Monthly Archives: May 2012


Last year, teaching a broadly differentiated algebra class, I had four to five tests per class. Cheating stopped being an option when kids didn’t even know what test their friends had, much less the person who happened to be sitting next to them.

This year, I did far less differentiation. My geometry classes formed a pretty tight abilities range; I have five or six kids who would have benefited from honors work and for whom my class as designed is just too easy. These kids learned a lot independently, but I didn’t frequently test them on it. I had another ten who were appropriately challenged by the highest difficulty problems in my normal class. Only two of my students were so far below the ability range that it was difficult for them to function; I’d had one of them last year and was able to help him be productive and keep learning math, which was much more important for him than learning geometry. The other one had health issues.

Of my 90 algebra II students, ten could have done more. The rest were sufficiently challenged with the “Advanced Algebra I with Algebra II Topics” course that I offered (to be fair, I covered all the state standards).

So I built two versions of each test and, in most cases, that was enough. Most of my low ability kids learn quickly that trying in class and doing their best on the test guarantees them a pass, and that’s sufficient for them. In this day and age, I find it heartwarming how many kids would rather be honest about their abilities and improve to the best extent they can, rather than continually try and cheat.

However, there are always a few who cheat. These kids are usually not the lowest ability students, but the liars. They do no work in class, then turn in a test with a whole bunch of right answers and no work on them. Some teachers demand that the students show their work and give them no credit for right answers without work. But many students—lots of them boys—do their work in their heads, and I see no reason to create a policy that makes them jump through extra hoops. Plus, the cheaters just invent work.

I am not a suspicious person, and identifying cheaters is hard for me. When I see a kid who has a lot of right answers and no work, I’m likely to blame myself first. Why didn’t I notice that this kid was a strong math student? This sounds incredibly naive, and I’m working to get past it.

I use multiple choice tests–not to save myself work, but because my top kids need the practice and my weaker kids have the opportunity to show knowledge without necessarily knowing all the steps needed to do the work. So the weak kid will get the zeros wrong on a parabola (picking 3 for the zero when the factor is x+3), but will be able to identify the basics for an exponential growth function, even if he doesn’t remember every step of the process. My quizzes are free response. This helps me confirm cheaters—a kid who aces the multiple choice test but has horrible math on a quiz? Cheater. (I’ve mentioned before that a number of kids have the opposite problem–they are strong at free response and terrible at multiple choice. Really.)

Anyway, at some point, I figure it out. My first response: I just don’t grade their tests. When they ask me about it, I say “You know, it’s the weirdest thing. You had a lot of right answers with no work. I’m thinking you should go to Vegas and try your luck. More likely, you should stop depending on the kindness of other students.”

“What, you’re saying I’m cheating?”


“You can’t prove it.”

“Don’t have to. I’m just not giving you a grade. Want to argue about it? Come show me how to do the work.”

They never take me up on the offer. Most of them get the hint and start to try. The rest of them get a free response test when everyone else is doing a multiple choice. They turn it in blank, I flunk them.

No, I don’t raise the issue with their parents. Life is too short. I am not morally scandalized by cheaters. I recognize cheating as an understandable response to the lack of choice students have at school. I don’t excuse it. I just don’t care one way or the other about the morality of it. The act of cheating will not, in and of itself, hurt a student’s grade. I will simply force the student to be graded on his or her actual ability, once I figure out that they are cheating.

In this state, teachers are legally prohibited from failing a student unless they got a D or lower at the second progress report. So if they have a C or higher at the progress report and stop showing entirely, they still can’t fail. If I have any students who stop working, therefore, I give them a D at the progress report even if their actual grade was higher (whether due to cheating or an earlier honest effort is irrelevant).

This year, I went longer than usual without any major tests—about 6 weeks from spring break to the state tests. I liked the results, as it gave me more time to focus on content. However, it meant that my cheaters got a break from scrutiny. Well, screw that. I don’t want to design my instructional strategy around cheaters and laggards.

So I told the kids that if they hadn’t been working or if in some way I were concerned about their abilities, they would be getting a D on the progress report—regardless of their actual grade. They were welcome to come in with their parents if they wanted to protest, and I could document their utter lack of work. (No one took me up on the offer).

This motivated a number of my cheaters to at least pretend to work after the second progress report, since they knew that they could be failed. And, happily, a couple did genuinely start to work.

Then, phase II. I test my kids weekly in May. I use the first two weeks to spot any additional cheaters and confirm the usual suspects. This year, I spotted two who I won’t be able to fail. But they’ll get a D-.

Those kids start getting free response tests instead of multiple choice. Which they turn in blank or with angry comments, because they have no clue how to do them. It’s kind of fun. “Hey, I don’t have the same test as other people!!” “And you know that because….?”

How many? I think it’s fewer than 10% of my 150 students who continue to cheat after I’ve made it clear I know what they’re up to.

So next week, when most of my kids are watching Rear Window, these kids will be taking an actual four page free response final. Not because I want to punish them. Heaven forfend. I just want to give them one last chance to pass.

I am certain that the students in question do not have any idea how to do any of the work. But if they do—if they convince me that they were just being lazy or trying to get a better grade than they could otherwise—they will pass. Like I said, I don’t care about the morality. I just want to grade them on their actual ability.

Teaching Algebra, or Banging Your Head With a Whiteboard

The Five Big Ideas of First Year Algebra:

  1. Identifying the slope and y-intercept of a line from a linear equation, and graphing a linear equation provided in slope-intercept form.
  2. Solving multistep, single-variable equations that involve distribution and combination of like terms.
  3. Using substitution or elimination to solve a system of equations.
  4. Binomial multiplication
  5. Factoring a quadratic equation (a=1)

(You middle-school algebra teachers are saying “Wait, what about graphing a parabola? What about point-slope and standard form for linear equations? What about….” Stop right there. I teach the kids who didn’t make it through your classes. Some winnowing is necessary. Furthermore, I said the five BIG ideas, not the ONLY ideas.)

I’ve taught some form of algebra every year. From my first year on, I’ve nailed factoring quadratics. I do it with the generic rectangle, which has the added feature of helping out when a > 1. The integrated method I use to teach both binomial multiplication and factoring really seems to help the students put it all together.

Towards the end of my first year teaching algebra, I noticed a weird thing. My kids were bombing multi-step equations, and I couldn’t see why. I’d taught them distribution, combination, and isolating—and they’d all done well. Then they were utterly discombobulated when faced with an equation like 3x + 5(2x-7) = 4. They’d add the 3x and 5 to get 8x, then multiply it by 2x, get 16x….it was insane. Yet when I walked them through the problem breaking down distribution and combination, they got each step individually.

Then, early in my second year, I saw the same problem. Kids who had shown solid mastery of distribution, combination of like terms, and solving for x were crashing and burning when I gave them a multi-step equation that mixed and matched everything. I suddenly got it. Multi-steps up the cognitive load considerably. The kids had to take each step in the context of a larger task, and they were losing track. They couldn’t look at the problem and break it down into parts.

So I created the Distribute-Combine-Isolate worksheet, one of the best worksheets I’ve ever done. First distribute, then combine, then isolate. It gave them a sequence to follow. The improvement was tremendous. My first year students, who had much lower incoming test scores than other classes, topped all the other classes in a course-alike assessment on the multi-step equation. This year, I used the same worksheet with any Algebra 2 students who struggled with multi-steps. Again, working multi-step equations has been a major success area; I don’t have to review it, and I can put a tricky question on a test and know that all the students will either get it right or make a few minor mistakes. I am pleased.

Systems: This is the lowest priority of the Big Five when I’m working with struggling students, but it’s a high priority item for my stronger students. I find the challenge comes in when I want them to recognize a system problem. They get the technique, but the overall solution approach is still iffy. But then, this is tough. I don’t feel any real frustration or energy about it.

Leaving linear equations and binomial multiplication, arguably the easiest of the Big Five, as the most challenging and mindbogglingly crazy-making. They get it and forget it. Get it and forget it. Over and over and over and over………….[bam bam bam bam bam]

Slope: You teach them how to plot points. You teach them to see the line. You use manipulatives, transparencies with lines on them, that they can use to match up two points and see how the slope and y-intercept change. You show them how different types of situations map to different slopes. And of course, you give them endless practice.

And then you sketch a line, clearly mark the slope and the y-intercept, and ask any kid who isn’t acing the class, “So, is the slope of this line positive or negative?” and wait, and wait, and wait, and wait and sure as a villain in a Bond film, the kid will say “Um, negative?” when the slope is positive and “Positive” affirmatively when the slope is negative.

So you teach them how to model equations quickly, which works a charm and gives them all sorts of new skills. You see them become much more proficient at word problems, at seeing an equation like 8x +3y = 24 and thinking “Burgers for $8, hotdogs for $3, total of $24” and by god, it’s awesome. All this keeps, beautifully; months later, they are still showing increased competency at word problems and linear equations. You also give them endless practice worksheets where all they have to do is identify + or – on a slope image—nothing more, and they do it cheerfully and successfully. You give them the “N” rule (negative slopes form an N).

And then, you give them a test, in which they have to identify a simple system of inequalities, and a student, a mid-level student calls you over and says, “I have no idea how to do these problems.”

“Well,” you say, “look at one of the lines in the system.” The student points to a line. “Positive or negative slope?” and wait and wait and wait and wait and sure enough, the student says “Positive” when it’s negative and “Um, negative?” when it’s positive and you gnash your teeth and try to figure out how to help them without making them feel hopeless.

And later, when the same thing happens again during the test review, and you start beating yourself over the head with a whiteboard (they make them student-sized, did you know? Like slates in Laura Ingalls’ day) and then you get up and say, carefully,

“Look. When you see me beating myself over the head with a whiteboard, it’s because I am wondering what other way I could teach you this HUGE, SINGLE MOST IMPORTANT idea in first year algebra, something that I’ve told you fifty times, and believe me when I say that I’m not angry or disgusted when people don’t get it. I just can’t figure out how to make it clearer. And I think the real problem is NOT that I can’t make it clearer, but that I can’t get you to stop and think about the many, many many ways to determine the direction of a slope. All you need to do is stop and think about it and remember what you’ve done. And for some reason, many of you don’t. Let me say it again: I am not blaming you. I don’t think you’re dumb. I JUST WANT YOU TO STOP DOING IT SO I WON’T HAVE TO BEAT MYSELF OVER THE HEAD ANYMORE.”

And the class laughs, and you remind them again to stop the minute they see a line. What is the direction? What methods do they have for making that determination? Do NOT simply look at it and say “Heads, positive. Tails, negative” and guess. Please?

Lather, rinse, repeat.

As bad as slope is–and it’s terrible, horrible, the single most frustrating thing about teaching algebra to kids who struggle with math–it doesn’t have the short sharp shock value of the Binomial Multiplication Middle Term Miss.

Last week I gave my kids a geometry test and one of the questions was:


BOTH CLASSES. Every single kid (except the top 6 students, who took a different test) took x2 + x + 9, meaning that they squared (x + 3) and got x2 + 9. WHY? WHY? WHY?

I tell them that this makes baby Jesus cry. It’s the math equivalent of clubbing cute little seals. THEY MUST STOP. It hurts. And we review it, with the rectangle, which they use for factoring and SHOULD MAKE IT CLEARER, DAMMIT! and they learn it again. But I know, very soon, they will forget. If only to make me crazy.

Still progressive. But smarter with it.

Yong Zhao, author and education professor, is asked to predict the education landscape after 5 years of Common Core:

Question: What will be different five years from now if the current plans go forward?

Yong Zhao: It’s always dangerous to predict the future. But if history is any indication, judging from the accomplishment of NCLB and Race-to-the Top, I would say that five years from now, American education will still be said to be broken and obsolete. We will find out that the Common Core Standards, after billions of dollars, millions of hours of teacher time, and numerous PD sessions, alignment task forces, is not the cure to American’s education ill. Worse yet, we will likely have most of nation’s schools teaching to the common tests aligned with the Common Core. As a result, we will see a further narrowing of the curriculum and educational experiences. Whatever innovative teaching that has not been completely lost in the schools may finally be gone. And then we will have a nation of students, teachers, and schools who are compliant with the Common Core Standards, but we may not have much else left.

He then goes on to argue that minorities do poorly because they need more government support, that tests aren’t accurate evaluators (really, what does it mean to be “good” at something?), and that students really should write about what they feeeeeeeeeel instead of what they think. In short, he’s a touchy-feely progressive who has all sorts of loopy ideas.

But hey. He’s right about the Common Core standards.

One of my great surprises of the past year or so is the emergence of progressives as, dare I say, realists on achievement, in contrast to the eduformers’ edging ever-nearer to a role as totalitarian dreamers.

Fundamentally Flawed

Mike Petrilli is puzzled by NAEP scores:

One of the great mysteries of modern-day school reform is why we’re seeing such strong progress (in math at least, especially among our lowest-performing students) at the elementary and middle school levels, but not in high school.

How is this not completely predictable?

Elementary school education focuses on fundamental skills and knowledge, increasing difficulty each year in a linear fashion. Each year, the students add to their existing base of knowledge. Start with adding and subtracting single digit integers, move to adding and subtracting two and three digit numbers. Eventually move into multiplication, then into division. With division comes the notion of non-integers, and so onto understanding fractions and decimals, then operations with fractions and decimals. Likewise, geometry starts with understanding shapes, then moving onto perimeter and area and then volume. Reading begins with decoding, and imperfectly expands vocabulary, voice complexity (irony, unreliable narrators, etc), while ideally adding content knowledge.

This is why elementary education is always assessed at “grade level”, because the curriculum assigns a particular amount of knowledge and achievement to be demonstrated at each grade. Principals looking to improve outcomes talk obsessively about how many students they can get to “grade level”, or better still, “above grade level”. Even the great content guru E. D. Hirsch talks about “what your Nth grader should know”–but only through 6th grade.

The rate at which students move through the “grade level” varies. Some students can effortlessly score at or above “grade level” without doing a bit of homework and watching 8 hours of TV a day. Other students will rarely achieve grade level without dramatic changes in instruction method and hours in school. Still others will never achieve “grade level”. These students are not distributed proportionately by either race or income, the two categories we monitor progress by because we don’t want to monitor by cognitive ability.

Because the rate of progress varies, and because the knowledge requirements are fundamental, education “reform” can achieve results. Students with poor reading skills can get up to their grade level or even beyond, if they are motivated and taught for longer hours in a few key subjects. But we don’t really know that they are actually achieving academic success or readiness for more demanding material. All we know for sure is they are acquiring that fundamental knowledge of the elementary school curriculum at a faster pace—and of course, in many happy cases, getting a higher level of that fundamental knowledge than they would otherwise acquire.

But then comes high school, where E. D. Hirsch has no advice, where no one talks about grade level, and where the knowledge transferred is anything but fundamental.

The cognitive demands of high school are not a linear step up from 8th grade. In fact, they so far outstrip elementary and middle school expectations that our insistence on ignoring this fact is just downright mindboggling. Kids whose teachers took 9 hours a day, chants, threats, exhortations, or College DAys to keep at or slightly above grade school level reading and math are, sadly, not at all ready to succeed at algebra, trigonometry, Shakespeare, and chemistry.

Of course, many students aren’t even getting to 8th grade level by 8th grade, meaning they are starting the onslaught of high school without even the bare minimum of fundamental knowledge. Given a decent and realistic high school education, some of these students could get those fundamentals by their senior year, but we won’t allow them, instead demanding they spend time in subjects they can’t understand.

“College prep” high school level curriculum, once the option of a few select students with the interest and ability to move to college, is now demanded of all students in the absurd notion that expectations–and, of course, good teachers—are all these students ever needed to achieve. When they don’t achieve despite expectations, who is left to blame but the teachers?

Genuine college prep work is simply beyond the cognitive ability of our lowest ability students. It doesn’t matter if their elementary school teachers got them to grade level on time, or even if they learned more as a “below level proficiency” student in fourth grade.

Why is this so hard to understand? You want to see better performance by our 12th graders? Make the test easier. That’s why the improvement is noticeable in the earlier grades. The 12th grade test doesn’t acknowledge the existence of students who aren’t capable of 8th grade complexity.

No one’s at fault. Our education system didn’t fail. The only problem is that everyone’s not smart enough for high school, much less college prep.

Now, eduformers and progressives both may find this idea offensive or unpleasant. They may just disagree. But to leave it off the list of possibilities entirely bespeaks a certain blindness that, sadly, pervades all levels of our educational policy debate.