Monthly Archives: February 2013

On Graduation Rates and “Standards”

Stephanie Simon has a piece out on the increasing graduation rate (while I’m at it, mad props to Simon for the charter school piece, which probably did a lot to alert the general audience to charter selections), and various tweets are hailing the good news but—and this is the funny part—expressing concern that this increase rate might be due to schools lowering standards. Checker Finn has also written disapprovingly of credit recovery.

hahahahahaha. This is me, laughing.

Imagine you have forty 18 year olds, who all read and calculate at the 6th grade level, and another group of forty who all read and calculate at the 10th grade level. They are all high school seniors in a state that requires graduation competency tests. Of this overall collection of eighty, the following distribution is entirely unexceptional (and of course, not the only one possible):

  1. Fifteen screwed around from the moment they entered high school, have a GPA in the tenths, and are currently in alternative high school filling out worksheets. No reason to worry about high school graduation tests, though, because they passed them first time out.
  2. Fifteen are, on paper, identical to the previous group, except they haven’t passed any of their graduation tests and so some of their high school time is spent in test prep instead of worksheet completion.
  3. Fifteen are far behind because they went to a charter school that prided itself on making kids repeat grades, and after two years of failure they went back to public school. They’ve passed the high school graduation tests, and have been doing well since they left the charter, GPAs of 2.0 or so. But they’re far behind, so are taking two hours every day to do online credit recovery.
  4. Fifteen are at a charter school, where they have a 4.0 GPA with a bunch of AP courses on their transcripts, (thanks, Jay Mathews and your horrorshow of a Challenge Index) but haven’t passed the high school graduation tests.
  5. Ten recovered from an early bad start, have a solid 2.5 GPA, but haven’t passed their state graduation tests. Half of them have IEPs and official learning disabilities (which means, of course, they aren’t in charters), and so they’ll just waive the requirement. The others will keep plugging away.
  6. Ten have a solid 2.5 GPA after an early bad start and have passed their state graduation tests.

(Note: In case it’s not clear, the kids who can pass the state grad tests are the ones with tenth grade abilities, the ones who can’t are the ones with sixth grade abilities).

Any diverse high school district in the country, surveying its population in comprehensive, alternatives, online campuses, and charters, could assemble those eighty kids without breaking a sweat.

On the lower half of the ability spectrum, grades and credits are utterly pointless differentiators. Once you accept that we graduate thousands of kids who can’t read, write, or add, there’s no reason to cavil at the method we use to boot them out of the schoolhouse.

No, don’t yammer at me about persistence or compliance or god spare me “grit” of illiterates plugging away at school and therefore being more deserving of the diploma than the lazy but somewhat smarter kid. The concern about the increase was not about persistence or compliance or grit, but academic ability.

And so, rest easy, people. We are already graduating illiterates. The increased graduation rate is not achieved by teaching more kids more effectively, nor is it achieved by shovelling through the bottom feeders and thus devaluing high school diplomas. We are simply taking kids, whether near-illiterate or low but functional ability, who fell off the path that our other near-illiterate or low but functional ability kids stayed on, and putting them on a different conveyor belt.

How? As Simon’s article makes clear, by spending lots and lots of money:

* Launching new schools designed to train kids for booming career fields, so they can see a direct connection between math class and future earnings

* Offering flexible academic schedules and well-supervised online courses so students with jobs or babies can earn credits as their time permits

* Hiring counselors to review every student’s transcript, identify missing credits and get as many as possible back on track

* Improving reading instruction and requiring kids who struggle with comprehension to give up some electives for intensive tutoring

* Sending emissaries door-to-door to hound chronic truants into returning to class

Notice that only one of the techniques used actually involved teaching the kids more—not that I’m in favor of forcing kids to give up electives for intensive tutoring (I still have nightmares). But most of the money spent involved forcing or coaxing the kids back to school—and while the kids are mostly low ability, they are no less and often considerably more intellectually able than kids who just happened to jump through the right hoops.

How does this happen, you ask? As I’ve said many times: grades are a fraud.

Or you could put it another way: the increased graduation rate is a triumph of administrators over teachers. Teachers, except those in majority minority urban schools, are flunking kids with little regard to ability and a whole bunch of regard to compliance, with no regard to administrative or societal cost. Administrators are spending money to work around teacher grades.

In this context, bleats about academic standards do seem a bit….well, silly, don’t they?

And now someone is going to say, “You’re absolutely right. We should be failing kids who don’t or can’t do the work, put teeth into the Fs. That’s the only way to raise academic standards.”

Sorry, that fool’s wrong, too. Higher standards are impossible. No, really. Common Core advocates, much like Mark Wahlberg at the end of Boogie Nights, are parading their favorite toy in front of a mirror in the desperate hope they’ll convince themselves, if no one else. (What, too much? Yeah, it’s late. I’m feeling bleak.) I very much doubt Common Core will ever be implemented (no test, no curriculum, baby), but if it is, nothing will change.

People assume that kids in the bottom half of the ability barrel are there because they suffered a deficit in environment, in parental attention and expectations, in teacher quality. Would that this were so.

Given all the money we’re spending on truancy officers, online credit recovery, counsellors to spot missing transcripts just to push kids through to a diploma, we might just want to consider teaching low ability kids less at a slower pace and stop pretending that they have a “deficit” that can be addressed by college level work and high expectations. We could create a hell of a curriculum for high school kids using nothing more than 8th grade math and vocabulary.

But we won’t do that for the same reason we won’t track, and for the same reason that adminstrators are spending a fortune coaxing kids back to school: namely, the racial distribution would make everyone wince.

Skills vs. Knowledge

E. D. Hirsch is all upset because teachers are deluded about the importance of knowledge (content), emphasizing skills such as critical thinking and written expression over content. A Common Core true believer, he is shocked, shocked I say! at the fact that most teachers think they are already implementing Common Core, but think its ability to impact achievement is minimal.

Fundamentally, the problem educators face is freeing themselves from the skills stranglehold. It is preventing them from understanding the Common Core standards, preventing them from meeting their own goals as professionals, and preventing them from closing achievement gaps between poor and privileged students.

We see evidence of it everywhere, especially in the MetLife survey. Nine in ten teachers and principals say they are knowledgeable about the Common Core standards, and a majority of teachers say they are already using them a great deal. At the same time, teachers, especially in later grades, are not all that confident about the effect the Common Core will have.
The fact that so many teachers (62%) say the teachers in their school are already using the Common Core standards a great deal shows that these “thought leaders” are correct: most educators remain unaware of the massive changes that fully implementing the new standards will require. But everyone has been talking about these changes for more than a year. Clearly, the message is not getting through.

I have no dog in this hunt; I emphasize content knowledge in all my teaching subjects, but think Hirsch, who believes the achievement gap can be closed, is a tad deluded himself on its magical qualities. I also agree with the utter invincibility of the teacher population when it comes to resisting changes they don’t want to make, and let it be known that I join with my brethren in this resistance, because a cold, cold day in hell it will be before, say, I teach literacy in my trig class or come up with a project-based implementation of the power laws.

But I thought this graph interesting.


Hirsch on this graph, from the 2010 Met Life Survey:

I’ll let the executives off the hook for not knowing that the problem-solving and critical-thinking skills they are after depend on the knowledge that they (largely) dismiss. The teachers ought to know better. That just 11% think knowledge of higher-level science and math are essential for college and career readiness is appalling.

Okay. So the executives AGREE with the teachers, and DISAGREE with the thought leaders. But never mind, he’ll be noble and overlook their stupidity, because they were taught using this horrible skills-based method and it apparently didn’t serve them well. Oh, wait.

And please. Can we stop pretending? Trigonometry, chemistry, physics, and calculus are utterly non-essential for success in the real world. They are only essential for signaling to colleges that the student is a smart cookie, and as Ron Unz and Chris Hayes both point out, the value in that varies based on the student race and family SES (including where Mom and Dad went to school).

So can we give it a rest on the pieties?

Of course, now that I think on it, E. D. Hirsch is a thought leader, so I guess it makes sense he’d back his own against teachers and Fortune 1000 executives.

Modeling Linear Equations, Part 3

See Part I and Part II.

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

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

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

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

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


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

Shrug. “I don’t know.”

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

Tentatively. “He could just buy 9 burritos?”

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


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

“Fifteen tacos. Why is it 15?”

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

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

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

“I don’t know.”

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

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

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

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

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

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

“And you know it’s $15 because….”

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

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

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

“Multiplying the burritos by five dollars.”

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

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

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


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

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

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


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

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

“So, guy over there. Yeah, you.”

“I wasn’t paying attention.”

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

Silence. I wait it out.

“Um. I can’t buy as many tacos?”

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

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

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

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

“Goes down.”

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

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

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

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

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


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

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

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

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


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

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

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


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

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

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

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

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

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


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

Here’s the boardwork for a systems models.

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

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


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


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

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

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

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

On John Quincy Adams and His Photograph

Of late there’s been notice that John Quincy Adams was photographed once or twice (the link is to Razib Khan’s thoughtful post). I’m not sure why the meme started; the picture has been around for decades. These articles seemed to have kicked off the trend.

The two original articles I read paid little attention to the man himself.

This is the man who authored the Monroe Doctrine declaring the US closed for settlement, who was probably the greatest Secretary of State in our history at a time when Sec of State was the second most important job in government. When Andrew Jackson overreached by invading Florida, Adams seized the opportunity to acquire the territory. When the US got involved in an ill-advised Second War for Independence and got mostly trounced, Adams negotiated the Treaty of Ghent, giving Britain next to nothing and keeping what the US had. This man, as both president and ex-President, strongly supported American nationalism and growth, but nonetheless had a relatively sympathetic policy towards native Americans and was one of the great early anti-slavery activists in Congress, routinely fighting the gag rule and, of course, defending the rights of African slaves in the Amistad case; both his opposition to slavery and his support of native Americans mark a profound difference from his much lionized successor, slaveowner and Cherokee remover Andrew Jackson.

Many who see the picture marvel that photography was around to capture a man with such a strong relationship to the founding fathers—how amazing it was that we have a picture of a man who knew Washington and Jefferson. One of the founding fathers was his dad, of course, and he knew many of the early leaders well. But arguably, he was a founding father, less than a decade younger than Alexander Hamilton. Washington—the man, not the city—considered him an extraordinary young diplomat after reading his series of articles in support of Washington’s Proclamation of Neutrality. Hamilton and Jefferson fought passionately over domestic policy and saw foreign affairs largely in terms of their own ideological preferences (not that this is a bad thing), while their boss listened and decided, usually in Hamilton’s favor. But neutrality was Washington’s own priority, one he developed from his own experience and values, decidedly rejecting both of his key advisers’ advice. That the first president appointed JQA to his first diplomatic position as US minister to the Netherlands on the strength of that series, read his letters home carefully, and used phrases from those dispatches in his Farewell Address speaks to Quincy Adams’ impact on early US foreign policy.

Like his father, JQA was a cranky misogynist and an unpopular but not necessarily terrible one-term president. He shared the founding fathers’ early vision of the country as a republic, in which an elite and well-informed minority undid the will of the impassioned people, hence his acquiescence to the original Corrupt Bargain.

He had a massive stroke in the House Chamber while voting against a resolution to award medals to Mexican-American War generals (he opposed the war and its treaty saying it would only exacerbate sectional tensions and lead to civil war, and by golly, he was right), dying two days later. Abraham Lincoln was part of the delegation escorting his body home to be buried in Massachusetts.

And while we’re mentioning photographs of people who knew the founding fathers, how about a shout out for this lady on the right?

The socialite Dolly Madison carved the path for the ceremonial but none-the-less essential role for first lady, serving in that capacity for the widow Thomas Jefferson as well as for her husband James. While she didn’t personally cut down Washington’s portrait, she was unquestionably the person who identified it for rescue, along with a copy of the Declaration of Independence. She refused to leave Washington until the last minute when the British army was a few miles away, and rushed back to the city as soon as possible, understanding the importance of her symbolic presence.

But I digress.

Is there any tidbit of information in this Atlantic article or the Daily Mail piece even a tenth as interesting as any one of the facts I’ve outlined above? Both pieces are more interested in minutiae about the pebble in the man’s eye, and maybe a brief mention of hey, he knew the founding fathers!

Yes, it is insanely cool that this man lived long enough to be captured by photography—not because of who he knew, but because of who he was.