Category Archives: philosophy

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 Gilliland, is a keeper.

What I Learned: Year 3

I want to continue my teaching retrospective, if only for my own edification. Year 3 in particular led to major changes in my curriculum and pacing.

To recap: my first year was spent in a very progressive school, where I taught algebra, geometry, and humanities, both literature and history. I loved teaching, didn’t much care for the school, and definitely wasn’t sufficiently of the left to stay there. Years 2 and 3 were at a Title I school, 65% Hispanic/ELL. As I’ve said before, year 2’s all algebra all the time schedule was my toughest schedule ever as a teacher; I do not expect to see its like again. Which is good, because I still get flashbacks. I have, in fact, never officially taught algebra 1 since that time although most people would consider what I teach in Algebra 2 to be, in fact, Algebra 1.

Year 3 was at the same school, but I was assigned Algebra 2 and Geometry. And that made all the difference.

Establishing Classroom Ambiance

My 65 geometry students included twenty I’d taught the previous year in Algebra I, students who knew and liked me.

First day, I started one class a bit early when in walked Robbie, redheaded, pale, anxious, diagnosed with Asperger’s but almost certainly a high functioning autistic. I told him to have a seat, and didn’t immediately realize that the little freshman was utterly aghast at the idea that he was late to class. He was murmuring “class starts at 9:15, I was here at 9:12” over and over again, slowly working up to a meltdown by the time I noticed. Before I could react Augustin, a junior, first student I’d met at this school the year before, leaned over from a desk in the same group.

“Relax. Teacher started early. Never cares about time anyway. You’re good.”

Meltdown over. Robbie was awestruck that a junior had deigned to notice him. He also remembered all year that I “never cared about time”, which did much to keep him balanced and happy with a teacher incapable of a predictable routine. I have always remembered Augustin for his offhand kindness to an odd kid.

My geometry classes gave me the feeling of being a known quantity, a teacher with student cred, something I’d long easily established in my Asian enrichment classes, as well as my Kaplan test prep, but never felt in a public school before. I’d always been a loose disciplinarian, an easy classroom controller, and this isn’t as easy in test prep as you might think—it’s why I got so much work. I knew that teaching outside of private instruction would be different, but I found the change more challenging than I expected.

For my first two years in public school, I struggled to recreate the friendly “we’re all in this together” atmosphere I expected to achieve easily. My first year, only my humanities class ever achieved the ambiance I took for granted in private instruction. Only two of my 4 algebra classes (one was a double block) had that cheerful noisiness that is now a trademark of my public school classes. I wasn’t a failure as a teacher; in many ways, I was doing exactly what I anticipated and dealing with expected obstacles. But I had secretly mourned the loss of my standing as a popular teacher. And now, suddenly, I had my mojo again.

My algebra 2 classes were more like my algebra 1 classes from the year before; I didn’t have yet the same easy rapport that I had with my geometry students. This gave me a chance to study the difference. Would I always need to have repeat students, or was there something I could do to establish the environment of easy fun with hard work–or at least some effort?

Over time, I learned that some students find me harder to understand than others. They often don’t grok my ironic asides. They do not understand that I “blast” without malice. They assume I hold grudges, that I count misdemeanors in a black book somewhere. They don’t understand I am often somewhat ruthlessly focused on one objective. As I’ve said before, teaching is a performance art, and the act of engaging students to convince them to learn is often an arduous mental task.

And so I’ve learned to explain this up front. That I am often sarcastic, and think attempts to ban this essential classroom management tool are Against God. That I’m not often annoyed, and usually harmless. But when I am annoyed I yell first, ask questions later when I remember to, which I often don’t. That I am unlikely to remember what I was mad about 20 minutes later, much less hold a grudge. In fact, the only behaviors that I remember are cruelty and cheating. That I love teaching, and like all of them. Except Joe. I can’t stand Joe. And frankly, I’ve never been a big fan of Alison. But except them. And Mario. Don’t care for Mario much. But everyone else. Really. (Yeah, see that, kids? Mild irony. Get used to it.)

I’ve also learned to reach out on things that don’t matter as much to me but I’ve realized matter much more than I realized to students. I’ve always been one to say “Hi!” in the hallways and chitchat for a moment with past and present students but in truth honestly don’t care about football games or sporting events. Still, kids really do like it when you show up at the games, or ask about the outcomes, or call out a student who had a great game or ran a PR. I ban the singing of Happy Birthday because the noise is unbearable, but after they beg, I give them a count of three and we all shout the phrase at once. And all my classes delight in realizing how easy it is to drive me off-topic by asking about food or politics.

All my ability to deliberately set a classroom environment came from the lucky break of teaching geometry to some of the same students I’d just passed in algebra.

Coverage vs. Comprehension


I sure hope Bud Blake got credit for this 1974 classic, reproduced daily in ed school and professional development lessons everywhere.

I used to take state tests more seriously, and was quite proud that my first year out, I “hit the dinger” in geometry and algebra. I hadn’t rushed, and even back then had deemed many topics non-essential, or at least far less important than others. My students were doing reasonably well on tests, which were free-response that year.

But towards the end of the year, I realized with a shock that many of my mid-tier students had forgotten most of the content. Students who understood the Pythagorean Theorem were now marking up triangles with SOHCAHTOA when they had two sides and just needed the third. Algebra students were plugging linear equations into the quadratic formula. Cats were sleeping with dogs. All was not right. It was as if they’d never been taught.

Year two, I was primed to look for learning loss but pacing was so impossible with the wide ability range that I instituted four levels of differentiation. I succeeded in slowing down instruction and letting students absorb more information.

But year three saw my first attempts to help Stripe learn to whistle.

In geometry, the first sign of change came in October. I’d explained transversals of parallel lines. I’d done a great job. Brilliant, even. Not content to simply lecture, I asked questions, prompted discussion, ensured students saw the connection and sketched the familiar representation.

And the lesson didn’t thud. All the students obediently worked the problem set. They asked reasonable questions.

So I don’t know, really, what compelled me to double check.

“Am I picking up a weird vibe? You all are working, but I have this sense that you’re still confused.”

Murmurs of agreement.

“How about everyone close their eyes and we’ll do a thumb check?” (I rarely use such obvious CFUs these days, but they’re still a great tool for uncertain situations.)

Most of the thumbs came up sideways.

So I told the kids I’d think about this for a while, and came back with an activity, one that required about $70 in materials that I still use to this day.



It worked. The transversal angle relationships were easier to understand with the physical representation, the students could see the inevitability, see how the angles “fit”. And from that point, they could easily see that unless the transversal was perpendicular, each transversal over parallel lines formed only two distinct angle measurements: an obtuse and an acute.

A nifty transversal lesson wasn’t the important development, even though my geometry students still enjoy the activity almost as much as they enjoy creating madcap patterns with the boards and rubber bands.

Sensing confusion despite a generally successful lesson, I had developed an illustration on my own to develop a stronger understanding. I was beginning to spot the difference between teaching and learning.

I still struggle with this. It’s very easy to get sloppy, particularly in a large class with ability ranges of 4 to 5 years, with kids in the lower ranges happily sleeping through classes, stirring themselves only enough to beg me for a passing grade. But ultimately, I circle back with yet one more pass through, coming up with an illustration or series of problems to shine a light on confusion.

I’ve written extensively of Year 3’s other major development. Faced with the reality that I’d wasted a semester covering linear equations and quadratics that students didn’t remember in the slightest, I decided to start over, beginning with modeling linear equations. Not only did I completely change my approach to curriculum, I also flatly punted on coverage from that point on, focusing on the big five for every subject. As I improve at introducing and explaining concepts, my students become capable of taking on more challenging topics; the interaction between my curriculum and student understanding is very much a positive feedback loop.

Ironically, my decision to abandon coverage was driven in part because Algebra 2 was a terminal course, meaning it was to be offered only to remedial seniors, students were not expected or in fact allowed to take any other math course. For this reason, I felt free to craft my own course to focus purely on getting the students ready for college math. But at least half of my students were juniors, and most of them took pre-calc the next year. This was my first exposure to Algebra 2’s dual nature. More on that later.

Mentoring Colleagues

For my first two years, I had almost no contact with colleagues. Year 3, two new math teachers joined and we instantly hit it off. Went for coffee on late start mornings, beers after work. I was their resource; both of them found me far more helpful than their assigned mentors. I still meet up with both of them four or five times a year at least.

I left that year for my current school, and went over two years again without any real colleagues. I missed it. Having spent most of my professional life working without colleagues that liked lunch, beer, coffee, whatever, I can map out the exception eras, and treasure them. Last year I began mentoring, and now have lunch, coffee, whatever with them individually and together.

I’m not chummy enough, much less normal enough, to bond easily with other teachers. But I’m a good mentor, and that seems to be how I make friends as a teacher.

Finally, Year Three taught me how to cope with genuinely unfair treatment, which I haven’t often had to deal with. I never go into details about it, but while I wasn’t crazy about the school, I didn’t want to look for jobs again. Being a fifty year old teacher without a job is a Very Bad Thing—of course, take out the word “teacher” and it’s still true, probably more so.

On the other hand, while I wasn’t crazy about that school, I am very happy at this one. What do they call that, perspective?

On the Spring Valley High Incident

So the Spring Valley High School incident is yet another case of a teenager treating a cop like a teacher. This is, as always, a terrible idea.

I watch the video and wonder about the teacher. I wonder if he’s wondering what I’d wonder in his shoes. Teachers aren’t just focused on the recalcitrant girl who refuses to comply, who hits the police officer, who gets arrested. Teachers notice the girl directly behind the cop and the defiant kid, the one who wasn’t a troublemaker, was just sitting in class doing her work and nearly gets clobbered by the flipped over desk. Or the other kids trying not to watch–suggestion, I think, that the shocking events aren’t a common occurrence. Teachers notice that the kids are working with laptops and hope none of them fly off a desk into another student. (Teachers probably also notice the photographer’s test has many wrong answers. Occupational hazard.)

He’s got to be wondering, now and forever, if he could have prevented this. One time, a student in my class inexplicably left her $600 iPhone on her desk during a class activity that involved working at boards, and it disappeared, which required a call to the supervisors and a full class search. I told them who I suspected, then left because I didn’t want to know. When I came back, they’d detained the strongest student in the class–not for stealing the phone, which was never found. I have decided it’s better not to say why, but it was one of those things that lots of kids do in violation of policy because they’re unlikely to get caught. But if they get caught, it’s bad. (No, not drugs). He was suspended for the maximum time period and had to worry about more than that, although more was mostly scare talk.

The point is, I felt absolutely terrible. The student who left the phone out was careless and silly, the student who stole the phone was a criminal, the student who got suspended was knowingly in violation of a major school policy without the slightest thought for his long-term prospects. But if I’d just seen the damn phone on the desk, none of this would have happened.

So when I look at this video, like many if not most teachers, I’m not thinking about whether the girl deserved to be flipped about, because that’s the cop’s problem. I’m wondering was there anything that teacher could have done to avoid having the cop there in the first place.

Reports say that the student initiated the event by refusing to turn over a cell phone—also offered up is refusal to stop chewing gum, which I find unlikely. However, it’s clear the student was refusing several direct orders that began with the teacher and moved up through the administrator and the cop.

Defiance is a big deal in high school. It must not be tolerated. Tolerating open defiance is what leads to hopelessness, to out of control classrooms, to kids wandering around the halls, to screaming fights on a routine basis. Some teachers care about dress code, others about swearing, still others get bothered by tardies. But most teachers enforce, and most administrators support, a strong, absolute bulwark against outright defiance as an essential discipline element.

Let me put it this way: an angry student tells me to f*** off or worse, I’m likely to shrug it off if peace is restored. Get an apology later when things have settled. But if that student refuses to hand me a cell phone, or change seats, or put food away, I tell him he’ll be removed from class if he doesn’t comply. No compliance, I call the supervisor and have the kid removed. Instantly. Not something I spend more than 30 seconds of class time on, including writing up a referral.

At that point, the student will occasionally leave the classroom without waiting for the supervisor, which changes the charge from “defiance” to “leaving class without permission”. The rest of the time the supervisor comes, the kid leaves, comes back the next day, and next time I tell them to do something, they do it. Overwhelmingly, though, the kids just hand me the phone, put away the food, change seatswhen I ask, every so often pleading for a second chance which every so often I give. Otherwise, the incident is over. Just today I had three phones in my pocket for just one class, and four lunches on the table that had to wait until advisory was over because I don’t like eating in my classroom.

We have a school resource officer (SRO), but I’d call a supervisor for defiance, and I’ve never heard of a kid refusing to go with a supervisor. If there was a refusal, at a certain point the supervisor would call an administrator, and it’s conceivable, I guess, that the administrator could authorize the SRO to step in. So assuming I couldn’t have talked this student down, I would have done what the teacher did, and called for someone else to take over—and long after I did something that should have been no big deal, this catastrophe could conceivably have happened.

I ask you, readers, to consider the recalcitrance required to defy three or four levels of authority, to hold up a class for at least 10-15 minutes, to refuse even to leave the classroom to discuss whatever outrage the student feels warrants this level of disruption.

Then I ask you to consider what would happen if students constantly defied orders (couched as requests, of course) to turn over a cell phone, or change seats, or stop combing their hair, or put the food away. If every time a student defied an order, a long drawn-out battle going through three levels of authority ensued. School would rapidly become unmanageable.

So you have two choices at that point: let madness prevail, or be unflinching with open defiance. Students have to understand that defiance is worse than compliance, that once defiance has occurred, complying with a supervisor is a step up from being turned over to an administrator, which is way, way better than being turned over to a cop. (Note that all of this assumes that the parents aren’t a fear factor.)

Some schools can’t avoid the insanity. Their students simply don’t fear the outcomes enough, and unlike charters, they are bound by federal and state laws to educate all children. If the schools suspend too many kids, the feds will come in and force you into a voluntary agreement. This is when desperate times lead to desperate measures like restorative justice, where each incident leads to an endless yammer about feeeeeeeeelings as teachers play therapist and tell their kids to circle up.

Judge the cop as you will. I can see no excuse for putting other students in danger; the fight could have seriously injured the girl sitting directly behind the incident. He could have cleared the area first, making sure all students were safe. I believe that’s his responsibility.

However, once the administrator asked the SRO to take over, the student was dealing not with a school official, but with a cop. At that point, she was disobeying a police officer’s order. On government property. And she is clearly hitting him, in this video.

And, like I said, disobeying a cop is a bad idea.

So the question is not what should the cop have done, but why did the administrator call the cop? And what would you have had the administrator do instead? Don’t focus on that single incident, because teachers, administrators, and cops don’t have that luxury. They have to handle it in such a way so that defiance doesn’t become a regular routine, that students customarily obey their teachers, maybe with some backtalk, maybe with ample opportunities to walk a bad mistake back. Ultimately, though, students have to comply. If a school backs away from that line, defiance gets contagious. It’s one thing for new, inept teachers to have trouble controlling their students, quite another for an entire school to give up.

I recently had an exchange with David Leonhardt on his NAEP scores article, and he asked me “I assume you agree school quality should be linked to amount students learn, yes?”

Well, not the way we currently measure it, probably not. But I do think school quality should be linked to established order and by “order” I don’t mean an Eva Moskowitz gulag. Control freaks like Moskowitz fail to allow for normal mood swings and eruptions from kids who are, after all, engaged in an involuntary activity for eight hours a day.

Schools that fail to establish order are those like Normandy High School, with out of control violence, open defiance of teachers and administrators, and students in constant danger of assault. Students should have the opportunity to learn, even if they aren’t mastering material at the rate our policy wonks would allow. Schools that can’t enable that are genuine failures.

The Moskowitz contingent point to schools like Normandy as rationale for their despotic rules. Look, they say. Let “these kids” think they can act however they like, and you end up with screaming, chaotic classrooms, truancy, assaults and fights on and between students, ineffectual teachers, and worst of all, low test scores. Teach them to behave respectfully, five times more compliant than suburban white kids, and you’re doing them a favor, saving them from “those schools”. Better Animal Farm than Clockwork Orange

Any school with a solid percentage of kids who’d really rather be somewhere else has to find a balance. Make enough kids want to comply so there’s room to expel the kids who routinely don’t. This isn’t achieved by Eva Moskowitz tyranny, but nor will restorative justice get the job done. It’s hard. There has to be limits. There has to be balance. Administrators who think they have the perfect mix are probably kidding themselves.

In the meantime, if, like Martin O’Malley or Chris Hayes, you’d be “ripped ballistic” if a cop did this to your kid, familiarize yourself and your children with the dangers of disobeying a cop and resisting arrest.

Handling the Teacher Perks

Before turning teacher, I spent all but five years as a temp worker, self-employed or contract. Unemployment? A hassle I didn’t bother with the few times I was eligible. Retirement? My very own funded SEP_IRA, no employer matching. Paid vacation and sick leave? Outside of those five years, I never had any.

Going from that life to public school teaching was kind of like Neal Stephenson’s description (excerpted from In the Beginning was the Command Line) of the guy who was raised by carpenters from early childhood with only a Hole Hawg as a drill and then meeting up finally with a puny homeowner’s version.

What the hell. With so much free stuff, how can you call this work?

From Veteran’s Day to the first week of the New Year, over three weeks off, the bulk of them from mid-December to early January. Five plus days off at spring break, and two months off in the summer. Eleven days of sick leave that accrue, and two “use it or lose it” days. I get the same amount of pay every single month. Guaranteed pension, already vested comfortably, probably to retire with 30%—not bad for a late entry.

Plus, I hear it’s hard to get fired.

I clearly remember watching the perks of corporate employment slowly be stripped away back in my twenties, perks that few people under 50 can even imagine. So it’s bizarre to have entered a profession where it feels like the 80s again.

Now, I’m wondering if I’m getting used to it.

In the previous five years of teaching, my collective time out of the classroom was 3 sick days and 6 mandated professional development days. This year, I was out of class for nearly 10 days of professional obligations: three days for an honest to god, out of state, education conference, two-plus days for mentoring and induction responsibilities, and 4 days of Common Core testing.

I felt very guilty about all this time off, and without question the absences impacted instruction time and coverage. So much so that when I came down with a really severe case of with food poisoning (you know those rotisserie chickens? Used to love them. Hope I eventually trust them again) during testing week, I came in anyway because I knew it would wreak havoc both on testing schedules for administration and my carefully scheduled coverage plans (I was missing alternate classes during the week). I went four days munching crackers and chugging that weird chalky pink stuff, previously unknown to me.

In retrospect this struck me as idiotic, so I went to the principal’s secretary and asked how to request time off. That’s when I learned formally I had 13 days a year, including two use or lose–which I’ve been losing for the past five years. I took a whole day and a half just for a family graduation 10 hours away, when I normally would have left Friday afternoon and come back Sunday night.

More evidence: for the first time in eight summers, six of them as a teacher, I decided to forego employment (part-time and no benefits, of course) at my favorite hagwon, where I usually act as chief lunatic for book club, PSAT prep, and occasionally geometry.

Why? I wanted more time off.

This wasn’t a sudden decision. Last year it finally sunk in that despite the easy hours and students, the elapsed time of my hagwon day clocked in at 9 hours: three on, three off, three on, for eight weeks. While this hadn’t seemed punitive with a 5 minute commute, the schedule lost much of its charm when I moved 45 minutes away. Meanwhile, the eight week schedule left just eight uninterrupted days off at the end of summer.

Yes. The four weeks I am granted throughout the year is not enough. I want more of the eight uninterrupted weeks. It shames me.

But there’s hope. If eight days seemed too little, two months off seemed….excessive. Years of temp work leaves me never entirely comfortable not knowing where my next dollar would come from. Long vacations make me nervous. Back in my tutor/test prep instructor life, my son and I took a long road trip one summer that culminated in a 6 week stay in another city. I notified a local Kaplan branch, got some SAT classes, put ads in Craigslist and got some private tutoring, making enough to offset the fuel and food expenditures for the trip.

I am not yet ready to abandon summer work altogether. I wanted a summer job. Just a different one, with a shorter work day, a shorter employment term, and higher hourly pay so I’d get more time off but the same dollars’ pay.

Normal people are thinking “Hah! And a pony.” Teachers are thinking “Duh. Just teach summer school.” Public summer school, that is. Six weeks at most in my area, higher hourly pay, out at 1:30.

I have very strong feelings about summer school, none of them positive. But public summer school it is, this summer. More of that later, assuming I can push through and finish this absurdly non-essential piece because family fun time and work are coming perilously close to giving me writer’s block.

As a side note, a transition marked: I’ve now left all three legs of my previous income behind. Private tutoring mostly gone over the past two years, the hagwon this last year, Kaplan since ed school.

A job change to get a longer summer break. Another worrisome trend?

But then, just when I began to worry about having been slowly sucked in, I learned what my preps for the upcoming year would be and nearly had a meltdown.

Every year, teachers are given a form to list their preferences for subject assignments (aka, “preps”). Every year, my form says “I’m happy to teach any academic subject I’ve got a credential for–but please don’t limit me to one prep a semester. Two is better, three is best.” Then I list three classes I haven’t taught in a while, or would like to do a second time. This year, I’d asked to teach at least one session of history, to build on my last year, pre-calc, which I hadn’t taught in a year, and any lower level class, just to keep myself humble. Again, this is in the context of teaching any other class as well.

I went into school after summer started to work on one of the professional obligations above, and as a thank-you, the principal showed us the master schedule board.

Semester One: Algebra 2, Trig. Two blocks of each.
Semester Two: Algebra 2, Trig. Three blocks total, two blocks Trig.

This schedule would be, to most teachers, a perk. Just two preps I’m familiar with. An easy year, after an extraordinarily demanding one in which I had two brand new classes, one of which was in a completely different academic subject for the first time in five years. Some might view the schedule as a form of thank-you, or maybe an acknowledgement that I’ve got more professional responsibilities so require a schedule with less planning or curriculum development.

I looked at the board and thought Christ, I have to quit this school, that’s awful, I love this school, but I have to get out of here. I need some time for job-hunting. I can’t quit summer school, it starts Monday. But I can jobhunt in the afternoons, it’s a Friday so I have some time to update my resume. Maybe I won’t have to leave the district, so I could keep tenure, and maybe I can talk to the administrator at summer school, hey, it’s actually good that I’m not at the hagwon this year, I just need to update my resume….

So not a perk, to me.

I tend towards extreme reactions, as alert readers may have noticed. Self-knowledge has led to compensatory braking systems. In years past, I would have just turned in my resignation on the spot. But my braking system kicked in, I remembered that quitting is just a symptom of my temporary worker mindset. I reminded myself how good it felt to get tenure, that my administration team likes me. Before I quit, I should perhaps consider other alternatives.

I will cover those alternatives, and my fears, in a follow-up post. No really, I promise.

So no, I’m not yet sucked in by the teacher perks. But I do want more free time during my 10 weeks off. Call me ungrateful.


Note: I will always value intellectual challenge over predictability for my own job satisfaction. But many teachers do an outstanding job teaching just one subject or the same two preps for thirty years. Outsiders, particularly well-educated folks with elite pedigrees, champion intellectual curious teachers with cognitive ability to spare as an obvious advancement over what they see as the “factory model” teacher turning out the same widgets ever year. But little evidence suggests that intellectual chops produces better results, much less better teachers. So please don’t interpret my rejection of predictability and routine as evidence of anything other than a fear of boredom.

Math isn’t Aspirin. Neither is Teaching.

First, congrats to Dan Meyer, who finished his doctorate at Stanford and just hired on as CAO for Desmos, a tremendously useful online graphing calculator. He persisted in the face of threatened failure, and didn’t give up even when he had an easy out into a great job. (Presumably Dan and most of the Math Twitter Blogosphere are still annoyed at my jeremiad about the meaning of his meteoric rise, in which Dan played the part of illustration.)

Dan has asked math teachers for ways to create “headaches” for which math can be considered aspirin:


And this interested me because the request completely, perfectly, captures the difference between our two philosophies, which I also wrote about a couple years ago:


The comparison is an instructive one, I think. Both of us find it necessary to build our own curriculum, rejecting the one on offer, and both of us, I think, tremendously enjoy the creation process. Both of us reject the typical didactic contract described by Guy Brouseau, setting expectations very different from those of typical math teachers: explain, work a few examples, assign a set. Both of us largely eschew textbooks for instruction, although I consider them completely unnecessary save as reference books that often provide interesting problems I can steal, while Dan dreams of the perfect digital textbook.

And yet we couldn’t differ more in both teaching philosophy and curriculum approach.

Dan’s still selling curiosity and desire for knowledge, assuming capability will follow. I’m still selling capability because I see confidence follow.

Dan still believes that student engagement captures their curiosity which leads to academic success, that the Holy Grail of academic success in math lies in finding the perfect problems that universally stimulate interest in finding answers, which leads to understanding for all. I hold that student engagement leads to their willingness to attempt what they previously thought was impossible but that the Holy Grail doesn’t exist.

Meyer thinks teachers skeptical of his methods are resistant to change and the best interests of their students. I advise teachers and recommend curriculum; if they find my advice helpful, great. I encourage them to modify or even reject my advice, to continue to see an approach that works for them and their students.

Dan wants to be “less helpful”. I want to teach kids to use their own resources, but given a kid who wants to give up, I’m offering help every time.

Meyer’s methods would probably need tremendous readjustment if he worked in a low-income school with a wide range of abilities. I’d probably be much “less helpful” if I taught at a school with a high-achieving, homogenous population obsessed about grades.

Meyer rose quickly in the rarefied world of rock star teachers. I aspire to the role of and indie with cult status.

Dan’s query: “Why did mathematicians think this skill was worth even a little bit of our time? If the ability to factor that trinomial is aspirin for a mathematician, then how do we create the headache?

My answer: You can’t.

The commenters, mostly teachers, took the question seriously, understanding that it was another way of looking at the students’ demand, “When will we use this?”. Answering this question clearly troubles most of the commenters—or they have an affirmative answer they’re satisfied with.

My answer to the student demand: “Probably never. But the more willing you are to take on challenging tasks you learn from, the more opportunities you’ll have in life, both professional and personal. Call me crazy, but I see this as a good thing.”

Dan Meyer is wrong, I believe, in looking for the Holy Grail that makes math “aspirin”1. But that’s not the point of my running through the Dan vs. Ed showdown.

Instead, consider the comparison yet another data point in my slowly developing thesis that ed schools need more flexibility and even less prescription. Few people understand the vast scale of values, philosophies, management and curriculum found in the teaching population.

Two teachers developing uncommon curriculum who agree on very little—yet both of us are considered successful teachers. (one has much more success selling his ideas to people with money, I grant you.) Take ten more math teachers likewise who build their own curriculum, have their own takes on philosophy, discipline, and even grading and they’re unlikely to change to suit another model. Take 100 more–ditto. Voila! an expanding population of teachers who have successful teaching approaches and curriculum design that they’ve developed and modified. None of them are going to agree on much. They have come to widely varying conclusions that they will continue to develop and enhance on their timeline as they see fit. No one will have anything approaching a convincing argument that could possibly convince them otherwise.

The point: the current push to “fix” ed schools, a fond delusion of reformers, progressives and union leaders alike. People as diverse as Benjamin Riley, Paul Bruno, Rick Hess and others believe we can find (or already have) a teaching knowledge base that can be passed on to novices.

Teachers are never going to agree.

Agreement or even consensus is impossible. Teachers and students form infinite combinations of interests, values, incentives and unlike reformers, teachers are going to value their experience and unique circumstances over anything ed schools tried to pretend was the only way.

Teaching, like math, isn’t aspirin. It’s not medicine. It’s not a cure. It is an art enhanced by skills appropriate to the situation and medium, that will achieve all outcomes including success and failure based on complex interactions between the teachers and their audience. Treat it as a medicine, mandate a particular course of treatment, and hundreds of thousands of teachers will simply refuse to comply because it won’t cure the challenges and opportunities they face.

So when the status quo has prevailed for the next 30 years, don’t say you weren’t warned.

1which isn’t to say I don’t plan on writing up the how and why of my quadratic equations section.

Grant Wiggins

Curriculum is the least understood of the reform efforts, even though parents have more day to day contact with curriculum than choice or accountability. This is in large part because curriculum advocates don’t agree to the degree that accountability and choice reformers do, but also because teachers have far more control over curriculum than most understand. As Larry Cuban explains, curriculum has multiple layers: intended, tested, taught, and learned. Curriculum battles usually involve the intended curriculum, the one designed by the state, which usually creates the tested curriculum as a manageable subset. (Much of the Common Core controversy is caused by the overwhelming difficulty of the tested curriculum, but leave that for another time.)

But intended and tested curriculum are irrelevant once the doors close, and in this essay, I refer to the taught curriculum, the one that we teachers sculpt, whether we use “the book” (actually just pieces of the district approved book), use another book we like better, or build our own.

To the extent most non-educators know anything about curriculum advocacy, it begins and ends with E. D. Hirsch, otherwise known as “the guy who says what my nth grader should know”, author of a book series he eventually transformed into a curriculum for k-6, Core Knowledge. Hirsch offers one Big Idea: improving student background knowledge will improve their reading comprehension, because only with background knowledge can students learn from text. But, the Idea continues, schools ignore content knowledge in favor of teaching students “skills”. To improve reading comprehension and ongoing student academic outcomes, schools must shift from a skills approach to one dedicated to improving knowledge.

Then there’s Grant Wiggins, whose death last week occasioned this essay as an attempt to explain that we’ve lost a giant.

The media proper didn’t give Wiggins’ passing much notice. Valerie Strauss gave his last blog sequence a good sendoff and Edutopia brought back all their interviews with him. Education World and Education Week gave him obits. It doesn’t look as if Real Clear Education noted his passing, which is a bit shocking but perhaps I missed the mention.

Inside education schools, that world reformers hold in considerable contempt, Wiggins’ work is incredibly influential and his death sent off shockwaves. Since 1998, Understanding by Design has been an essential component in preparing teachers for the professional challenge of deciding what to teach and how to deliver the instruction.

Prospective teachers don’t always understand this preparation will have relevance to their lives until their first year in the classroom. Progressive ed schools would never say anything so directly as “You will be faced with 30 kids with an 8 year range in ability and the textbooks won’t work.” Their ideology demands they wrap this message up in hooha on how insensitive textbooks are to the diverse needs of the classroom. Then, their ideology influences the examples and tasks they choose for instruction. Teacher candidates with an instructivist bent thus often tune out curriculum development classes in ed school, rolling their eyes at the absurd examples and thinking keerist, just use the textbook. (Yeah. This was me.)

Usually, they figure out the relevance of curriculum instruction when they get into the classroom, when they realize how laughably inadequate the textbook is for the wide range of abilities and interests of their students. When they realize the book assumes kids will sit patiently and listen, then obediently practice. When they realize that most of the kids won’t bring their books, and that all the well-intended advice about giving consequences for unprepared students will alone result in failing half the class, never mind the problems with their ability. When they realize that many kids have checked out, either actively misbehaving or passively sitting. Worst of all the teachers experience the kids who are eager to learn, try hard, don’t get it, and don’t remember anyway. Then, even after they make a bunch of adjustments, these teachers realize that kids who do seem to be learning don’t remember much—that is, in Cuban’s paradigm, the learned curriculum is wildly different than the one taught (or in the Wiggins universe, “transferred”).

The teachers who don’t quit or move to charters or comprehensives with a higher SES may remember vaguely hey, there was something about this in ed school (hell, maybe that’s just me). So they go dig up their readers and textbooks and suddenly, all the twaddle about diversity and cultural imperialism fades away and the real message becomes legible, like developing invisible ink. How do you create a learning unit? What are your objectives? How will you assess student learning? And at that point, many roads lead to Wiggins.

Grant Wiggins was impossible to pigeonhole in a reform typology. In 1988, he made 10 proposals for high school reform that leaned progressive but that everyone could find some agreement with. He didn’t think much of lecturing, but he wrote a really terrific analysis of lectures that should be required reading for all teachers. (While I also liked Harry Webb’s rejoinder, I reread them in preparation for this essay and Grant’s is far superior.) He approved of Common Core’s ELA standards, but found the math ones weak. In the space of two weeks in 2013, he took on both Diane Ravitch and E. D. Hirsch, and this is after Ravitch flipped on Hirsch and other traditionalists.

Grant Wiggins was more than ready to mix it up. Both his essays on Hirsch and Ravitch might fairly be called broadsides, although backed with research and logic that made both compelling, (perhaps that’s because I largely agreed with them). His last two posts dissected Hirsch supporter Dan Willingham’s op-ed on reading strategies. While he listened and watched teachers intently, he would readily disagree with them and was rarely gentle in pointing it out. I found his insights on curriculum and instruction absolutely fascinating, but rolled my eyes hard at his more excessive plaints on behalf of students, like the nonsense on apartheid bathrooms and the shadowing experience that supposedly revealed the terrible lives of high school students—and if teachers were all denied the right to sarcasm, as he would have it, I’d quit. He didn’t hesitate to say I didn’t understand the lives that students lead, and I told him right back that he was wrong. More troubling to me was his conviction that most teachers were derelict in their duty and his belief that teachers are responsible for low test scores. But what made him so compelling, I think, is that he offered value to all teachers on a wide range of topics near to our needs, whether or not we shared all his opinions.

I knew him slightly. He once linked to my essay on math philosophies as an example of a “learned” teacher, and read my extended response (do I have any other kind?) and took the time to answer. Then, a few months later, I responded to his post on “teacher job descriptions” with a comment he found worthy of pulling out for a post on planning. He then privately emailed to let me know he’d used my comment and asked me to give feedback on his survey. That was a very big day. Like, I told my folks about it.

In the last week of his life, Grant had asked Robert Pondiscio to read his Willingham critique. Pondiscio, a passionate advocate of all things content knowledge, dismissed this overture and declared his posts on both Willingham and Hirsch “intemperate”. Benjamin Riley of Deans for Impact broke in, complimenting Grant and encouraging the idea of debate. The next day, Daniel Willingham responded to Grant on his site (I would be unsurprised to learn that Riley had something to do with that, and kudos to him if so). Grant was clearly pleased to be hashing the issues out directly and they exchanged a series of comments.

I had been retweeting the conversation and adding comments. Grant agreed with my observation that Core Knowledge advocates are (wrongly) treated as neutral experts.

On the last day of his life, Grant favorited a few of these tweets, I think because he realized I understood both his frustration at the silence and his delight at finally engaging Dan in debate.

And then Grant Wiggins died suddenly, shockingly. He’ll will never finish that conversation with Dan Willingham. Death, clearly, has no respect for the demands of social media discourse.

Dan Willingham tweeted his respect. Robert Pondiscio wrote an appreciation, expressing regret for his abruptness. If the general media ignored Grant’s passing, Twitter did not.

I didn’t know Grant well enough to provide personal insights. But I’m an educator, and so I will try to educate people, make them aware of who was lost, and what he had to offer.

Novices can find plenty of vidoes on his “backwards design” with a simple google. But his discussions on learning and assessment are probably more interesting to the general audience and teachers alike—and my favorites as well.

Reformers like Michael Petrilli are experiencing a significant backlash to their causes. Petrilli isn’t wrong about the need for parent buy-in, but as Rick Hess recently wrote, the talkers in education policy are simply uninterested in what the “doers” have to offer the conversation.

Amen to that. The best education policy advocates—Wiggins, Larry Cuban, Tom Loveless–have all spent significant time as teachers. Grant Wiggins set an example reformers could follow as someone who could criticize teachers, rightly or wrongly, and be heard because he listened. If he disagreed, he’d either cite evidence or argue values. So while he genuinely believed that most teachers were inadequate, teachers who engaged with him instantly knew this guy understood their world, and were more likely to listen.

And for the teachers that Grant found inadequate—well, I will always think him in error about the responsibility teachers own for academic outcomes. But teachers should stretch and challenge themselves. I encourage all teachers to look for ways to increase engagement, rigor, and learning, and I can think of no better starting point than Grant Wiggins’ blog.

I will honor his memory by reading his work regularly and looking for new insights to bring to both my teaching and writing.

If there’s an afterlife, I’m sure Grant is currently explaining to God how the world would have turned out better if he’d had started with the assessment and worked backwards. It would have taken longer than seven days, though.

My sincere condolences to his wife, four children, two grandsons, his long-time colleague Jay McTighe, his band the Hazbins, and the many people who were privileged to know him well. But even out here on the outskirts of Grant’s galaxy of influence, he’ll be sorely missed.

Group Work vs. Working In Groups

I sit my kids in groups. But I don’t like “group work”.

No, that’s not a paradox. Sitting in groups isn’t “group work”.

Group work is an activity that falls under the larger rubric of “collaborative learning”, an organizing bubble to collect techniques and strategies like “Think Pair Share”, jigsawing, peer tutoring, and the like. The most fully-realized collaborative learning pedagogy is probably complex instruction, which was developed by Elizabeth Cohen. (That’s CI, not CISC.) To illustrate, CPM curriculum is based on complex instruction, whereas Everyday Math is not.

Complex Instruction had been in development for over 20 years, by the time it caught on  in the early 90s. Jeannie Oakes’ book Keeping Track, a broadside against any sort of ability grouping.  Oakes accused parents and schools of racial discrimination, an argument that found favor with many schools and teachers. Those schools that weren’t favorable to the argument faced lawsuits or the threat of one. A good chunk of the 90s was wasted as districts and states desperately tried to win her approval, and adopting the CI method was often adopted as the strategy. Fortunately, they all ultimately learned it was easier to disappoint her.1

Complex Instruction was also developed by tracking opponents, but opponents who nonetheless cared about learning. It’s explicitly designed to give schools a tool for the havoc that results when kids with a 3 to 8 year range in abilities are put in the same room, and thus was grabbed at by many schools back in the early 90s. Many CI concepts are also found in “reform math”—Jo Boaler’s Railside study on San Lorenzo High School was all about Complex Instruction. Carlos Cabana and Estelle Woodbury, who just co-authored Mathematics for Equity, a book on teaching math with Complex Instruction, both worked at San Lorenzo High School during Boaler’s study.

So start with the theory, articulated here by Rachel Lotan, the late Cohen’s key associate. You should watch this, or at least fast forward through parts, because Lotan clearly articulates the admirable goals of complex instruction minus the castigation, blame, and fuming ideology. Or, Complex Instruction’s major components in written form:


Both Lotan and the writeup offer much that is problematic. Reducing the ability range: not good. Creating busywork tasks (writing down questions, getting supplies) to let everyone feel “smart”: not good.

The write up mentions “status problems”. Lotan gives a great account of an absurdly pretentious term, “mitigating status” that is something every teacher in every classroom–no matter how they are seated—should take seriously. Lotan does a better job of explaining it, but since many won’t listen to the video, here’s a written version:

CI targets equity and, in particular, three ideas: first, that all students are smart; second, that issues of status—who is perceived as smart and who is not—interfere with students’ participation and learning; and third, that it is teachers’ responsibility to provide all students with opportunities to reveal how they are smart and develop/recognize new ways of being smart. The complex instruction model aims to “disrupt typical hierarchies of who is ‘smart’ and who is not” (Sapon-Shevin, 2004) by promoting equal status interactions amongst students so that they engage with tasks that have high cognitive demand within a cooperative learning environment.

(emphasis mine)

Ed schools wanting to hammer home how putting kids in groups doesn’t by itself address status usually show this video, but brace yourself. I tell myself that the ignored kid is probably a pest all the time, that everyone in the class is tired of his nonsense, that we’re just seeing a carefully culled selection to maximize the impact of exclusion and of course, race. It doesn’t matter. It’s still hard to watch.

And the video also reinforces the practical message that CI advocates are pushing, as opposed to the theory. In theory, status can be unearned by anyone of any gender or color. In practice, most CI advocates expect teachers to shut down white males. In theory, kids learn that everyone is smart. In practice, kids still know who’s “smart” and who’s not.

But then, CI advocates have their own frustrations. In theory, they’d put teachers in PD designed to indoctrinate them into realizing the error of their racist ways. In practice, teachers who haven’t already drunk the Koolaid either politely fake it until they can find an exit or get really annoyed when they’re called racists, as an excerpt for Mathematics for Equity makes clear:

Cite: Mathematics for Equity1

Complex Instruction done well is pretty interesting and often thought-provoking. Cathy Humphreys is a long-time advocate of “reform math” and complex instruction. She was at the center of one of those “rich educated parents” meltdowns that you saw over reform math back in the 90s. Humphreys represented the reform side, of course, and further endeared herself to parents by proposing to get rid of tracking at a Palo Alto, CA middle school. That went over like a water balloon down a balcony, she quit, worked as a math coach for a while, and then taught for years at a diverse high school in the Bay Area that had ended tracking. She also teaches at Stanford’s education program, according to her bio. Carlos Cabana, one of the co-authors of Mathematics for Equity, has also been teaching complex instruction for a long time; he’s one of the teachers at Railside, Jo Boaler’s pseudonym for San Lorenzo High School.

You can see both Humphreys and Cabana here at a website put together by the Noyce Foundation to promote the 8 essential practices. (Notice the link between “reform math” and supporting “common core”? As Tom Loveless says, Common Core is a “dog whistle” for reform math. Humphreys and Cabana are teaching high school math in the videos. You can also see Humphreys teaching at what I assume is the middle school that melted down. Humphreys and Cabana are much better demonstrations of complex instruction than the absurdly flashy promos that Jo Boaler puts out.

When I began teaching, I thought sitting kids in groups was absurd. I remember being pleased one of my mentoring teachers put kids in rows. But my primary student teaching assignment required me to sit kids in groups, as we were using CPM, a reform text that requires groups. I adjusted and liked it much more than I thought I would, especially when I took over the class and could group by ability. But my first year out, I happily put my desks in rows, thinking that groups were good, but now I could finally run my class the way I wanted.

Four weeks later, I put the kids in groups. It just….felt better. Year 2, I was teaching all-algebra, all the time, and thought rows would make more sense. The rows lasted 2 weeks and since around September of 2010, the only time my kids sit in rows is for tests.

I have….mixed feelings about CI. When promoted by the fanatic adherents, it’s both Orwellian and despicable. Teachers have to squelch kids who know the answer, force kids who understand the concept to explain, endlessly, to the kids who don’t, and then grade the kids who know the answer not on their demonstrated knowledge but on the success of their explanation and their willingness to do so. Teachers have to pretend to their students that asking a good question or taking notes is just as important as understanding the math (no, say the fanatic adherents, teachers aren’t pretending. These tasks are just as important!).

But while no student is ever going to believe that everyone is smart, “issues of status” do absolutely impact a students’ willingness to participate. Let the “smart kids” talk, everyone thinks, and sits back and zones out.

However, in my opinion and experience, CI methods often achieve exactly what they are defined to avoid, precisely because of their Orwellian insistence on ignoring reality. Kids know who is smart. They shut down if the smart kid is in their group, and go through the motions when the teacher walks by.

Ironically, I “mitigate status” by violating Complex Instruction’s most sacred tenet. Complex Instruction holds that student groups must be heterogeneous. Organization can’t be based on the rigid, academic version of “smart”. But I group my kids by ability as the most effective way of “mitigating status”.

I don’t want the weakest students in my class feeling as if any success short of an “A” is irrelevant. I also don’t want to try and convince them they’re just as “smart” as students who don’t struggle with the same material. That way, my students know that they can talk about math, what they need to know, what questions they have, knowing that other students probably have similar issues.

I don’t want to make it sound as if “mitigating status” is the only reason I sit kids in groups. Groups allow me to differentiate tasks slightly (or extensively) and enables me to quickly give help or new tasks. Groups allow kids to work together, discussing math, developing at their own speed with peers who have similar abilities.

But whether it’s status or some other curricular reason, when I sit them in groups, they start working and talking about math. They discover they are working with peers who won’t make them feel stupid, and they start to have discussions. Should we do this or this? They call me over to adjudicate. They try things. They check their notes, engage in all those excellent student behaviors. Not always, of course. But many times. They are less likely to sit passively and wait until I come by to personally tutor them through problems.

Moreover, because they are working with students of their own ability, they don’t feel alone or stupid. They work to improve. Maybe not great, maybe not good. But better.

Sitting kids in groups is not group work. But sitting kids in groups based on ability and giving them achievable tasks makes them more likely to work, and as math teachers often know, that’s no small thing.

1 I was thinking crap, I don’t want to have to look up the whole history of the ebb and flow of tracking and then went hey, Tom Loveless has to have something on this and by golly he does: The Resurgence of Ability Grouping and Persistence of Tracking covers the whole era, Oakes included. I would only quibble slightly with this sentence: Although the call to detrack was not accompanied by conventional incentives—the big budgets, regulatory regimes, and rewards and sanctions that draw the attention of policy analysts—detracking was, in a field famous for ignored or subverted policies, adopted by a large number of schools.

Loveless appears to forget the biggest incentive of all: lawsuit avoidance. Detracking lawsuits were the rage in this time period. Unlike new curriculum or teaching styles, detracking is achieved by executive fiat by district superintendents. No training, no carrots needed. Shazam! But leaving aside that minor quibble, a great piece documenting the move to and then the move away from heterogeneous classrooms (de-tracked).

Teaching: My Retrospective

Okay, I’m rolling along on my task of drawing clear lines of demarcation between my particular brand of squish and traditional progressive education (heh–traditional progressive. Get it?). First up was my new no homework policy.

I then decided to take on sitting my kids in groups (as opposed to group work), which led me to look back at some old post, which forced me to look back at my practice over the years, and that’s been a trip. So much of a trip that I decided to do the retrospective first.

The introspection kicked off when I reread one of the first posts I ever wrote on this site, over 3 years ago, halfway through my third year of teaching. Some key observations:

  1. I focused almost entirely on classwork, even then. The essay doesn’t even mention homework which, at that time, I assigned in much the way I describe in my last essay.
  2. At that time, the school I worked at used a traditional schedule of 60 minute classes, so the 3 day span per lesson is about two days at my current school. Additional evidence I was focused primarily on what kids learned in class, although as I said, my original homework policy goes back even further than this post.
  3. Here’s a real change. Me on low ability students three years ago: lowabilstds3yrs
    I’m so cheered to realize how much I’ve improved. I had good student engagement back then, but in rereading this I can remember how many students I had to nudge endlessly, how I had to constantly pick up pencils and hand them to kids to get them to work. Recall I was teaching algebra and geometry, and had just begun what is now my bread and butter class of Algebra 2. So my experience at the time of writing those words was with a lower level of math class, which will always mean lower engagement. Nonetheless, that simple paragraphs reminds me of the struggles I had to get total engagement. I’ve come a long way. Yay, me.

  4. Interesting to see my off-hand mention of EDI. No one seeing my teaching would think of me as using the direct instruction mode, but in fact I always, at some point, give kids specific, explicit instructions on the concept at hand.
  5. While I talked about differentiation and my need to challenge top students, I have actually moved away from different assessments for different students. At that time, I was just three months of teaching out from year two, all-algebra I-all-the-time, and I basically taught 4 different classes. I’d tentatively planned on continuing this approach, but learned that year (and confirmed in later years) that this wouldn’t work for any class but algebra I.

I wrote this post on January 8, 2012, at almost exactly the same time I began an experiment that utterly transformed my teaching. I speak, of course, of Modeling Linear Equations, which I’m amazed to realize I wrote just one week after the “How I Teach” post. So shortly after I began this blog and described my teaching method, I started on a path that took my existing teaching approach–which was pretty good, I think–and gave it a form and shape that has allowed me to grow and progress even further.

I haven’t really read this post in over two years—I tend to link in Modeling Linear Equations, Part 3, written a year later (two years ago today!), when I’d realized how much my teaching had changed. So reading the original is instructive. I talk about the Christmas Mull, something that stands very large in my memory but don’t remember quite as described here:


The part that’s consistent with my memory: Christmas 2011, I was depressed by the dismal finals in my three algebra II classes. In the first semester, I had gone through all of linear and quadratic equations, including complex numbers, at a rate considerably slower than two colleagues also teaching the course. Yet the kids remembered next to nothing. Every single person failed the multiple choice test–the top students had around half right. I had experienced knowledge fall-offs in algebra and geometry, but nothing that had so sublimely illustrated how much time I’d wasted in three months. So I came out of the Christmas break determined to reteach linear and quadratic equations, because to continue on teaching more advanced topics with these numbers was purely insane. And I wasn’t just going to reteach, but come up with an entirely different, less structured approach that allowed my students to use their own understanding of real-life situations.

What I hadn’t remembered until reading this closely was my rationale for ignoring the regular curriculum requrements. At the time, Algebra 2 was considered a “terminal” class; students weren’t expected to take another course in the college-prep sequence. This has changed, of course–these days, algebra 2/trig is, if anything, experiencing a fall-off in favor of a full year of each course. But at the time, I justified my decision to go off-curriculum based on the student needs. These students’ primary concern, whether they knew it or not, was what happened to them in college. How much remediation were they going to need? Could the best of them escape any remedial work and go straight onto credit bearing courses? This, of course, still remains my priority–I’d just forgotten how linked it was to my initial decision to try something new.

Also interesting that I described this approach by the specific method I used for linear equations–using “inherent math ability”. That’s not how I describe my approach these days, but I can see the germination of the idea. At the time I wrote this, I had no idea I would go beyond linear equations and use this approach consistently throughout my instruction.

I think the best description I’ve come up with for my approach is modified instructivist, which comes in one of two forms: “highly structured instructivist discovery, and classroom discussions with lots of student involvement”.

As for the latter: I don’t lecture, with or without powerpoints. When I do explanations, they are classroom discussions, and you can see this demonstrated in all my pedagogy posts. However, I am constantly migrating my classroom discussions to structured discovery.

What’s structured discovery? Imagine a teacher and students on a cliff, with a beach below. There’s a path, but it’s not visible.

In a traditional lecture or classroom discussion, the teacher shows them the path and leads them down to the beach.

In a discovery class, the teacher doesn’t even tell them there’s a path or even a beach. In fact, to the discovery/reform teacher, it doesn’t matter whether there’s a path or not—the kids will all find their own way down. Or maybe they’ll just find some really cool flowers and stop to examine their biology. Or maybe they’ll just kick back and have a picnic. It’s all good, in reform math. (sez the skeptic)

In what I call structured discovery, the kids are given a series of tasks that use their existing knowledge base and find the path themselves. They may not yet know there’s a beach. They may not know what the path means. But they will find the path and recognize it as a consistent finding that makes them go “hmm”. In some cases, an interesting finding. In other cases, just something they can see and understand.

Sometimes the path they’ve found is the concept–for example, modeling linear equations or exponential functions, or finding gravity in projectile motion problems.

In other cases, the model just introduces an inevitable observation that leads to the new concept. For example, I teach my kids about function operations when we do linear equations–adding and subtracting are good models for simple profit and loss applications.

So I kick off quadratics by asking my students to multiply linear functions, which they can see clearly as an extension of adding and subtracting them. This is an activity they can start off cold, with no intro (I haven’t written it up yet). I designed this because parabolas just don’t have a natural “real life” model other than area, which gets kind of boring. Plus, I need to cover function operations anyway, so hey, synergy. In any event, the kids are seeing an extension of a concept they already know (function operations) and seeing a new graph form consistently emerge. Then we can talk about factors (the zeros) and realize that we are looking at products of two lines. Could a parabola exist without being a product of two lines? Well, this is algebra 2 so they are fully aware that parabolas don’t have to have zeros. But what does that mean in terms of multiplying lines being factors of parabolas? Well, they must not have factors. So are all parabolas the product of two lines? And we go from there.

Understand that my classes still have lots of practice time where kids just factor equations and graph parabolas, learn about the different forms, and so on. But rather than just saying “now we’ll do this new thing called a parabola”, I give them a task that builds on their existing work and leads them into the new equation type. I don’t define the path. But nor do I let them go off on their own. I give them something to do that looks kind of random, but is in fact a path.

And all of this came from the results of the Great Christmas Mull. The previous Christmas had been productive, too–it’s when I came up with differentiated instruction for my algebra class.

So what can I say about my teaching, 5.5 years in? What’s consistent, what’s changed?

  1. I never lectured. I always explained, with increasing emphasis on classroom discussion.

  2. I have always been focused on student work during class, emphasizing demonstrated test ability above everything, and minimizing (or now eliminating) homework.
  3. I have always tried to move the student needle at all ability levels, from the no-hopers to the strugglers to the average achievers to the top-tier thinkers. I’m not always successful, but that’s consistently my stated priority.
  4. I have always designed my own curriculum and assessments.

  5. My teaching was transformed Christmas of 2011, when I realized I could introduce and teach topics using existing knowledge, forcing students to engage immediately with the material and start “doing” right away, increasing engagement and understanding. I have evolved from a teacher who mostly explains first to a teacher who only occasionally explains first. And that is a huge change that takes a lot of work.
  6. The observer might think that this change makes my classes student-centered, but I disagree. My classes are definitely teacher-centered, and let’s be clear, I’m the star of my teaching movie.
  7. Thanks also to the Great Christmas Mull, I’ve become far less concerned about curriculum coverage than I was in my first two years of teaching.
  8. I have always been a teacher who values explanation. It’s the heart of my teaching. I’ll explain through discussion or demonstration, but I’m not a reformer letting kids “construct” the meaning of math. I’m there to tell them what it all means.

I have plenty of development areas ahead. I’m working on tossing in the occasional open-ended instruction, just to see if I can come up with ideas that don’t waste hours and have some interesting learning objectives. I still have many concepts waiting to be converted to a “path to the beach”. And I’m now teaching something other than math, which gives me new challenges and more opportunities to see how to construct those paths without running off the cliff.

I Don’t Do Homework

Our school had its second Back to School Night. Attendance was spotty. I don’t judge. As a parent, I rarely attended.

But boy oh boy, could four sets of parents generate some excitement. I had a genuine culture clash.

It all began when I was going through my brief dog and pony show for my second trig class.

“Student grades are 80% tests and quizzes, 20% classwork. But I don’t grade classwork. Students get a B or A- just for showing up and working, which bumps their grade slightly.”

Until recently, I weighted homework for 10% and classwork for 15%–but not really. More accurately, if a student did most of his homework in a relatively timely manner, he’d get a little more of a boost. He couldn’t get the boost by “making up” missed homework; nor could he get the boost for just a couple homework completions. But if he didn’t do the homework at all, no harm no foul.

A few of my students got the boost, and they came from all points on the ability spectrum. I always remembered to assign homework through the first semester, then I’d fall off. For my first five years of teaching, homework had always completely stopped at some point in the third quarter.

“But last term, I suddenly realized that the end of the first semester was weeks away, and I hadn’t been assigning homework for a very long time.”

Remember my mentioning it had been a busy first term? Well, yeah.

“Most of my kids don’t do homework. So this realization just reinforced my awareness that I was only engaging in the homework ritual because I didn’t want to stray too far off the beaten path in comparison to my colleagues. But once I’d given up homework by accident, it seemed natural to make it official.”

The fact that I got that glorious tenure email and didn’t have to worry too much if my colleagues complained may have played a teensy, tiny part.

“So if you’ve got one of those kids who gets an A on tests but pulls his grade down by ignoring all homework, he–and it’s a usually a he–has probably mentioned it by now, and worships at my feet. I accept Starbucks cards or sixpacks of Diet Coke in tribute.”

One parent raises his hand.

“But don’t you find that homework ensures the students will get more practice? They need practice, just as we did when we were kids. I think it’s best for students to genuinely learn the math with practice.”

Uh oh. I take a deep breath.

“My students have always been graded overwhelmingly by what they do in class and the learning they demonstrate on tests. Homework was always optional, and I didn’t assign enough of it for students to practice fluency.”

“But I want my son to have practice material.”

“Well, I use the book pretty regularly, and there’s plenty of relevant practice material in there.”

“But do you think that’s how we all learned math?”

“Well, we weren’t all required to take advanced math. Look, I want to be clear: my method is the ultimate in hippy dippy squish.” Two parents laughed.

“I’m not trying to pretend that it’s normal for a math teacher to abandon homework. The whole homework ediscussion is basically a religious issue–and I don’t mean Muslim, Christian, and Jewish. People have strong ideological beliefs about the best way to achieve academically. However, the research on the intellectual impact of homework is very weak. But no research has shown that doing homework is the cause of comprehension.”

Another parent spoke up with a, er, very pointed tone. “I am so happy that you grade based on their work in class. So much better than to have them confused with nothing more than busy work after school. They can’t ask questions, they feel lost, and then they get discouraged.” Another parent nodded.

Original parent: “But the confusion is part of learning. Then they can come in the next day and ask for help.”

“They learn in class. If I take the bulk of one class to explain something, then they spend the next day working on that concept. I ensure students demonstrate their understanding, to the best of their ability. They won’t be able to copy the work from someone else; if I spot them not working, I work with them until I can see them understand it. If they’re talking or goofing around, they move to a different seat. My kids work math while they’re here. And ninety minutes of working or thinking about math is plenty.”

“But shouldn’t the students be practicing at home? Couldn’t you go through the course much quicker if they did?” the original parent is not to be discouraged.

“Again, they are welcome to work additional problems of their choice. But in my experience, students forget a lot of what they ‘go through’. My goal is to ensure that if they do forget material in this course, at least they really did understand at the time, rather than just follow through on some algorithms.”

“Exactly. I want them to understand the math.” said another parent.

“One last thing: I follow my students’ progress in subsequent classes. For the most part, they are keeping up and doing fine. I teach some of those subsequent classes, and so am able to compare my students to those given a more traditional course, and they’re doing fine. Many of my students go to junior college or local public universities, and I track their placement results as well. They, too, are ending up just as I’d expect. The weakest ones need some small amount of remediation, but most are placing in college credit courses. Meanwhile, they have far more accurate GPAs and weren’t forced to retake courses and slow down their progress simply because they didn’t do homework.”

And….the bell rang. Saved!

The original parent came up to me and asked, “You will assign my son additional homework?”

I smiled at the dad and the son. “All he has to do is ask.”

(He hasn’t.)

I decided to describe my policy change thusly because, well, the story happened and it was fun. All parents were respectful; I did not feel insulted or bothered by the first parent’s concerns. If I have in any way seemed contemptuous of the parents involved it’s unintentional. That said, ethnic stereotypes will prove helpful in deciphering the anecdote. The reason for the change is as described—I was busy, suddenly realized I had stopped assigning homework, decided it was time to cut the cord.

I usually just pick holes in everyone else’s arguments, but math homework is a teaching issue I have strong feelings about. Grading homework compliance is hurting a lot of kids, and all it does for those who comply is give them higher grades, not better academic skills.

Administrators understand this more than most, as they’re the ones putting additional math sections on their master schedule to accommodate all the kids with reasonable test scores who nonetheless flunked for not doing their homework. That’s the impetus behind all those stories you read of a district limiting homework’s percentage on the grade.

So as I wave goodbye to homework, let me take this opportunity to urge my compatriots to consider a similar policy, particularly if their classes look something like this:

The class opens with a warmup, designed to either review the previous material or introduce a new concept. Teacher reviews the warmup problem, then lectures or holds a class discussion on a new concept, works a few problems, has the class work a few problems, assigns a problem set, and those problems are called “homework”. Your basic I tell, I do, we do, you do.

The kids have the rest of the period to work on the problems, while the teacher is available to answer questions. If they finish in class, no “homework”! If they don’t work in class or do work for some other teacher, no big deal. It’s just time-shifting. They’ll turn in the work tomorrow, maybe do it with their tutors, maybe just copy it from friends who did it with their tutor.

Or they won’t do the problem set, either because they don’t understand, can’t be bothered, or just forget. The teacher will encourage them to come in and ask for help, or go to after school tutoring. Some of them will. Many of them won’t show up. Then they’ll get a zero, or turn it in late for a reduced grade, or stop doing homework altogether until they flunk. Or maybe their parents will call a conference and the teacher will be persuaded to accept a bunch of late homework to help the student pass the class.

How many high school math classrooms does this describe, with the occasional variation? A whole lot.

Notice that it’s only “homework” for those who can’t finish the work in class. The kids who don’t understand the material have to struggle at home. The students who really understand the material and could use more challenge get the night off.

High school teachers borrowed this method from colleges fifty years ago or more, a method designed for highly ambitious 20-somethings with demonstrated ability and interest. Today, our well-meaning education policy forces everyone into three years or more of advanced math, regardless of their demonstrated ability and interest. The college model is unlikely to work well with many students.

So go ahead and sneer at me for being a softie who skips homework, but understand that my students work to the bell. More often than not, my introduction is 10-20 minutes or even less, so the students are working the entire class period, taking on problems of increasing challenge. On those occasions where I have to explain something complicated, they focus on the relevant concepts for another day or more. But all my students are getting 60-90 minutes each day actively thinking and working about math, and my student engagement level has always been high. Strong students who finish early just do more problems. The student who treats my class as a study hall for her other homework because she has a tutor will experience teacher disapproval, often for the first time, and I’m a cranky cuss. She rarely makes the mistake twice.

When I did assign homework, I didn’t just continue from the same classwork problems, but created or selected much easier problems, designed for students to determined if they understood the basics of that particular concept.

Most education debates are tediously binary and thus wholly inaccurate. And so the math homework debate becomes “teachers who want to challenge their kids assign demanding homework” vs. “teachers who want to coddle their kids neglect their responsibility to prepare kids for college.”

In my classroom, kids are working pretty much non-stop, usually much harder on average than in the classrooms where kids are left to their own devices to finish their work. But somehow I’m the squish because I don’t engage in the great morality play known as homework. Are there teachers who don’t assign homework and also allow their kids to discover their pagh? Sure. That’s why the binary debate is a waste of time. The reality of classroom activity requires many additional points on a compass–not a bi-directional spectrum.

Finally, none of this really has anything to do with the actual teacher quality. Many teachers are doing a great job explaining math in those I do, etc lessons. Nor would any observer consider me hippy dippy or squish, which is why the comment always gets a laugh.

I was going to end with a joke about being a Unitarian in a Calvinist world. But hell, that plays right into the wrong sterotype.

Troubling Students

My classes are easy to pass, hard to do really well in. I’m a pushover for a D, but think three or four times about giving out an A. I didn’t fail a single kid last year. Save for Year Two, All Algebra All the Time, I’ve failed fewer than six kids a year, and even Year Two I had the second lowest fail rate of the math teachers.

I teach mostly math at a comprehensive high school, and the previous paragraph is very near heresy. Some math teachers cheer me on as a brave, admirable soul, but I spot them making the Mano Pantea while they walk away, just in case the Overlord is Watching. Others think I’m What’s Wrong With Education Today. These teachers hold as gospel that math standards could be upheld if we teachers were just willing to fail 60-70% of our students. In contrast to Checker Finn, who thinks teachers like me are spreading out two years of math content over three years of instruction because we can’t be bothered, these folks don’t think I’m lazy. They think I’m soft. They think I’m damaging their ability to cover all the course content they could get through if there weren’t all these kids who shouldn’t be there.

I became a lot less conflicted about my high pass rate–not that I ever lost sleep over it–after teaching precalc and discovering that a third of the kids had forgotten how to graph a linear equation and half couldn’t graph a parabola. These were kids that those other teachers had, teachers who had covered everything. Meanwhile, my kids do well in subsequent classes, so I’m not doing any harm.

But I digress. The students who trouble me aren’t the strugglers. I can take a kid who hates math, doesn’t want to be in class, and get him (it’s usually a him) to try. I can get that kid to attack a projectile motion problem and, even while making multiple small mistakes, beam with pride because by god, he kind of gets this and who ever would have thought? Kids like that, I can pass with nary a qualm.

The worrisome ones pretend they understand, but don’t have a clue. They cheat whenever they can, and not just on tests. They copy classwork in the guise of “working together” or “getting help”, and do their best to sit next to strong students. I group students by ability and, unless they can cheat on my assessment test, they are outed and placed up front, where I can keep an eye on then. They will then ask if they can sit next to John, or Sally, or Patel, their friend, because “they explain it so well”. I say no.

But if they cheated on the test, they can sometimes escape notice for a while. I circle constantly, watching kids work, changing seating when I see too much “consulting” with little discussion. Still others are more clever, and it takes a while before I realize they’ve been cheating not only in classwork, but on the tests–even when I create multiple tests. As a new teacher, I would sometimes miss these kids through the first semester. My success rate at pegging them early has improved.

This isn’t a big group, thank god. I might run into one or two a year. They have a telltale bipolar profile: for example, failing English entirely one year, and passing it the next year with Bs. Passing algebra with straight As, failing geometry completely–and failing the mostly pre-algebra and algebra state graduation test with a spectacularly low score. They aren’t fooling all of the teachers all of the time.

These kids are not your Stuyvesant cheaters, conspiring with others to satisfy demanding parents and create a fraudulent resume to get into a good school. Nor are these the low achievers who just want to get a passing grade in these time units called classes organized into a larger time period called school that others apparently view as a place of learning but they see as little more than a community network in which they have invested considerable social capital.

In fact, they’re almost worse than identified low incentive low achievers, cheating or otherwise. These kids almost seem incapable of learning. I can’t get them to slow down. They often resist help from me. Typical conversation:

Me, stopping by: “Okay, let’s start this again. You’ve plotted these points….”

Student: “Oh, yeah, I see.” Frantically erases.

Me: “Well, hang on, I want to be sure…”

Student: “I got it I got it I got it.” Starts to plot a point, then pauses.

I realize the student is waiting for me to say where to plot it in order to say “Yes, I know, I know.” So I wait. The student takes a deep breath and plots the point then lifts his pencil. “No, that’s not right, duh…”

Me: “You aren’t sure how to plot points.”

Student: “Yes, I am.”

Me: “Great. Plot (7,-7).”

Student plots (-7, -7).

Me: “Stop there.” I go grab a handout I have specifically for these situations, a simple handout that explains plotting points with some amusing activities to drive the point home.

Student: “I don’t need this. I know how to do it!”

Me: “Great. Then it should just take you a few minutes.”

At this point, I get a variety of reactions. Some students become furious. Others get sulky. Still others do the handout, making many mistakes, all the while assuring me that this is easy. I obligingly correct the mistakes, make them do it correctly. The ones that get furious, I shrug and let them continue.

Regardless, within a day, they are making the same mistakes. Nothing sinks in. Don’t get overly focused on plotting points; the problem could be anything–factoring, solving multi-step equations, working with negatives, exponential properties, fractions, whatever. Or a new concept. They have absolutely no clue, and can’t do much of anything.

Yet they don’t have the profile of a low ability student. Test scores, yes. Profile, no. They often have As, win praise from teachers for their teamwork and effort. They are heavily invested in appearing “normal”. Serious control freaks. Sometimes, but not always, with parents who expect success. More often, but not always, Asian. All races. Both genders.

I haven’t taught freshmen since oh, lord, fall 2012.1 I teach relatively few sophomores these days, running into them only in Algebra 2.

That matters because when I taught freshmen and sophomores, I would go full-scale intervention. I might talk to a counselor to see if they should be assessed for a learning disability. I would insist that they stop lying to me and themselves. I had no small success at getting some of them to acknowledge their desperate attempts at fraud, get them to work at their actual level, deal with the discomfort. They didn’t make much progress, but it was real progress, and they had skills to move forward. I ran into some of them again the next year, and we could start on an honest basis and make additional progress. Those who didn’t acknowledge their issues were among the few students I failed.

But that’s a lot harder to do when dealing with juniors taking trigonometry or, god forbid, precalc. Should I fail them? They will probably do better in a class with teachers who give “practice tests”, study guides that have exactly the same questions as the eventual real test but with different numbers. They will definitely do better with teachers who actually grade homework and count it as 25% of the overall.

A small problem. This approach turns my grading policy into: work hard and honestly acknowledge your ignorance and I’ll pass you. Lie and do your best to cheat with similar ignorance and I’ll fail you. I’m comfortable with holistic grading at the bottom of the scale, but I don’t like morality plays.

Then I remember that kids who honestly acknowledge their inability in a trig or pre-calc class are usually seniors, off to junior college and a placement test that will accurately put them in remedial math. I’m only ensuring they are learning as much as possible for free before paying. If they are juniors, I always have a talk with them about their next steps, telling them not to take the next course in the sequence but maybe stats or something else that will keep them working math, but not out of their league.

The kids who cheat and fake it in trig and precalc are usually juniors, and they will be going onto another course. They will not listen to me when I tell them under no circumstances should they continue into pre-calc or, god forbid, calculus. I might be teaching that course, which just gives me the same problem again. Or they’ll be cheating their way through with another teacher–or, that teacher will do what I should have done and flunked them.

This quandary doesn’t make any sense unless you realize that in my view, these kids are pathologically terrified of facing reality, the sort of thing that some of them, forced to face up, might not survive in good form. These aren’t blithe liars gaming the system to look good. Then I remind myself that they’ve been caught before, they’ve flunked other classes, they’ll survive. But I still don’t like the quandary, because these are kids who literally can’t learn. (And remember, I’ve seen them in my non-math classes, too). By junior year, given their denial and fear, does it do any good to make them aware of this? They’re going to be able to point to any number of teachers who disagree with my assessment, and have all sorts of excuses for why they got those Fs. Besides, they just don’t test well. It’s always been a problem.

At times like this, I envy my colleagues who never notice the cheating, or who focus purely on achievement and aren’t interested in the distinctions I’m making.

But these are the students who trouble me.

1Holy Crap. That’s an amazing realization. New math teachers doing your time in the algebra/geometry trenches, take heed. If you want variety, it will come.


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