Modeling Linear Equations, Part 3

See Part I and Part II.

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

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

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

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

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

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

Shrug. “I don’t know.”

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

Tentatively. “He could just buy 9 burritos?”

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

“None.”

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

“Fifteen tacos. Why is it 15?”

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

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

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

“I don’t know.”

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

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

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

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

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

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

“And you know it’s $15 because….”

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

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

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

“Multiplying the burritos by five dollars.”

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

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

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

“Five.”

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

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

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

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

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

“So, guy over there. Yeah, you.”

“I wasn’t paying attention.”

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

Silence. I wait it out.

“Um. I can’t buy as many tacos?”

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

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

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

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

“Goes down.”

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Here’s the boardwork for a systems models.

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

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

Phew.

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

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

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

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

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

Push the Right Buttons

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

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

“Are you Ms. L’s Econ class?

“Class was cancelled.”

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

“Who are you?”

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

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

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

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

“We don’t know what to do.”

“Here.” I produce the assignment handout.

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

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

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

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

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

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

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

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

“Naw, I’m just finding shoe prices.”

“Shoe shopping?”

“For the team.”

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

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

“What are you staffing?”

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

I look at them expectantly.

“What?”

“Nothing comes to mind?”

“She didn’t tell us what to do.”

“But you’re pricing shoes.”

“Yeah.”

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

“Okay. Who will be wearing the shoes?”

“The players. Oh.”

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

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

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

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

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

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

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

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

Boaler’s Bias (or BS)

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

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

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

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

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

(page 13)

Phoenix Park, on the other hand:

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

(emphasis mine) (page 18)

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

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

(page 64, 65)

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

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

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

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

(page 73) (emphasis mine)

Meanwhile, the Amber Hill girls were miserable:

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

(page 139)

(all emphasis mine)

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

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

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

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

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

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

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

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

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

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

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

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

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

Best Movie About Teaching. Ever.

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

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

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

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

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

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

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

Best Movie about Teaching Ever: The Browning Version

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

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

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

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

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

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

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

The false god of elementary school test scores

Rocketship Academy wants to go national. Rocket Academy is a hybrid charter school chain that focuses solely on getting low income Hispanic elementary school students to proficiency. (Note: Larry Cuban has some excellent observations from his visit to a Rocketship Academy.)

First things first: I’ve checked the numbers every way I can think of, and Rocketship’s numbers are solid. They don’t have huge attrition problems that I can see. They are, in fact, getting 60% or higher proficiency in most test categories, and the bulk of their students are Hispanic, many of them not proficient in English. Of course, that brings up an interesting question–if they are proficient on the ELA tests, why aren’t they considered proficient in English? But I digress.

The larger point is this: getting high test scores on California’s elementary school math tests ain’t all that much to get worked up about. Here’s some data from the 2011 California Standards test in math:

I used two standards, because the NCLB obsession with “Proficient and higher” is, to me, moronic. I prefer Basic or higher. The blue line is the percentage of all California in grades 2-9 scoring Basic or higher in General Math, the red is the percentage of same scoring Proficient or higher.

So it gets a bit tricky here, because after 6th grade, the entry to algebra varies. In order to simplify it slightly, I’m ignoring the seventh grade algebra track (call it “accelerated advanced” path), which is about 40,000 students this year, fewer in previous years.

I combined 8th and 9th grade students in General Math and attached that result to the red and blue lines.

Then I separated two groups–the ones who took algebra in 8th grade, and the ones who took algebra later than that. The first group are those who entered algebra in 8th grade, passed it, and continued on the “average advanced” course path, culminating with Calculus senior year. The second group are those who took algebra for the first, second, or third time in high school and then continued on. For each group, I calculated percentages for Basic + and Proficient +.

Notes:

1. Through grade 6, the scores represent all students. In grade 7 and 8/9, general math scores reflect only those students who haven’t moved onto algebra. That’s probably why the proficiency levels drop to 50% and lower for the last two groups.In other words, the green and purple lines represent the advanced track students–most, but not all of the the strong algebra and higher math students. The turquoise and orange lines represent the weaker students taking algebra and higher.
2. Roughly 80% of all students test at Basic or higher from second through sixth grade.
3. Over 70% of the strong studentws test at Basic or higher from algebra through “summative math” (taken for all subjects after Algebra 2).
4. The percentage of students testing proficient from second through sixth grade starts at 65%, rises slightly, and then drops steadily.
5. In no course do more than 50% of the strong students in algebra and higher achieve a score of Proficient or higher.
6. In no course do more than 50% of the weaker students in algebra or higher achieve a score of Basic or higher.

So the chart reveals that all California second through sixth graders, high and low ability, averaged higher scores on their tested subject than the strongest high school students did.

I used 2011 scores, and I may have made a minor error here or there, but the fall off has been in the scores for several years now, and it’s easy enough to check.

What could cause this? Why are California’s elementary school students doing so phenomenally well, and then fall apart when they get to high school? Let’s go through the usual culprits.

California’s high school math teachers suck.–Well, in that case, there’s not much point in demanding higher standards for math teachers, because California’s high school math teachers have had to pass a rigorous content knowledge test for over 20 years. California’s elementary school teachers have to pass a much easier test–which is much harder than anything they had to pass before 2001. In other words, try again.

The teachers aren’t covering the fundamentals! So when the students get to algebra, they aren’t prepared.–But hang on. Elementary school kids, the ones being taught the fundamentals, are getting good test scores. What evidence do you have that they aren’t being taught properly?

Well, they’re only getting good test scores because the tests are too easy!—dingdingding! This is a distinct possibility. Perhaps the elementary tests aren’t challenging enough. Having looked at the tests, I’m a big believer in this one. I think California’s elementary math tests, through seventh grade, are far less challenging to the tested elementary school population than are the general math and specific subject tests are to the older kids. (On the other hand, the NAEP scores show this same dropoff.)

However, while that might explain the disparity between the slower track math student achievement and elementary school, it doesn’t adequately address why the students in the “average advanced” track aren’t achieving more than 50% proficiency, does it?

Trigonometry is harder than memorizing math facts–We should take to heart the Wise Words of Barbie. Math achievement will fall off as the courses get more challenging. Students who excelled at their times tables and easily grasped fractions might still struggle with complex numbers or combinatorics.

So if you ask me—and no one does. Hell, no one has even really noticed the fall-off—it’s a combination of test design and subject difficulty.

Whatever the reason, the test score falloff has enormous implications for those who are banking on Rocketship Academy, KIPP, and all those other “proven” charters that focus exclusively on elementary school children.

Elementary school test scores are false gods. We have no evidence that kids who had to work longer school days simply to achieve proficiency in fifth grade reading and math will be, er, “shovel ready” for algebra and Hamlet. KIPP’s College Completion Report made no mention of its college students SAT scores, or indeed made any mention of demonstrated ability (e.g., AP tests), and color me a cynic, but I’m thinking they’d have mentioned both if the numbers were anything other than dismal.

So let’s assume that those Rocketship scores are solid (and I do). So what? How will they do in high school? Where’s the follow through? Everyone is banking on the belief that we can “catch them early”. Get kids competent and engaged while they are young, and it all falls into place.

Fine. Just let me know when the test scores back up that lovely vision.

I am very Barbie about math*

Why are High School Teachers Convinced that White Girls Can’t Do Math?: apparently, high school teachers rate white girls lower than boys when they are matched in grades and test scores. Therefore, sez the Forbes article and the study it reports on, high school math teachers are biased against white girls.

Okay, so first off, the headline and the study are absurdly biased, which is a tad ironic in an article about bias.

Using the Forbes article as a guide to the study, the study didn’t establish bias. What it established was that high school teachers consistently rate white girls a tad lower in ability than boys with the same grades and test scores. That’s not the same thing.

First off, throw out grades. I’ve written on this before, but it bears repeating: if a teacher counts homework as a significant part of the grade, the grade simply isn’t accurate. Moreover, grades skew dramatically based on population. Suburban schools with lots of high-achieving kids have a tougher grading standard than Title I schools.

I am hoping that this study focuses primarily on test scores. Let’s assume that teachers rate boys higher in ability than girls, even though they have the same test scores. Is that necessarily a sign of bias?

Hell, no. Girls do more homework than boys. Girls are, as a group, more worried about grades than boys. Girls, as a group, work harder than boys.

So suppose you have two students. One of them turns in every bit of homework, asks questions purely about methodology and algorithsm, works very hard to “get it”. The other student doesn’t do homework, asks questions about process and concept, and always grasps everything without any particular effort involved. They both get the same test scores.

Who will the teacher say is “better” at math?

That’s not bias. That’s a totally rational inference about the ease of understanding, grasp of concept, and underlying aptitude.

I have a good number of students who are entirely obsessed with grades and utterly uninterested in math. Most, but not all, of these students are girls. I have a large number of students who are fascinated by how math works, often praise an “interesting” question, ask all sorts of conceptual questions because they want to know how things work. Most, but not all, of these students are boys.

Spare me the sturm und drang. It’s not bias. Teachers are longing to find female students who are strong at math. They aren’t ignoring white female students with math talent. They are just less likely to confuse “slog” with “talent” than, say, your average researcher.

*I used to say that all the time, until I became a math teacher. I hate it when my cultural references start to date. (They should stay single for life. Hyuk.)

Discovery Doesn’t Work

I had trouble in ed school because (well, at least in my view of it) I openly disdained the primary tenets of progressive education. I am pro-tracking, anti-constructivist, and pro-testing, all of which put me at odds with progressives. Here is the irony: I mention often that I am a squishy teacher (squishy=touchy feely). I am not just squishy for a math teacher, I’m the squishiest damn math teacher from my cohort at the elite, relatively progressive ed school that made my life very difficult. My supervisor, who knew me first as a student in a curriculum class, was genuinely shocked to learn that I didn’t talk at my kids in lecture form for 45 minutes or more, given my oft-expressed disagreement with discovery. Even my lectures are more classroom back and forth than me yammering for minutes on end. (In fact, my teaching style did much to save me at ed school, but that’s a different story.)

Here is what I mean by squishy: My kids sit in groups, not rows. When I set them to practicing, which is usually 20-35 minutes of class, they are allowed to work independently, in pairs, or as a group of four. I often use manipulatives to demonstrate important math facts. My explanations are, god help me, “accessible”. I don’t just identify the opposite, adjacent, and hypotenuse and then lay out the ratios. No, I’ve been mentioning opposite, adjacent and hypotenuse for weeks, whenever I talked about special rights. I introduce trig by drawing a line with a rise of 4, a run of 3, and demonstrate how every right triangle made in which one leg is 3 and the other 4 (that is, have a “slope” of .75) must have the same angle forming it. I spend a great deal of time trying to think of a way to help kids file away knowledge under images, concepts, pictures, anything that will help them access the right method for the problem or subject at hand. (For more info, see How I Teach and The Virtues of Last Minute Planning.)

However, I am not in any sense a constructivist as progressive educators use it. I use discovery as illustration, not learning method. I don’t let kids puzzle over a situation and see if they can “construct” meaning. I explain, give specific instructions, and by god, my classroom is teacher centered. I am the sage on stage, baby. And that’s why I got in trouble in ed school, despite my highly accessible, extremely concept-oriented teaching style; I routinely argued against constructivist philosophy, and emphasized the importance of telling kids what to do.

Anyway. I was incredibly excited to read an article that openly states the obvious: Putting Students on the Path to Learning: The Case for Guided Instruction. This article is just so dead on right. To pick one of many great excerpts–click to enlarge, but why can’t I copy text from pdf files any more?:

Yes. Low ability kids like discovery; it is less work for them, yet they feel they are doing something important—but in fact, they aren’t learning very much. High ability kids tend to be “for chrissake, give me the algorithm”, when they would be better off puzzling through the math for themselves.

The article talks about the importance of worked-out examples. I read the article this morning and had a worked out example on the board the same day—step by step factoring of a quadratic. Here’s the weird thing: the kids who need the help with factoring had to be prompted to use the example, but the kids who got factoring were clamoring for worked examples in the area they had trouble with.

This would be a great thing for notebooks. But how do you get the kids who need help to keep the notebooks?

Great article, that changed my teaching immediately. How often does that happen?

Teaching Humanities, Part I

I got lucky my first year out and was able to teach math and humanities.

Were I ever to get a full-time job teaching either English or history, I would feel guilty for abandoning math and taking the easy way out. That’s how much easier it is to teach either subject. Do not picture me as short-timer, stuck teaching math as some sort of dead-end, droning on praying for the day that I get to teach my true passion. No. I find teaching math, constantly struggling to find a way to make abstract concepts understandable to uninterested students with no inherent ability, to be one of the most fascinating tasks invented. I’m hooked. But teaching English or history is a hell of a lot easier, and my lord, is it fun.

I taught 9th grade humanities at an extremely progressive school. (You’re wondering why on earth they hired me. They were desperate and got rid of me as soon as they decently could.) I planned out the curriculum with two other teachers for most of the year. As a rule, I did tests, grammar, and history (I was the only certified history teacher of the three of us, had considerable experience with standardized tests and–also as a result of my test experience–a lot of background in teaching grammar). They did literature and most of the actual planning (weeks spent on each section and so on). It was all collaborative and lots of fun–I learned a lot about planning out a book, and they gave me great feedback on how to simplify a history lesson for freshmen, while I taught them how to keep the rigor in even if the vocabulary is simplified.

We did the history of India, history of the Philippines, a brief history of Russia from the freeing of the serfs to the Russian Revolution, and the Age of Enlightenment, Age of Discovery, and revolutions industrial and agricultural. The kids read Nectar in a Sieve, a book on post-colonial India, a choice of three books on the Philippines (can’t remember their names), Animal Farm, and Twelfth Night. They also did some sort of project on natural disasters, which interested me not at all except I learned a good deal about geography, and some sort of personal narrative.

There was a great deal of indoctrination in the course material from years past, but I convinced the other teachers to dump a lot of it (without ever mentioning my opinion of the indoctrination) and the rest of it I just cut from my lesson.

The kids’ reading abilities ranged from 6th grade to college level, as did their writing. We were supposed to do “sustained silent reading/writing” each day for 20 minutes (it was a block class of 100 minutes) and it was supposed to be based on student choice, but I realized that most of them were just sitting around doing nothing. So I instituted my own hand-made SRA program of three levels. I just went to the bookstore, picked out some enrichment materials at various levels (and yes, bought them with my own money), and made copies for the kids. The kids read nifty little passages on all sorts of subjects, answered questions, did crossword puzzles with new vocabulary, and had little tests at the end of each unit that I checked on.

The kids gained tremendously in content knowledge at all levels. My favorite example: when we were talking about Russian history and Trotsky, I mentioned in passing that Stalin hunted down Trotsky even after he fled to Mexico, where he lived for a while with two famous artists, Diego Rivera and Frieda Kahlo.

One of my weakest readers perked up. “Is that how you say her name?”

I laughed. “I think so, but I’m not sure. I’ll go look it up later. You know about Frida Kahlo?”

“She was in the reading packets. She was famous for painting herself, or something.”

“Self-portraits?”

“Yeah, that was it. She wasn’t happy with the Diego guy, right?”

“Oh, is that the one that had a car accident?” pipes up another weak reader.

Another boy pops in. “She crashed into something.”

A strong reader (who therefore had not read about Frida) was interested. “When was this?”

“Yeah, this was like….it was after 1900. I think it was a lot after 1900, but not like 1950 or anything.”

“I think her car crash was in the 1920s, but don’t quote me,” sez I. (It was.) “That was very useful information, and thanks for the interesting details. Back to Trotsky and the axe.”

Content knowledge, baby.

Anyway, the segment on Twelfth Night we were expected to do was designed by a student teacher, and it was all about identity and examining their own navel—exactly the kind of nonsense I don’t like to do. By now, I knew I wouldn’t be back next year and this was the last segment of the year, so I went off the reservation. I did two weeks on Twelfth Night and two weeks on Elizabethan theater, and every minute of it was joyous fun. For the kids sometimes, too.

This post is getting long, so I’ll put the lesson plans and a story about in subsequent posts.

Getting Engagement

I once got in a bit of trouble with someone who, after an observation, expressed deep concern that my students had been “off task”. Had I any sense, I would have nodded sagely and agreed, asking for suggestions and methods to improve my students’ engagement level.

Alas, I lay claim to a fair amount of brains but no sense at all, and so, fatally, I looked askance at the comment. It was a Thursday afternoon, sixth period, and the kids had been on task. No, not every second, not every student, but I’d set them a difficult and challenging assignment and they’d known I was being observed. They’d jumped into a high-octane class discussion, jumping in with questions and answers, asking for clarifications, bursting with enthusiasm on cue. A fantastic performance (in the acting sense). Then they worked studiously on the handout. When they got stuck, naturally (sigh) they chitchatted until I came by and answered their questions, and then went back to work. I’d wanted the class to finish at least a quarter of the handout, to finish it up the next day. Three students finished the entire handout, over half the class finished half of it, and everyone finished the quarter I’d planned. I’d spent the last ten minutes of class, as always, going round one last time to ensure everyone was at a good stopping point for the next day. Then I went back up front, praised them for a good day’s work, reminded them of the key ideas, and told them to be ready to finish up the next day.

And so, instead of asking for suggestions and methods, I demurred—and here’s the really stupid part—told the observer truthfully that a certain amount of off-task is normal, particularly with kids who struggle with math. The trick was, I said nonchalantly, to keep them moving and minimize the off-task behavior by ensuring the students feel capable of doing the work. Like I said: no sense. No worries, though: It all ended well, if not without bloodshed.
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One might think—truth be told, people who know me often do think—that I am incapable of accepting criticism. Unfair. My objections were well-founded. In fact, other teachers hearing of that conversation are filled with horror that any observer would expect 100% engagement or anything close to it with a heterogeneous class of mostly math strugglers.

But more to the point, I take all suggestions in, even as I am giving out with my objections and rationales, and they sit in the back of my mind, waiting to pounce.

Improving engagement levels is one of my top two or three concerns as a teacher, and something I think about constantly. Those who don’t worry about it teach in tracked classes, the lucky dogs. My objection to that observer was precisely on this point: the observer didn’t understand how high the engagement level was, given the audience, and how the engagement level and activity was the primary focus of the activity I’d chosen.

Much later, when I was reviewing midpoint and distance with my Algebra II kids on another Thursday, I’d noticed they were moving too slowly. I’d given them a double worksheet with lots of problems because I wanted them to gain fluency. Some of them were working hard, but too many had lollygagged through five or six on Day One, which meant my original plans for Day Two, finishing the worksheet—on a Friday no less—would be an exercise in herding cats, feeling relieved if over half the class did a few problems lackadaisically. It wasn’t so much the number of problems as the time spent focused on the work. What I needed was some sort of activity that would keep them on task the whole time.

And just then, the observer’s suggestion pounced. Not the literal suggestion made that day, of a competition during the last 20 minutes of class. Totally unworkable in my classrooms, But I’d still filed the idea away. Somewhere in that notion was a nugget I instinctively knew was useful—no, not the competition, not the last 20 minutes, god knew (just organizing it would take 10 minutes), but….and so I came up with Switch and Stay.

Flashing neon sign goes here: Despite my immediate pushback and valid objection, I did not slough off the suggestion but rather morphed it into something I could use to be more successful at engagement. See? I do listen to criticism.

But onto larger issues:

Progressives think the key to engagement is “relevant curriculum”. They’re mostly wrong, but not entirely. Eduformers think a lack of engagement is the teacher’s fault. They’re mostly wrong, too.

What most people fail to understand is that engagement, in and of itself, doesn’t lead to learning. It seems like it ought to–that’s what happens in all the feel-good teacher movies. Get the kids to care through rap and poetry slams, and suddenly they’re spouting Shakespeare and writing award-winning blogs.

Larry Cuban spells out the engagement assumptions:

1. Motivated students will engage in academic work that teachers assign.

2. Engaged students are attentive, participate in classroom activities, and complete assigned work.
3. Because students pay attention, participate, and complete work, they acquire academic knowledge and skills from teachers and peers that result in classroom and school rewards further strengthening engagement.
4. Expanded school-based knowledge and skills produce academic improvement as measured by teacher grades and standardized tests.

Cuban is focusing on engagement via high tech in the classroom, but his larger point is worth remembering: there is little evidence that supports this chain of reasoning.

I am reasonably certain that engagement leads to increased individual achievement on the margin. I am equally near certain that engagement has little to do with low test scores. I know many hardworking, engaged kids who don’t do as well as kids who tune out and show up periodically but learn a lot more in half the time with half their attention.

I just need engagement because otherwise, the kids don’t do the work. If they don’t do the work, they won’t have a chance at achieving fluency or creating memories they can access later. But all engagement gives them is a chance at that, not the guarantee of increased understanding and achievement.

Switchers and Stayers

I often read suggestions about competitions and other games to get students working “to speed”. But direct competition doesn’t work unless you’ve got truly homogenous abilities–the same students always win and the rest won’t try. Likewise, games like trashketball suffer from the problems this blogger describes: either the high ability students dominate or the low ability students just sit there and let them do all the work.

But I’ve been noticing that any time I give my students a lot of problems to practice automaticity, a lot of students will just do a few, slowly–the same way they work new material. And that’s not the point. So, thanks to a comment by an observer, I’ve spent quite some time mulling how to do some form of high-octane activity. This one was successful enough I’m going to try it again.

I put the desks all about the room in pairs, instead of the usual quartets. As the algebra II students came in the room, I told them to get out a notebook and pencil and put their backpacks on the counter, out of the way (something I really should do more often), and then stand at the back of the room. I told them that I’d seen they hadn’t been productive the day before, and I wanted more focused work, that this activity would ensure they completed a number of problems. I took from half the class and put them one to a pair of seats. They were the Stayers. The rest of the class, at the back of the room, were the Switchers.

The activity: Each Switcher sits next to a Stayer. I put up a problem, each pair works it–separately or together, but both of them must have the correct answer in the allotted time in order for them each to get a chip. Then the Switchers move to a different Stayer, and it starts again. The more chips, the better the classwork grade. I made them practice Switching, which they thought insane (and said so) and made sure the Stayers knew they were to stay put. Switchers must switch–no sticking to the same person. I also made sure that certain students were both Stayers or both Switchers, so they’d never be together. Such a clever teacher am I.

Do not imagine that I gave these instructions to a happily compliant group of productive teenagers, eagerly awaiting a new activity from a beloved teacher. Surly, sarcastic, grousing, rude, and close to rebellious, the students made the opening very difficult, and their constant interruptions added a good five minutes onto my instructions–in each class! Sixth period, as always, was the best.

But then, my god. They worked like dogs to get those chips. They set up their own internal competitions to see who could finish first and the team that finished first started asking if they could pick the color of chip they got. I gave them five minutes for the first problem in each category, then four, then three. If I noticed a student struggling with a concept, I’d ensure he or she was paired with a high ability student in the next round and signal that I’d like the second one to give a thorough explanation–and they always complied. The students worked 9-10 problems through the period, with very little downtime. Very few students missed any problems, although I made sure a few didn’t get chips the first time round just to show it could happen.

In short, it worked beautifully after the Great Battle of the Beginning.

But it’s very unsettling, their instant hostility when I do something different. Third and fourth period Algebra II act like I’ve whipped them daily and just stopped yesterday; hackles ready to rise if I raise a hand again. Sixth period is much more relaxed. And then other classes are completely on my side. My geometry kids, particularly second period, would walk off a cliff if I asked them. It was the same thing last year. Second period algebra seemed to hate my guts, while my intervention and sixth period algebra students were likewise ready to try anything on my word. You just never know.