Modeling Exponential Growth/Decay Interspersed with a Reform Rant

Quadratics have become my new nadir, which is cheerier news than it sounds since it means I’ve kicked linear equations into obedient submission. For the first two and a half years of my teaching career, I felt good about quadratics because if nothing else, most kids remembered how to factor, and remembered that factors had something to do with zeros on the graph. Which was a big step up compared to what they retained of linear equations. But then, last year, I cracked linear equations in a big way, which is great except now I just feel bad about quadratics, because as I develop as a teacher I realize the suckers are absurdly complicated and don’t model very easily. The kids learn a lot, but at their level of ability I’d need to do two months to have them internalize quadratics the way most of them internalize linear equations. And I don’t have two months. I just tell myself they still learn a lot. Consequently, I am relieved to see quadratics in the rear view as I move them onto the third of the models that define second year algebra (at least, as I teach it).

Exponential functions are awesome. First, they’re absurdly simple compared to both lines and quadratics. Second, they model actual, honest to god, real life situations. I’m not a big teacher for “Hey, this is something you’ll use again” but automobile depreciation or interest payments are, in fact, something they’ll use again. Third, they provide a memorable and again, useful, reason to review (or learn for the first time) percentage increase and decrease. Finally, they present a situation in which any kid who has even somewhat grasped the course essentials can see hey: Given y, I can’t solve for x. This leads beautifully and meaningfully into logarithms.

So like linear equations, I can kick off the unit with a modeling activity and get the kids moving easily into the math.

I begin with a brief lecture reminding them of the two previous models.

No. Quadratics aren’t repeated multiplication. Exponential functions involve repeated multiplication, as they’ll see in the lesson.

Then I review percentage increase and decrease. I am of two minds about this review. On the plus side, it’s immediately relevant, easy to apply, and gives them a good reason to remember it long term. The downside: the kids never remember what I taught them when they get to the percentage problems. So I explain it up front, knowing that 90% of the kids will forget everything I said just 20 minutes later, when they get to the first percentage exponential increase.

So I explain it, go round the room asking “So, if I want to increase a number by 8%, what do I multiply it by, Jose?” “1 point…..8?” “Watch that leading zero!” “Oh, 1.08.” “Right.” Do that with five or six times, think everyone gets it, and set them to working on models. This is one side of the worksheet, crunched for space so I could “snip” it.

And sure enough, the kids work through the models, making great progress, and stop cold at the third one.

“I can’t do this. How do you increase by a percentage?”

“Excuse me while I beat myself on the head with this whiteboard.”

“What?”

“Nothing. Do you remember me just talking about percentages?”

“Yeah.”

“Do you see it on the board there? All the stuff about turning it from two steps into one step, and why you need to do that?”

“Yeah.”

“DO YOU SEE ANY POSSIBLE CONNECTION BETWEEN THAT CONVERSATION AND THIS PROBLEM?”

“Man, I don’t see why you’re so mean.”

“Read what it says on the board. Right there. In red.”

“Increase x by a%.”

“Yes. Can you read problem 3 and tell me what you think might possibly qualify as x?”

“The population?”

“Yes. And do you see the value that might possibly qualify as a%?”

“Um.” Long pause as the student stares at the problem, and finds the ONLY OTHER VALUE MENTIONED. “Twenty percent?”

“Indeed.”

“Okay.”

I repeat that four or five times to four or five groups and then, miracle of miracles, find a student with a full table of five values for the population problem. There is a god.

“Great.”

“But I don’t know how to find the equation for this one like I did the first two. This one isn’t repeated multiplication. I had to take 20% of 250 and then add it….why are you hitting yourself on the head?”

“We need a function. We need an operation in which we can plug in x—do you have any thoughts on what x might be?”

“How many months?”

“How is it you know that, you smart child, and yet make me go through this torture? Yes. We need an operation that we can plug in the number of months (x) and get the population (y).”

“Right. But this is like three steps.”

“And we need only one.”

“Right.”

“Wouldn’t it be cool if there were a way to increase a number by a given percentage in just one step?”

“How do you do that?”

“LOOK AT THE BOARD!”

“Oh, is that what you were talking about? I was already doing the worksheet.”

And still, the lesson is largely a success. Kids are absolutely freaked out at the cell growth caused just by doubling and yes, I bring up the million dollar mission example, but at the end of the lesson, not as part of it. Most of the kids correctly graph the models, although a few end up with lines that I correct. The flip side of the handout is a blank graph, which they use to take notes on the basic exponential growth model.

Total Amount = Initial Amount * Rate^time

Initial Amount > 0
Rate > 1

One thing I mull over—the book, and the state test, go through the exponential equation (basically, Initial Amount = 1), along with the transformation model (f(x) = a^x-c +- k. I haven’t focused on this in previous classes, because in my experience the kids don’t even get tranformations of lines and quadratics. But I’m going to give it a try on Monday.

Anyway. Day 2 is exponential decay, but I start by going over percentage decrease. I am nothing if not optimistic.

“So if I take away a third of something, how much is left?”

Pause. Pause some more. Pause still more. I grab three whiteboard pens.

“Rhea, decrease these pens by a third.” Rhea obediently takes one pen.

“Class, how much is left after she decreased the pens by 33%, or a third?”

“TWO!!!”

“Two……?” I wait. No. I sigh, and grab three more pens, getting the one back from Rhea as well.

“Paul, take away a third of these six pens.” Paul takes two pens.

“Class, he’s taken away 33% of the pens. How much is left?”

“FOUR!”

“AUUGGGGHHH!”

It all works out. Seriously. By the end of the exercise, most of the class is shouting back the correct answers as I ask “I take away 30%, how much is left?” 35%? 23%?” and the only mistakes they make are place errors—that is, 100-23 does not, in fact, equal 87.

The second day is always better, because it has slowly permeated their skulls that I’m serious about this percentage nonsense, that it has some relationship to the worksheet. So when they ask questions, it’s more of the “could you run this whole percentage decrease by me again? If they take away a third, I have two thirds left? But what’s two thirds as a decimal?” and trust me, this is a big step up for my blood pressure. Well, a step down. And they do the decay modeling and notes with no small degree of interest:

They have the model graph on the back, too, for exponential decay:

Total Amount = Initial Amount * Rate^time

Yes, it’s the same equation, so what’s different?

Initial Amount > 0
0 < Rate < 1

By day’s end, they have registered the import of the realization that Estefania has 95 cents left after ten days, and they’ve figured out that Jose is right, that his car is worth more than Stan’s after five years, which they managed by using an equation they built themselves, by golly, rather than decrease 25,000 by 5% 5 times.

You notice, of course, that I’ve spent most of this post talking about the percentage issue, something the kids learned were first taught back in middle school, than the exponential growth/decay functions, the actual new material. This should not come as a shock to regular readers.

Back in March, there was much fuss about a study revealing that algebra and geometry classes aren’t rigorous enough.

Of course the classes aren’t rigorous enough. They can’t be. I refer you again to the false god of elementary school test scores and the Wise Words of Barbie.

This twitter debate between reformers Mike Petrilli and Rishawn Biddle is typical of reform debates about “rigor”. Petrilli wants end of course exams to stop us teachers from pretending to teach a subject. Biddle wants more of the same, just shout louder and MANDATE instruction, particularly to those disenfranchised black and Hispanic youth who are being let down by lousy teachers with low expectations.

Both of them assume that the problem is ineffective teaching, that all us math teachers could actually teach percentages and fractions to all seventh graders if we were just smarter and better. Or maybe they just think we take the easy way out, that it’d be really really hard to teach the kids properly, and what the hell, we get paid no matter what and behind close doors it’s easier to just go through the motions. Well, sure.

Petrilli’s proposal, end-of-course exams, would trigger a bloodbath. People really don’t seem to understand how I’d be all in favor of that, if the result were a rethinking of expectations. But of course, what would actually happen is that we’d end the end-of-course exams. That’s what always happens whenever a state or district tries to enforce higher standards (cf Oklahoma and now Texas). And of course, that’s what’s going to happen with Common Core standards, assuming that anyone actually takes them seriously after the testing bloodbath this year. But I’d be all for end-of-course testing if reformers would accept responsibility for the 80% decrease in graduation rates among blacks and Hispanics who would never get past algebra I and understand, finally, that they believe in a myth.

But I digress. And I’m still going to like exponential functions, at least until I crack quadratics. Because you know what? The kids do make progress in understanding percentages, and they learn for the first time not only about exponential functions, but about asymptotes, as I explain Zeno’s Paradox. I don’t use Achilles and the tortoise as an example, but instead talk about how I could throw a stapler right at BTS’s head and know that the stapler would never draw blood because it wouldn’t reach his noggin, so I couldn’t get fired. Or that I could walk to the door and never get there. I do get to the door, of course, and alas, the stapler would eventually crack BTS’s skull. But even though we know that this is true, the tools for proving the paradox false, as opposed to demonstrating it, don’t come around until calculus. They get a kick out of that.

If all that’s not fun enough, I see genuine, honest-to-god intellectual curiosity among most students as they realize that they don’t have the tools to isolate x in the equation 8 = 3^x. That for all these years they’ve been getting along fine with addition/subtraction, multiplication/division, nth power/nth root, but none of those will work here. Which sets us up beautifully for both logs and a proper discussion of inverses, leading into inverse functions. Yes, their skills are still basic, but I can see the glimmering of understanding of the underlying concepts. If the damn state tests would just ask questions about those underlying concepts instead of demanding underlying concepts and advanced operations, I might even be able to get the kids to show that understanding.

And in writing up this essay, I am struck by the obvious solution to the percentage problem on day one: I need a worksheet. They fill it out, and not until they are done with that do I give them the worksheets on growth and decay. Naturally, this solution is again a lowering of expectations, a realization that a clear explanation on a blackboard that they can refer to isn’t enough, that I need to give fifteen to seventeen year olds an activity so the information will sink in and they use the method right away without asking me to explain it all again group by group. But to hell with expectations. It will be much better for my bloodpressure.

Modeling Linear Equations, Part 3

See Part I and Part II.

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

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

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

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

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

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

Shrug. “I don’t know.”

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

Tentatively. “He could just buy 9 burritos?”

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

“None.”

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

“Fifteen tacos. Why is it 15?”

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

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

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

“I don’t know.”

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

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

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

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

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

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

“And you know it’s $15 because….”

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

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

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

“Multiplying the burritos by five dollars.”

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

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

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

“Five.”

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

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

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

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

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

“So, guy over there. Yeah, you.”

“I wasn’t paying attention.”

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

Silence. I wait it out.

“Um. I can’t buy as many tacos?”

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

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

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

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

“Goes down.”

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Here’s the boardwork for a systems models.

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

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

Phew.

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

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

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

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

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

Algebra 1 Growth in Geometry and Algebra II

Last September, I wrote about my classes and the pre-algebra/Algebra 1 assessment results.

My school covers a year of instruction in a semester, so we just finished the first “year” of courses. I start with new students and four preps on Monday. Last week, I gave them the same assessment to see if they’d improved.

Unfortunately, the hard drive on my school computer got wiped in a re-imaging. This shouldn’t have been a problem, because I shouldn’t have had any data on the hard drive, except I never got put on the network. Happily, I use Dropbox for all my curriculum development, so an entire year’s worth of intellectual property wasn’t obliterated. I only lost the original assessment results, which I had accidentally stored on the school hard drive. I should have entered the scores in the school grading system (with a 0 weight, since they don’t count towards the grade) but only did that for geometry, the only class I can directly compare results with.

My algebra II class, though, was incredibly stable. I only lost three students, one of whom got a perfect score—which the only new addition to the class also got, so balance maintained. The other two students who left got around 10-15 wrong, so were squarely in the average at the time. I feel pretty comfortable that the original scores didn’t change substantially. My geometry class did have some major additions and removals, but since I had their scores I could recalculate.

	Mean	Median	Mode	Range
Original	just above 10	9.5	7	22
Recalculated	just below 10 (9.8)	8	7	22

I didn’t have the Math Support scores, and enough students didn’t take the second test that comparisons would be pointless.

One confession: Two Algebra II students, the weakest two in the class, who did no work, scored 23 and 24 wrong, which was 11 more than the next lowest score. Their scores added an entire point to the average wrong, increased the range by 14 points, and you know, I just said bye and stopped them from distorting the results the other 32 kids. (I don’t remember exactly, but the original A2 tests had five or six 20+ wrong scores.)

So here’s the original September graph and the new graph of January:

The geometry class was bimodal: 0 and 10. Excel refused to acknowledge this and I wasn’t sure how to force it. The 10s, as a group, were pretty consistent—only one of them improved by more than a point. The perfect scores ranged from 8 wrong to 2 wrong on the first test.

In short, they learned a lot of first year algebra, and that’s because I spent quite a bit of time teaching them first year algebra. In Algebra II, I did it with data modeling, which was a much more sophisticated approach than what they’d had before, but it was still first year algebra. In geometry, I minimize certain standards (proofs, circles, solid shapes) in favor of applied geometry problems with lots of algebra.

And for all that improvement, a still distressing number of students answered x² + 12 when asked what the product of (x+3) and (x+4) was, including two students who got an A in the class. I beat this into their heads, and STILL some of them forget that.

Some folks are going to draw exactly the wrong impression. “See?” these misguided souls will say, nodding wisely. “Our kids just aren’t being taught properly in early grades. Better standards, better teachers, this problems’s fixed! Until then, this poor teacher has to make up the slack.” In short, these poor fools still believe in the myth that they’ve never been taught.

When in fact, they were taught. Including by me—and I don’t mean the “hey, by the way, don’t forget the middle term in binomial multiplication”, but “you are clubbing orphan seals and making baby Jesus cry when you forget the middle term” while banging myself on the head with a whiteboard. And some of them just forgot anyway.

I don’t know how my kids will do on their state tests, but it’s safe to say that the geometry and second year algebra I exposed them to was considerably less than it would have been had their assessment scores at the beginning of class been the ones they got at the end of class. And because no one wants to acknowledge the huge deficit half or more of each class has in advanced high school math, high schools won’t be able to teach the kids the skills they need in the classes they need—namely, prealgebra for a year, “first year” algebra for two years, and then maybe some geometry and second year algebra. If they do okay on the earlier stuff.

Instead, high schools are forced to pretend that transcripts reflect reality, that all kids in geometry classes are capable of passing a pre-algebra test, much less an algebra one test. Meanwhile, reformers won’t know that I improved my kids’ basic algebra skills whilst still teaching them a lot of geometry/algebra II, because the tests they’ll insist on judging me with will assume a) that the kids had that earlier material mastered or b) that I could just catch them up quickly because after all, the only problem was the kids’ earlier teachers had never taught them.

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!

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?

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.

Algebra and the Pointlessness of The Whole Damn Thing

The whole algebra debate kicked off by Hacker’s algebra essay has…..well, if not depressed me, then at least enervated me.

A recap:

Hacker:

We shouldn’t make everyone take algebra. No one needs algebra anyway; we never really use it. Statistics would be much more useful. Algebra is the primary obstacle to high school success; millions of kids are failing because they can’t manage this course. If we just allowed students to have an easier time in high school, more of them would graduate successfully and go on to college.

Outraged Opposition:

Algebra is essential to college success and “real life” and one of many obstacles to high school success. No one is happy with the current state of affairs, but it’s clear that kids aren’t learning algebra because their teachers suck, particularly in elementary school. We need to teach math better in the lower grades, rather than lower our standards. Besides, the corollary to “not everyone should take algebra” is “some people should take algebra” and just how are you planning to divide up those teams? (Examples: Dan Willingham, Dropout Nation)

Judicious Analysis:

Sigh. Guys, this is really a debate about tracking, you know? And no one wants to go there. While it’s true that algebra really isn’t necessary for college, colleges use success in advanced math as a convenient sorting mechanism. Besides, once we say algebra isn’t necessary, where do we stop? Literature? Biology? Chemistry? But without doubt, Hacker is right in part. Did I say that no one wants to go there? Or just hint it really, really loudly?
Examples: Dana Goldstein, Justin Baeder Iand II.

Voldemort Support:

Well, of course not everyone should take algebra, trig, or calculus. Or advanced literature. Or science. Not everyone has the cognitive ability or the interest. We should have a richer and more flexible curriculum, allowing anyone with the interest to take whatever classes they like with the understanding that not all choices lead to college and that outcomes probably won’t have the racial distributions we’d all prefer to see. Oh, and while we’re at it, we should be reviewing our immigration policies because it’s pretty clear that our country doesn’t need cheap labor right now.

Hacker, Outraged Opposition and Judicious Analysis to Voldemort Support:

SHUT UP, RACIST!

So really, what else is left to say? The Judicious Analysis essays I linked above were the strongest by far, particularly Justin Baeder II.

Instead, I’m going to revisit a chart I updated from the last time I posted it:

These are California’s math scores by grade and subject, the percentage scoring basic/proficient or higher on the CST. Algebra entry points differ, so the two higher (and slightly longer) of the four short lines are the percentages of “advanced” students with those scores—those who took algebra in 7th or 8th grade. The lower, shortest lines represent the scores of students who began algebra in 9th grade.

Notice that advanced students don’t match the performance of the entire elementary school population through 5th grade. Notice, too, that the percentage of advanced students scoring proficient or higher is just around half of the population. When I just considered algebra students who began in 8th grade (see link above), the percentage never tops 50. Notice that around 40% of the kids who started algebra in 9th grade achieved basic or higher.

NAEP scores show the same thing—4th grade math scores have risen, while 12th grade scores stay flat. In fact, Daniel Willingham, who declares above that we’re doing a bad job at teaching elementary math, was considerably more sanguine about teacher quality back in December, citing the improved elementary school math performance shown in the NAEP. So the strong elementary school performance, coupled with a huge dropoff in advanced math, is not unique to California.

These numbers, on the surface, don’t support the conventional wisdom about math performance: namely, that elementary school teachers need improvement and that the seeds of our students’ failure in higher math starts in the lower grades. Elementary students are doing quite well. It’s only in advanced math, when the teachers are much more knowledgeable, with higher SAT scores and tougher credentialling tests, that student performance starts to decline dramatically.

What these numbers do suggest is that as math gets harder, fewer and fewer students achieve mastery, or anything near it. . What they suggest, really, is that math knowledge doesn’t advance in a linear fashion. Shocking news, I know. We have all forgotten the Great Wisdom of Barbie.

Break it down by race and the percentages vary, but not the pattern. I skipped Asians, because California tracks Asians by subcategory, and life’s too short. I’m going to go right out on a limb and predict that Asians did a bit better than whites.

(Note: I know it’s weird that in all cases, 9th graders in general math have nearly the same percentages as 9th graders in algebra, but it’s easily confirmed: whites, blacks, Hispanics).

Whites in the standard math track perform as well as advanced math blacks and just a bit worse than advanced track Hispanics. Sixty to seventy percent of blacks and Hispanics on the standard track fail to achieve a “basic” score.

Some people are wondering how poverty affects these results, I’m sure. Let’s check.

Hey! Look at that! The achievement gap disappears!

Just kidding. This chart shows the results of blacks and Hispanics who are NOT economically disadvantaged and whites who ARE economically disadvantaged. You can see it on the legend.

So that’s how to make the achievement gap disappear: compare low income whites to middle class or higher blacks and Hispanics and hey, presto.

And that’s all the charts for today. I’m not detail-oriented, and massaged this all in Excel. You can do your own noodling here. Let me know if I made any major errors. The 2012 results should be out in a couple weeks.

Anyway. With numbers like these, it’s hard not to just see this entire debate as insanely pointless. In California, at least, tens of thousands of high school kids are sitting in math classes that they don’t understand, feeling useless, understanding deep in their bones that education has nothing to offer them. Meanwhile, well-meaning people who have never spent an hour of their lives trying to explain advanced math concepts to the lower to middle section of the cognitive scale pontificate about teacher ability, statistics vs. algebra, college for everyone, and other useless fantasies that they are allowed to engage in because until our low performers represent the wide diversity of our country to perfection, no one’s going to ruin a career by pointing out that this a pipe dream. And of course, while they’re engaging in these fantasies, they’ll blame teachers, or poverty, or curriculum, or parents, or the kids, for the fact that their dreams aren’t reality.

If we could just get whites and Asians to do a lot worse, no one would argue about the absurdity of sending everyone to college.

Until then, everyone will divert themselves by engaging in this debate—which, like many kids stuck in the hell of unfair expectations, will go nowhere.

Learning Math

Jessica Lahey’s In Defense of Algebra, in which she describes her adult triumph over math, reminded me of my own experiences learning math and how they might be relevant. This is long, but if I make what I did sound too easy, people will get the wrong idea.

I struggled in math during high school, but was simply too clueless to quit. I got As and Bs algebra in 8th grade, got Cs and Ds in the subsequent courses until senior year, when I held onto a B the entire year in AP Calculus and, in what remains one of the great academic shockers of my life, passed the AP Calc test with a 3. Many years later, I took the GRE for my first master’s. I spent a month slogging through math, relearning enough of it to get by, and was very pleased with my 650 quant score, which was the 65th percentile on the GRE. Good, but not great, which is pretty much how I did on the SAT many years earlier. (Verbal, which I spent no time on, was 790).

My math turnaround began after I started grad school, when my son was failing geometry. After reading his book and working dozens of problems, I was able to help him (he went on to pass both the AP Stats and Calc AB tests, and is stronger in college math than I am). Needing a part time job for grad school, I auditioned for a job at Kaplan.

I originally hired on to teach GRE classes, but within a month I was working close to 40 hours a week teaching the high school tests. Two months later, I was teaching Math 1c and Math 2c, which was ridiculous. I had to learn the SAT math by rote at first, and the SAT Subject math tests required trig and second year algebra. I protested to no avail, and my manager’s decision was prescient—within a few months, parents were emailing me for private homework tutoring in high school math. It turns out I’m very good at explaining things; when I didn’t know how to do a problem in the early days, which was often, I’d go look it up or ask a student who did understand the problem. Ah, that’s the part I missed! Anyone else do that? I could see students nodding. I made the process very transparent, showed students where the glitches in my comprehension were, helped them find their own glitches and, it turns out, students would rather be tutored by someone who knows where the understanding glitches are.

I told parents I didn’t have a clue what a log was if it wasn’t in an Ingalls story, but even after raising my rates as an attempt to scare them off, I was getting lots of work. Still, for my own peace of mind, I decided I needed to learn more math instead of literally learning it while teaching others.

I started with test math, my strength. The REA and Kaplan test prep books served as my educational foundation. Between the two books, I learned how to recognize the subject and how to solve a number of common problems. When I was tutoring students, I would recognize the type of problem it was. Then, using their textbook, I’d help them work problems by example and explain. In explaining the math, I learned a great deal. My students learned and got improved grades, more confidence, and better test scores. Me, I kept getting more clients.

After four years of this, I had an excellent understanding of high school math through pre-calc, and routinely scored 800 when taking practice Math 2c tests (which I did for the first few years to keep me alert to weak spots). I not only knew the material cold, for the most part, but was by this point extremely familiar with the curriculum and sequencing for pre-algebra through algebra II/trig, and somewhat familiar with the same for math analysis (pre-calc) and calculus.

Seven years after my first GRE and untold decades after high school, I aced two of the three teacher math credential tests, and passed the third, in calc, with what can be called a gentleman’s C—the conceptual questions, the trig and the math history questions pulled me through. I scheduled my ed school GRE with two days’ lead, and got an 800 on the quant—by that time I was getting 800s on the practice test without a pencil, sitting around the Kaplan office with 15 minutes to kill. (Verbal: 780, but I was distracted and finished in 12 minutes.)

As I was studying for the calc credential test (which I took twice), I suddenly had an epiphany about why I was now able to learn math, when I’d done so poorly in high school, and for this, boys and girls, I must go back even further, back to the dark old ages of the mainframe and the earliest days of my previous career:

I was unfocused after college, having done well in English lit classes and nothing else. I started as a data entry clerk, but got a contract at IBM using a mainframe product that everyone in its customer base hated, but had to use. While entering the data, I noticed a few short cuts, asked for a manual, and began customizing the product in ways very few people knew was possible. The boss was impressed and made me responsible for product demonstrations, showing my work and explaining how the product could be customized. I got a job from one of the companies that came to the demo, a major brokerage firm. It was my first real job after five years of temping in admin jobs, during and after college. (Irony alert: I’d taken, and nearly flunked, one computer programming course in college and was convinced it wasn’t the career for me. I was three years into my new job before I felt comfortable saying I was an applications programmer, much less a computer programmer.)

And the first day I got to my new job, my boss said to me, “So I need to put you on a different project first. Our systems management application needs to be installed on the NYC machine.We have this automated process that moves the source code out of Panvalet, automatically builds the compile JCL based on the program—you know, if it’s CICS or batch, IDMS or DB2, and so on, and compiles it, linkrefs it, and installs it in the right library. It’s already working here; you just need to transfer all the code, change the libraries, and so on. You know CICS, right?”

“No.”

“Oh, it’s a transaction server environment. We mostly use COBOL here, but we’ve got some assembler routines. You know COBOL or assembler? No problem, I’ll sign you up for a class. The code itself is mostly ISPF calls with EXEC, although I’d like to upgrade to REXX.”

It’s not just that I didn’t know COBOL, and had no idea what CICS was. It’s that I didn’t know what compile meant, or source code, or batch, or JCL, much less IDMS or DB2. And I didn’t have the foggiest clue what REXX or EXEC was, and ISPF to me was an application, not something I programmed with.

Baby, I brought that motherf***er in on time. Two and a half months. I got a bonus, too, because by the time it went live, my director had figured out a small fraction of how much I didn’t know, and was extremely impressed. She never grasped the sum total of my ignorance, thank god. Nor did she ever realize that I still didn’t know what the difference between CICS and batch was and didn’t realize that (at the time) a program couldn’t be both, only vaguely understood that “compile” meant translating code I could read into weird symbols I couldn’t read but presumably the computer could, never did learn COBOL, and only vaguely understood what JCL was or what it did. All that was in my future. The only thing I was pretty confident about after those two and a half months was that I was pretty darn good at EXEC and ISPF dialogs, and that these things weren’t what the brokerage products ran on.

So my epiphany was this: Working with computers had taught me how I learned. How I learned. Which was not like most people. When books don’t work as a learning tool, then I have to learn by a particular type of doing. Explanations won’t help. Learning in a vacuum won’t help. I need to learn by trial and error. And then, I learn like Wile E. Coyote traverses the desert; I just keep on going until something blows up in my face. Go this way? Boom! Okay, that way doesn’t work. File it away. Go that way? Two steps, yes, then BOOM! Okay, the two steps, file away, then don’t go that way because BOOM! how about this way? Tiptoe, tiptoe, try this, ha! It worked! Done. On to the next. Make sense of the chaos, bit by bit, understanding the rules by the reaction.

When I’m learning something, I neither know nor care about why. Understanding will usually come. So just as I ultimately understood CICS only several months after I started making changes to a mission critical CICS transaction, I didn’t bother with understanding what, exactly, trigonometry was. Two years after I first learned how to work trig problems, I read that trigonometry was the study of the ratio of right triangle sides, and I was like Holy Crap, that’s exactly what trig is. What a trip. One day soon I’ll internalize the fundamental theorem of algebra, but give me time. If not, I’ll wave the dead chicken over the problem because that’s what worked the last time. And it usually works again. (Note: Ironically, as a math teacher, I am big on explaining why, but that’s because I’ve realized that most people aren’t like me.)

Of course, theory, whether it be math or computers, is usually beyond me. And yet, in both math and computers, I am capable of occasional insights that please actual mathematicians and computer scientists, even though most of the time I don’t care, as they are working on things that I find utterly incomprehensible.

Why did I struggle with math in high school? The usual reasons don’t apply. I was an A student, and remained one in English and history. Unlike Jessica, I never felt labelled, nor did I give up. I had excellent math teachers, all of whom knew that my intellect was considerable and took the time to reach out—each one sat me down at some point and asked why I wasn’t doing better, given my obvious brains.

I just know that some people are going to read my post, as they read Jessica Lahey’s, and conclude that, by golly, we prove that anyone can learn math, and that labelling kids based on early progress is cruel and wrong and demoralizing. Lahey herself clearly holds this position.

Well, no. Lahey and I are both extremely bright and I know I say this a lot, but that’s because people persist in ignoring the relationship between “smart” and “academic achievement”. Lahey clearly has excellent verbal skills, strong at writing and foreign language (she’s a Latin teacher at an elite middle school), whereas I’m a hybrid who, in addition to excellent verbal skills, tested high on every computer aptitude test that came out back when I was in college. On the other hand, I can’t speak any foreign languages, and I suspect Lahey isn’t as strong on logic and pattern recognition, which is why she needed an algebra teacher to get through first year algebra, whereas I self-taught myself the entire high school curriculum.

What Lahey and I both demonstrate is that it’s possible to be well above average in smarts, yet still struggle in math when later experience proves that we were entirely capable of grasping it.

Why, then, does an otherwise smart person struggle with math?

I have a theory, involving my layman’s understanding of IQ, which I’ll go into briefly.

Two visual aids to categorizing or measuring intelligence: Wechsler Adult Intelligence Scale subscores and subtests and Cattell-Horn-Carroll theory. In both, you can see what most people know on a casual level: intelligence has a verbal component and a visual/spatial component (known as performance in Wechsler). Logic seems to cut across the categories. It’s not terribly controversial to point out that advanced math, even that found in high school, requires more visual spatial and logic ability. I don’t know, specifically, how my intelligence maps to these categories. But I’ve always known that my verbal abilities were very high, my pattern recognition and decision processing equally so, and my visual-spatial relatively weak.

Imagine smart kids who has really strong verbal skills but unknown weaknesses in either logic or visual spatial abilities. These kids would coast easily through elementary school, where the skills needed are almost exclusively verbal—reading and arithmetic. By the end of of 8th grade, they’re bored out of their minds. Most of elementary school is time spent teaching them things they already know and developing social skills.

So for 8 years, this type of smart kid hadn’t ever had to struggle to learn something—in fact, learning itself is pretty alien to smart kids. (This, parents of smart kids, is why you should make sure your kids have to struggle with something—cooking, art, horseback riding, making nice with other people, whatever.)

And then, math. Algebra and beyond can come as a big shock. When school has come easily all your life, it’s hard to even know what “learning” is, much less how it applies to you. I’ve talked to countless people who qualify as Really, Really Smart, and every one of them has agreed that at some point in their lives, they realized that they had no idea how to learn. Some figured it out in high school or college. Some, like me, didn’t figure it out until they got a job that required them to learn something. Others, like Lahey, had access to a math teacher and went back and learned it because they wanted to fill in a gap that they felt was necessary for parenting.

So the otherwise smart kids who struggled with math did so in part because their particular intelligence was strong on verbal, and lighter on the spatial and (probably) visualization skills that are helpful in math. Plus, math was, to quote the Great Barbie, tough. And these kids had never once faced “tough” in a school subject. Many folded.

This was particularly true in the era before we began demanding higher math of all students (say about a decade or fifteen years ago). Before the 90s, math teachers weren’t held responsible for their students’ failures, and we accepted that not everyone was “good at math”. Kids with strong verbal abilities just took less math—in those days, it didn’t end your hopes of college, even of elite college. It was quite normal to get into an excellent school with a strong performance in history or English with little more than second year algebra on a transcript.

Today, of course, any white or Asian kid who wants to go to an elite college has to have advanced math, so the smart verbal kids who don’t have the requisite math skills have a much stronger incentive to either compensate or fake it, with or without a tutor. Moreover, math teachers these days are far more likely to reward effort over ability, so it’s easier for a student to get As in math by religiously doing homework and extra credit, even if they do poorly on tests. And of course, math teachers are also less likely to dismiss the effort involved in teaching math to those who aren’t necessarily strong in the subject. Thus, for many reasons, smart kids today with primarily verbal skills are less likely to have given up on math and are at least willing to fake it. Many learn to compensate, as I did much later in life, by using their other cognitive abilities to make up for their relative weakness.

(Compensation: Picture a circle inscribed in a square. Can you estimate the ratio of the circle’s area to the square’s? Or would you, like me, be completely incapable of a reliable guess and so calculate the difference by creating a radius of length r and work it as an algebra problem? I contend that people who can make an accurate estimate and do the algebra have an easier time in math than the people who can only do it with the algebra.)

Back in the day, the kids whose intelligence was strong in the math aspects but weak in verbal got a similar shock when they were expected to write an analytical essay on Hamlet and offer some original insights. Unfortunately, as has been noted before, our history and English curriculum has never recovered from the dumbing down it suffered through in the interests of multi-culti, giving kids who are strong at math little reason to ever struggle to access their verbal abilities.

This has obvious implications for the big algebra controversy kicked off by Hacker. But it involves recognizing that underlying cognitive ability is a huge determinant when deciding who should, or perhaps shouldn’t, take algebra.

Appropriate math questions

I had an interview for a middle school math position (why they called me, I dunno, I’m sure I was just filler), and had to bring a sample lesson plan.

I always create worksheets when asked to bring lesson plans, since it gives a better idea of how I teach. I read the standards and came up with this cartoon as a starting point:

I use buying decisions all the time in math classes, because the phrase “Pretend that it’s your money” has a near-magical effect on kids who would otherwise ignore word problems.

So I’d kick off a discussion by asking the kids to identify the difference between the two questions. Most kids of any ability will realize that the girl’s question has one answer, while the boy’s question has several.

From there, I drill down to the boy’s question. (click on the image for a slightly better view). Don’t get too quibbly on the definitions—math purists are insanely annoying.

Now, here’s the thing: this is not a lesson on functions. This is a lesson on variables and generating tables. It’s seventh grade math, not pre-algebra or algebra.

But in order to get kids to think about variables and tables, I find it’s helpful to start with a familiar situation and lead it back to math. In order to explain the difference between the girl’s question and the boy’s question, I have to introduce functions, which leads us to variables.

In my opinion, teachers don’t spend enough time leading kids through the process of identifying the definition of the unknowns (because trust me, if you say “identify the unknowns”, half the class says “Three Cheeseburgers!” and suddenly you’re in a Bill Murray skit.) and generating a table of solutions to the problem. Teachers see these as obvious, as mere activities that we use to solve problems. But low to mid ability kids see nothing obvious about defining variables and generating tables. I’ve been slowly realizing this over time, and it crystallized into my data modelling unit in my Algebra II class. This activity, which I drummed up for an interview (but will undoubtedly use at some point), was my attempt to move it back to a pre-algebra state.

Many teachers who only work with high ability kids would see this handout as ludicrously easy and in fact, I’d hand this worksheet to my top seventh graders and tell them to go it alone, while I talked the rest of the class through. Top kids should figure this out fairly quickly, and on day 2, when the rest of my kids are still practicing simple situations, they’d be on to more advanced scenarios (giving two points and figuring out the rule, giving them a table, and graphing).

But many teachers who work with lower ability kids work hard to find meaningful questions and yet miss the mark by making the question too complex. I was talking to one of my curriculum instructors, and he asked me about a lesson he was planning to give to middle school students in a low income district. He wanted to give kids fractions like 7/20, 8/25, 9/30 or 8/30, 9/35, 4/15 and have them just ballpark whether the fraction was greater or less than the obvious value nearby. So is 7/20 greater than or less than 1/3? and so on.

I told him that he was dramatically overestimating their ability, and recommended he start by doing the same thing with halves. So a list of numbers: 2/5, 9/17, 5/11, and so on. Greater than or less than one half? The kids who can do that quickly, move onto numbers around 1/4. Only after a few iterations should he give the strongest kids the exercise he was thinking of, which most of the kids wouldn’t be able to do, full stop. But he could build the weaker kids ability and fluency simply by getting them to think about greater or less than one half. I advised starting with the much simpler exercise. If I was wrong, he could have the more challenging activity ready to follow up, no harm done.

He took my advice and reported back. Most of the kids never got beyond the over/under 1/2, finding it challenging and meaningful. Most of the teachers he was working with had been sure the activity would be too easy.

I do think it’s important to set open questions for all kids, not just the top students, to let them tussle with before you settle into working problems for practice. However, the questions must be tailored to student ability, and math teachers dramatically overestimate the ability of their kids to work with these questions in a meaningful manner.

It gets old to be told that acknowledgement of low cognitive ability is “setting expectations low”. One of the comments on my “myth” post was “One question that I have is the extent to which students are playing Whack-a-Mole in order to get easy classes for a week or two.” My original response was rather rude, so I changed it to a simple “No”. It boggles my mind that anyone would think kids are faking it.

At this point, some teachers are remembering the time they taught their algebra intervention kids about logarithms, or complex numbers, or trigonometry cycles. I’m not saying you can’t teach it. They just won’t remember it. It will be as if they’ve never been taught.

The myth of “they weren’t ever taught….”

A year or so ago, our math department met with one of the feeder middle schools to engage in a required exercise. The course-alike teachers had to put together a list of “needed skills” for each subject, to inform the teachers of the feeder course of the subjects they should cover.

One of the pre-algebra teachers looked at the algebra “needed skills” list and said, “Integer operations and fractions! Damn. Why didn’t I think of that?” and we all cracked up. End of Potemkin drill.

All teachers working in low-ability populations go through a discovery process.

Stage One: I will describe this stage for algebra I teachers, but plug in reading, geometry, writing, science, any subject you choose, with the relevant details. This stage begins when teachers realize that easily half the class adds the numerators and denominators when adding fractions, doesn’t see the difference between 3-5 and 5-3, counts on fingers to add 8 and 6, and looks blank when asked what 7 times 3 is.

Ah, they think. The kids weren’t ever taught fractions and basic math facts! What the hell are these other teachers doing, then, taking a salary for showing the kids movies and playing Math Bingo? Insanity on the public penny. But hey, helping these kids, teaching them properly, is the reason they became teachers in the first place. So they push their schedule back, what, two weeks? Three? And go through fraction operations, reciprocals, negative numbers, the meaning of subtraction, a few properties of equality, and just wallow in the glories of basic arithmetic. Some use manipulatives, others use drills and games to increase engagement, but whatever the method, they’re basking in the glow of knowledge that they are Closing the Gap, that their kids are finally getting the attention that privileged suburban students get by virtue of their summer enrichment and more expensive teachers.

At first, it seems to work. The kids beam and say, “You explain it so much better than my last teacher did!” and the quizzes seem to show real progress. Phew! Now it’s possible to get on to teaching algebra, rather than the material the kids just hadn’t been taught.

But then, a few weeks later, the kids go back to ignoring the difference between 3-5 and 5-3. Furthermore, despite hours of explanation and practice, half the class seems to do no better than toss a coin to make the call on positive or negative slopes. Many students who demonstrated mastery of distributing multiplication over addition are now making a complete hash of the process in multi-step equations. And many students are still counting on their fingers.

It’s as if they weren’t taught at all.

But teachers are resilient. They redouble their efforts. They spend additional time on “warm-up” questions, they “activate prior knowledge” to reteach even the simple subjects that have apparently been forgotten, and they pull down all the kaleidoscopic, mathy posters and psychology-boosting epigrams they’d hung up in their optimistic naivete and paper the walls with colorful images formulas and algorithms.

They see progress in the areas they review—until they realize that the kids now have lost knowledge in the areas that weren’t being taught for the first time or in review, much as if the new activity caused them to overwrite the original files with the new information.

At some point, all teachers realize they are playing Whack-a-Mole in reverse, that the moles are never all up. Any new learning seems to overwrite or at best confuse the old learning, like an insufficient hard drive.

That’s when they get it: the kids were taught. They just forgot it all, just as they’re going to forget what they were taught this year.

All over America, teachers reach this moment of epiphany. Think of a double mirror shot, an look of shocked comprehension on an infinity of teachers who come to the awful truth.

End Stage One and the algebra specificity.

Stage Two: At this point, some teachers quit. But for the rest, their reaction to Stage One takes one of two paths.

Blame the students: The transformation from “these poor kids have just never been taught anything” to “These kids just don’t value education” is on display throughout the idealistic Teach for America blogs. It’s pretty funny to watch, since on many sites you have the naive newbies excoriating their kids’ previous teachers for taking money and doing nothing, while on other sites the cynical second-years are simultaneously posting about how they hadn’t understood the degree to which kids could sabotage their own destinies, or some such nonsense. Indeed, I once had a conversation with a TFAer at my school, and she said this to a word: “I’ve realized I’m a great teacher, but my students are terrible.”

Not that this reaction is unique to TFAers. Many experienced teachers who began their careers in a homogenous, high-achieving district that transformed over time into a Title I area with a majority of low income blacks or Hispanics have this response as well.

It’s easy to denounce this attitude, but teaching has taught me that easy is never a good way to go. These teachers are best served at a place like KIPP, where the kids who don’t work are booted. It’s not that the kids learn more, but at least the ones that stay work hard, and that allows the blamers to reward virtue. At comprehensive schools, the teachers who saw their student body population change over time respond by failing half or more of their classes.

These teachers please both progressives and eduformers, because they have high expectations. Their low-achiever test scores, however, are often (but not always) terrible.

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

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

Which is the best approach? Well, I’m an accepter. Not that I was ever particularly naive, but despite my realism, I was caught off-guard by just how much low ability students can forget. But as I’ve said before, that’s the challenge I see in teaching.

I could go into more on this, but this post is long enough. Besides, I don’t want to lose sight of the opening story and the pre-algebra teacher’s mockery of the entire point of the exercise. Of course they were teaching integer operations and fractions. Of course they were doing their best to impart an understanding of exponents and negatives. They didn’t need the list. They knew their job.

Teachers know something that educational policy folk of all stripes seem incapable of recognizing: it’s the students, not the teachers. They have been taught. And why they don’t remember is an issue we really should start to treat as a key piece of the puzzle.