of college math professors about students with strong math transcripts but limited math knowledge.

I’ve seen zombies in tutoring through calculus, in my own teaching through pre-calc. In lower level classes, I’ve stopped some zombies dead in their tracks, often devastating them and angering their parents. The zombies, obviously, are the younger students in my classes, since I don’t teach honors courses. Most of the zombies in my school don’t go through my courses.

Whether math zombies are a problem rather depends on one’s point of view.

There are many math teachers who agree with G&B, who rip through the material, explaining it both procedurally and conceptually but focus on procedural competence. They assign difficult math problems in class with lots of homework. Their tests are difficult but predictable. They value students who wrote the didactic contract with Dolores Umbridge’s nasty pen, etching it into their skin. They diligently memorize the cues and procedures, and obediently regurgitate the procedures, aping understanding without having a clue. There is no dawning moment of conceptual understanding. The students don’t care in the slightest. They are there for the A and, to varying degrees, play Clever Hans for math teachers interested only in correctly worked procedures and right answers. Left as an open issue is the degree to which zombies are also cheating (and if they cheat are they zombies? is also a question left for another day). For now, assume I’m referring to kids who simply go through the motions, stuffing procedures into episodic memory with nothing making it to semantic, all to be forgotten as soon as the test is over.

Math zombies enable our absurd national math expectations. Twenty or thirty years ago, top tier kids had less incentive to fake it through advanced math. But as AP Calculus or die drove our national policy (thanks, Jay Mathews!) and students were driven to start advanced math earlier each year, zombies were rewarded for rather frightening behavior.

G&B and those who operate from the presumption that math can easily be mastered by memorizing procedures, who believe that teachers who slow down or limit coverage are enablers, don’t see math zombies as a problem. They’re the solution. You can see this in G&B’s devotion and constant appeal to the test scores of China, Singapore, and Korea, the ur-Zombies and still the sublime practitioners of the art, if it is to be called that.

For those of us who disagree, zombies create two related problems. First, their behavior encourages math teachers and policy makers to raise expectations, increase covered material, accelerate instruction pace. They allow schools to pretend that half their students or more are capable of advanced, college level math in high school while simultaneously getting As in many other difficult topics. They lead to BC Calculus pass rates of 50% or more (because yes, the AP Calc tests reward zombie math). Arguably, they have created a distortion in our sense of what “college math” should be, by pretending that “college math” is easily doable by most high school students willing to put in some time.

But the related problem is even more of an issue, because the more math teachers and policies reward zombies, the more smart, intellectually curious non-zombies bow out of the game, decide they’ll go to a state school or community college. Which means zombie kids just aren’t numbered among the “smart” kids, they become the smart kids. They define what smart kids “are capable of”, because no one comes along later to measure what they’ve…well, not forgotten, but never really learned to start with. So people think it really is possible to take 10-12 AP courses and understand the material (as opposed to get a 5 on the AP), and that defines what they expect from all top rank students. Meanwhile, those kids–and I know many–are neither intellectually curious nor even “intelligent” as we’d define it.

The Garelick/Beals piece is just a symptom of this mindset, not a cause. They don’t even know enough to realize that most high school math is taught just the way they like it. They’d understand this better if they were teachers, but neither of them has spent any significant time in the classroom, despite their bio claims. Both have significant academic knowledge in related areas–Garelick in elementary math pedagogy, which he studied as a hobby, Beals as a language expert for Asperger’s—which someone at the Atlantic confused with relevant experience.

Such is the nature of discourse in education policy that some people will think I’m rebutting G&B. No. I don’t even disagree with them on everything. The push for elementary school explanation is misguided and wasteful. Many math teachers reward words, not valid explanations; that’s why I use multiple answer math tests to assess conceptual knowledge. I also would love–yea, love–to see my kids willing to work to acquire greater procedural fluency.

But G&B go far beyond their actual expertise and ultimately, their piece is just a sad reminder of how easy it is to be treated as an “expert” by major publications simply by having the right contacts and backers. Nice work if you can get it.

And the “zombie” allusion, further developed by Brett Gilland, is a keeper.

Illustrating Functions

Function definitions aren’t usually tested on either the SAT or the ACT and since I never worked professionally with math, functions were something I’d barely considered in algebra a billion years ago. So for the first few years of teaching, I kind of went through the motions on functions: unique output for each input, vertical line test, blah blah. I didn’t ignore them or rush through them. But I taught them in straight lecture mode.

Once I got out of the algebra I ghetto (which really does warp your brain if that’s all you do), I accepted that a lot of the concepts I originally thought boring or unimportant show up later. So it’s worth my time to come up with the same third way activities and lessons for things like functions or absolute value or inverses that I do for binomial multiplication and modeling linear equations and inequalities.

So every year I pick concepts to transfer from pure lecture/explanation to illustration. Sometimes it’s spur of the moment, other times I plan a formal curriculum change. In the case of functions, the former.

Last year I was teaching algebra II/trig and–entirely in passing–noted a problem in the Holt book that looked something like this:

and had two simultaneous thoughts: what a boring question and hey, I could really do something with that.

So the next day, I tossed this up on the board without comment.


I’ve given these instructions three times now–a2/trig, trigonometry, algetbra 2–and the kids are always mystified, but what the heck, the activity seems simple enough. No student ever reads through the entire list of instructions first. They spend a lot of time picking the message, with many snickers, then have fun translating the code twice.

But then, as they all try to translate someone else’s message using the cell phone code, bam. They realize intuitively that translating the whole-alphabet code would be an easy task. And with a few moments of thought, they realize why the cell phone code doesn’t offer the same simple path. They don’t know what it means, exactly. But the students all realize that I’ve demonstrated a difference that they’d never considered.

From there, I graph the processes, which is usually a surprise as well. The translation process can be graphed?



At this point, I can usually convince kids to remember the Vertical Line Test, which they were taught in algebra I. At that point, I go through the definitions for relation, function, and one-to-one function, using a Venn diagram (something like this with an addition inner circle for one to ones). Then I go back through what the students vaguely remember about functions and link it to the correct code example.

Thus the students realize that translating the message into code is a function in either code key. I hammer this point home, because the most common misconception kids get from this is that all functions must be one to one. Both are functions. Each letter has one and only one number assigned, and the fact that one translation key puts several letters to the same number is irrelevant for its determination as a function. Reversing the process, going from numbers to letters, only one of them is a function.

Then I sketch parabolas and circles. Are they both functions? Are either of them one-to-one functions?

In Algebra 2, I do this long before the inverse unit. In Trig, I introduce it right before graphing the individual functions as part of an overview. In both classes, the early intro gives them time to recognize the significance of the difference between a function and the more limited case of the one-to-one function–particularly in trig, since the inverse functions are very limited graphs for exactly the reason. In algebra II, the graphs reinforce the meaning of the Horizontal Line Test.

I haven’t taught algebra I recently, but I’d change the lesson by giving them a coded message and ask them to translate with the cell phone code first.

This leads right into function and not-function, which is all they need in algebra I.

I have periodically mentioned my mixed feelings about CPM. Here’s a classic example. The CPM book introduces functions with the following example.

Okay. This is a terrible example. And really boring. Worst of all, as far as this non-mathie can tell, towards the end it’s flat out wrong. A relation can be predictable without being a function (isn’t that what a circle is?). But just looking at it, I got an idea for a great test question (click to enlarge):


And I could certainly see some great Algebra I activities using the same concept. But CPM just sucks the joy and interest out of the great starting ideas it has.

Anyway. I wanted to finish up with a push for illustrations. What, exactly, do the students understand at the moment of discovery in this little activity? I’ve never seen anyone make the intuitive leap to functions. However, they do all grasp that two tasks that until that moment seemed identical…aren’t. They all realize that translating the message in the whole-alphabet code would be a simple task. They all understand why the cell phone code translation doesn’t lend itself to the same easy translation.

I look for illustrative tasks that convince kids to think about concepts. As I’ve said before, the tasks might kick off a unit, or they might show up in the middle. They may demonstrate a phenomenon in math, or they might be problems designed to lead the students to the next step.

The most common pushback I get from math teachers when I talk about this method is “I love the idea, but I don’t have enough time.” To which I respond that pushing on through just means they’ll forget. Well, they’ll probably forget my lessons, too, but–maybe not so much. Maybe they’ll have more of a memory of the experience, a recollection of the “aha” that got them there. That’s my theory, anyway.

There’s no question that telling is quicker than illustrating or letting them figure it out for themselves. Certainly, the illustration should be followed by a clear explanation with much telling. I love explaining. But I’ve stopped kidding myself that a clear explanation is sufficient for most kids.

That said, let me restate what I said in my retrospective: The tasks must either be quick or achievable. They must illustrate something important. And they must be designed to lead the student directly to the observations or principles you want them to learn. It’s not a do it yourself walk in the park. Compare my lesson on exploring triangles with this more typical reform math example. I resist structure in many aspects of my life, but not curriculum.

In researching this piece, I stumbled across this really excellent essay Why Illustrations Aid Understanding by David Kirsh. I strongly recommend giving it a read. He is only discussing the importance of visual illustrations, whereas I’m using the word more broadly. Kirsh comes up with so many wonderful examples (math and otherwise) that categorize many different purposes of these illustrations. Truly great mind food. In the appendix, he discusses the limitations of visually representing uncertainty.


On reading this, I felt like my students did when they realized the cell phone message was much harder to translate: I have observed something important, something that I realize immediately is true and relevant to my work–even if I don’t yet know why or how.

The Release and “Dumbing it Down”

I’ve said before I’m an isolationist whose methods are more reform than traditional. I try to teach real math, not some distorted form of discovery math, but I also try to avoid straight lecture. I want to make real math accessible to the students by creating meaningful tasks, whether practice or illustration, that they feel ready to tackle.

I can’t tell you that students remember more math if they are actively working the problems I give them. Research is not hopeful on this point (Larry Cuban does a masterful job breaking down the assumptions that chain from engagement to higher achievement.) Will my students, who are often actively engaged in modeling and working problems on their own, retain more of the material than the students who stare vacantly through a lecture and then doodle through the problems? Or six months from now, are they all back to the same level of math knowledge? I fear, I suspect, it’s the latter. I think we could do better on this point if we gave students less. Not Common Core “less”, in which they just shovel the work at the students earlier. But a lot less math, depending on their ability and interest, over the four year period of high school.

Four plus years of teaching has given me a lot more respect for the sheer value of engagement, though. I believe, even if I can’t prove, that the kid who works through class, feeling successful and capable of tackling problems that have been (god save me for using this word) scaffolded for his ability, has learned more than the kid who sits and does nothing. Even if it’s not math.

Anyway. There comes a moment when the teacher says to the students, “go”. Best described as release of responsibility, whether or not a teacher follows any particular method, it’s when the teacher finishes the lecture, the class discussion, or simply handing out the task the students are supposed to take on without any other instruction.

It’s the moment when novices often feel like Mork. Done poorly, it’s the lost second half of a lesson. Done well, it’s the kind of moment that any observer of any philosophy would unhesitatingly describe as “good teaching”.

I started off being pretty good at release, and got better. That is, as a novice using straightforward explanation/discussion (I rarely lecture per se) or an illustrating activity, I could usually get 30% of the class going right away, another 40% doing a problem or two before asking for reassurance, and convince most of the remaining 30% to try it with explicitly hand-crafted persuasion. And for a new teacher, that’s nothing to sneeze at. Sure, every so often I let them go to utter silence, or a forest of raised hands, but only rarely. (And every teacher gets that sometimes.)

I remember pointing out to my teacher instructor, however, that I spent a lot of time re-explaining to kids. He said “Yeah, that’s how it works. You’re going to get some of them during the first explanation, some of them while helping them through the first task….” and basically validated the stats I just described in the previous paragraph. I still think he’s right about the fundamental fact: teachers can’t get everyone right away.

But all that re-explaining is a lot of work, and it leads to kids sitting around waiting for their personal explanation—and no small number of kids who then decide why bother listening to the lecture anyway, since they won’t get it until I explain it to them again, with of course the stragglers, the last 30%, screwing around until I show up to convince them to try. Of course, I went through (and still go through) the exhortation process, telling them to ask questions, “checking for understanding”, and so on.

And it absolutely does help to make the “release” visible to the kids, “Okay, let’s be clear–we are wrapping up the explanation portion, it’s time for work, and I WILL NOT BE HAPPY if you shoot your hand up right after I say ‘go’ and whine about how you don’t get it.”

This works. No, really. Kids say “Could you go through it one more time?” before I release them, particularly after I’ve put them “on blast mode” for saying “I don’t get it” when I show up at their desk to see where they are.

But I focused on release almost immediately as an area for my own improvement. As I did so, I began to understand why release is so hard for teachers, particularly new ones.

We overestimate. We think, “I explain it, they do it.” We think, “I gave them instructions they can follow.” We think, “This is the easy part” and are already mapping out how we’ll explain the hard part.

And then we say “Fly, be free!” and the class drops with a splat. Burial at sea. Wash away the evidence.

We aren’t explaining enough. Or they aren’t listening. We aren’t giving clear instructions. They don’t read the instructions. “Too many words.”

What I have discovered, over time, is that I must halve or even quarter what I think students can do, and then deliver it at half the pace. With this adjustment, I can release them to work that they will find challenging, but doable. This is the big news, the news that I pass on to all new teachers, the news they invariably scoff at first and then, reluctantly, acknowledge to be true.

But what I have begun to realize, again over time, is that by first “dumbing it down”, I have slowly increased the difficulty and breadth of coverage I can deliver. Not a lot. But some. For example, I now teach the modeling of inequalities, modeling of absolute values, and function operations, in addition to modeling linear equations, exponentials, probability, and binomial multiplication. I don’t think my test scores have increased as a result, but it makes me feel better about what my course is called, anyway.

In mulling this development, I have concluded, tentatively, that I’ve become a better teacher. Or at least a better curriculum developer. That is, I don’t think “dumbing down” itself has led to my increased coverage or my students’ ability to handle the topic. But I’ve gotten better at the “release”, at developing explanations and tasks that allow the students to engage in the material.

It’s possible I’ve been unwittingly participating in a positive feedback loop. As I get better at the release, at correctly matching their ability to my tasks and explanations, the students are more likely to listen, to try to learn, to dig in to a new task and give it a shot. So I get bolder and come up with ideas for more complex subjects.

I dunno. Here’s what I do know: effective release requires willing students. The able students are willing by default. The rest of them need something else.

Put it another way: the able students have trust in their own abilities. The kids who don’t trust in their own abilities need to trust me.

No news there, that trust is an essential part of teaching. But I’m only now considering that my lesson sequencing and content might be an essential element in building the trust the students need to take on challenges.

Eighteen months ago, I wrote an essay that captured the moment when teachers realize that their students don’t retain learning. They demonstrate understanding, they pass tests demonstrating some ability, and then two weeks, three weeks, a couple months later, it’s gone. (Every SINGLE time I introduce completing the square, it’s a day.)

The “myth” essay describes what happens after release. That is, after the teacher realizes that students didn’t understand the lecture, didn’t understand the worksheet, are goofing off until the teacher comes around to give one on one tutoring, after the teacher does the additional work to get the instruction out, the kids seem to get it. And then forget it all completely, or remember it imperfectly, or rush at problems like stampeding cattle and write down anything just to have an answer.

So consider this the companion piece: the front end of classroom teaching to the myth’s back end.

But in fact, it’s all part of the same problem. And, as I said in the first essay, teachers tend to react in one of two ways: Blame or Accept. Many accepters just skedaddle to higher ability students. I’m teaching precalc this year and have some interesting observations on that point. But leave that for another essay.

I’m an accepter:

Acceptance: Here, I do not refer to teachers who show movies all day, but teachers who realize that Whack-a-Mole is what it’s going to be. They adjust. Many, but not all, accept that cognitive ability is the root cause of this learning and forgetting (some blame poverty, still others can’t figure it out and don’t try). They try to find a path from the kids’ current knowledge to the demands of the course at hand, and the best ones try to find a way to craft the teaching so that the kids remember a few core ideas.

On the other hand, these teachers are clearly “lowering expectations” for their students.

And that’s me. I lower expectations. I do my best to come up with intellectually challenging math that my students will tackle. I don’t lecture because the kids will zone out; instead, I have a classroom discussion in which the kids live in some terror that I might call on them to answer a question, because they know I won’t ask for raised hands. So they should maybe pay attention. I have no problem with students taking notes, but for the most part I know they don’t, and I don’t require it. I give them a graphic organizer with key formulas or ideas (or they add them). I periodically restate the critical documents they should save, tell them I designed the documents to be useful to them in subsequent math classes, double check them periodically to see if they have the key material.

Dan Meyer sees himself as a math salesman. I see myself as selling….competence? Ability? A sense of achievement?

Whatever. When you read of those studies showing that math courses don’t match the titles, you’re reading about courses I teach. I teach the standards, sure, but I teach them slowly, and under no circumstances are the kids in my algebra II class getting anything close to all of second year algebra, or the geometry students getting anywhere near all the geometry coverage. That’s because they don’t know much first year algebra, and if you’re about to say that the Next New Thing will fix that problem, then you haven’t been paying attention to me for the past two years.

But at some point, maybe we’ll all realize that the issue isn’t how much we teach, but how much they remember.

Or not.

Be clear on this point: I do not consider myself a hero, the one with all the answers. I am well aware that many math teachers see teachers like me as the problem. Many, if not most, math teachers believe that kids can learn if they are taught correctly, that the failings they see are caused by previous teachers. And I constantly wonder if they are right, and I’m letting my students down. While I sound confident, I want to be wrong. Until I can convince myself of that, though, onwards.

I began this essay intending to describe a glorious lesson I taught on Monday, one in which I released the kids and by god, they flew. But I figured I’d explain why it matters first.

Dan Meyer and the Gatekeepers

I have at least one more post on reform math, but I got distracted while looking for examples of Dan Meyer’s teaching (as an example of his math in action) then realizing that many of my regular readers wouldn’t know Dan Meyer, and so started to construct a brief bio. In doing so, I got distracted again in considering Meyer’s quick-yeast rise and what it says about the gatekeepers in the education racket and access to microphones.

This may seem like insider baseball, but I hope to illustrate that Dan Meyer is an unobjectionable guy with a good idea, whose unhesitating adoption by the elites represents a real problem with educational discourse in this country. I will probably overstate and paint a picture that suggests plan and intent by those causing the trouble, when in fact it’s fuzzy and reactive with only big picture general directions, but probably not to the extent that Diane Ravitch (or, indeed, Dan Meyer) commit that particular sin.

Dan Meyer, 31, is in the process of becoming a celebrity math teacher (hey, it’s a small group). Much of his rapid trajectory upward can be explained by his message, which involves a digital curriculum that will (he says) instantly engage and perplex kids and thus resolve all classroom management issues (more on this later), a message tailor-made to appeal to both techies, since it implicitly attacks all teachers, and progressive educators, since it is inherently constructivist.

Most of the rest of his said trajectory can be explained by his excellent luck in his early audience—not only were they progressives and techies, but they were influential progressives and techies–Chris Lehmann, O’Reilly Publishing folk like Kathy Sierra, Nat Torkington and Tim O’Reilly himself, Brian Fitzpatrik of Google, and Maggie Johnson of Google and Stanford.

A teeny-tiny bit–ok, maybe more–of that trajectory can be explained by the Great White Hope factor. As I’ve written many times, every corner of education is desperate for young teachers, particularly young male teachers, most especially young white male teachers. Smart young white male pushes technology-based teaching, implicitly or explicitly declaring that all those old teachers (mostly white female grandmas) are doing it wrong. Hard to resist. So attractive message, demographic felicity, and luck. Not bad.

I’m going to summarize what I see as the relevant points of Meyer’s career thus far, but go straight to the source: Meyer describes his teaching career in this excellent video, which I recommend watching to instantly “get” his appeal. Go watch. I’ll wait.

He taught his first year at a Title I school in Sacramento, CA and, as he says above, was both miserable and ineffective, which he blames on his failure to create a “classroom ethos”. The improvement in classroom ethos began during his second year at San Lorenzo Valley High. It apparently never occurred to him to wonder whether the “classroom ethos” improvement at his second school, was helped along by a student demographic that was 87% white. Meyer actually noted the novelty of a non-English speaking Hispanic student which is the only time he ever mentions a minority student on his blog, best I can see.

While he made numerous videos that ended with the tagline, “I like to teach”, he in fact wasn’t all that attached to teaching. At the beginning of his third year, he was already predicting he’d be in school for either an administrative credential or doctorate by the end of the next year (he was off by two), because “I’m just keenly aware how much of my strength as a teacher derives from my ability to relate to student culture, to talk like they talk and dress like they dress” and his awareness that he feels “obliged to entertain”. He often implied that he’d mastered the technical aspects My personal favorite::

I am at a place, for example, where classroom management no longer challenges me. Not that every day is all smiles and hard work, just that I have identified the mix of engaging instruction, mutual respect, and tough love that eluded me for years.

Four ENTIRE years, this eluded him! This meme runs throughout his blog and is, in fact, the seed of his image-based curriculum. Meyer states time and again that he worked hours on end to keep from boring his students, thinks student approval as essential to improved learning outcomes and thus presents his curriculum as a better way to entertain kids, to perplex them in a way they will value, and once entertained and perplexed, they will learn.

Then, at the end of five years, he declared he was quitting teaching because he’d been transformed from “miserable to happy, incompetent to competent” (astonishing, really, how few of the commenters openly laughed at his hubris). He originally planned to attend UC Santa Cruz’s PhD program but his aforementioned contacts got him a year as curriculum fellow at Google, and he taught part-time for one more year. While at Google, he made his first TV appearance on Good Morning America (probably via Google) to discuss his theory as to whether regular or express grocery lanes were faster.

At some point after that, he pulled a TEDx invitation—very nice work if you can get it—which got him onto CNN and good lord, how could Stanford let him get his doctorate at Santa Cruz, after all that publicity? So now he’s at Stanford in year 3 of his doctoral program.

A star was born.

Like most teachers, Meyer’s a good talker; unlike many teachers, he’s good with any audience. He’s a bright guy and his videos are genuinely entertaining (go to the end to catch his early work), and I say that as someone who disagrees with very close to all of his primary assertions. As a young white male teacher he could demand nearly anything, and he nonetheless stayed in algebra and geometry, rather than push for advanced classes that his principal, eager to keep him, would easily grant. I suspect that some of his willingness to stay in low status classes was caused by his short-timer’s attitude, while another part of it was caused by his 70-hour work week. Anyone working that hard and long on classes he’s been teaching for years is unlikely to embrace new subjects. But those stated priorities nonetheless reveal a guy who is well beyond committed and flat out obsessed with doing a good job.

He’s hard to pin down ideologically not because he’s an original thinker but because he was, and is, profoundly uninterested in education policy. So no coherent philosophy, but Me Like, Me No Like. He would disavow the charge that he is on the “reform” side of the math wars, although less vehemently than a few years ago—Jo Boaler, High Priestess of Reform Math, is his adviser, after all. But even now, a few years after starting a Stanford PhD program, he’s very foggy about the specifics of major debates in math education. So he’s been trying to consolidate his positions, but he’s not always sure what the right ones are. In his earlier iterations, for example, before he became well-known, he often adopted strong education (not math) reform positions—he had an “educrush” on Michelle Rhee. In the early years of the blog, he dripped contempt on most teachers, particularly older ones, including coworkers. Early on he harped often on the need for professionalism, and asserting we’d be better off without teachers that do it just for love. But once Stanford put him on the doctoral payroll, he’s become more typically math reform, which means he’s disavowing education reform positions and doing his best to walk all that talk back. Well, not all of it–here he is on a forum last year talking about the need to train teachers on Common Core:

I think if you’ve taught for thirty years under a particular style of teaching, it has to distort what your perception is of math and how it should be taught. It’s unavoidable, to be steeped in that for so long. So to realign yourself, I imagine, is a very difficult thing. So PD that involves problem solving, involves reasoning, argumentation, that’ll be essential going forward.

So the nastiness to older teachers, still there. I don’t blame anyone who wonders if promoters consider that a bug or a feature.

Meyer’s writings never describe his “classroom action” in detail relative to other math bloggers (e.g., Fawn Nguyen, Sam Shah, Michael Pershan, Kate Nowak and, okay, me). He rarely describes the success or failure of a particular lesson, or gives any kind of walkthrough. He never describes a lesson in full detail, down to the worksheets and responses. He often went to the data collection well, and just as often failed, as in his two-month long “Feltron” project in which half the class dropped out early during data collection and had to be given other tasks, or this similar project

Meyer and metrics aren’t a natural fit. A few years ago, he was, comically, shocked by news of California’s Hispanic achievement gap. Dude, didn’t you get the memo? He never blogs about it, never discusses it, then out of the blue: Damn! We’ve got an achievement gap! And then he rarely mentions it again, save for this recap of Uri Treisman’s speech. He almost never discusses his student’s test scores and when he does, they are usually not great, although he mentions once in passing that his algebra students beat the department (no data, though). He cheerfully talked about standards-based grading for a year or two and blew off the commenters who wondered if the students were retaining the skills they’d “mastered” twice in a week. When he did finally get around to looking for that data, the answer was no, and it’s quite clear that he’d never before wondered about this essential element of success. So while I suspect that Meyer was a popular teacher who convinced a lot of kids–mostly white boys–to work hard at math, there’s little evidence of that in his written history of his years teaching.

I can find little evidence of intellectual achievement in education once he left teaching, either. At Google, he and three other curriculum fellows worked for a year on computational thinking projects. When his project shipped he wrote, somewhat obscurely, “Near as I can tell, of the sixty-or-so modules listed, only one of them ….is mine. I always admired Google’s lack of sentiment in deciding when to invest itself and when to divest itself. Still it’s strange to see a year of work reduced to a single entry in a long list.” (emphasis mine)

At Stanford, his qualifying paper was not hailed as an instant masterpiece:

The criticism I remember most vividly: a) my weak review of the literature, b) the sense that I wasn’t really taking myself anywhere new with the study, and c) a claim about equity that had me reaching beyond my data.

In short, he didn’t set the curriculum world on fire at Google, and the critiques of his qualifying paper suggest an analytical lightweight—which is pretty typical for salesfolk. So thus far Meyer has established himself as a stupendous salesman, but not much of an intellectual—at least, not of the sort that Google and Stanford like to pretend they invest in. He was even wrong in the GMA segment. Unsurprisingly, he was unflustered.

Realize that I know all of this because of Dan Meyer’s blog, so he’s not hiding anything. Hell, he doesn’t need to.

But he was brought to Google and then to Stanford and then Apple gets involved, and now we’re talking three of the most elite institutions in the country are pushing him not because they have any evidence of his ability to close the achievement gap, or even whether his digital curriculum works, but simply because he’s Dale Carnegie, and boy oh boy, is that a depressing insight into their motivations, just as his success is indicative of the desires of the larger educational world. It’s not “go develop your ideas and expand and prove them” but “here’s a bunch of elite credentials that will make your sales job easier”. So they dub Dan an “expert” and give him a microphone—which makes it a whole lot easier for a largely ignorant general media to hear him.

No, I’m not jealous. My karmic destiny demands that I enter new communities with neither warning nor fanfare and utterly polarize them within a month, usually without any intention of causing trouble. Lather, rinse, repeat. I gave up fighting that fate fifteen years ago. I have attended two elite institutions in recent memory; one of them ignored me desperately, the other did its best to hork me up like a furball. I don’t want to go back. Academia isn’t for me. And if a corporation handed me money to sell my message, they’d be facing a boycott. My blog has fifty times the readership and influence that I ever imagined, and I love teaching. I am content.

Previous paragraph notwithstanding, this essay will be interpreted by some as an attack on Dan Meyer, who is largely unfamiliar with anything short of worshipful plaudits from eager acolytes (he occasionally heeds polite dissenters, but only occasionally) since he began his blog. But while he’s a dilettante as a teacher, I think his simplistic curriculum ideas have interesting potential in teaching certain demographics, and I wish him all success in developing a coherent educational philosophy. Oh crap, that was snarky. I wish him all success in his academic and business career.

Dan Meyer’s rapid rise isn’t the problem. Dan Meyer himself isn’t the problem. The problem lies with the Gatekeepers: with Stanford, who knows that Dan’s not the solution, with Google, Apple, and publishing companies like Shell Centre (well, they’re in England) and Pearson. That intersection between academia and business, the group that picks the educational platitudes and pushes them hard, while ignoring or banishing dissent. They’re the ones granting Meyer the credentials that cloak him in the illusion of expertise. And I believe that, at least in part, they grant those credentials with a clear eye to the attributes that are diametrically opposed to the attributes they pretend to focus on. It’s no coincidence that Dan Meyer is a young white male. It’s the point. It’s not a fluke that he primarily taught white kids, many of whom were obviously sent to him with strong skills by teachers who valued homework above ability. It’s the only way he could have come up with his curriculum. Yet his message is adopted and embraced by elites who castigate education, particularly teachers, for failing black and Hispanic kids. I don’t know if they do this consciously or if they genuinely believe that all teachers are just meanspirited morons who don’t know math and deliberately deprive certain kids of meaningful math experiences. Ultimately, it doesn’t matter.

I suspect Meyer and others will ignore this essay (Meyer snarked obscurely at my reform piece, assuming this tweet means what I think it does), but whether that’s because he doesn’t like dissent or, more probably, because he subscribes to the Voldemort View, I couldn’t tell you. But maybe this piece will make reporters and educational wonks a bit more wary about the backgrounds of the “experts” they quote, and the gatekeepers who create them.

Who I Am as a Teacher

As I thought about writing specific disagreements I have with reformers, I realized that time and again I’d be having to break off and explain how my values and priorities differed. So I thought I’d do that first.

At my last school’s Christmas party (Year 1 at that school, Year 2 of teaching), the popular, widely respected “teacher at large” showed up an hour late. A PE teacher whose credential had been disallowed by NCLB-wrought changes, he was at that point responsible for coming up with plans to help “at-risk” kids.

“Yeah, I was having all sorts of fun reviewing the Lists.”

“The Lists?” asked another teacher.

“Top 25 Discipline Problems, Top 25 Kids On Probation for Felonies, Top 25 Absentee/Truancy Students, Lowest 25 GPAs, you name it. I look for the kids who aren’t on more than one or two lists and try to reach them before they qualify for more lists.”

“I think I have a few of those kids.” groused a teacher.

“A few? Pity Ed here.” He nodded at me. “Half the kids on each list are in one of your classes.”

I think he made up the felonies list. I hope.

Fall of year two was about as tough a time as I’ve ever had as a teacher.

Getting quiet for teaching was job one. I’d separate inveterate chatters, then I’d move the worst offenders to the front groups, and then, if one of them still didn’t shut up, I’d pull the desk forward all the way to a wall (with the kid in it). The rest of the class would snicker at the talker—at, not with.

“It’s not like I’m going to pay attention to you up here. I’ll just go to sleep,” one of them said, defiantly.

“You say that as if it were a bad thing.”

He or she often did go to sleep, which gave me some quiet from that corner, anyway. Otherwise, I wrote a referral. I also wrote referrals when they called me a f***ing [noun of your choice, profane or not], a sh**ty f**ing boring teacher (boring! I ask you), when they threw things, when they got up and wandered around the room refusing to sit, when they texted in open view and refused to give over the phone, when they left the room without permission, when they howled I HAVE TO PISS at the top of their voices (usually one at a time), and so on—all during the time that I was trying to teach the lesson “up front”. Once I released them for work it got easier, as I wasn’t trying to maintain order and some notion of what I’d been doing before the last interruption, but rather walking around the room helping students and telling others to shut up.

As bad as I make it sound, every senior teacher I worked with was astonished at how well I did, given the pressure; all the previous teachers stuck with all algebra all the time had routinely lost control of the classes and had supervisors posted. Administrators didn’t approve of my approach, alas; since my kids were mostly Hispanic, my referrals were, too. So I was caught between an administration who would really rather I’d have flailed ineffectually than kick kids out for order, and the bulk of my students, who opined frequently that I should boot students more often and earlier.

The beginning of the way back up that year began in second period when I’d thrown out the third kid of the day, and Kiley said “Could you toss out Elijah, while you’re at it?” and much of the class laughed. Elijah stood up and said “Yeah, send me, too! I don’t want to be here! Let me go!”

I tend to stay pretty focused on teaching; rarely do I give A Talk. Today, I have no idea why I made an exception.

“Why don’t you want to be here, again?”

“Because I hate math? F***ing duh.”

“What is it you think I want?”

“You want me to shut up.”

“Well, yeah. But why?”

“So you can teach!”


“Because it’s your job!”

Because I want everyone to pass this class.” And to this day, I thank all that’s holy that I caught the class’s sudden silence and realized that my remark had an impact.

“Maybe I need to make that clearer. I want every single person in here to pass algebra and move onto geometry. Remind me again, how many people have taken algebra more than once?” Almost everyone in the class raised a hand, including Elijah.

“Yeah. Don’t raise your hand, but I know at least ten students in here are taking it for the third time, including some people who get tossed out of class regularly. I don’t kick kids out for fun. I kick them out because I need to teach everyone. I have kids who want to excel in algebra. I have kids who would like to get better at algebra. I have kids who simply would like to survive algebra, although many days they think that’s a pipe dream. And I have kids who don’t want to be here at all. I figure, I kick kids out from the last group, I’m meeting everyone’s goals but mine.” I actually get a couple laughs; they’re listening.

“But make no mistake, that’s my goal. I want everyone in here to pass.” I looked at Elijah, who’d slipped back into his chair, his eyes fixed on me.

“You could tell me about your troubles, and I’ll give you an ear, but here’s a basic truth: there’s not a single situation in your life that gets worse if you pass algebra. And there’s a whole bunch of things that improve.”

“I could get a work permit, for one thing,” Eduardo muttered.

“Get back on the football team,” said DeWayne.

“And now I know some of you are thinking sure, there’s a catch. No. I didn’t say I want you to like algebra. I didn’t even say I want you to understand algebra, although I guarantee that trying will improve your understanding. I’m making a simple commitment: show up and try. You will get a passing grade. No catch.”

The rest of second period, the toughest class, went so well that I decided to repeat that little speech for every class, and in every class, I got utter quiet. I don’t say that all the problems were solved that day, but from that point on far more of the kids “had my back”. Psychologically, their support made it much easier for me to develop a strategy to teach algebra in the face of these challenges.

Here’s how I taught it, and here’s how they did. I only failed 10 kids out of the final 90, or 11%. (Elijah had left. Eduardo got his permit, and DeWayne made it back onto the football team.) That’s the highest failure rate I’ve ever had, but then it’s the last time I taught algebra I. It’s easier to work with kids in geometry and algebra II—they’ve got skin in the game, and graduation becomes a real objective as opposed to the remote possibility it presents to a sophomore taking algebra I for the third time.

The wise reader can infer much about my students and a great deal, although certainly not all, about my values and priorities as a teacher from that tale.

First, I mostly teach kids from the lower third to the middle of the cognitive ability spectrum, with a few outliers on each end. That’s who takes algebra in high school. No more than 10% of my students in any year are capable of genuinely comprehending an actual formal math course in geometry or algebra (I or II). Another 30-50% of the rest are perfectly capable of understanding geometry, algebra and even more advanced topics in applied math, even if they couldn’t really master a formal math course, but they’d have to try a lot harder and want it much more. About a quarter of my students each year are barely capable of learning basic algebra and geometry well enough to apply it in simple, rote situations. A much smaller number can’t even manage that much.

For other teachers, the percentages are skewed heavily to the first and second categories; some of them don’t even know there’s a third and fourth category. A teacher covering precalc and honors algebra II/trig in high-income or Asian suburb, teaching mostly freshmen and sophomores, would have a much higher percentage of students who could master a formal course; their notion of “struggling kids” would be those who aren’t working hard enough. But that’s not my universe—and it’s not the universe I signed up for, although I wouldn’t mind visiting occasionally.

Until this year, my assignments weren’t deliberate. I was just an unimportant teacher who schools didn’t care about losing. In fact, the following year at that same school the administration assigned Algebra II/Trig classes to a teacher who was not qualified to teach the subject while I, who was qualified, was given the lower level Algebra II classes. The administration knew full well about the distinction, which necessitated a “your teacher is not highly qualified” letter to some 90 kids, but that teacher was more valuable than than I was, and so it goes. I’m not bitter, and I’m not marking time until I get “better” kids. I’m doing exactly what I want to do. But every teaching decision I make must be considered in light of my students’ cognitive abilities and, related to that ability, their motivation.

Second, I am a teacher who doesn’t overvalue any individual student at the expense of the class, which means I have no compunction about kicking kids out for the day. You run into these teachers philosophically opposed to removing kids from class; how can these students learn if they aren’t in class, they bleat. These teachers never seem to worry about how all the other kids learn with a disruptive hellion wreaking havoc because, they strongly hint (or outright assert), the right curriculum and caring teachers would eliminate the need to disrupt.

I ask these teachers, politely, do you have kids with tracking bracelets and/or probation officers? Do you have students who have fathered two kids while wearing that tracking bracelet, or gave birth to one? Do you have students who have been suspended or expelled for putting other students in the hospital, or for having a knife in their backpack? Do you have students who routinely tell you to f*** off and don’t bother me? Do you have all of these students plus twelve more who have just enough motivation that, given no distractions, would be able to learn some math but with a distraction will readily jump over to the side telling you to f*** off? And with all that, are you math teachers trying to help students with a four-year range in skills figure out second year algebra? Because otherwise, you can go sing your smug little songs of no student left behind to someone with kids who really shouldn’t be kicked out of the classroom. Okay, maybe not politely.

Come back the next day or even the same day, hat in hand, and no harm, no foul. I don’t only act like it didn’t happen, I have completely forgotten it happened. But get out of my class if you won’t shut up or can’t consider the day a success unless you’ve sucked in three other kids with your distractions.

The biggest pressure on teachers like me these days is the huge pushback they get from administration, district, and state/federal education agencies when they try to maintain an orderly classroom. And charter schools’ ability to a) have none of these kids to start with and b) kick moderately ill-behaved kids back to public school when they act out can’t be overstated as factor in their “success”.

That’s a shame. Because invariably, the bulk of my unmotivated rabble-rousers realize that I really mean it about that whole “passing” thing, if they would just shut up and give the class a shot. And so they do.

Next, I am a teacher who explains. I don’t mean lecture; my explanations always take the form of a semi-Socratic discussion, leading the kids through a process. But when I start to talk, the conversation has a direction and that directed conversation, to me, is the heart of teaching. One of my favorite memories of an ed school classmate came about as we were driving to our placement school.

“I’m really enjoying working on aspects of my teaching that I don’t like. For example, explanations. I hate doing that.”

“Um, what? You hate explanations?”

“Yeah. I’d rather never explain anything.”


“What is teaching, if it’s not explaining things?”

I thought it was a rhetorical question. I was wrong. He went on and on about other aspects of teaching: curriculum, motivation, role modeling, assessing students, and so on. Huh. Interesting. Eye-opening. It’s not that I disagreed, but how can you be a teacher if you don’t like to explain things?

And as I began to develop, I realized that teaching is not synonymous with explaining. Still. It’s my go-to skill, it’s what I do best, it’s a big part of my success with low ability students, and it’s why I prioritize getting my students to shut up while I’m teaching up front.

Next, the story reveals that I adopt my students’ values and goals, rather than insist they adopt mine. The kids were shocked into silence when they realize that my most heartfelt goal was to pass everyone in the class.

I learned a key lesson I still use every time I meet a new class, and make it clear I want to help them achieve their goals, which usually involve surviving the class. I do not understand why so many teachers set out objectives based on the assumption that they will successfully re-align their students’ value systems.

And in a related revelation, you can see how I frame my task. In his TED talk, Dan Meyer asks the audience to imagine:

“you really loved something…and you recommended it wholeheartedly to someone you really liked…and the person hated it. By way of introduction, that is the exact same state in which I spent every working day of the last six years. I teach high school math. I sell a product to a market that doesn’t want it but is forced by law to buy it.”

All math teachers can relate to this statement; it’s clever, funny, and does a good job of introducing the fundamental dilemma of high school math teachers: most kids hate math and are required to take it. Many dedicated math teachers would not only relate, but agree with Dan’s framing of his task as a sales job, regardless of their teaching ideology. When I say I disagree, it’s not because Meyer is wrong but because we approach our jobs in fundamentally different ways. I don’t love math, and I’m not selling a product.

Victoria: I’m terrible. I know I’m terrible. I look at the mirror and I’m ashamed. Maybe I should quit. I just can’t seem to do anything right.

Joe Gideon: Listen. I can’t make you a great dancer. I don’t even know if I can make you a good dancer. But, if you keep trying and don’t quit, I know I can make you a better dancer. I’d like very much to do that. Stay?

Were it not for the unfortunate plot point about Joe Gideon’s motives for hiring Victoria for the show (he was a hounddog who had her in bed an hour after they first met), I suspect more math teachers would reveal that they can quote this scene from All That Jazz verbatim.

And those math teachers mostly would agree with me. Teaching math, for us, isn’t about creating mathematicians. It’s only occasionally about working with kids who want to be engineers, doctors, or architects. Mostly, it’s about giving kids enough math skills to pass a college placement test so they won’t end up spending a fortune on remedial math classes and never get any further—or at least enough skills so they’ll pass a remedial math class and move on. Or giving kids enough math so they look at a trade school placement test and think, “Hey, I can do this.” Or just giving kids the will to pass the class and keep them out of mindless credit recovery in alternative institutions, letting them feel part of the educational system, not a failure who couldn’t cut it at normal high school.

We don’t promise miracles. We do promise “better”.

Finally, though, the story indicates that I am acutely aware of all my students’ motivations, that not all my students just want to pass. I have bright kids in almost every class, I have highly motivated kids, I have kids with specific objectives, most of whom want to learn as much as they can. I never forget them, and if I can’t dedicate my entire teaching agenda to meeting their goals, it’s only because I owe allegiance to all my students. I never stop looking for better ways to give these kids what they need while still ensuring I meet my overall responsibility. Many other teachers say these kids should come first. I always worry they might be right. But as I said above, I do not overvalue any individual kid over the needs of the entire class.

This tale doesn’t tell much about how I teach, but that particular topic gets plenty of coverage in other essays.

Anyone who is familiar with reform math can probably infer not only my teaching values and priorities, but also a lot of reasons why I’m not crazy about reform math. But I’ll go into details in the next post.

Modeling Linear Equations, Part 3

See Part I and Part II.

The success of my linear modeling unit has completely transformed the way I teach algebra.

From Part II, which I wrote at the beginning of the second semester at my last school:

In Modeling Linear Equations, I described the first weeks of my effort to give my Algebra II students a more (lord save me) organic understanding of linear equations. These students have been through algebra I twice (8th and 9th grade), and then I taught them linear equations for the better part of a month last semester. Yet before this month, none of them could quickly generate a table of values for a linear equation in any form (slope intercept, standard form, or a verbal model). They did know how to read a slope from a graph, for the most part, but weren’t able to find an equation from a table. They didn’t understand how a graph of a line was related to a verbal model—what would the slope be, a starting price or a monthly rate? What sort of situations would have a meaningful x-intercept?

This approach was instantly successful, as I relate. Last year, I taught the entire first semester content again in two months before moving on, and still got in about 60% of the Algebra II standards (pretty normal for a low ability class).

So when I began intermediate algebra in the fall, I decided to start right off with modeling. I just toss up some problems on the board–Well, actually, I start with a stick figure cartoon based on this lesson plan:


I put it on the board, and ask a student who did middling poorly on my assessment test, “So, what could Stan buy?”

Shrug. “I don’t know.”

“Oh, come on. You’re telling me you never had $45 bucks and a spending decision? Assume no sales tax.”

Tentatively. “He could just buy 9 burritos?”

“Yes, he could! See? Told you you could do it. How many tacos could he buy?”


At this point, another student figures it out, “So if he doesn’t buy any burritos, he could buy, like,…”

“Fifteen tacos. Why is it 15?”

“Because that’s how much you can buy for $45.”

“Anyone have another possibility? You? Guy in grey?”

Long pause, as guy in grey hopes desperately I’ll move on. I wait him out.

“I don’t know.”

“Really? Not at all? Oh, come on. Pretend it’s you. It’s your money. You bought 3 burritos. How many tacos can you get?”

This is the great part, really, because whoever I call on, and it’s always a kid who doesn’t want to be in the room, his brain starts working.

“He has $30 left, right? So he can buy ten tacos.”

“Hey, now, look at that. You did know. How’d you come up with ten?”

“It costs $15 to get three burritos, and he has $30 left.”

So I start a table, with Taco and Burrito headers, entering the first three values.

“And you know it’s $15 because….”

He’s worried it’s a trick question. “…it’s five dollars for each burrito?”

I force a couple other unwilling suckers to give me the last two integer entries

“Yeah. So see how you’re doing this in your head. You are automatically figuring the total cost of the burritos how?”

“Multiplying the burritos by five dollars.”

“And, girl over there, in pink, how do you know how much money to spend on tacos?”

“It’s $3 a taco, and you see how much left you have of the $45.”

“And again with the math in your head. You are multiplying the number of tacos by 3, and the number of burritos by….”


“Right. So we could write it out and have an actual equation.” And so I write out the equation, first with tacos and burritos, and then substituting x and y.

“This equation describes a line. We call it the standard form: Ax + By = C. Standard form is an extremely useful way to describe lines that model purchasing decisions.”

Then I graph the table and by golly, it’s dots in the shape of a line.


“Okay, who remembers anything about lines and slopes? Is this a positive or a negative slope?”

Silence. Of course. Which is better than someone shouting out “Positive!”

“So, guy over there. Yeah, you.”

“I wasn’t paying attention.”

“I know. Now you are. So tell me what happens to tacos when you buy more burritos.”

Silence. I wait it out.

“Um. I can’t buy as many tacos?”

“Nice. So what does that mean about tacos and burritos?”

At this point, I usually get some raised hands. “Blue jersey?”

“If you buy more tacos, you can’t buy as many burritos, either.”

“So as the number of tacos goes up, the number of burritos…”

“Goes down.”

“So. This dotted line is reflecting the fact that as tacos go up, burritos go down. I ask again: is this slope a positive slope or a negative slope?” and now I get a good spattering of “Negative” responses.

From there, I remind them of how to calculate a slope, which is always great because now, instead of it just being the 8 thousandth time they’ve been given the formula, they see that it has direct relevance to a spending decision they make daily. The slope is the reduction in burritos they can buy for every increased taco. I remind them how to find the equation of a slope from both the line and the table itself.

“So I just showed you guys the standard form of a line, but does anyone remember the equation form you learned back in algebra one?”

By now they’re warming up as they realize that they do remember information from algebra one and earlier, information that they thought had no relevance to their lives but, apparently, does. Someone usually comes up with the slope-intercept form. I put y=mx+b on the board and talk the students through identifying the parameters. Then, using the taco-burrito model, we plug in the slope and y-intercept and the kids see that the buying decision, one they are extremely familiar with, can be described in math equations that they now understand.

So then, I put a bunch of situations on the board and set them to work, for the rest of that day and the next.


I’ve now kicked off three intermediate algebra classes cold with this approach, and in every case the kids start modeling the problems with no hesitation.

Remember, all but maybe ten of the students in each class are kids who scored below basic or lower in Algebra I. Many of them have already failed intermediate algebra (aka Algebra II, no trig) once. And in day one, they are modeling linear equations and genuinely getting it. Even the ones who are unhappy (more on that in a minute) are getting it.

So from this point on, when a kid sees something like 5x + 7y = 35, they are thinking “something costs $5, something costs $7, and they have $35 to spend” which helps them make concrete sense of an abstract expression. Or y = 3x-7 means that Joe has seven fewer than 3 times as many graphic novels as Tio does (and, class, who has fewer graphic novels? Yes, Tio. Trust me, it’s much easier to make the smaller value x.)

Here’s an early student sample, from my current class, done just two days in. This is a boy who traditionally struggles with math—and this is homework, which he did on his own—definitely not his usual approach.


Notice that he’s still having trouble figuring out the equation, which is normal. But three of the four tables are correct (he struggles with perimeter, also common), and two of the four graphs are perfect—even though he hasn’t yet figured out how to use the graph to find the equation.

So he’s doing the part he’s learned in class with purpose and accuracy, clearly demonstrating ability to pull out solutions from a word model and then graph them. Time to improve his skills at building equations from graphs and tables.

After two days of this, I break the skills up into parts, reminding the weakest students how to find the slope from a graph, and then mixing and matching equations with models, like this:


So now, I’m emphasizing stuff they’ve learned before, but never been able to integrate because it’s been too abstract. The strongest kids in the class are moving through it all much faster, and are often into linear inequalities after a couple weeks.

Then I bring in one of my favorite handouts, built the first time I did this all a year ago: ModelingDatawithPoints. Back to word models, but instead of the model describing the math, the model gives them two points. Their task is to find the equation from the points. And glory be, the kids get it every time. I’m not sure who’s happier, them or me.

At some point in the first week, I give them a quiz, in which they have to turn two different models into tables, equations, and graphs (one from points), identify an equation from a line, identify an equation from a table, and graph two points to find the equation. The last question is, “How’s it going?”

This has been consistent through three classes (two this semester, one last). Most of the kids like it a lot and specifically tell me they are learning more. The top kids often say it’s very interesting to think of linear equations in this fashion. And about 10-20% of the students this first week are very, very nervous. They want specific methods and explicit instructions.

The day after the quiz, I address these concerns by pointing out that everyone in the room has been given these procedures countless times, and fewer than 30% of them remember how to apply them. The purpose of my method, I tell them, is to give them countless ways of thinking about linear equations, come up with their own preferred methods, and increase their ability to move from one form to another all at once, rather than focusing in on one method and moving to another, and so on. I also point out that almost all the students who said they didn’t like my method did pretty well on the quiz. The weakest kids almost always like the approach, even with initially weak results.

After a week or more of this, I move onto systems. First, solving them graphically—and I use this as a reason to explitly instruct them on sketching lines quickly, using one of three methods:


Then I move on to models, two at a time. Last semester, my kids struggled with this and I didn’t pick up on it until a month later. This last week, I was alert to the problems they were having creating two separate models within a problem, so I spent an extra day focusing on the methods. The kids approved, and I could see a much better understanding. We’ll see how it goes on the test.

Here’s the boardwork for a systems models.

So I start by having them generate solutions to each model and matching them up, as well as finding the equations. Then they graph the equations and see that the intersection, the graphing solution, is identical to the values that match up in the tables.

Which sets the stage for the two algebraic methods: substitution and combination (aka elimination, addition).


Last semester, I taught modeling to my math support class, and they really enjoyed it:


Some sample work–the one on the far right is done by a Hispanic sophomore who speaks no English.

Okay, back at 2000 words. Time to wrap it up. I’ll discuss where I’m taking it next in a second post.

Some tidbits: modeling quadratics is tough to do organically, because there are so few real-life models. The velocity problems are helpful, but since they’re the only type they are a bit too canned. I usually use area questions, but they aren’t nearly as realistic. Exponentials, on the other hand, are easy to model with real-life examples. I’m adding in absolute value modeling this semester for the first time, to see how it goes.

Anyway. This works a treat. If I were going to teach algebra I again (nooooooo!) I would start with this, rather than go through integer operations and fractions for the nineteenth time.

Push the Right Buttons

Since my school only has four block periods, teachers often sub for each other during their prep periods. Cheaper than hiring an outside sub, and with an experienced classroom manager running things, there’s an outside chance that the class won’t be a complete waste of time, which is normally the case for all but honors and AP courses when teacher’s not around.

Today I subbed for an economics teacher during fourth block, last class of the day. She told me to go to the library, that she’d put a note on her classroom door telling the students where to go. Ten minutes after the late bell, no students. I wander over to her classroom, where about half the students are milling around trying to get the nerve to leave. No note on the door. (I suspect someone pulled it down.)

“Are you Ms. L’s Econ class?

“Class was cancelled.”

“Yeah. What is this, college? Go to the library and work on your business project.”

“Who are you?”

“I’m the person telling you to go to the library and work on your business project. Come on, move. You’re late.”

(Note to new teachers, particularly you little twenty-somethings bleating about relating to students: it goes much smoother if you expect obedience. Better yet if you demand compliance. And like John Travolta says in Get Shorty, what you’re not doing is feeling about it one way or the other.)

They all trudge to the library and get to work on their project with little fuss; most of the remaining third trickle in over the next 25 minutes.

Last in, half an hour late, are two African American young men who clearly took the opportunity to pick up a second lunch at Wendys, and tried to get out of a tardy by telling me that they’d been waiting by the door “the whole time!” I laugh at them and tell them to get to work.

“We don’t know what to do.”

“Here.” I produce the assignment handout.

“Oh, yeah, the sports project. But there aren’t any computers open, they’re all full.”

“You two?” I speak to two kids on the end row of computers. “You’re not in this class?” They try to ignore me. “Are you in this class?” They still ignore me. “Let me find your name on the class list.”

“Okay, we’re not in this class. But we have homework.”

“Good for you. Do it later. Bye.” They are not happy, but they leave. (The computers are booked for the class, in case you just think I’m mean.)

“Look, guys! Two computers! Put the food away, and get to work.”

It’s a trip watching these two. They try. I watch them page through the assignment, puzzling over it, telling each other man, this is f**** up, she doesn’t tell us what to do. “It’s just a list of things we need to have done! But what do we do, man?”

After five minutes, I see them both paging through search lists, getting “Site blocked” messages. I wander over.

“Please tell me you aren’t googling porn sites in open view of a teacher and librarian.”

“Naw, I’m just finding shoe prices.”

“Shoe shopping?”

“For the team.”

“Oh, so you’re pricing shoes for your business. How much are you spending on personnel?”

“We don’t know how to do that. She doesn’t say how.”

“What are you staffing?”

“We have to staff a professional high school basketball team for $100,000.”

I look at them expectantly.


“Nothing comes to mind?”

“She didn’t tell us what to do.”

“But you’re pricing shoes.”


I wait, to see if enlightenment will dawn. But it is still well before midnight.

“Okay. Who will be wearing the shoes?”

“The players. Oh.”

“Indeed. So maybe start with people instead of shoes. How many players on a professional basketball team?”

“Twelve. And we’ll need a manager, and at least three coaches….”

“and what about a guy to manage endorsements at the team level?”

And now they are cooking with gas, telling me what they need to staff the team, what prices seem reasonable for a “basketball farm team, these are high school players, they’ll be chill with maybe $500 a season”. I walk away, and ten minutes later they come back to me asking if I know how they can put this “in a file or we’ll lose this paper with our notes.” I bring up Excel, they carefully enter all their data.

A bit later, one of them comes over to me and says “I took this class on Excel when I was a freshman and you could, like, add things up?” So I look at what they have, and suggest that they create one column for count and one for price, so they could then change values and automatically change the sums and they caught on and by god if they didn’t have a damn spreadsheet with personnel variables and starting costs entered, ready for tweaking. They were brainstorming equipment and facility costs by the end of class.

Do not imagine, dear reader, that these two boys will come on time to class tomorrow, ready to jump in and pick up where they left off. More likely, they will cut class one day, get pulled out for some activity on another, and by the time they get back to it will have forgotten the file name and where they emailed it. Still, for 45 minutes, these kids did productive work, thinking about what they’d need to create a small business. Count it in the win column, once I pushed the right buttons.

There’s a larger point in this story somewhere, about the degree of scaffolding low ability, low incentive kids need to do any sort of project based work. The teacher had put together a good lesson, too: achievable, relevant, and interesting. But many of the kids would need far more support—leading questions and a project broken down into key milestones, Excel templates for business plans and budgets, and so on. And as always, I am boggled by the gap between the idiots calling for project based learning which is, to their thinking, essential to modern education, and the actual students who simply don’t have the ability or motivation to meaningfully engage in the learning required for the projects as they are currently envisioned.

But it’s late and I’m pleased with the win, so I’ll stop here.

Boaler’s Bias (or BS)

I began this piece a week ago intending to opine on the Boaler letter. However, I realized I have to confess a strong bias: I read Boaler in ed school and nearly vomited all over my reader. And that will take a whole post.

Experiencing School Mathematics: Traditional and Reform Approaches to Teaching and Their Impact on Student Learning

Boaler, a Brit who has held math education academic positions in England as well as at Stanford, performed a three-year study of two English schools, matched up in demographics and test scores. Phoenix Park believed in progressive, student-centered instruction, whereas Amber Hill taught a traditionalist method—more than traditionalist, they taught math by rote and drill, which is by no means required for teacher-centered instruction.

Boaler was ostensibly investigating the two instruction methods, but the fix was clearly in. Despite Boaler’s constant assurances that the Amber Hill teachers were dedicated and caring, the school presents as an Orwellian fantasy:

One of the first things I noticed when I began my research was the apparent respectability of the school. Walking into the reception area on my arrival, I was struck by the tranquility of the arena. The reception was separated from the rest of the school by a set of heavy double doors. The floors were carpeted in a somber gray; a number of easy chairs had been placed by the secretary’s window and a small tray of flowers sat above them. …Amber Hill was unusually orderly and controlled. Students generally did as they were told, their behavior governed by numerous enforced rules and a general school ethos that induced obedience and conformity. All students were required to wear a school uniform, which the vast majority of students wore exactly as the regulations required. The annual school report that teachers sent home to parents required the teachers to give the students a grade on their “co-operation” and their “wearing of school uniform.” The head clearly wanted to present the school as academic and respectable, and he was successful in this aim at least in terms of the general facade. Visitors walking around the corridors would see unusually quiet and calm classrooms, with students sitting in rows or small groups usually watching the board. When students were unhappy in lessons, they tended to withdraw instead of being disruptive. The corridors were mainly quiet, and at break times the students walked in an orderly fashion between lessons. The students’ lives at Amber Hill were, in many ways, structured, disciplined, and controlled

(page 13)

Phoenix Park, on the other hand:

…had an attractive campus feel. The atmosphere was unusually calm—described in a newspaper article on the school as peaceful. Students walked slowly around the school, and there was a noticeable absence of students running, screaming, or shouting. This was not because of school rules; it seemed to be a product of the school’s overall ambiance. I mentioned this to one of the mathematics teachers one day and she agreed, saying that she did not think she had ever heard anybody shout—teacher or student. She added that this was particularly evident at break times in the hall: “The students are all so orderly, but no-one ever tells them to be.”…. Students were taught all subjects in mixed-ability groups. Phoenix Park students did not wear school uniforms. Most students wore fashionable but inexpensive clothes such as jeans, with trainers or boots, and shirts or t-shirts worn loosely outside. A central part of the school’s approach involved the development of independence among students. The students were encouraged to act responsibly—not because of school rules, but because they could see a reason to act in this way.

(emphasis mine) (page 18)

And yet, while the Amber Hill students were well-behaved little automatons, the Phoenix Park kids–the ones who simply behave well by choice and idealism, not some lower-class aspiration to respectability–ran amok:

In the 100 or so lessons I observed at Phoenix Park, I would typically see approximately one third of students wandering around the room chatting about non-work issues and generally not attending to the project they had been given. In some lessons, and for some parts of lessons, the numbers off task would be greater than this. Some students remained off task for long periods of time, sometimes all of the lessons; other students drifted on and off task at various points in the lessons. In a small quantitative assessment of time on task, I stood at the back of lessons and counted the number of students who appeared to be working 10 minutes into the lesson, halfway through the lesson, and 10 minutes before the end of the lesson. Over 11 lessons, with approximately 28 students in each , 69%, 64%, and 58% of students were on task, respectively [the corresponding numbers at Amber Hill were in the 90%s].
More important than either of these factors, however, is that the freedom the students experienced seemed to relate directly to the relaxed and non-disciplinarian nature of the three teachers and the school as a whole. Most of the time, the teachers did not seem to notice when students stopped working unless they became very disruptive. All three teachers seemed concerned to help and support students and, consequently, spent almost all of their time helping students who wanted help, leaving the others to their own devices.

(page 64, 65)

But far from criticizing the school for abysmal classroom management, Boaler blames the students.

However, this freedom was also the reason the third group of students hated the approach. Approximately one fifth of the cohort thought that mathematics was too open, and they did not want to be left to make their own decisions about their work. They complained that they were often left on their own not knowing what to do, and they wanted more help and structure from their teachers. The students felt that the school’s approach placed too great a demand on them—they did not want to use their own ideas or structure their own work, and they said that they would have preferred to work from books. What for some students meant freedom and opportunity, for others meant insecurity and hard work. There were approximately five students in each class who disliked and resisted the open nature of their work. These students were mainly boys and were often disruptive— not only in mathematics, but across the school. (page 68)

In every mathematics lesson I observed at Phoenix Park, between three and six students would do little work and spend much of their time disrupting others. I now try to describe the motivation of these 20 or so students, who represented a small but interesting group. The students who did little work in class were mainly boys, and they related their lack of motivation to the openness of the mathematical approach and, more specifically, the fact that they were often left to work out what they had to do on their own. …..Many of the Phoenix Park students talked about the difficulty they experienced when they firststarted at the school working on open projects that required them to think for themselves. But most of the students gradually adapted to this demand, whereas the disruptive students continued to resist it.

In Years 9 and 10, I interviewed six of the most disruptive and badly behaved students in the year group: five boys and one girl. They explained their misbehavior during lessons in terms of the lack of structure or direction they were given and, related to this, the need for more teacher help. These students had been given the same starting points as every-body else, but for some reason seemed unwilling to think of ways to work on the activities without the teacher telling them what to do. This was a necessary requirement with the Phoenix Park approach because it was impossible for all of the students to be supported by the teacher when they needed to make decisions. The students who did not work in lessons were no less able than other students; they did not come from the same middle school and they were socioeconomically diverse. In questionnaires, the students did not respond differently from other students, even on questions designed to assess learning style preferences. The only aspect that seemed to unite the students was their behavior and the fact that most of them were boys. The reasons that some students acted in this way and others did not were obviously complex and due to a number of interrelated factors. Martin Collins [one of the Phoenix Park teachers] believed that more of the boys experienced difficulty with the approach because they were less mature and less willing to take responsibility for their own learning than the girls. The idea that the boys were badly behaved because of immaturity was also partly validated by the improvement in the boys’ behavior as they got older .

(page 73) (emphasis mine)

Meanwhile, the Amber Hill girls were miserable:

All of the Amber Hill girls interviewed in Years 9 and 10 expressed a strong preference for their coursework lessons and the individualized booklet approach, which they followed in Years 6 and 7, as against their textbook work. The girls gave clear reasons why these two approaches were more appropriate ways of learning mathematics for them; all of these reasons were linked to their desire to understand mathematics. In conversations and interviews, students expressed a concern for their lack of understanding of the mathematics they encountered in class. This was particularly acute for the girls not because they understood less than the boys, but because they appeared to be less willing to relinquish their desire for understanding…..Just as frequently, I observed girls looking lost and confused, struggling to understand their work or giving up all together. On the whole, the boys were content if they attained correct answers. The girls would also attain correct answers, but they wanted more. The different responses of the girls and boys to group work related to the opportunity it gave them to think about topics in depth and increase their understanding through discussion. This was not perceived as a great advantage to the boys probably because their aim was not to understand, but to get through work quickly. These different responses were also evident in response to the students’ preferences for working at their own pace. In chapter 6, I showed that an overwhelming desire for both girls and boys at Amber Hill was to work at their own pace. This desire united the sexes, but the reasons boys and girls gave for their preferences were generally different. The boys said they enjoyed individualized work that could be completed at their own pace because it allowed them to tear ahead and complete as many books as possible….The girls again explained their preference for working at their own pace in terms of an increased access to understanding. The girls at Amber Hill consistently demonstrated that they believed in the importance of an open, reflective style of learning, and that they did not value a competitive approach or one in which there was one teacher-determined answer. Unfortunately for them ,the approach they thought would enhance their understanding was not attainable in their mathematics classrooms except for 3 weeks of each year .

(page 139)

(all emphasis mine)

So in each school, there were students who really hated the teaching method used. But Boaler blames the complex-instruction haters at Phoenix Park (of course, it’s just a coincidence they are mostly male), for their immaturity and disruption, because they didn’t like the open-ended discovery method she so vehemently approves of. Meanwhile, she not only sympathizes with the Amber Hill girls, poor dears, who didn’t like the procedure-oriented teaching method at their school, but continually slams the Amber Hill boys who do enjoy it because those competitive, goal-driven little twerps aren’t interested in learning math but just doing more problems than their pals.

It was at this point I threw my reader across the room.

Moreover, reading between the lines of Boaler’s screed shows clearly that both schools are doing what I would consider an utterly crap job of teaching math. Boaler also mentions Phoenix Park is the low achiever in its affluent school district, and both schools have dismal test scores (which, let me be clear, could be true even if both schools were doing an outstanding job in math instruction).

Indeed, Boaler’s entire thesis—that the “reform” approach leads to better test scores—is poorly supported by her own data. Boaler received special permission to evaluate the students’ individual GCSE scores. She coded problems as either “procedural” or “conceptual”.

Amber Hill, of the dull, grey school and the dreary uniforms, actually outscored Phoenix Park, the progressive’s paradise, on procedural questions. While Phoenix Park outscores Amber Hill on conceptual problems, it wasn’t by all that much.

Like any dedicated ideologue, Boaler misses the monster lede apparent in these representations: Phoenix Park’s score range is nearly double that of Amber Hill’s, suggesting that discovery-based math helps high ability kids, while procedural math helps low ability students. Low ability students lost out at Phoenix Park, because they couldn’t cope with the open-ended, unstructured approach. Boaler didn’t give a damn about those kids, because they were boys. Meanwhile, high ability kids do better with an open-ended approach, gaining a better understanding of math concepts.

This finding has been well-documented in subsequent research—at least, the research done by academics who aren’t hacks bent on turning math education into a group project. I wrote about this earlier.

Here, too, is a takedown of some of the specifics in her research. You can read the whole thing, but here are the primary points in direct quotes:

  • “Also these scores are very similar. A notable difference is that rather a lot of students at Amber Hill fail, whereas more students at Phoenix Park get the very low grades E,F,G. Boaler sees this as a positive thing about Phoenix Park. A possible explanation (which Boaler does not give) has to do with the fact that the GCSE is actually not one exam, but three exams….. it is perfectly conceivable that at Amber Hill many students aimed higher than they could achieve and failed. Note that it is essential for further education to receive at least a C, so that participating in the basic exam is virtually useless. The figures show that nonetheless at Phoenix Park at least 43.5 percent of the students (the Fs and Gs) participated in this exam and by doing this gave up their chance at higher education without even trying.”
  • “This indicates that, compared to the nation, the students at Phoenix Park did worse on the GCSE than they did on the NFER. So Phoenix Park seems not to have done its students a lot of good. The same is of course true for Amber Hill, which performed very similarly to Phoenix Park. I also took a look on the internet at typical average scores of schools on the GCSE. It seems that Phoenix Park and Amber Hill are just about the schools with the worst GCSE scores in the UK. I cannot help but think that Amber Hill was specifically chosen for this fact.”
  • “Boaler doesn’t say anything about the GCSE scores of Amber Hill at the moment that she decided to include this school in her study, but there is not reason to believe that it was markedly different from the above mentioned scores for Amber Hill. If that is the case, then Boaler seems to have been stacking the deck in favor of Phoenix Park and its discovery learning approach to mathematics teaching.”
  • “Boaler also doesn’t mention that the grades for the GCSE at both schools are lower than one would expect given the NFER scores. She seems determined to interpret everything in favor of Phoenix Park. ”

If you’ve read anything about the Boaler/Milgram/Bishop debate, some of these Boaler critiques may sound a tad familiar. But don’t get them confused. This is a different study. Which means Boaler has pulled this nonsense twice.

It was reading horror shows like Boaler that made me loathe progressive educators. It took me a while to acknowledge that they weren’t all dishonest hacks bent on distorting reality. Not all progressives are determined to create an ideological force field that repels all sane discussion of the genuine advantages and disadvantages of different educational approaches, and an honest acknowledgement that student cognitive ability—which appears unevenly distributed by both race *and* gender, at least as we measure it—is a factor in determining the best approach for a given student population. And ultimately, I find myself slightly more sympathetic to progressives than reformers because at least progressives (and here I include Boaler) actually know about teaching, even if they often do it with blinders on.

So getting all this out of my system means I’m not writing—yet—about Boaler/Milgram/Bishop. But then, I imagine my opinion’s pretty clear, isn’t it?

Ironically, I know people who know Boaler, and assure me she’s quite nice. But then, she’s British. It’s probably the accent.

Best Movie About Teaching. Ever.

Cheery news: Won’t Back Down had a hideous opening. Here’s a hint, folks: teachers are a big piece of the audience for simplistic, feel-good teacher movies, so it’s a terrible idea to make a simplistic feel-good teacher movie suggesting that most of them suck.

I, however, am not a fan of simplistic, feel-good teacher movies: Dangerous Minds, Lean on Me, Mr. Holland’s Opus, or Freedom Writers, are tripe. (But the best of that group by far is Holland.)

I occasionally enjoy movies about flamboyant teachers for whom students function primarily as an audience (Prime of Miss Jean Brodie, Dead Poets Society)—and in my enrichment classes, I fear I am that sort of teacher—but they send the wrong signal and thus, I deny them official status as teacher films. They are “idiosyncratic adult who happens to be a teacher opens the eyes of his appreciative audience” movies.

Stand and Deliver is overrated, but Lou Diamond Phillip’s performance covers up a lot of sins. The story’s a big lie alas, and the students did cheat.

Up the Down Staircase, written by Bel Kaufman—still enjoying life at 101, Holla!—is far superior to To Sir with Love, which had the bigger star and English accents, so the first film has been mostly forgotten. It’s worth a look for its honesty and refusal to portray simplistic success. Staircase, like Kindergarten Cop, a guilty pleasure, and the delightful Goodbye Mr. Chips, does a nice job of focusing on classroom management, so essential to teaching inner city kids, wild suburban kindergartners, or British boarding school brats.

Searching for Bobby Fischer is a beautiful film about parenting and teaching; both Vinnie and Mr. Pandolfini are exemplars of their individual approaches. School of Rock is sublimely silly, but at its heart is a similar film; specialist teachers (the arts, chess, what have you) have all the fun, sometimes.

There has been much in the news lately about the importance of teaching writing, which reminded me of an odd, lesser, film for both Doris Day (another Holla!) and Clark Gable, Teacher’s Pet. Day is quite gorgeous as a journalism professor who thinks rough, tough (and far too old) newspaper editor Clark is actually a journalism student with great talent. Gig Young has a great role as the intellectual boyfriend (no holla for Gig, alas). It’s no great shakes, but has two or three excellent scenes about the “how” of writing, particularly towards the end, when Clark tells a young Nick Adams how much time he had to spend learning to write.

Best Movie about Teaching Ever: The Browning Version

But the most perfect movie ever made about teaching focuses, paradoxically, on a failed teacher. Written by Terrence Rattigan, The Browning Version explores the last days of classics teacher Andrew Crocker-Harris, who is leaving a mid-tier “public school” post from which he has been prematurely retired. It’s the kind of play with a few parts, the type about which one says “the TV version has Ian Holm as the Crock, Judy Dench as the wife, and Michael Kitchen as the lover” and anyone familiar with the play goes oh, great cast! Albert Finney played the Crock in the 1992 remake, but Michael Redgrave offers the definitive version in Anthony Asquith’s 1951 film.

To describe the plot is to unnecessarily depress the unprepared. One must witness four or five scenes of brutal psychological cruelty and then blink away tears at moments of extraordinary kindness. Rattigan was gay when homosexual activity was a crime, and that may be why that in the pantheon of Brit Lit, Crocker-Harris’ wife is ranked second only to Lady Macbeth as the Ultimate Evil Female, from whose clutches Crocker-Harris must be rescued by a sympathetic male friend if only to view the wreckage of his failed life from a safe distance.

The Browning Version examines that failed life through the prism of the Crock’s status as a failed teacher. His failure lies not in his ability or knowledge, but in his failure to teach with joy and passion and, most importantly, in his failure to show his students that he cared for them (although it’s clear that privately, he did). Faced with students who didn’t care about his subject, he gave up. Eduformers talk about such teachers with cheap abandon and no understanding; Redgrave, a theater legend in the best of his few film roles, does nothing on the cheap, and his pain, which rarely cracks his stiff British reserve, is ever present. If you’re up for it, watch the Himmler scene, and see what eduformers miss about these failing teachers.

But if we must bear witness to the Crock’s failure, we also are given the relief of his redemption in the film’s great insight: students bear a responsibility to their teachers, too. Thanks to the glorious accident of a young man who normally loves science but thinks the classics a bit of a bore, Crocker-Lewis learns that he is, still, a teacher who can find and inspire passion for his subject, given a willing student. Of course, if one teaches Greek and Latin—or algebra II and math support— willing, engaged students are about as thick on the ground as dodos. In the early scenes, we see Crocker’s class paralleled with the science teacher’s (who is also Crocker’s wife’s lover). The science teacher, who has an easy, informal rapport with his students, also has a way cooler subject and offers up a whiz bang experiment. Crock has nothing but old plays and conjugations. How much of a teacher’s ability to hold on to enthusiasm is dependent on the subject he teaches? How much easier is it to hold onto your own motivation when most of your students are actually interested in your subject?

I’ve been at three schools now, all of them with a high percentage of low ability students, and the math teachers are always on the outside looking in. They aren’t the ones the principals thank profusely at the end of the year for inspiring the students. When math classes have a 40-60% failure rate, math teachers don’t make “favorite” or “best” lists. They are the ones who are on the hook for test scores, the ones who are simultaneously expected to keep standards high but not fail too many students, the ones most likely to see students two years in a row in the same class. I became a teacher knowing full well this was in my future, knowing that most of my students, at best, would think of me as someone who makes a horrible hour and hated subject marginally bearable. Yet even with that hardnosed realism, I still often end the day feeling a tad beat down. I cope with the knowledge by continuing my work in private instruction and tutoring, where my kids think I’m the bomb. Many teachers don’t have this out, and leave for schools with higher ability kids–or leave teaching altogether—unable to stand the dreary hatred reflected back at them class after class.

The Browning Version assigns all blame to the teacher for his failure, but at the same time shows how little it takes to put the Crock back on his game. All the man needed was one student who cared; he responded tentatively and then more openly, as the teaching relationship gelled. We are left with the impression that Crocker-Lewis, reminded of what teaching feels like when students care, will go to his new post with a determination to at least show his kids he cares, and search for the very few who might be engaged. That is, we trust and believe he’ll do his job.

The Browning Version is neither easy nor feel-good. It will thus add nothing to the current educational policy debate. But every teacher should watch it, if only to remind themselves that giving up damages souls, their own even more than those of their students.