Tag Archives: factoring

Coaching Teachers

In 2011’s Personal Best, Atul Gawande recounts his desire to “up his game”, by hiring a retired surgeon who had once trained him, Robert Osteen, to act as a coach.  I often reread the article just for the best passage in an already great piece: when  Osteen gives Gawande feedback for the first time.

Prior to his own coaching experience, Gawande explores the difference between “coaching” and “teaching” in the teaching career itself. He sits in on a lesson and coaching session with  an 8th grade math teacher. One of the coaches was a history teacher, the other a math teacher who’d given up teaching to work at the district. While Gawande implies coaching is unusual, many school districts have coaching staffs, usually made up of history teachers and middle school math teachers, just like this one.

Everything that crackles and glows when Gawande describes Osteen’s observations falls with a thud in the teaching section. The lesson on simplifying radicals sounded fairly traditional, but seemed dull in the telling. The coaching feedback was similar to what I’ve experienced–banal platitudes. Socratic questioning. “What do you think you could do to make it better?” (Translated: I personally have no idea.) Not the same assertive advice Osteen gave Gawande, but carefully scripted prompts. Critzer seemed to like the “feedback”, such as it was, but I found the whole exchange extremely antiseptic. In no way were the two coaches “operating” (heh) on the same level as Osteen’s expert.

In 2011 I was a newbie. Now I’m edging towards a full decade of teaching and have now mentored  three teachers through induction and one student teacher. I’m better prepared to think about coaching, both as provider and recipient, and the stark differences in those two passages keep coming back to me.

My ed school supervisor , a full-metal discovery proponent, gave me one of the great learning experiences of my entire life. She never tried to convert me or push particular lesson approaches.  I can still remember the excitement I felt as she pushed me to think of new methods to achieve my goals, while I realized that regardless of teaching philosophy, teaching objectives remain resolutely the same: are the kids engaged? Are they learning, or parroting back what they think I want to hear? Am I using time effectively?  Osteen’s feedback reminded me of those conversations, and as I moved into a mentor role, she became my model.

A couple weeks ago, a district curriculum meeting ended early and I went back to school just in time for fourth block to observe my newest induction mentee.  This was an unscheduled observation, but she welcomed me into her pre-algebra class for a lesson on simplifying fractions prior to multiplication. Through the lesson, the students worked on this worksheet. The concepts involved are not dissimilar from the ones in Jennie Critzer’s lesson.

Here’s my feedback, delivered immediately after the bell rang.

“Okay, I’m going to split my feedback into three categories. First up are issues involving safety and management that you should take action on immediately. Everything subsequent is my opinion and advice  based on my teaching preferences as well as what I saw of your teaching style. I will try to separate objective from method. If you agree with the objective but not the method, then we’ll brainstorm other ideas. If you disagree with the objective, fine! Argue back. OK?” She agreed.

“For immediate action, make students put their skateboards under that back table, or in a corner completely away from foot traffic. The administration will support you in this in the unlikely event a student refuses to obey you, I’d also suggest making all the students put their backpacks completely under the desk. It’s like ski week around here, you nearly tripped twice. Now for the suggestions…”

“Wait. That’s the only mandatory change? My classroom management is good?”

“Yes. Kids were attentive and on task. But I want you to move about the room more, as you’ll see, and the way your kids strew their stuff around the floor, you’ll kill yourself.”

“I was worried about management because the students often seem…slow to respond.”

“We can talk more about your concerns before our formal observation so I can watch that closely. I’d like more enthusiasm, more interest, but that’s a subjective thing we’ll get into next. They listen to you and follow your requests. They’re trying to learn. You’ve got buy-in. You’re waiting for quiet. All good.”

“Phew. I’m relieved.”

“Now, some opinions. I’d like you to work more on your delivery and pacing.  You are anchored to the front of the class during your explanation time. Move about! Walk around the room. Own it. It’s your space.”

“I am never sure how to do that.”

“Practice. When you have a few sentences nailed down, just walk to the back by the door,  stand there for a minute or so, then move to another point, all while talking. Then go back up front. Do that until it feels comfortable. Then ask a question while away from the front. Then practice introducing a new topic while away, and so on.”

“I didn’t think of practicing. I thought it would come naturally.”

“I’m as big a  movie star teacher as they get, and what I just described is how I escaped the front-left cellblock.”


“Next up: you’re killing the flow of the lesson.  Here’s what you did today: give a brief description of method, work an example, assign two problems, go around the room looking at student work, come back up, work the problems. Then assign two more, go around the room looking at student work, come back up, work the problems. Lather, rinse, repeat. This precludes any concentrated work periods and it’s hurting your ability to help your top students. It’s also really boring.”

“Yes, many of my students have worked all the way through the handout. But I have to help the students who don’t get it right away and that takes time, right?”

“Sure.  So give a brief lecture with your own examples that illustrate two or three key concepts–NOT the ones on the worksheet. And while that lesson is going on, my advice is to insist that all students watch you. Right now, the strong students are completely ignoring your lesson to work the handout–and from what I can tell, occasionally getting things wrong.”

“Yes, they don’t know as much as they think they do in every case. But it’s good that they’re working, right? They’re interested?”

“Not if they aren’t paying attention to you. You are the diva. Attention must be paid.”

“But if they know it all…”

“Then they can finish it quickly after your lesson–as you say, they sometimes make mistakes you covered. So do an up front lesson of 15-20 minutes or less, depending on the topic. Then release them to work on the entire page or assignment. Let them work at their own pace. You walk around the room, giving them feedback. Don’t let the stronger kids move ahead in your packet. Have another handout ready that challenges them further You might have an answer sheet ready so kids can check their own work.”

She was taking notes. “How do I get these more challenging handouts?”

“Ask other teachers. Or I’ll show you how to build some. I know you’re using  someone else’s curriculum, but you can have additional challenges ready to keep your top kids humble. Math gets much harder. They need to be pushed.”

“So then I teach upfront and give them 30-45 minutes to do all the work, giving the kids who finish more work. Maybe a brief review at the end.”


“Got it. I’m going to try this.”

“Last thing on delivery: you’ve got a Promethean. Use it. It will free you from the document camera.”

“I don’t know how. I asked the tech guy for guidance and he said you were one of the most knowledgeable people on this brand.”

“Well, let’s do that next. Now, onto the much more difficult third topic: your curriculum. I could see you often backtracking from your own, authentic instruction method to return to the worksheet which forcefeeds one method: find the Greatest Common Factor or bust.  I could tell you didn’t like this approach, because you kept on saying ‘they want you to use GCF’, meaning the folks who developed the worksheet.”

“Yes, I kept forgetting to avoid my own method and  support the worksheet’s method.”


“Well, I have to use that worksheet.”

“Toots, you don’t have to use a thing. You’re the teacher. They can’t require you to teach it. I don’t dislike the curriculum, but that particular worksheet is flawed. As I walked round your room, I saw kids who just cancelled the first factor they saw, and then had an incomplete simplification. So 9/27 became 3/9 because the kid turned 9 into 3×3 and 27 into 9×3.”

“Yes, that’s what I saw, too. They didn’t realize it wasn’t fully simplified, because they weren’t realizing the need to find the GCF.”

“That’s because the method isn’t as important as the end result.  Who cares if they use that method? That’s what the one student said who challenged you, right? You were trying to push her to find the GCF, and she pushed back, saying ‘what difference does it make?’ and you were stuck because you agreed with her, but felt forced into this method.”

“God, that’s so right,” she groaned.

“But you weren’t giving them any plan B, any way to see if they’d achieved the goal. How much advanced math have you taught? Algebra 2, Trig, Precalc? None? You should observe some classes to see how essential factoring is. I talked to many of your students, and none have any real idea what the lesson’s purpose was. Why do we simplify at all? What was the difference between simplifying fractions and multiplying them?  What are factors? Why do we use factors?  I suggest returning to this tomorrow and confess that the student was correct, that in the case of simplifying fractions by eliminating common factors, there are many ways to get to the end result. Acknowledge you were trying to be a good sport and use the method in the handout, but it’s not the method you use.”

She wrote all this down. “And then I need to tell them how to know that they have fully simplified.”

“Exactly. Here’s what I saw as the two failures of the worksheet and your lesson: first, you didn’t tell them how they could test their results for completeness. Then, you didn’t tell them the reason for this activity. Namely, SIMPLIFY FIRST. When using numbers, it’s just an annoying few extra steps. But when you start working with binomials, failing to factor is disastrous for novices.”

“OK, but how can I circle back on this? Just tell them that I’m going to revisit this because of what I saw yesterday?”

“Yes! I recommend a simple explanation of  relatively prime. That’s the goal, right? The method doesn’t matter if that’s the end result.  And then, here’s a fun question that will startle your top kids. Given “two fourths”, why can we simplify by changing it to 2×1 over 2×2 and ‘canceling out’ the twos, but we can’t simplify by changing it to 1+1 over 1+3 and ‘cancel out’ the ones? Why don’t we tell them to simplify across fractiosn when adding? ”

“Wow. That’s a great question.”

“Yes. Then come up with a good, complicated fraction multiplication example and show them why all these things are true. Make them experience the truth by multiplying, say, 13/42 and 14/65. They might not retain all the information. But here’s what’s important, in my view: they’ll remember that the explanation made sense at the time. They’ll have faith. Furthermore, they’ll see you as an expert, not just someone who’s going through a packet that someone else built for her.”

“Ouch. But that’s how I feel.”

“Even when you’re going through someone else’s curriculum, you have to spend time thinking about the explanation you give, the examples you use. This isn’t a terrible curriculum, I like a lot of it. But fill in gaps as needed. Maybe try a graphic organizer to reinforce key issues.  Also, try mixing it up. Build your own activities that take them through the problems in a different way. Vary it up. You’ve got a good start. The kids trust you. You can push off in new directions.”

I then gave her a brief Promethean tutorial and told her I’d like to  see a lesson with some hands on activities or “cold starts” (activities or problems with no lecture first), if she’s interested in trying.


Mid-career teachers, like those in any other profession, are going to vary in their desire and interest in improving their game. Twitter and the blogosphere are filled with teachers who write about their practice.  Perusing social media is a much better form of  development than a district coach that isn’t experienced in working with the same population and subject. Conversations with motivated colleagues interested in exploring their practice, but hared to find the time or interested participants.

But  unlike other professions, we teachers are given ample, and often paid, opportunity to be coaches, and not the weak-tea district sorts. Induction and other new teacher programs give us a chance to push others to find their best.  I find these activities also lead me to review and improve my own practice.

If you’re tasked with helping beginning teachers, then really dig in. Challenge them. Encourage them to push back, but do more than ask a few questions. They’ll thank you later. Often, they’ll thank you right away.


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.