Algebra Terrors

A day or so before the school year began, I went to an empty classroom that had a supply cabinet. This classroom was way better than mine. It was 3 or 4 feet wider, had shelving and a smart board. Now I didn’t care much about the smart board, but all smartboards have document cameras, which my room does not.

“Hey. This is a great room. Who gets it?”

“I guess the other new teacher.”

“When’s he coming?”

“I don’t think he’s even hired yet. You know, you should ask for this room! You’re here first.”

“Yeah, I think I will. It can’t hurt.”

So off I went to find an administrator, and the first one I I ran into was the AVP of discipline and scheduling. (As a sidenote, the conversation recorded below is the one of only two I’ve ever had with him.)

“Hi. Please don’t view this as a complaint of any sort, but I really like to teach with document cameras, and I notice that room 1170E has a smartboard. Hank (not his real name) suggested I ask if I could switch rooms?”

“To 1170E? Oh, yes, that’s for Ramon. I suppose I could switch rooms, but we’d need to change schedules as well. You see, we got the Promethean smartboard funding as part of our algebra initiative, and we committed to give those boards to teachers teaching Algebra I at least 60% of the time. If you’re interested….”

You know in Terminator 2, when Linda Hamilton has just finally broken out of her padded cell, broken Earl Boen’s arm, beaten the crap out of three security guards and is waiting for the elevator? Freedom is there, baby. She can taste it. She can get her son, escape to Mexico, stop the machines, save the world. All she needs is the ding of the elevator door.

Ding!

….and out of the elevator steps

Algebra I Arnold

NOOOOOOOOOOOOOOOooooooooooooooooo!
“I was told that you had expressed a strong preference to teach geometry and intermediate algebra. But I’m always happy to find interested algebra I teachers….”

“No, no, sir, no really. It’s fine. I do have a very strong preference to teach geometry and intermediate algebra, you were correctly informed, and I am happy with my current room. It’s fantastic. I can deal without a camera, it’s fine.”

“You’re sure?” Clearly, this man is an evil sadist. “You really do seem to like the document camera, and we prefer that the rooms go to teachers who will use them…”

No! No! I’ll stop! The machines can win! Take my son! Just don’t make me go back!!!

****************************************

It was just a bad scare. I’m teaching geometry and algebra 2. Well, Math Support, but even though the kids are weaker, I’d rather teach Math Support than Algebra I.

Math teachers think this story is very funny.

In retrospect, my second year of teaching was my most brutal, thanks to my schedule of all algebra I, all the time. I learned a lot. I never want to go back. Oh, sure, I’d like to teach one class of Algebra I, particularly to see if my data modeling lessons they work as well in algebra I as algebra II. But I do not want to be an “algebra I specialist”, and never, ever, EVER want to devote anything more than a class a year to algebra I. I’ve said it many times, but I’m always ready to bore folks: high school algebra I classes should convince anyone—from loopy liberal progressive to anti-teacher union tenure hating eduformer—that our educational policy is twisted and broken beyond all recovery.

So why bring it up now? Because until I saw this article, I’d forgotten the very worst part: A Double Dose of Algebra (ht: Joanne Jacobs).

Yes, I didn’t just teach straight algebra I classes. I taught a double class of Algebra Intervention. Let’s switch T2 characters for just a moment, shall we?

This is what happens when I’m reminded of that intervention class without time to prepare myself.

What is Algebra Intervention, or “double dose algebra”? Well, it’s this brilliant strategy of identifying kids who are really weak in math and increasing their hours of torture.

The best study of this approach, by Takako Nomi and Elaine Allensworth, examined the short-term impact of such a policy in the Chicago Public Schools (CPS), where double-dose algebra was implemented in 2003. …. Nomi and Allensworth reported no improvement in 9th-grade algebra failure rates as a result of this intervention, a disappointing result for CPS. The time frame of their study did not, however, allow them to explore longer-run outcomes of even greater importance to students, parents, and policymakers. (emphasis mine)

So double dose algebra didn’t work. Did that stop them? Hahahahah! Of course not! They just commissioned another study! One that would allow them to explore “outcomes of even greater importance” to students, like “will I make an extra $50,000 a year to compensate me for the time I spent in this tortuous hell?”

Using data that track students from 8th grade through college enrollment, we analyze the effect of this innovative policy by comparing the outcomes for students just above and just below the double-dose threshold. These two groups of students are nearly identical in terms of academic skills and other characteristics, but differ in the extent to which they were exposed to this new approach to algebra. Comparing the two groups thus provides unusually rigorous evidence on the policy’s impact.

Wait. You checked the kids just below and just above the threshold? So you only compared the strongest intervention students with the weakest regular students? Well, golly. Did you, perchance, check how the weakest regular students did compared to the weakest intervention students? Was it substantially different from the gap between the strongest intervention and weakest intervention?

The benefits of double-dose algebra were largest for students with decent math skills” but below-average reading skills, perhaps because the intervention focused on written expression of mathematical concepts.

Guys, half of all regular high school algebra students can’t add fractions or work with negative numbers—that is, they do not have decent math skills. So what the hell is relevant about progress made by intervention students with “decent math skills”?

With the new policy, CPS offered teachers of double-dose algebra two specific curricula called Agile Mind and Cognitive Tutor, stand-alone lesson plans they could use, and three professional development workshops each year, where teachers were given suggestions about how to take advantage of the extra instructional time.

Eight days of PD. EIGHT DAYS! In three plus years of teaching, I’ve taken 1.5 days off for being sick. In one year of teaching algebra and algebra intervention, I was required to leave the classroom for 8 days. The PD was utterly useless. The lunches with the other math teachers, good—lots of conversations, sharing of lessons, venting, and so on. We would do better to just give us money and an extra half hour every month for lunch.

CPS also strongly advised schools to schedule their algebra support courses in three specific ways. First, double-dose algebra students should have the same teacher for their two periods of algebra. Second, the two algebra periods should be offered consecutively. Third, double-dose students should take the algebra support class with the same students who are in their regular algebra class. Most schools followed these recommendations in the initial year. In the second year, schools began to object to the scheduling difficulties of assigning the same teacher to both periods, so CPS removed that recommendation.

It wasn’t just the schools that objected, I’m betting. I taught intervention the first year it was offered by the school. Of the three intervention teachers, one (a TFAer) turned in her resignation in January purely because she felt beaten down by intervention. Another teacher, an algebra specialist, a near-phlegmatically calm Type B, burst into tears when she met with the principal to make absolutely sure she wasn’t given an intervention class the next year.

I was the third. I never complained. I was under continual pressure because I wouldn’t tolerate three kids who were deliberately disrupting the class. The administration hinted I was racist, that I was exaggerating their behavior, and only relented on the pressure when my induction adviser witnessed a middling incident of blatant misbehavior and blew a gasket when the AVP of discipline shrugged it off until he learned that she’d seen it. Admin got a long letter from her and started to make the kids’ lives hell.

The following year, the school dropped the requirement for consecutive periods and allowed two teachers to split the course, rather than requiring the same teacher to do both sections. That same year, the intervention teachers got called into a room by the district and were given a blistering come to jesus meeting in which they were informed that their pass rates better go way, way up until they were as good as the pass rates from last year, which were clearly a goal to be attained. Of course, that last year, when they were dumping all that pressure on me, they never said “Yeah, too many referrals but hey, your pass rate is awesome. You’re only failing two kids, who never show up. Great job!”

This year, the school has dropped the requirement that the students all be in the same class. Hey. That sounds familiar, doesn’t it?

The pressure on the teachers is tremendous. So the schools try to find a way to pay lip service to the method—we’re offering intervention for our weak students!—without all their teachers quitting on them simultaneously. Intervention is brutal on teachers.

The recommendation that students take the two classes with the same set of peers increased tracking by skill level. All of these factors were likely to, if anything, improve student outcomes. We will also show, however, that the increased tracking by skill placed double-dose students among substantially lower-skilled classmates than non-double-dose students, which could have hurt student outcomes.

In addition to the strain on teachers, intervention is a huge hassle for administration and has an unintended consequence that escapes the notice of people who haven’t talked to an AVP responsible for the master schedule. But the reality is that a group of kids who must take two classes back to back end up taking most, if not all, of their classes together.

Say a school has 10 freshmen English classes but only three of them are double block remedial and (please note, this will come up again) many of the kids who take double block algebra also require double block English. The intervention freshmen are in periods 3 and 4 for algebra intervention, and the only other double block English class they can take is 5 and 6, leaving periods 1 and 2 open. Only one freshman PE class available in period 1, so science (bio or general) has to go in period 2. All done. So all the kids in algebra intervention periods 3 and 4 who are also in double block English take all their classes together. For every intervention class, some 20-30 underachieving, low incentive kids are moving through their entire day together, in non-remedial and remedial classes both. Of course, since most intervention kids are weak in all their subjects, this means that their classes have a disproportionately high number of low achievers—all of whom spend their entire day together, socializing. Or planning ways to wreak havoc. The troublemakers in my class arranged signals that they would use to disrupt classes—all their classes. They’d pick a code word, and whenever the teacher said that word, they’d all start laughing loudly, or squeaking their shoes, or sneezing.

I’m a big fan of tracking. I am vehemently opposed to taking a group of low achieving kids who are already buddies, already with next to no investment in school, already really annoyed at having to take a double dose of math—and give them every single class together, so they can reinforce each other in noncompliance and have an entire school day to socialize.

And then this section, which caused more flashbacks:

Overall, 55 percent of CPS students scored below the 50th percentile and thus should have been assigned to double-dose algebra, but only 42 percent were actually assigned to the support class. In addition, some students took double-dose algebra, even though they scored above the cutoff on the exam.

You’re thinking, wait. Some of the weak kids didn’t get intervention, and some of the strong kids did? That’s a weird fluke, isn’t it?

And so, another anecdote.

My strongest intervention kids had taken Algebra I the year before. Each of these six kids had scored higher on their state test than my average score for all my non-intervention algebra students. Yes, you read that right. Six of my intervention kids were good enough for the top half of my non-intervention algebra class. Not just better than my worst. Better than HALF the 100 students in my non-intervention classes. Two of them had actually achieved Basic on the previous year’s test. I got them to bring in their test scores and show them to the AVP of Instruction, demanding they be put into normal algebra (leaving me out of it, of course). One of them was put into my regular algebra class, and got an A-. The other four missed Basic by just a few points and despite my asking on their behalf, were required to take intervention. This despite the fact that I had over a dozen non-intervention freshmen who’d scored Below Basic or Far Below Basic. None of it mattered.

I was so foolish as to write the AVP of Instruction saying randomly, casually, something like “Hey, okay, so I can’t move strong students out. But so long as I’m teaching an intervention class for really really weak students, could I move some of my weak non-intervention class IN? Some of them are even Resource (sped) students, so they could substitute the intervention class for their guided studies class, so it wouldn’t create a scheduling disaster (sped kids get a study hall). Here’s a list.”

AVP of Instruction wrote back, in all caps, “THESE STUDENTS ARE SOPHOMORES. SOPHOMORES CAN’T TAKE INTERVENTION.”

Three weeks later, as God is my witness, the AVP of Instruction sends me a note, “I’m moving Fred McInery [not his real name] into your intervention class. He is a weak math student who needs more support.”

I look up Fred. He is a sophomore. I am very excited, because I am a moron, and send her a note. “Hey, great! We’re putting sophomores in intervention now? Could we revisit my list? I really think it will help these extremely weak students succeed in math.”

She writes back in all caps, “THESE STUDENTS ARE SOPHOMORES. SOPHOMORES CAN’T TAKE INTERVENTION.”

WE HAVE ALWAYS BEEN AT WAR WITH EASTASIA.

That story is an amusing flashback, even if it is crazy-making. Here is a horrible one:

We shall call her Denise. She is a doll. She was in my intervention class and had extremely weak skills and was the propaganda child for intervention, the one that everyone is thinking of when they propose it, because she worked her ass off and actually became better at math. She was not “just below the cutoff point”, either, but an FBB child who surreptitiously counted on her fingers to add 4 + 2. But conceptually, she got it. In the first semester, she did so poorly she was one of my contract students. She improved dramatically on her test, missing Basic by just a point. She had the third highest state test score of my intervention students, and passed my class.

Not only didn’t the school move her on to Geometry, but they put her into intervention again. (Yes. Now they had a sophomore intervention class.) When Denise told me this, I went quietly berserk and emailed the AVP of Instruction. It is not the same AVP. This one is worse.

Keep in mind, this second year at the same school, I am teaching Geometry, thank all the gods, and have two of my last year’s intervention kids taking my class even though they received slightly lower state test scores than Denise. Five others of my kids have also moved on to geometry with lower state test scores. Denise and three others were kept behind. I email the AVP of Instruction—a different one, as last year’s AVP has been promoted to principal of another school. This AVP is much worse–and point out these facts. I do not point out that I can discern no organizing principle behind this decision, that I suspect a very disorganized AV principal behind it. I am very polite; hey, this is just some oversight? Want to make sure it gets cleared up.

I write three notes, all very polite, and finally, a month after school starts, Denise gets moved….to a regular Algebra class. I gnash my teeth, but Denise is thrilled and thanks me profusely.

I see Denise at the year-end, and ask how she’s doing. “Great. But I failed the first semester of geometry, so I’ll have to go to summer school.”

“What? They put you in geometry?”

“Yeah, they said you advised it. But they didn’t move me until, like, November, so I failed. But that’s okay. I did good second semester, and I’m going to pass it over the summer.”

I guess it worked out okay, ultimately. But had she been put in the geometry class originally, she’d have had her summer.

Double-dosing had an immediate impact on student performance in algebra, increasing the proportion of students earning at least a B by 9.4 percentage points, or more than 65 percent. It did not have a significant impact on passing rates in 9th-grade algebra, however, or in geometry (usually taken the next year). Double-dosed students were, however, substantially more likely to pass trigonometry, a course typically taken in 11th grade. The mean GPA across all math courses taken after freshman year increased by 0.14 grade points on a 4.0 scale.

(emphasis mine)

Clearly, most students did not do all that well. As the study acknowledges, the low-achieving students did not benefit at all from the intervention; the students most likely to benefit were those who just missed the cutoff. More on that later.

Here’s what the study doesn’t make clear: many high school algebra students never make it to trig. They take it twice in high school, then take geometry twice. Or they take algebra once, geometry twice, and algebra 2 without trig (that’s the class I teach). So are they only counting the students who made it to trig?

The more meaningful stat would be the percentage of double-dosed kids who made it to trig vs. the non-double-dosed kids who achieved same. Reading this passage, the study appears to be saying that all the kids made it to trig and hahahahahaha, no. Not happening.

And since that’s not happening, then who, exactly, is being compared in the GPA? All the kids, or just the ones that made it to trigonometry? Presumably, just that set, because otherwise, the GPA number isn’t worth much. Hey, the double dose kids who flunked algebra twice and made through geometry by their senior year had a GPA .14 points higher than the single dose kids who made it through trig. Whoo and hoo.

It is important to note that many of these results are much stronger for students with weaker reading skills, as measured by their 8th-grade reading scores. For example, double-dosing raised the ACT scores of students with below-average reading scores by 0.22 standard deviations but raised above-average readers’ ACT scores by only 0.09 standard deviations. The overall impact of double-dosing on college enrollment is almost entirely due to its 13-percentage-point impact on below-average readers (see Figure 3). This unexpected pattern may reflect the intervention’s focus on reading and writing skills in the context of learning algebra.

(emphasis mine)

Oh, yes. That’s what we do in these algebra intervention classes. We focus on reading and writing! We’re given a bunch of kids who add 8 to 6 on their fingers, and we figure their struggle comes from not being able to read the word problems. So we put up a word wall and teach them five new terms and suddenly their reading skills skyrocket wildly.

Or—and this is just a wild, random, thought—perhaps my last school isn’t the only school in which Set A = {names of students taking double dose algebra} and Set B = {names of students taking double dose English} and is a Venn diagram in the two circles largely overlap?

You say oh, don’t be silly, ER. Of course they’d account for the possibility that the double dose algebra kids are also getting a double dose of reading intervention! And then not mention it! And I say, you don’t read much educational research, do you?

Because keep in mind the conclusion of this research:

As a whole, these results imply that the double-dose policy greatly improved freshman algebra grades for the higher-achieving double-dosed students, but had relatively little impact on passing rates for the lower-achieving students.

Apart from that, Mrs. Lincoln, how’d you like the friggin’ play?

Look. None of my outraged noise makes any sense at all if you don’t realize that, in the world of high school math, the kids who benefited, according to this study, kids achieving just below the passing standard, are WAY ABOVE AVERAGE for that population, particularly in a Title I school. Intervention exists because these schools have dozens, if not hundreds, of algebra students who have taken the course three times and still score Far Below Basic. It does not exist to help kids just below the 50% mark in math get better scores in reading, marginally higher grades and ACT scores, and better Trig scores—if they get to trig, which the normal intervention kid does not.

What people fondly imagine algebra intervention to do is this: kids are just a little behind, you know? They just need some extra time learning integer operations and fractions. They didn’t learn it the FIRST FIFTEEN TIMES they were taught it, so all they really need is another hour or so a day and they’ll be right up there with the rest of them, all right? And if they aren’t, well, it’s those damn teachers who just don’t want to work with “those kids”, and we’ll just have to find more teachers who really, really care about these kids who just need a few hours more help than the others. (Yes. This is the myth of “They’ve never been taught…..”)

Meanwhile, forty percent of the freshman class comes in having taken algebra once and scored far below basic or barely below basic, and are randomly assigned to double block or no double block using a dartboard, from what I can see. The teachers are dealing with the same lack of basic skills in both double and single block algebra, and rapidly realize (if they didn’t know already) that the kids who don’t know integer operations and fractions have this gap because they aren’t terribly bright. They can’t come up with an intervention vs. non-intervention approach, because some kids in the intervention class don’t need support while some kids in the non-intervention class do. But in the non-intervention classes, the teachers only have to deal with 3-8 kids with low skills, while in the intervention classes it’s 14-15 out of 20. So the only thing different about the intervention classes is monstrously bad behavior and more time in hell.

All this, mind you, so that we can do research that reveals no real improvement in outcomes.

But I’m out of it, baby. It’s enough to make me believe in god. Death to algebra intervention.

But the nightmares, they won’t stop until it’s destroyed!

100 Posts

I started this blog on January 1 with two primary goals. As I mentioned in my initial post, I’d gone a whole year without writing anything for publication (under my real name, which is not Ed). I wanted to initiate more and respond less, and I wanted my writing here to spur me to write more under my real name. As a second goal, I hoped to reflect the full spectrum of my views on education and teaching—and nothing more. I wanted to write both about teaching and educational policy. I teach a great many subjects, and the joys of teaching composition, literature, American history, and text prep ideally needed some of my writing attention, while I expected to write primarily about the challenges of teaching math (yes, I am saying there are relatively few joys in teaching math, but that doesn’t mean I’d give it up.) When the subject turned to educational policy, I expected to focus on the degree to which Voldemortean avoidance prevents us from sane, realistic objectives, but I also intended to discuss the very real problems I saw with both eduform and progressive math philosophies.

Thus far, the blog has exceeded my goals. I had one very successful piece go out under my own name, and to the extent I haven’t written more it’s been because of time constraints. I have plenty of ideas, which was not the case a year ago, when I felt hamstrung. I also think my posts fairly reflect all my teaching interests.

What I didn’t expect, and has been deeply satisfying, is the degree of attention many of my posts have had. Here are the top six posts:

1. Algebra, and the Pointlessness of the Whole Damn Thing

2. The myth of “They weren’t ever taught…”
3. Teacher Quality Pseudofacts, Part II
4. Why Chris Hayes Fails
5. The Gap in the GRE
6. Homework and Grades

I wrote all but one of these hoping they’d get a big audience—Homework and Grades is the exception; while it got a nice bump when it first came out with a link from Joanne Jacobs, most of the activity has been from consistent attention over time. People refer to it a great deal, for some reason. (Actually, half of my big pieces got link love from JJ, and a host of smaller ones as well, which means a lot because she’s the best pure education blogger out there. Other bloggers who contributed a lot of readers to the above pieces: Steve Sailer and Gene Expression.)

Three policy pieces that I was personally pleased with, audience or not: Why Chris Christie Picks on Teachers, On the CTU Strike, and The Fallacy at the Heart of All Reform.

Three teaching pieces that are regularly linked to or used as references by teachers: Modeling Linear Equations, Teaching Algebra, or Banging Your Head with a Whiteboard, and Teaching Polynomials.

Total views at the time of this post: 38,000

I’d also like to shout out to my commenters and Twitter readers. Thanks for your great feedback.

So on to the next 100. I’d say I’ll try to keep them brief, but that’s a big lie.

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.

Mapping Real Life with Coordinate Geometry

Yesterday, I wanted to close off the coordinate geometry section (distance, midpoint) before I moved into logic. Rather than put a few random problems on the board, I came up with a map description.

Han is a driver for Harley’s Restaurant Supplies, making his Monday morning route.

1. He went due north for three miles, dropping off supplies for diNardo’s.
2. He then cut northeast along Steep Street to Patel’s Naan and Curry shop. He could have gone due east for 8 miles along Grimley, and then two miles due north along Freeman, but he wanted the shortest route.
3. He then drove southwest along Morespark, back past Harley’s, all the way down to Bob’s Burgers. Harley’s is exactly halfway between Patel’s and Bob’s.
4. Next stop, 17 miles due east along AutoBahn Boulevard to Andy’s Noodle Shop.
5. Then it was northwest along Bracken Drive to Tomas’s Taqueria, which was just 6 miles due north of Harley’s.
6. Back to Harley’s for his lunch of his noodles, naan, tacos, and burgers before he started out for the afternoon.

A. Create a map of Han’s route, including street, restaurant names, and coordinates. Suggest using (0,0) for Harley’s.
B. How far did Han travel?

All the students had their own whiteboards, so they could sketch and erase as needed. Step A, the sketch, went really well. As I expected, step 2 gave students the most difficulty, but a third of the class understood it without assistance, and the rest had drawn the two descriptions as two different locations.

First student finished with the sketch on whiteboard:

(Yeah, no street names. He put them in after I took the picture).

As students began moving from the sketch to calculating the distance, I brought it back up front. What was the difference between finding the distance from Bob’s to Andy’s (due east) and Andy’s to Tomas’s (northwest)?

This is the final product—I forgot to take a picture mid-lesson. But see how some trips are starred, and some have a plus. The class identified the stars, which required further calculation to get the difference. I stress the “slope triangle” in all aspects of coordinate geometry (slope, midpoint, distance), and you can see my light colored sketches of the three relevant triangles.

Later, the class identified the missing distances, and then we added it all up. Final instructions: transfer all of this to a quality sketch in your notes. Use color to identify the triangles.

When I teach the Big Three of Coordinate Geometry (slope, midpoint, distance), I emphasize the triangle because for so many students, the formulas are just one more reason to get negatives and subtraction all hosed up. Sketch in the triangles, and you’ve got a backup. Does this answer make sense? Yes, it’s fine if they use the formulas. I will forgive them. Provided they don’t muck up the math. And remember, knowing the formulas is essential. I want them to recognize the format of each formula, even if they never use them.

This took about 45 minutes? Wrap up and transition, maybe 55 minutes.

A few days ago, an a**l obsessive overly rigid teacher called me lazy for not having weeks of lesson plans written in advance. I am usually pretty nice to commenters (which is, like, so not me) but while I don’t object to teachers who plan, I vehemently object to teachers who confuse planning with teaching, and this guy is a prime example of the moralizing putz who never got over his potty training and wants everyone else to suffer his pain.

But here’s the thing: I built this lesson in the fifteen minutes before the day started. I do not think my ability to do so is an essential aspect of good teaching. But it’s a part of teaching I really enjoy, the combination of a) my understanding of my kids’ immediate need and b) my strength at creating interesting lessons on the fly. Forcing me to put together a schedule weeks in advance would either make a liar of me or take away that essential piece of my teaching. I’d become a liar, of course. But why go through the farce?

Teaching Math vs. Doing Math

Justin Reich of EdWeek (not to be confused with Justin Baeder of EdWeek) wrote enthusiastically of a new study, asking What If Your Word Problems Knew What You Liked?:

Last week, Education Week ran an article about a recent study from Southern Methodist University showing that students performed better on algebra word problems when the problems tapped into their interests. …The researchers surveyed a group of students, identified some general categories of students’ interests (sports, music, art, video games, etc.), and then modified the word problems to align with those categories. So a problem about costs of of new home construction ($46.50/square foot) could be modified to be about a football game ($46.50/ticket) or the arts ($46.50/new yearbook). Researchers then randomly divided students into two groups, and they gave one group the regular problems while the other group of students received problems aligned to their interests.

The math was exactly the same, but the results weren’t. Students with personalized problems solved them faster and more accurately (emphasis mine), with the biggest gains going to the students with the most difficulty with the mathematics. The gains from the treatment group of students (those who got the personalized problems) persisted even after the personalization treatment ended, suggesting that students didn’t just do better solving the personalized problems, but they actually learned the math better.

Reich has it wrong. From the study:

Students in the experimental group who received personalization for Unit 6 had significantly higher performance within Unit 6, particularly on the most difficult concept in the unit, writing algebraic expressions (10% performance difference, p<.001). The effect of the treatment on expression-writing was significantly larger (p<.05) for students identified as struggling within the tutoring environment1 (22% performance difference). Performance differences favoring the experimental group for solving result and start unknowns did not reach significance (p=.089). In terms of overall efficiency, students in the experimental group obtained 1.88 correct answers per minute in Unit 6, while students in the control group obtained 1.56 correct answers per minute. Students in the experimental group also spent significantly less time (p<.01) writing algebraic expressions (8.6 second reduction). However, just because personalization made problems in Unit 6 easier for students to solve, does not necessary mean that students learned more from solving the personalized problems.

(bold emphasis mine)

and in the Significance section:

As a perceptual scaffold (Goldstone & Son, 2005), personalization allowed students to grasp the deeper, structural characteristics of story situations and then represent them symbolically, and retain this understanding with the support removed. This was evidenced by the transfer, performance, and efficiency effects being strongest for, or even limited to, algebraic expression-writing (even though other concepts, like solving start unknowns, were not near ceiling).

So the students who got personalized instruction did not demonstrate improved accuracy, at least to the same standard as they demonstrated improved ability to model.

I tweeted this as an observation and got into a mild debate with Michael Pershan, who runs a neat blog on math mistakes. Here’s the result:

I’m like oooh, I got snarked at! My own private definition of math!

But I hate having conversations on Twitter, and I probably should have just written a blog entry anyway.

Here’s my point:

Yes, personalizing the context enabled a greater degree of translation. But when did “translating word problems” become, as Michael Pershan puts it, “math”? Probably about 30 years old, back when we began trying to figure out why some kids weren’t doing as well in math as others were. We started noticing that word problems gave kids more difficulty than straight equations, so we start focusing a lot of time and energy on helping students translate word problems into equations—and once the problems are in equation form, the kids can solve them, no sweat!

Except, in this study, that didn’t happen. The kids did better at translating, but no better at solving. That strikes me as interesting, and clearly, the paper’s author also found it relevant.

Pershan chastised me, a tad snootily, for saying the kids “didn’t do better at math”. Translating math IS math. He cited the Common Core standards showing the importance of data modeling. Well, yeah. Go find a grandma and teach her eggsucking. I teach modeling as a fundamental in my algebra classes. It makes sense that Pershan would do this; he’s very much about the why and the how of math, and not as much about the what. Nothing wrong with this in a math teacher, and lord knows I do it as well.

But we shouldn’t confuse the distinction between teaching math and doing it. So I asked the following hypothetical: Suppose you have two groups of kids given a test on word problems. Group 1 translates each problem impeccably into an equation that is then solved incorrectly. Group 2 doesn’t bother with the equations but gives the correct answer to each problem.

Which group would you say was “better at math”?

I mean, really. Think like a real person, instead of a math teacher.

Many math teachers have forgotten that for most people, the point of math is to get the answer. Getting the answer used to be enough for math teachers, too, until kids stopped getting the answer with any reliability. Then we started pretending that the process was more important than the product. Progressives do this all the time: if you can’t explain how you did it, kid, you didn’t really do it. I know a number of math teachers who will give a higher grade to a student who shows his work and “thinking”, even if the answer is completely inaccurate, and give zero credit to a correct answer by a student who did the work in his head.

Not that any of this matters, really. Reich got it wrong. No big deal. The author of the study did not. She understood the difference between translating a word problem into an equation and getting the correct answer.

But Pershan’s objection—and, for that matter, the Common Core standards themselves—shows how far we’ve gone down the path of explaining failure over the past 30-40 years. We’ve moved from not caring how they defined the problem to grading them on how they defined the problem to creating standards so that now they are evaluated solely on how they define the problem. It’s crazy.

End rant.

Remember, though, we’re talking about the lowest ability kids here. Do they need models, or do they need to know how to find the right answer?

Teaching Students with Utilitarian Spectacles

In my last post, commenter AllaninPortland said, of my Math Support students, “Their brains are wired a little too literally for modern life.”

James Flynn, of the Flynn Effect:

A century ago, people mostly used their minds to manipulate the concrete world for advantage. They wore what I call “utilitarian spectacles.” Our minds now tend toward logical analysis of abstract symbols—what I call “scientific spectacles.” Today we tend to classify things rather than to be obsessed with their differences. We take the hypothetical seriously and easily discern symbolic relationships.

Yesterday I gave my math support kids a handout on single step equations similar to the one in the link.

“Oh, I know how to do this,” said Dewayne. “Just subtract six from both sides.”

“You could do that,” I said. “But here’s what I want people to try. I want everyone to read the first equation as a sentence. What is it saying?”

“Some number added to six gets fourteen,” came from Andy.

“Excellent!”

“You mean, you don’t want us to subtract, add, do things to get x by itself?” asked Jose.

“That’s called ‘isolation’. You are ‘isolating’ x, getting it all by itself as you put it. Who knows how to do that?” Over half the class raised their hands. “Great. You can do that if you want to, but I’d like you to try seeing each equation just as Andy described it. Put the equation you see into words. This will help make it real, and will often give you the answer right away. For example, what number do I add to six to get 14?”

“Eight.” chorused most of the room.

“There you go. Now, remember, what did I say a fraction was?”

“Division.”

“So instead of saying ‘x over 5′, you’re going to say….”

“X divided by 5″ came back a number of students.

“Off you go.”

This worked for most of the students, but one student, Gerry, sat at the back of the room drawing, as he often does. After watching him do no work for 10 minutes, I called him up front. (Normally, I am wandering the room, but every so often I call them up for conversations instead.)

“So you aren’t working.”

“Yeah. I can’t do this.”

“Remember yesterday, when we were doing those PEMDAS problems? You were on fire!”

“Yeah, but it didn’t have the letters in it. I can do math when it doesn’t have letters. And yesterday, when you showed us how to just draw pictures for the word problems? That was cool. I think I can do those now.”

“You need to look at these problems from a different part of your brain.”

“A different what?”

“This is a really, really easy problem. Way easier than the math problems you solved in your head yesterday. But you don’t see this as the same kind of problem, so we have to fool your brain.”

“How do we do that?”

“Read the first problem aloud.”

“X + 6 = 14. This is when you have to do stuff to both sides, right? I can’t do that.”

“Read it again. But instead of saying x, say ‘what’.”

“Say ‘what’?”

“Yep.”

“You crazy.”

“Definitely. Try it.”

“What plus 6 = 14? 8.”

“There you go.”

He was sitting in one of my wheeled chairs, pushing it back and forth with his feet. This stopped him cold.

“Eight’s the answer? Holy sh**.”

“Try another. Without the language.”

“What minus 3 = 7. That’s nine…no, 10. Ten? Really? No f**k….no way.”

“And this one?”

“Oh, that’s a fraction. I can’t do those.”

“What did I tell you fractions were?”

“Division. Oh. What divided by 5 is 9? Forty five? No way?”

“So. I want to see you do this whole handout, 1-26, and every time you see an x, call it ‘what’. Remember to sketch out subtraction questions on a numberline and think about direction.”

“Okay. Man, I can’t believe this.”

Fifteen minutes later, Gerry was done with the entire set. Only three minor errors, all involving negative numbers.

“I feel like a math genius,” he said with a wry grin.

I sat down next to him. “It’s like I said. We have to ask your brain a different question. So instead of tuning me out, next time I come up with some goofy idea using pictures or tiles or different words, give it a shot. And tell me if it works to give your brain the right question. Some of my ideas will work, some won’t. And some things, we won’t be able to fool your brain to answer a different way. But you know a lot more math than you think you do. You just have to figure out how to ask the question in a way your brain understands.”

Back to Flynn:

A greater pool of those capable of understanding abstractions, more contact with people who enjoy playing with ideas, the enhancement of leisure—all of these developments have benefited society. And they have come about without upgrading the human brain genetically or physiologically. Our mental abilities have grown, simply enough, through a wider acquaintance with the world’s possibilities.

But not everyone is capable of understanding abstractions to the same degree. Some people do better learning the names of capitals and Presidents and the planets in the solar system. They’d learn confidence and competence through interesting, concrete math word problems and situations, and enjoy reading and writing about specific historic events, news, or scientific inventions that helped society. Instead, we shovel them into algebra, chemistry, and literature analysis and make them feel stupid.

Students’ names have been changed. They are all awesome kids. Do not say mean things about them in the comments, which I can control, or other blogs, which I cannot.

Math fluency

My Math Support class, for students who haven’t yet passed the state graduation test, is the most challenging of my preps. In many ways, though, the class offers the dream scenario for any math teacher who longs to focus on fundamentals.

I owe no allegiance to a curriculum. I’m not teaching arithmetic in and around an algebra course; arithmetic and a tiptoe into algebra is all the test requires. I only have 18 students (16 boys) in a 90 minute class, so I have tons of time to work one on one. While the kids probably wouldn’t strike the average observer as motivated, they are juniors and seniors who want to pass the test, so by their internal standards, motivation is high. Many (but not all) of the kids are acknowledged classroom challenges at the school. However, this school’s notion of a serious classroom challenge is something around the 30% mark of the students I taught for the last two years, so my basement has moved way, way up the stairs.

So I have a small class, a meaningful curriculum, motivated kids with low abilities, and, for that population, no significant management challenges. I was, and am, enthusiastic about the opportunity. However, please take renewed notice of the blog name. I am not under the impression that these students have merely been waiting for The Messiah, after years of suffering through false prophets (aka bad teachers). I was eager to see which of my assumptions played out, and which didn’t, and I wanted to test, anecdotally at least, some commonly held wisdoms that hadn’t, in my limited experience, borne out.

For example, I have long suspected that the received wisdom about math fluency has holes in it:

Educators and cognitive scientists agree that the ability to recall basic math facts fluently is necessary for students to attain higher-order math skills. Grover Whitehurst, the Director of the Institute for Educational Sciences (IES), noted this research during the launch of the federal Math Summit in 2003: “Cognitive psychologists have discovered that humans have fixed limits on the attention and memory that can be used to solve problems. One way around these limits is to have certain components of a task become so routine and over-learned that they become automatic.”

The implication for mathematics is that some of the sub-processes, particularly basic facts, need to be developed to the point that they are done automatically. If this fluent retrieval does not develop then the development of higher-order mathematics skills — such as multiple-digit addition and subtraction, long division, and fractions — may be severely impaired. Indeed, studies have found that lack of math fact retrieval can impede participation in math class discussions, successful mathematics problem-solving, and even the development of everyday life skills. And rapid math-fact retrieval has been shown to be a strong predictor of performance on mathematics achievement tests.

I used to accept this as a given until seven years ago, when I ran into my first kid who knew his math facts cold but couldn’t solve 2x + 7 = 11, unless I asked him what number I could multiply by two and add seven in order to get 11 and got the correct response almost before I finished the sentence. By that time, I’d already met a few 600+ SAT students who growled in frustration and reached for the calculator when it came to knowing 6 x 9. I’ve also tutored a dozen or more ISEE/SSAT (private school test) fifth and sixth grade students who went to precious little snowflake schools and knew none of their math facts with any fluency yet easily mastered fractions, ratios, and solving for unknowns and scored in the top 90% of a highly skilled population.

I’ve long since abandoned the notion that fluency might be necessary, but not sufficient, given the last group. Kids who can abstract can cope without fluency. What’s troubling is that fluency might be irrelevant.

None of this means we shouldn’t emphasize fluency. But plenty of solid math students don’t have fluency and—here is the important part—many exceptionally weak math students have strong fact fluency.

Every week, I get an extra 20 minutes with each of my classes. In Math Support, I use this time for drill competitions. The kids pair up and get a selection of MDAS flash cards. I set the timer and holler “GO!” First kid holds up cards for the second kid and go through the cards as fast as they can—correct answers in one pile, missed in the other. I stress that the “miss” is determined in 2-3 seconds for most kids (more on that in a minute). If the kid hesitates, it’s a miss.

I originally set the timer for 2 minutes, but all but two of the kids get through a whole pile of 30 cards in one minute, so I dropped it down to a minute.

The kids’ fluency falls into one of these zones:

• High: I mean, 7×12, 6×9, 7×8 high. 121/11, 96/12 high. 7+9 and 15-8 high. No hesitation, no pauses. The five students in this group all struggle with abstractions, although two of them have solid arithmetic competency and excellent estimation skills. The rest struggle in every area of math. All of them test poorly, all are seniors.
• Solid: Fluent except the usual suspects: higher 12s, the cross sections of 7, 8, and 9 and a few hard to remember addition/subtraction facts. Many of these kids have told me that this activity is improving their recall of their problem facts. All of my overall strongest students are in this category, the rest are average. Seven in total.
• Weak: Say about 50% mastery. Four students, not noticeably different otherwise from the “average” students in the solid category. I haven’t yet noticed any improvement, but they’d likely take longer.
• Non-existent: I have two kids who can’t quickly recall their 2 multiplication facts, struggle with basic addition. Clearly some sort of memorization issues. These two are given 6 seconds per card before it’s counted as a miss.

One of the two students in the non-existent zone is, hands down, the strongest procedural algebra student in the class. She can solve multi-step equations and identify linear equations from a graph. I have explained fractions and ratios to her on several occasions, and it all escapes her instantly. So no fluency, no proportional thinking, but algebra procedures and linear equations. If she can operate by rote, she’s fine. I haven’t checked yet, but I’d bet she can master the quadratic formula (with a calculator) more easily than factoring binomials. My strongest overall students, while not as solid on algebra procedures, are much stronger at proportional thinking, more capable of thinking abstractly, and are all in either geometry or algebra II. (Why yes, you can get to algebra II without passing the state math graduation test. Happens constantly.)

All of my students easily manage multiple digit addition and subtraction. A few of them are completely unfamiliar with long division. Fractions are a struggle for most of them. All but a few understand and use distribution. Combination of like terms, not so much. They all do quite well simplifying exponential expressions and have a solid grasp of scientific notation.

What does this mean? Beats me.

Assertion: Students who are categorically failing in math are almost certainly not doing so because of math fluency. They may or may not be fluent, but fluency is not the condition holding them back.

Tentative hypothesis: The rationale for math fluency (quoted above) does hold for many students who are moving through the math curriculum without ever achieving genuine proficiency, who would certainly be able to learn and hold onto more information if they weren’t spending so much of their time trying to remember what 6 x 3 is, particularly in algebra.

So go ahead and drill. Just remember that the kids it will help the most aren’t the ones you’re worried about, and many of the ones you’re worried about won’t need the drill.

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.

Geometry: Starting Off

The first day or two of geometry is always point line plane. We never really use it again. Geometry has mostly been subordinated to algebra in high school, as I’ve written before, and my geometry class is best thought of as algebra applications with geometry. Or is it the other way round? Purists see geometry as the medium for introducing proofs, logic, and construction. To which I say pish tosh. Most of them are never going to see those subjects again. “But if they don’t learn rigorous logic in geometry, they won’t be able to learn advanced math!” Yeah, that’s moronic nonsense. What is “solve for x”, if not a proof?

But I love history, so I always start by telling them to put their pencils down and just listen as I explain the significance of Euclid’s Elements and the wonder of a book written 2300 years ago. Three hundred years ago is older than our country. Euclid wrote Elements 300 years before the birth of Christ, so Christ’s contemporaries (the educated ones) thought of Euclid much the way we think of Alexander Hamilton or George Washington. Take seven additional chunks of people looking back 300 years and here we are. A book written that long ago was “in print” over 1000 years before “print” existed, and since then, is second only to the Bible in published editions—not just in the English language, which had to wait another 100 years after the Latin version was published, but in all languages.

As to writing another book on geometry [to replace Euclid] the middle ages would have as soon thought of composing another New Testament.–Augustus de Morgan

Why? Because he* nailed it. For over 2000 years, his model met the world’s requirements, and when the world finally found limits to his model, it wasn’t because he was wrong.

Euclid was nagged by his “fifth postulate”, which is easier to sketch than describe:

That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

If you’re not a mathematician—and I am not—you’re like, um, duh? What else is going to happen? The lines will meet up. But Euclid and other early mathematicians knew the fifth postulate wasn’t the same as the other four, and that’s almost certainly why he established the first 28 theorems without reference to it. For the next couple millennia, mathematicians tried to prove the fifth postulate using the other four, and failed. This collected history of effort around that single postulate ultimately led to the realization that there other, non-Euclidean geometries, many of which (if I understand this properly) begin with the negation of the fifth postulate. This discovery rocked the world, robbed it of a truth previously assumed absolute, and ultimately contributed a bit to Einstein’s theory of relativity.

And 2300 years ago, Euclid needed the fifth postulate to complete his model, furrowed his brow and said, “Yeah, hmm. Something’s not right about that one.”

I tell my students that I’m not a mathematician, and that they don’t need to be, either, in order to realize what a stunning achievement Elements is, and to realize the significance of math in our world that thousands of years ago, mathematicians knew enough to be bothered by a postulate that seemed obvious but was yet somehow different from the others needed for the model. That they, my students, are studying an incredibly old math, one that holds up for our ordinary requirements to this day, but also created the foundation for deeper, more complex models. That if they don’t like math, don’t like geometry, to at least appreciate it from a historical standpoint.

I am probably fooling myself a little bit, but the kids always seem interested. Which is all I’m looking for. Just to show I’m not making this up, here are my board notes:

(Yes, my board work sucks. It’s something I build as I go through a lecture most of the time, a document in progress. I’ve started taking pictures of my boardwork to get a better sense of what I said, what I emphasized, and what I could do to improve boardwork next time.)

Then I go onto undefined terms—not just the terms for geometry, but the meaning of undefined terms. Here, again, Euclid nails the building blocks for his model. (Geometry books give point, line, and plane as the three undefined terms, but I also spend time on “congruence” and “between”.)

Then I show how the building blocks of the undefined terms allow us to define everything else in the Eucliden model. I usually use ray, segment, and angle just to give the the idea.

This year, I decided I wanted to do more with 3-dimensional graphing (xyz) and introduced it as part of this lecture. First, the students learned to represent three dimensional planes without a coordinate system, and see for themselves what happened when two planes intersected. The kids had fun with that; here’s one of the best:

Then we went into more formal xyz graphing. I’m including more 3-d graphing this year to help prepare students for 3 variable systems next year, and also to give the students more variety in visualizing images. Click on the board work below to see that I draw in the rectangular prism, which helps students grasp the difference between 2-axis graphing, in which any two points are a diagonal in a rectangle, and 3-axis graphing, in which any two points are the vertices of a rectangular prism. I heard a lot of “ahas” as I went through this. Not sure what the next 3-d graphing activity will be, but I think I’ve started with a good foundation.

So that was the first day, really. Then I went into the meat of unit one: angle types, angle pairs, perimeter and area formulas, and as always, using these relationships to set up equations and solve with the ever loved algebra.

Here’s the first test. I think I caught all the glitches after I captured this. But I’m sure I missed something; I have a pathological tolerance for typos.

*I’m assuming it was just Euclid. More fun that way.

Binomial Multiplication and Factoring Trinomials with The Rectangle

I was reading about Joseph Nebus’s factoring method….

(okay, a brief note. Early in his writeup, Nebus (Joe? Joseph?) writes: “It’s a method for factoring quadratic expressions into binomial expressions, and I must admit, it’s not very good. It’s cumbersome and totally useless once one knows the quadratic equation.”

Many, many math teachers have expelled much breath on the uselessness of factoring, as the skill is completely nullified by the quadratic formula (which I think what he means here). But when they make this comment, they are thinking as mathematicians, not as teachers. Mathematicians work with math to solve problems. Teachers teach math so their kids can demonstrate their knowledge on tests*—not just state tests, but college admissions tests and placement tests. And on these tests, the questions are designed for either factoring or the quadratic formula—and far more the former than the latter. All students must learn to factor trinomials if they are to escape remediation. The quadratic formula is optional. And if the test pragmatism isn’t enough of an argument, please know that students with limited integer operations skills will do better with factoring than the formula because they rarely have squares memorized and please, please believe me when I say that they will ALWAYS subtract 4ac from b squared, even if c is negative. End brief note.)

There are teachers who think this is a science, and teachers who know it’s an art based largely on the audience. And which kind of teacher you are is a religion, or an expression of personality (which I often think is the same thing).

So when I say that the method Nebus describes sounds extremely convoluted, I am simply a Jehovah’s Witness expressing doubt about the utility of the Amish rumspringa.

But many math teachers aren’t even aware of the box/diamond method, and many others who do use it don’t teach it in a fully integrated manner. So for the teachers of the artist mindset looking to find the right method for certain audiences, here’s an outline of the method.

I got the approach from CPM. I don’t know if it originated with CPM, so apologies if the original idea goes back earlier. CPM’s curriculum is insanely irritating: text heavy, lousy examples and insufficient practice. But in many cases, its approach to a topic provides a beautiful, fully integrated, and consistent framework that I steal without shame.

Factoring out common terms

I always introduce the generic rectangle when introducing or reviewing simple factoring (pulling out common terms). The area model uses the fact that the rectangle’s area is both the product of the length and width and the sum of the individual areas. You break up the side of the rectangle into as many different segments as there are terms.

So 8x + 18 is the sum of two areas, both created by a product of length and width. One side is used for both areas, so it must be a factor common to both areas. In other words, what is the greatest common factor of both terms?

Once you find the GCF, work backwards. What do I multiply by 2 in order to get 8x? Most students do well on this, but if they struggle, I show them how to divide in order to find the answer.

I don’t stress its use here, as I do during binomial multiplication; I just want students to be familiar with my use of it. At the same time, I always find a few students who struggle with factoring common terms and find the approach helpful.

Binomial Multiplication

I do not mention FOIL, although most of my students have learned it at one time or another. While I don’t require my students to use any particular method for tests, I require them to use the area model method for binomial multiplication at least for a day or two so they understand the underpinnings of the factoring method.

So obviously, binomial multiplication is the opposite of factoring; the terms go on the outside of the box and generate the individual areas. (x+2) is a segment of length of x and 2, (x+3) a segment of x and 3. I always point out that the lengths don’t need to be accurate or drawn to scale.

I demonstrate this method several times, up front. I explain the area concept again and how multiplication of length times width for each smaller rectangle is the same as the area of the larger rectangle. I don’t really expect my students to be able to repeat it back to me. What I expect, or hope, they will think is “Oh, okay, that makes sense”. Because from this point on, when they think of this method, I want them to remember that the method made sense to them, even if they don’t remember the specifics. That’s also why I don’t write any of this down—most of my students will toss any documentation, anyway. I work a variety of examples (at this point, a=1), picking students to walk through the process.

The kids have a handout with 20-30 problems (this is one of the few topics that kuta software doesn’t have a good handout for), but I don’t have my usual handout online. The original problems would all be a=1, b and c of all signs, because I want them to work dozens of problems and see the pattern, if they are able to. Then, on day 2, 3, and 4, I introduce difference of two squares (what happens to b?), a>1, and 2×3 or 3×3 polynomial multiplication—which works really well with this model, as the kids just make a bigger rectangle.

I wish I could say that this method eliminates the problem of (x+2)(x+3) = x² + 6. Alas. However, when a student makes the mistake and I scowl and draw the rectangle, with no other explanation, 9 out of 10 kids making the mistake go “Oh, yeah.” That’s the win, such as it is.

Factoring Trinomials

So after a week or so of multiplication, I point out something interesting about the completed rectangle:

This is particularly interesting when we consider the two “middle” terms that add up to bx. We now know that they add to bx and multiply to the same product as ax² and c.

Interesting, yes, but also useful. I remind the kids that distribution is the inverse of factoring, that distribution converts a product into a sum, while factoring turns a sum into a product. So if they are faced with a quadratic equation in ax² + bx + c—say, for example, x² + 9x + 14—how could they turn this sum back into a product?

I ask the kids, if I’d multiplied two binomials to get x² + 9x + 14, what would have been in the box?

Factoring trinomials is the task of finding the numbers for the other half of the rectangle.

And thanks to the properties of the generic rectangle, we know that the terms we are looking for add to 9x and multiply to 14x², the product of the first kittycorner.

So we use the “diamond” as a visual tool to help find those terms.

No matter what method a teacher chooses to teach, factoring comes down to that question: What do I multiply to get ac that I add to get b?

I teach the students to write out the factors in pairs, starting with 1 and the number itself (otherwise they tend to forget) and working up from there. Remember that I teach students who will often have a tough time remembering all the factors of 24, and pause on each term to remember the pair.

So once you find the terms that meet the requirements, you put them in the box. It doesn’t matter which goes where. I repeat that phrase a lot. I sometimes wonder if I should create a rule for where the terms go, just so I won’t get the question any more.

It’s worth stressing to your students that, while you’ve found the missing terms, you still have one more step! They’ll still forget, and this will bite them back when they start factoring trinomials in which a>1.

The last step involves finding the GCF for each row. This is where I get the payoff for introducing “single row” factoring much earlier. The students are familiar with the process; they’ve seen me explain that the outside terms are the GCF for a month or more, even if they didn’t use it themselves.

Again, I work five or six problems with the class as a group each day. The kids have a page of 20-30 problems they work through; if they finish one page, I give them another with more complex problems. Anyone who can do the work peels off from the class discussion and works independently from the beginning, the rest are “released” after the class discussion. I put worked examples all over the whiteboards to give them models to follow. Many of my struggling students don’t move past a=1. Some of the weakest will only be reasonable competent at c>0 in the first go-round and struggle with finding the difference of two terms for a while. So over the next two-three days, the kids work on factoring at their own pace. The strongest kids are working a>1 by the last day (and their third page of problems), and working problems like x² – 9x = 10, learning to set it equal to 0 and factor.

Here’s an example with c<0:

Here’s a>1—and this, by the way, is where anyone can benefit from the generic rectangle. Any other method of factoring a>1 trinomials is a pain in comparison:

I return to factoring throughout the year. Every so often I’ll declare it time to build on existing skills, so kids who had just gotten competent at c>0 can get more practice time on c1, and then the strongest kids start to identify patterns—identifying perfect squares, difference of two squares, and so on.

As time goes on, I give fewer worked examples and just the general outline below, to see how they do at moving from general to specific:

Next Steps

I have traditionally gone from this to completing the square and quadratic formula, then onto graphing parabolas. I am going to reverse these two topics this year. Teach factoring trinomials, then graph parabolas. Get that going well, and then move onto completing the square, quadratic formula, and then graphing those cases. See how that goes.

Finally, I can’t stress this enough: a quarter or more of my algebra classes are low ability kids, so if you’re thinking Jesus, two weeks or more for multiplying and factoring quadratics? then you aren’t teaching low ability kids or you’re just ignoring the fact that they’re flunking your class. My top kids are doing in depth work on the topic or, in some cases, moving onto another topic entirely.

I’ve been getting some people lately asking, or complaining, that “low ability” is vague. I’m sorry, but it’s not. Potter Stewart was right: you know it when you see it. If you want a specific metric, it’s a kid with cognitive abilities measured at the 50th percentile (say, IQ from 95-105, but that’s a guess). In other words, kids that are perfectly functional in the real world, but simply don’t have the interest or ability for advanced math. Kids with cognitive abilities any lower than that aren’t, as a rule, going to be able to even fake it in algebra, much less anything past that. There are always exceptions.

It’s the delusion of eduformers and progressives, one and all, that if teachers find the right approach, a low ability kid is transformed into a competent high ability kid. In reality, success in teaching low ability kids comes when they start to feel a sense of competence at some level of math. When a kid goes from staring blankly at a trinomial to thinking “Oh, yeah” when I draw the rectangle, that’s a big goddamn win. I believe a lot of kids in this category could learn specific high level math in the context of a concrete task, although I have no evidence of this. But we’d have to sort kids into different groups and sorting’s just one big no-no.

However, this method is helpful for kids of all abilities. High ability kids get a real kick out of seeing the link between the area model and the algebra, and I’ve rarely met a kid who didn’t appreciate the utility of this method for a>1.

I don’t have a handout per se for this whole method; what I’ve just laid out is 8-10 days of practice, followed by days interspersed here and there throughout the year. However, if I get a kid who comes in late, or who wants a specific tutorial, I have a document that I really need to rework, which is why I spent some time creating images for this writeup. But remember, all of this is religion and on factoring, I’m in a state of epistemic closure. Convert or live life as a heretic. I was going to say “Die, infidel”, but really, the current insanity in the mideast takes all the hyperbole out of that statement and thus all the fun.

*If you are a mathematician who is also a teacher, stop hyperventilating. It’s true. You know it is. Embrace the reality we live.