Tag Archives: teaching

Every so often….

I got into teaching to work with struggling kids. I’d enjoy working with an entire class of motivated and able kids, but it would also come as a shock. Most of my time I spend pushing kids up the hill and praying they don’t roll backwards.

But I do have exceptionally bright kids in my classes, too. My geometry classes in particular are a joy, since even the low ability, low incentive kids know they’ve finally escaped algebra and aren’t eager to repeat that experience by repeating geometry. So it’s a well-behaved group with decent motivation. On the other hand, 80% of my Algebra II classes scored Far Below Basic or Below Basic in Algebra two years earlier and a few of them are much harder to keep in line, even though the bulk of them are well-behaved and want to learn, not out of any love of math, but because they understand, thanks in part to my frequent rants, that they will be taking a placement test in a year or two and the only good thing about my class is that it’s free.

But I do have some students who chose an easier course because they are athletes, or because they struggled in geometry, or just because someone somewhere made an odd placement decision, and that are in fact quite strong in math—and of course, there are always a few students who finally “get it” and who actually start to grasp the material (this is extremely rare, and one major reason why value added is a problem in high school teacher evaluations).

Anyway, on Thursday, I had two interactions with bright students that stay with me because of their infrequency. I’m not complaining. It was just fun.

Incident #1

A solid geometry student had done horribly on her last test. She doesn’t often engage with me and remains a bit distant, but after I turned back the tests I checked in with her.

“I do not get special right triangles. I don’t understand how they work. I get similar triangles, but I can’t ever remember the ratios and I can’t see how it works.”

I drew a right triangle on the board with the two legs labelled “x” and the hypotenuse labelled “hyp”. “What kind of triangle is this?”

“Isosceles right.”

“Okay. Create an equation to solve for the hypotenuse using the Pythagorean Theorem. I’ll be back in a bit.”

I came back, and she was working through it, a classmate tossing in advice and argument.

“No, it’s square root of two! You took the square root of both sides!”

“Oh, that’s right.” and she had it solved, the hypotenuse equal to x * sqrt 2.

I pointed to the class notes which were on the board. The isosceles right triangle was labelled x, x, x sqrt 2.

“Oh! I get it.”

I then sketched out a 30-60-90, asked her what it was and she correctly said it was a half of an equilateral triangle. I told her to label the sides, making sure she put the x opposite the 30. Then I told her to solve for the second leg. When I came back, she’d finished that up and was in a great mood.

Had I taught the Pythagorean method before? Yes. Several times. Sometimes they just aren’t ready to get it until they’re ready to get it. But only a strong student can grasp the algebra of the Pythagorean proof and see how that knowledge can help her remember the ratios.

Incident 2

One of my strongest Algebra II students was struggling with synthetic division, which I introduced as a method of testing quadratic values (more on that later). She asked me to explain it to her again—not just the how, but the why and the what.

I began from the top and ran through it all. When she grasped that the division process revealed not only the quotient but that the remainder was the equivalent to evaluating the function at the divisor value, she said “Wow. You might almost think that was planned.”

I laughed in unexpected pleasure. I am not a believer in God nor a mathematician, nor am I a proponent of the “math is everywhere, math is beauty” propaganda that true math lovers preach. But the Remainder Theorem, like the Fundamental Theorems of both Algebra and Calculus, is indeed enough to make even me wonder if there’s some Grand Design. That a student of mine should reach that conclusion after a largely utilitarian but comprehensive explanation by yours truly was an unlooked for joy.

I was telling this to a colleague, and he reminded me of the famous quote: “God exists because Arithmetic is consistent. The Devil exists because we can’t prove it.”


I teach composition and book club on Saturdays at an SAT academy, which is codespeak for a near-entire Asian (ethnic Chinese, Korean, Indian, and Vietnamese) first generation immigrant parent clientele who want a place to send their kids from 7th grade on. My kids spend three hours every Saturday with me for a year, then I teach summer school, often to some of the same kids. Understand that most of them live in Asian enclaves in which they rarely run into white people, much less black or Hispanic. The public schools they go to are 80% Asian, then they go off to public or private universities that are 40-50% Asian, they (thus far) marry other Asians and will eventually form additional enclaves and renew. I always start off every new year by asking the kids to estimate the percentage of the American population that is Asian–the lowest guess ever has been 15%. Most of them guess 30%.

I love it. I have my doubts about the impact this same population is having on public schools and college admissions, but my affection for the kids themselves knows no bounds.

It’s a book club, and the primary emphasis is on vocabulary, reading comprehension, and writing in different forms. But I nonetheless include a great deal of instruction on “white people world*” and most of them soak it up eagerly. I am often the first person they’ve met who has told them that watching more TV is actually helpful, that good grades are nice but only if they are accompanied by actual knowledge and achievement which is not the same thing, and who understands but gently mocks their parents’ demands. I can only be satisfied by them thinking for themselves, and there are no grades—a topsy turvy world for these kids.

Each class quickly grasps that I will mention things that they’ve never heard of, and that they should know of, and that I think it’s a problem, or at least a deficit. And periodically, the deficit will be so significant that I immediately act to remedy it.

Which is what happened today, when we were going over the news of the week. They all knew that Whitney Houston had died. It took me a while to realize that none of them could identify a single song of hers.

“Seriously? I Will Always Love You? Never heard of it? Hmmf.”

“Was she really popular?”

“Oh, hugely so for about 15 years back in the 80s and 90s. She came from a talented family. You’ve probably never heard of her mom, and probably wouldn’t know Dionne Warwick, but Aretha Franklin was her godmother, and…..” I see the blank looks.

“Oh, come ON. You do too know Aretha.” They all shake their heads. “You have too. What’s annoying is that you’ve heard her and just didn’t know it was her, and you SHOULD. So I’m going to play her most famous song, you’re going to go ‘oh, yeah, I know that song!’ and from now on you are going to know who Aretha Franklin is.” I am thumbing through my Android as they assure me they have no idea who Aretha Franklin is. Their assurances last through the opening of “RESPECT” and then , as her voice comes on, sure enough….”

“Oh, is THAT Aretha Franklin! I know that song!” and they are all cracking up because they are doing exactly what I told them they’d do.

“Now. Never forget who Aretha is, okay?” They nod.

I then play two Whitney songs. Not only had they heard them before, but one of them had “I Wanna Dance With Somebody” on her Ipod.

Don’t worry, parents, we talked about art and Asher Lev, too.

*Yes, I know, there’s a certain irony in my calling it “white people world” when I’m explaining Aretha and Whitney.

Modeling with Quadratics

After my success (I hope) with linear equations, I started a unit doing the same thing with quadratics.

Days 1-3:

“A triangle’s height is three feet longer than its base. Create a table linking the height to the area. Graph.”

“A rectangle’s length is twice as long as its width. Create a table linking the width to its area.”

“A rectangle has sides of X+3 and X-2. Create a table linking X to the rectangle’s area.”

Just as with linear equations, the students really improved at generating values. They also, I think, quickly grasped that generating data for quadratics is considerably more complicated than linear equations. More than one pointed out to me that they couldn’t just “add three each time” as they could with linear equations.

I taught them how to break it down into parts and assign each part to a column. For example, the first triangle problem:



Base * Height

Half BH (Area)







The stronger students could see how this led to the equation; even the weaker students could see that each equation had steps, and they started to get suspicious if it got too easy. One struggling student called me over to tell me he must be doing something wrong because “look, it’s going up by the same amount”. He was linking length to width, rather than length to area.

Even if they don’t get much stronger in working with quadratics generally, this exercise clearly helped them gain competence at working through a problem. Next step: generating values quickly from an equation in standard form. Hello, synthetic substitution.

Homework and grades.

The NY Times rewinds the typical homework debate. The post gets predictable pro and con responses: “homework is ruining my kid’s life” vs. “homework is a necessary component to learning”.

As is often the case, the situation at hand involves middle and elementary school students. High school homework rarely gets much scrutiny, unless it’s high achieving students complaining (with a lot of justification) about the huge amount of work they have to stay on top of to stay competitive.

But outside the top 10%, homework’s impact on high school students is a much neglected issue, and it shouldn’t be. Few people seem to understand the inordinate influence homework has on student transcripts—and the results, for the most part, are near-fraudulent.

High school students are far less likely to do assigned homework and the consequences for non-compliance are much higher, because students who don’t do homework often fail—not for lack of demonstrated subject matter skills, but simply for not doing their homework.

Here’s a chart that pretty much any teacher in the country could produce, comparing achievement (test scores) to classwork and homework effort.

(from Reflections of a First Year Math Teacher)

Don’t be distracted by the positive correlation. This is for individual grades, so the individual points are what matter. Notice how many students work hard, yet have failing test results, and how many students don’t work much at all, yet ace the tests.

This graph reflects the reality I point out ad infinitum: effort is only tangentially related to achievement, and then only at the individual level. Students who try harder don’t do better than students who don’t try at all. The lurking variable, of course, is ability.

For this reason, teachers should not include homework as a significant part of a grade, and should never allow missing homework to lower a grade. (This means, English teachers, that if a student doesn’t do an assigned essay, you find another way to assess the student.)

But of course, teachers routinely include homework as 25, 35, 50% of the grade. Happens all the time, and no one calls teachers on this behavior because it’s so damn cheap and easy to argue that homework is essential, good for both discipline and achievement. Never mind that there’s no real evidence for the latter, and the former should not influence grading.

The “homework proxy for effort” skew is understandable, given that teachers really can’t grade students purely on demonstrated ability. Teachers would fail too many students if they set an absolute ability standard. (See the above chart again if you need reminding.)

Teachers tend to value effort anyway—it makes them feel needed. So this preference, coupled with the real dilemma imposed by teaching and assessing students whose skills are far below the required ability level, gives them license to reward effort, to some degree.

But the degree matters.

Boosting hardworking students’ grades just a bit (say from one grade’s “+” to another grade’s “-“) is fine. While some may raise an eyebrow at the idea of giving a failing student a D- because he shows up and tries, I not only forgive this, but engage in the practice frequently.

Giving a student with mediocre math skills an A or B simply because they work hard and finish all their homework is quite another matter and worst of all, giving a low grade to students with excellent test performance—in many cases even failing the student—is outright fraud.

This happens every day, although it’s drowned out by all the middle class parent whining about how much work their middle schoolers have.

In high school, teachers are assigning homework, students aren’t doing it, and teachers are giving lower grades—often failing students completely—even though their skills are strong, simply because they don’t do their homework. Teachers are a moralizing lot, by and large, and they are far more comfortable giving low grades, or outright failing, kids who don’t try and aren’t compliant than they are doing the same to hardworking kids with low skills.

This leads to astonishingly bizarre grade results. Two students might each have very weak algebra skills but one gets an A, because she goes to a school that weights homework as 50 or more percent of the grade and does extra credit, while the other fails at the same school simply for not doing his homework. Students who can read at basic proficiency can fail English for not doing their essays, while functionally illiterate students who earnestly string together sentences on books they didn’t understand get Bs.

Five of my algebra sophomore students last year scored Basic on their state tests–but failed algebra for not doing their homework. One of my best geometry students failed geometry last year for not doing his homework—at least, he was one of my best students until he left for alternative high school because he’d failed so many classes (all by not doing homework) that he can’t graduate on a normal schedule. Several of my top Algebra II students this year took Algebra II/Trig last year and scored basic—but, yes, failed for not doing their homework. Meanwhile, I have colleagues teaching AP Calculus to students who scored Below Basic on all their math state tests up to that point. How can that happen?

State universities don’t use test scores for basic admission, but grades. Which explains why remediation is such a huge problem, doesn’t it?

Of course, at this point in a conversation someone will say, condescendingly, that the students just need to learn how to put in some effort, go through the motions, and I have to fight the urge to go find a baseball bat. Really? We’re talking about a nationwide problem and some idiot treats this as a cheap sermon on morality and obedience? Seriously?

I mean, never mind the fraud that teachers are engaging in, failing competent students while giving good grades to functional illiterates. Consider the massive waste of money thrown away because so many teachers confuse homework obedience with academic achievement. And of course, because our nation is convinced that all kids must be on the college track, there’s so little room for error that one or two Fs ensure that a student is off-track and just marking time until alternative high school is an option.

Districts desperate to stop teachers from indiscriminately failing otherwise competent kids (compared to the kids who are passing) institute those policies that annoy eduformers and earn them lots of mocking catcalls—Fs can’t be less than 50%, homework can’t be more than 10% of the grade, and so on—but these policies make perfect sense when considered in light of the money districts lose to dropouts and quick credit factories that allow students to collect enough credits without learning a thing—even less than they learned from the teacher who gave them an F.

Never forget: grades are a fraud. And in homework, stop wondering about how much is given, and start asking about how much it’s worth to the grade. Because if it’s more than 5%, it’s too much.

Modeling Linear Equations, Part 2

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?

The assessment confirmed my hunch that I haven’t been wasting my time. I tried to focus on problems they could solve in multiple ways—up to and including plugging in the answers. I wanted them to be able to approach a problem “as if it were their money”, as I kept telling them when they were figuring out how many power bars and gatorade they could buy for $20.

Here’s the assessment, with the percentage of 83 students who answered correctly (click to enlarge).


  • Questions 1, 3, 8, and 10 were at or just above the “random guess” percentage. Everyone screwed up the first question, flipping rise and run (answering -2). 3, 8, and 10, however, were answered correctly by students who received a C or higher.
  • Question 11 makes me want to beat my head with a student whiteboard. Only 62% of the class knew what the slope of y=-10x + 7? Really? Some of the students who got it wrong then went on to accurately identify the slope from a table and pick the right equation in slope intercept form. It is to weep. But anyway, my guess is that 20% of the kids who got it wrong actually know it, leaving 20% who still aren’t sure. And that’s bad enough.
  • Very proud am I of the results for question 13, since we hadn’t done anything like that in class. Apart from the “read carefully” hint, I gave no assistance. Of course, I didn’t include the midpoint for AM, which would have cut the success rate in half. But it still shows the students are thinking and not giving up.
  • While the system of inequalities questions are barely at 50%, that’s a huge improvement over the semester final.
  • The students had trouble with question 19. This suggests that many of them are still just plugging the values into the equation rather than reading the slope from the table, since the slope is a fraction—and at least a third of the class still can’t multiply fractions.

I curve my multiple choice questions on a 15 point scale, so 85 and up is an A, 70-84 a B, down to 40-54 a D. A student could pass with a D- by answering 8 questions; I think getting 40% of a test that would not (for those students) include any gimmees is worth a low passing grade. On that scale, the average score was 70%—and that’s a first. Every other assessment, including the two on linear equations last semester, had average scores in the D range.

Only two As and four Bs. Many students who usually nail Bs got Cs, while the students who usually failed also got Cs or solid Ds. This too makes me think I’m on the right track. It tells me that some of the B students are used to memorizing methods—and since I didn’t give them one clear-cut method, they had a bit of trouble. The D and F students, who couldn’t memorize or even really understand the methods, were really able to benefit from the instruction and had solid results. (For many students, I consider a passing grade of D a great achievement, and tell them so.)

Several of the B students who did poorly came up to talk to me about it, and that, too, was revealing. Most of them understood realized that this new approach was exposing a weakness on their part and, instead of complaining, talked about their difficulties and asked how they could improve.

We’re doing quadratics in the same way—just spent three days learning how to create table values from descriptions. For some reason, the only quadratic word equations I can think of involve geometry—but then, most of my kids need four or five seconds to remember the formula for a triangle, so the review is win-win.

Grading Tests

It kills me to say this, but any honest description of my grading would have to include the word “holistic”.

This tendency is getting worse. My normal method for a quiz: I assign points before hand, weighting the important problems heavily, and then grade the tests. I do not curve, but if I discover all students really tanked an important problem, I go back and re-weight, with a growl and a sigh.

Today, I was grading the data modelling quizzes I described in an earlier post, and just didn’t feel like assigning points.

Here’s the quiz (Click to enlarge):

Yes, yes, some of you will say “But this is algebra I material! Pre-algebra, in fact!” Newsflash: many, many students still don’t understand this. So get over it. The students had to create a table of values, graphs, and linear equations for four word models, and then four table of values and graphs for given equations. I included one more difficult equation (a difference equalling a constant).

I am usually pretty good at timing tests–I’d say 1 out of every 10 tests, I am genuinely surprised when my students don’t finish. In this case, I was certain that some students wouldn’t finish, but I was interested in fluency. How many students would be able to finish the whole thing? But even so, I would have been better off with three questions in each section.

Anyway–I wasn’t really interested in finely tuned grades here. So I created four categories before looking at the student tests:

A–finished 6 of 8 problems accurately or with minor errors. Identified the equations and came up with reasonable word models in most OR completed and graphed the difference equation.

B–did all of one side correctly and clearly didn’t finish the second half (I’d given them the option to come in at lunch or after school to finish), or did parts of both sides correctly.

C– To get a C, students had to have done 2-3 problems correctly in full (table of values, graph) OR done one part of several questions correctly (e.g. table of values done for most problems, no graph).

D/F–Very little completed, with a range of 1-2 problems done somewhat correctly to clearly had no clue.

So then I reviewed the tests and put them into those categories without any markings. I got a nice heap of Bs and Cs, more Ds/Fs than I’d like, but still within reason (only 2-3 absolutely no clue), and about 10 As. Tonight I’ll go through them and point out errors.

Points, schmoints.

Modeling Linear Equations

“I have a certain amount of nickels and dimes that add up to $2.10.”

“Sam bought a number of tacos and burritos. The tacos were $2 and the burritos were $3. Sam spent $24.”

“Janice joined a gym with a sign-up fee of $40 and a monthly rate of $25.”

I put these three statements on the board and told my Algebra II students to generate a table of values and graph the values.

Some students were instantly able to use their “real-life” math knowledge to start working. Others needed a push and were then able to start. Some needed calculators to work out the values, but all 90 students were, with minimal prompting, able to use that part of their brain that had nothing to do with school to create a list of possibilities. Only a few questioned the lack of an “answer”, and none needed the explanation more than once.

We did this for three days. Over half my students were able to determine the slope of the line and link it back to the word problem without my expressly teaching it. Some of them improved as time went on; few of them are completely incapable of linking the word model to an equation. With discussion, most students began to realize that models with two changing numbers adding up to a constant (nickels and dimes, tacos and burritos) had a negative slope, because an increase in one led to a decrease in the other.

Then, for two days, I gave them problems in equation form:

“y = 3x+2”

“2x + 5y = 45”

“3x – 2y = 6”

“y = -.5 + 50”

They had to generate a table of values and graph the line. When some of them had difficulties, I pointed out the links to word models (what if you were buying burritos for $2 and 6-packs for $5–how much money did you have to spend?) and they got it right away. The subtraction models were the most difficult (which I expected, and didn’t emphasize).

And then, for a couple days, I mixed it up–gave them word models and equations.

I began writing the answer to the most common question in huge letters on the white board: YES, YOU CAN JUST PICK ANY NUMBER.

We’ve just spent two days doing the same thing with word models that provide two points but no relationship.

“Janice joined a gym that had a monthly rate and a signup fee. After three months, she’d paid $145. After 6 months, she paid $190.”

“Brad buys grain for his livestock on a regular cycle and re-orders it when he runs out. After 3 days, he had 72 pounds left. After 7 days, he had 24 pounds left.”

So they have to create the table, find the slope from the table and graph it. Day 2, they had to do that while also answering questions (“What was the signup fee? What was the monthly rate? How much grain did Brad use daily? When would he have to reorder?)

At no point during these two weeks did I work problems algebraically. Some students did, but in all cases I encouraged them to think about other ways to work the data.

They improved dramatically. I gave them a quiz, and all but a few students were able to do the problems with a minimum of questions, although they needed lots of reassurance. “Okay, I think I know what’s going on, so there must be something wrong.” A few kids gave me the “I have no idea how do to this” and I was pretty brutal in my lack of sympathy because they were the kids who don’t pay attention.

I spent the entire first semester teaching them linear and quadratic equations–graphing, systems, solving, factoring, the works. Algebra II is a course designed for students who don’t want or aren’t ready to move to Algebra II/Trigonometry–or who failed that and need a third year. So the first semester is a rehash of Algebra I. I covered it, they all learned a lot–and yet, the semester final was dismal. Some of it was Christmas crazy, and then I wasn’t happy with the test. But nonetheless, they should have done better.

So I mulled this over Christmas break. All but a few of my students are juniors or seniors. Some will be taking a proficiency exam in a few months. Others will be taking a proxy for the exam in their state tests. I’ve always been more focused on their college tests than their knowledge of second year algebra. I want my students to test out of remedial math, or spend as little time as possible in it.

That’s why I’ve decided to spend a month helping them use their math knowledge–the knowledge they see as entirely separate from algebra and geometry, their “real-life” knowledge–to model data.

If this works properly, the strongest students will have a much deeper understanding of the equations and how they relate to the data. The weaker students will be able to work problems using their inherent math ability, rather than struggling to turn the problem into an abstract representation they don’t recognize.

I’m going to finish up linear equations with maps, to give them a better understanding of midpoint, distance, parallel and perpendicular.

For more samples and boardwork, see Modeling Linear Equations, part 3
Then it’s onto quadratics.

Teaching Geometry

I taught two geometry sections my first year at a different school, and while I didn’t do a particularly good job (the classroom management problems were horrible for a different reason, and the curriculum was CPM–ick), I came away with useful insights that have really improved my execution this year.

Geometry Then and Now

Back in the dark ages, we used to say “There are two sorts of people in the world: those who prefer algebra and those who prefer geometry.” This mindset comes from a time when advanced students took algebra in 8th grade, most of everyone else college bound took it in 9th or 10th grade, and then followed it up with geometry, Algebra II, and precalc if there were enough time. Students who were really bad at math took Basic Math or Business Math and maybe took algebra their senior year. If this sounds familiar to you, fine–but it’s not like that anymore.

“Students who were really bad at math” were not representing our nation’s racial balance, and research unsurprisingly showed that students who went beyond Algebra II in high school had higher college completion rates. Naturally, this meant that everyone should take algebra as early as possible, cognitive ability or readiness be damned. The resulting carnage of this policy did not lead to re-evaluation, but rather to the determination that pre-algebra preparation should start earlier–and, of course, kids who fail algebra need to take it again.

Consider the effect of this policy on the average, “okay at math” kid today. Starting in sixth grade, it’s All Algebra, All the Time. By the time they get to geometry, “math instruction” and “algebra” are virtually synonymous–and they don’t even realize it. Kids have spent three, four, or even five years with algebra preparation or instruction. Specifically, using processes to solve for an unknown.

And then: geometry. Good god, what fresh hell is this? Facts. Vocabulary. Relationships. And then, in some weird way, you use these facts and vocabulary and relationships to come up with more facts and vocabulary and relationships. There’s no solving. There’s not even an answer. Half the time the book gives you the answer but then expects you to explain it using, god help us all, facts and vocabulary and relationships.

This is a whole galaxy away from “I like algebra better than geometry.” First off, all but 10-15% of my students found algebra completely unmanageable, so they aren’t looking back fondly at an easier subject. They’re trying hard not to curl up in a fetal position at the realization that math gets worse than the horror of the past three years.

Geometry teachers would do well, I think, to acknowledge this confusion. I tell my students some version of what I’ve just explained above and I see the light dawn. They get it. They might not get geometry, yet, but they get why they feel so lost. And that helps them move forward.

Try, try again
Two years ago, I could see that many of my students weren’t getting it. I retaught, thought of other ways to explain things, but I didn’t understand the degree of their lostness until relatively late in the first semester. I adjusted my teaching more, but I still hadn’t figured out why they were so lost.

This year, I was teaching parallel lines and transversals in week 2 or so and I suddenly realized that most of my students didn’t get it. They weren’t complaining, they weren’t acting out, they were just lost. I recognized the look from two years ago, and was now able to distinguish a furrowed brow of mild confusion from a blank look of utter nihilistic despair.

So at the end of day 2, I did a thumb check. “Okay, guys, my sense is a lot of you are feeling lost. Thumbs up if you feel confident, sideways or down if you’re kind of or totally lost.” And sure enough, most of the class was sideways or down.

I told the class I would come up with a different way to explain it. The next day, I used Geoboards, rubber bands, and little wooden geometric shapes to create a visual image of corresponding, alternate interior, and so on. (I’ll write that up some time). The lesson was very effective in helping students understand the angle relationship. But more important, the students recognized that I had stopped everything, rethought the lesson, came up with a radically different way of explaining the concepts–and had gone through this effort because I could see they were lost. The feedback from the new lesson was very enthusiastic, kids felt much less lost–but more importantly, they felt like I understood their confusion and was willing to spend time and effort helping them out.

This created a lot of good will, and since then they’ve been very trusting of my oftentimes bizarre way of building visual images to help them grasp geometric concepts.

It’s easier to do this in geometry than algebra, since geometry is new to everyone. Even my top students appreciate the occasional visual exercise, and I always have extra challenges for them. In algebra, some kids are lost right from the beginning, and it’s impossible to reteach everyone. (Which means, now that I think about it, that if I differentiate immediately after my assessment test in algebra, I might have an easier time. Hmmm.)

But fundamentally, it’s important to understand that time spent at the beginning, pacing be damned, will really pay off in student investment. I now realize that many of my geometry students two years ago had checked out because none of it made sense, and I didn’t pick up on that early enough to intervene. I took five days to explain parallel lines and transversals rather than two, but every minute of it was well spent.

De-emphasize what they won’t use

Most college graduates think proofs, logic, and construction as quintessential geometry subjects. That’s because we never use them again. We don’t spend any time in formal logic, never do formal proofs, and as for construction, forget it.

So I mostly dump them (which is how I make up the five days on transversals). Not completely. My Holt text starts with these five chapters:

  1. Foundations (Undefined Terms, Segments, Angles, area/perimeter formulas, coordinate geometry, transformations)
  2. Geometry Reasoning (logic and proofs)
  3. Parallel and Perpendicular Lines
  4. Triangle Congruence
  5. Triangle Properties

I dumped transformations entirely. I then took coordinate geometry, proofs, and logic and broke them up into tiny digestible chunks (coordinate geometry was review), rather than cover them all at one time, and covered about 20% of the material. So rather than an entire section on proofs, I introduced algebraic proofs at a natural pause point, when I had a day or two between major sections. I just introduced it; my goal was familiarity and recognition but not competence. Then, after introducing congruent triangles, I introduced two column proofs, and the students used congruence shortcuts to create two column proofs. This was much more successful than introducing a whole chapter on proofs when they were still in the WTF stage.

Yes, I know, the purists out there, assuming anyone is reading, is shocked. What? Proofs and logic introduce an invaluable way of thinking logically and methodically! Yep. But ask geometry teachers in heterogeneous classrooms if their kids understand proofs, and they will sigh. There’s just no way to get the lower ability half of the population to understand proofs and they’ll never use it again. I could spend lots of time trying, but I have better things to do with their time.

Ideally, I’d love to make my top students go through rigorous proofs, but it would take more instruction time than I can manage in differentiation. I hope to figure this out at some point, but I’m not as practiced at teaching geometry as I am at algebra.

So you’re thinking my class is too easy, right? Well, we just had our semester final and the geometry teachers agreed to start with a common assessment, built by a traditional geometry teacher who had covered far more material than I had in the first semester. I considered the test a little too easy and more picayune than I would build, so I substituted some harder questions. I didn’t dump more than two or three of the questions that we didn’t cover, because I felt pretty confident the students could figure it out–and, for the most part, they did.

Always remember where they are going
Geometry is just a brief respite. The next year, it’s back to algebra II, another course that causes a lot of carnage. Half of my class has extremely weak algebra skills, half of the rest are adequate, and the top students were rarely challenged with tough material. They need the practice. My sophomores will be taking the algebra and pre-algebra intensive state graduation test and my juniors are taking the SAT. Algebra is a big part of their testing load this year.

So I teach my geometry course as Applying Algebra with Geometry Facts. My students will never again need to prove that triangle ABC is congruent to triangle XYZ, but they will always need to know how to find the angle measures of a triangle whose angle ratio is 2:3:5. They will never use a compass again, but they will need to know what to do if Angle A and Angle B are supplementary and Angle A = 4x+ 13 and Angle B = 2x + 17 and they need to solve for x.

The problem is that the state tests tend to emphasis more traditional geometry. Aggravating, really, given that the state has clearly de-emphasized traditional geometry in its overall curriculum, but so be it.


I had told all my students this year that if they showed up and worked, they’d pass with a D-. In my Algebra II course, several students did not in any way demonstrate understanding of the material we covered that year (not for lack of trying, in most cases), but I kept my promise.

But in my Geometry classes, my D students were genuinely Ds. They struggled, but got Ds or “respectable Fs” (50% or higher) on all the tests and quizzes. On the 100 question final (40 correct is a D-, 15 point grade scale instead of 10), 15 students failed. All but two had “respectable” Fs (answered 30 or more questions correctly, and those two were just below 30. The distribution was pretty close to normal, the average score and the mode were C. So far, so good.

Teaching Algebra I

This is the first year I have been completely uninvolved with “first year” algebra. I use quotes because in high school, almost all–say, 80%–have taken the class at least once, and a good 20-30% have taken it twice. A fraction have taken it three times.

I wake up each day grateful that I’m not teaching algebra I, despite the fact that I’ve spent more time thinking about how to teach algebra than any other subject. But I rejoice nonetheless. No class is designed more perfectly to slap you in the face with the insane inadequacies of our educational policy. See Tom Loveless’s study,The Misplaced Math Student, for great research on the idiocies of putting unprepared students in eighth grade algebra.


Schools with majority URM (under-represented minorities, aka black and Hispanic) can simply pretend to teach algebra. I’m speaking here of charter schools and urban comprehensive schools with no high-achieving population to worry about. Pretending to teach algebra doesn’t do much for test scores, but it’s a lot easier for classroom management if you can give struggling students something they know how to do.

But many Title I schools are in the suburbs, which aren’t as economically homogeneous as they used to be and these schools have it much tougher. First, they can’t track, because their majority minority population sued back in the 90s. So all their classes are “heterogeneous”, progressive-speak for “put functionally illiterate/innumerate kids, struggling but not completely unskilled kids, ready-to-learn kids and highly skilled kids all in the same classroom and yammer about differentiation”. Second, because they have kids who are ready to learn algebra, they have to actually teach algebra. So they have to figure out a pace that doesn’t lose the middle and doesn’t bore the top. The bottom is largely left out of the equation. Unless you differentiate, and differentiation is a lot of work.

Last year, I taught all Algebra I: three regular courses and one Intervention, a double period course. Because the best thing to do with kids who are terrible at math is give them twice as much time to feel inadequate.


I slowly moved to a differentiation model. First, I started giving the top kids separate lessons, which was partially successful. It gave them more challenges and they did very well, but they felt neglected because I didn’t always get back to close up with them. By the end of the first semester, I’d seen a second group, skilled but slightly less motivated, start to break away from the pack. At the same time, I had about 20 students who simply had not demonstrated understanding of the first semester basics–and I’m very flexible in demonstrating understanding. These students, I decided, would go “on contract”. I would give them a D- if they demonstrated they could graph a line, factor a quadratic, solve a complicated multi-step equation, and make a reasonable stab at solving a system.

So I had four groups, which I called Purple, Black, Gold, and Blue. They had different objectives, different assessments, but not different standards. A Gold student could get, at best, a B-. If any gold student started doing better than that it was time to move them up. It worked beautifully, if “beautifully” can be used to describe teaching kids who still struggled with math. By setting up the groups, I was able to formalize the process of working with them–“Okay, golds, you’re clear on what you’re doing? Good, I have to check in with the Purples”. I could give myself time to close up with the top kids, who all agreed they were not neglected in the second half. I designed lessons in three day chunks and staggered them so that I could introduce a lesson to Gold while Purple, Black, and Blue (heh) were on day 2 or 3 of their lesson. The contract kids in particular did very well; I still lost nearly half of them to chronic absence and/or refusal to work, but three of them moved out of Contract entirely and the rest of that half passed with a D-. I had a low failure rate.

Most people who saw my class in action said it looked like an enormous amount of work, and it was. But you pick your stress levels. I found it far more stressful to teach one lesson to 30 kids with a four year range in ability levels than I did to design four lessons for smaller groups.


Did the kids learn algebra? I was able to compare the scores of 200 students who took algebra two years in a row. I had by far the most students of any algebra teacher. My students incoming test scores were 15 points below two of the teachers, and 30 points below one of them. In other words, I got most of the low-scoring kids, and I had no students with Advanced or Proficient scores on their previous year’s test, unlike the other teachers. When I broke down the comparison by incoming ability level, I did roughly as well as all the other teachers. Specifically, my students improved at a slightly lower rate than the other three teachers, which makes sense when you consider I had more low ability kids, but my standard deviation was, along with one other teacher, much lower.

For example: Teacher A, who had a class average 30 points higher than mine, had spectacular results with a couple of the 10 Far Below Basic students she had, but the others didn’t do as well. So the standard deviation of her new average was huge, whereas mine was pretty tight. Translation: I improved more students by smaller amounts, but had few huge wins. My improvements, like to like, were on par with the best teacher in the department, an algebra specialist who I very much admire, although our teaching methods couldn’t be more different. In my opinion, my differentiation allowed me to be a better teacher. But I don’t think it works for everyone.

I had no kids with Advanced scores; I attribute this in part to my failure to really challenge my top students. On the other hand, my top students all came in with Basic scores the year before, and almost all of them moved to Proficient (a few moved to high Basic). A number of the contract kids had Below Basic rather than Far Below Basic scores, which was a good sign.

We had a very good year last year, and our year was widely considered successful because of Algebra results. I taught more algebra students than anyone else, so in addition to specific results, it’s fairly evident I did a decent job. Nonetheless, I was moved to Geometry and Algebra II this year. As I said, I give thanks daily.

Classroom Management

So if my kids learned well, why am I so thankful to be out of algebra? Because classroom management difficulty correlates directly with the percent of the class with no skin in the game, and in my classes that was 10-20%. I would regularly check my students’ grades in all their classes; all my disruptive students were failing all their classes. For the most part, they are just waiting out the clock until they can go to alternative schools of some sort. They don’t mind going to school; it’s fun. It’s social. It’s their life. Learning, behaving, and engaging–not so much.

I say this to all new teachers: getting this population under control is imperative. Start by moving kids away from each other. I have often pulled a desk (or two) way up front and made a talking kid (or kids) move to that seat. The disruptor will usually then go to sleep. Fine by me. Many teachers are philosophically opposed to sending disruptive kids out of the classroom. Get over it. If moving seats didn’t work, or multiple warnings didn’t work, give them the boot. Ignore the pressure your management will put on you to stop sending them out of the classroom (although you will, inevitably, think 8 times before doing it). It’s incredibly hard to be patient and controlled enough to send kids out of the classroom until and you won’t always win the battle, but you must fight. Sing me no sad songs about these kids and their problems. They are making it impossible to teach the others, and you owe the others every bit as much.

What is essential, though, is to always hold to the rule no harm, no foul. I kick a kid out one day, he wants to come in the next day and works, I give him my full attention. And behavior never affects the grade. I had more than one student who showed functional understanding of algebra despite being a monumental pain who was often sent out. Those kids passed.

And it’s also essential to reach out in every way you can. I told every one of these students that I wanted to pass them, that I understood this was a course they didn’t like and didn’t want. I also turned to differentiation because I hoped that it would give the unmotivated strugglers a sense of possibility–and it did work, for a lot of them. You never want to take their behavior personally. But you also don’t want to tolerate it.

So don’t bleat ineffectually at your students, and don’t teach to a room that isn’t quiet.

Curriculum Paths and Classroom Management

Survey math teachers and you will find that Algebra I classes have the most severe classroom management challenges, quite apart from being difficult to teach content-wise. We put kids in classes they don’t understand and make them take that same class two or even three times. By the time they are sophomores, these kids know they can’t graduate. They are failing all their other classes, too, but in math, a lot of the high achievers get winnowed out into geometry, many before they even get to high school. The proportion of nohopers is insanely high.

By Geometry and Algebra II, most of your kids have skin in the game. Even if they don’t like math, they have something to lose. Classroom management moves from being a nightmare to a manageable challenge.

I’ve been working with a colleague on a math sequence that will find a path for the no-hopers. Remember, from a purely pragmatic standpoint, that the more kids who fail algebra, the more kids who will leave your school for a credit-generating factory–and take their attendance dollars with them.

New Year Resolution

I went a year without writing anything for publication (or attempted publication) because I felt sure that anything I’d write would either be deemed too controversial or too specialized or too opinionated for someone who wasn’t an expert. I kept tossing around ideas but nothing seemed to pass that barrier.

But then, I did write anyway–in the comments sections of a hundred different blogs, spouting my opinions,  telling people they are idiots, whiners, or unrealistic dreamers, throwing in inconvenient facts.  As a commenter I am not nice and am often disrespectful, two qualities (“not nice” and “disrespectful”) that aren’t given nearly their due in online discourse.  Over time, snark and sarcasm with decent data can change a lot of minds. But while I’m a mean and disrespectful commenter, I am not, in fact, a mean and disrespectful person. No, really. And comments don’t leave much room to initiate ideas, to talk about the fun side of teaching, or bring up things that no blogger noticed.  So as a New Year’s resolution, I decided to try blogging.

It’s very dangerous for teachers to engage in any online discourse. I’ll take the usual precautions, but I wish there were clear rules about what teachers can and can’t do. Right now the rule is “If your administrator finds out and doesn’t like it, you’re in a lot of trouble”. I’m also not a natural blogger; I like the many to many discourse format of forums much better than blogging. Hence the resolution to blog, to keep me focused on writing something daily, or close to it.

Education is filled with unpleasant realities that “experts” routinely ignore. Some realities are ignored because the experts have a policy idea they want to sell (literally). Other realities are ignored because it’s ideologically inconvenient to everyone. Still others, however, are ignored because our world is constructed in such a way as to make those realities illegal, or at least actionable.  The National Association of Scholars published an anonymous article by a teacher who called some of these realities The Voldemort View–The View That Must Not Be Named. Hence I will call these realities Voldemortean and, well, name them anyway.

Many Voldemorteans speak with what almost seems like glee. They don’t mean it that way; it’s more a “Hah! got you!” to the ignoramuses who refuse to even acknowledge what must not be named. I will not. I don’t see the Voldemortean realities as good or bad. They just are. And we won’t get anywhere until we start focusing on what these realities mean.

But education has all sorts of other realities, particularly the realities of teaching, and I’ll write about those, too. I love teaching. I do it as my job, I do it in my spare time. For most of my life, I’ve been paid for providing information and giving advice–and, for most of my life, my clients ignored me, even though they agreed with me. In contrast, my student “clients” listen to me. Not every day, not every class, not all of them. But the percentages are much higher than I ever saw in corporate America. I’m hooked.