Tag Archives: multiplying fractions

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.”

“OK.”

“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.”

“Bingo.”

“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.”

“Why?”

“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.

 


The Test that Made Them Go Hmmmm

So school has begun and despite my palpitations about the boredom of only two familiar preps, I’m pleasantly busy. Last year was a hell of a lot of work, and given the nosedive that my writing time took, I should maybe not be so eager for a less…familiar schedule. So instead of demanding new classes, I accepted the first semester, threw a minor temper tantrum when no one listened about second semester and all is well. Algebra 2 in particular is proving a delightful challenge, given my new emphasis on functions.

In no small part because of this planning breathing room (is anyone noticing I’m saying my panic was a total overreaction?), the senior Water Park Day registered in my awareness ahead of time. In prior years, I didn’t heed the warnings that half my class would disappear, and so would be forced to dump my lesson plan on the Day itself, when the smaller classes would just have a day to practice. But thanks to this old, familiar schedule that gives me more time, I anticipated the impact.

So for the first time, I was able to give serious thought to having a day to pursue math without regard to subject matter or schedule. I could have a “math day”! Then I remembered Grant Wiggins’ challenge to math teachers everywhere in the form of a conceptual knowledge quiz.

hmmquiz

Grant proposed this as an actual test: I will make a friendly wager: I predict that no student will get all the questions correct. Prove me wrong and I’ll give the teacher and student(s) a big shout-out.

What math teachers think their kids would know the answers? I certainly didn’t. In some cases, they probably were taught, but in others, I doubt an elementary school teacher would ever think to bring them up. But even if all the concepts were taught by fifth grade, how many kids of that age could really appreciate the questions?

Most of the questions tease at the paradox….wrong word? tension? between the functional day-to-day applications of arithmetic, and the amazing truths that underlie them. John Derbyshire wrote, in Prime Obsession, that “arithmetic has the peculiar characteristic that it easy to state problems in it that are ferociously difficult to solve.” (I was rereading Prime Obsession last night; there’s tons of useful thought material for math teachers. I need to go get his book on algebra.)

Arithmetic looks easy. (And certainly in the last twenty years, the rush to shove everyone into calculus has led to a certain contempt for “basic arithmetic” classes.) But even if elementary school age children are capable of understanding its ideas fully (and most of them aren’t), they haven’t experienced several years’ utility of arithmetic. They haven’t had time to get bored of the routine rules that they are expected to remember (mind you, many don’t, but leave that for another day.) Yeah, yeah, invert and multiply. Yeah, yeah, you can’t divide by zero. Wait, what the hell do you mean multiplication isn’t repeated addition?

To really enjoy this test, to be fascinated by the underlying truths–or misconceptions–behind certain everyday math tools, requires familiarity with “the rules”. Time spent in the trenches of doing math just because.

That’s when a teacher can spend an enjoyable hour taking the kids back through a re-examination of the basics and what they really know. I’d much rather discuss these concepts with adolescents who have survived two or three years of high school math than try to force sixth graders to “demonstrate conceptual understanding” of dividing by zero.

I had no real expectations—no, that’s wrong. I had hopes. My sense was the students would be interested in the exploration, if I didn’t take on too much or dive in to the wrong end of the pool. But which end was the wrong end?

So for each of my four classes–two Algebra 2, two Trigonometry–I gave them the test and 20 plus minutes to write down their thoughts. I was alert to the possibility that kids would use five minutes to doodle and fifteen to giggle, but in each class the bulk of students asked for and got an additional five minutes to finish up. I collected their answers and will share some of them in later posts; they were often detailed and thoughtful.

After the writing time, the students had a few minutes to “share out” in their groups, so they could learn what questions puzzled their classmates—and also as reassurance that they weren’t alone in their befuddlement. Again, this seems different from Grant’s intent; he considered it a real test that the students would either answer correctly or leave blank in confusion. I listened in on many conversations; they were rich with exchange as the students realized they weren’t alone in their uncertainty.

But certain questions also sparked genuine debate and interest. More than a few students offered up multiplying negatives as an example of multiplication being something other than repeated addition. In every case I witnessed, their group members, who had written something to the effect of “isn’t it always repeated addition?” instantly recognized the roadblock that negative numbers posed to their definition. I came across more than one group arguing whether multiplying by zero counted as repeated addition (“yes, it does. If I have zero groups of five, I have zero!”). Interestingly, no one came up with the roadblock I was interested in, and I’d never once considered negative numbers until my students brought it up.

Their discussion time was about ten minutes. My goal wasn’t to have them determine the answers; rather, I wanted them all to have a shared experience before we discussed them as a class, and I gave them the “answers” (to the extent I knew them). That way, there’d be more of a sense of “we”–yeah, we thought of zero, too! yeah, we all have 3F=Y–that’s not the answer? yeah, we think dividing by zero gives you zero–it doesn’t?

So then we went through the answers as a group.

I had taken a subset of Grant’s list, ignoring the last three items. Doing it again, I would have swapped out question 2 for question 11 “appropriately precise”), because while question #2 is good, it really requires its own day. The rest of them are easily covered and discussed in at most 15-20 minutes each.

The questions I really wanted to spend time on, to explain in at least introductory depth, were 1, 3, and 5. From a practical standpoint, I wanted to be sure everyone understood why they got questions 4, 6, and 8 wrong, assuming most missed at least one of them. I was genuinely interested to see what they had to say about 7 and 9 but was going to take most of my lead from them. Question 10, I wanted to know if the trig students knew it; obviously, my algebra 2 students learn about imaginary numbers for the first time.

My trig classes are quite different in nature. Both are small, just 25 in each. Both are doing quite well; I have no kids who simply shouldn’t be there, as I did last year. My first block class is stronger, on average, but has more surly kids who mouth off. It’s very irritating, frankly, since the five or six kids giving me quite nasty sass are seniors who are doing relatively well (Bs and Cs), and who openly acknowledge that they think I’m a hell of a teacher. Two of the surlies had me last year for algebra 2, when they were much less trouble, and had been switched into my class because they were failing with another teacher. But these other teachers, who they didn’t like (and often failed, forcing them to retake a fake summer school course if they couldn’t switch to my class), didn’t get nearly the lip. I’m a tad flummoxed. My second block class has more kids who are amiable and interested but not taking the class as seriously as they should, so several more low scores on the first test. First block has a stupendous top tier, but it’s just three or four kids. Second block has a top tier of close to eight, but they aren’t quite as strong.

Anyway, I was expecting more interesting conversation from second block, and I had it backwards. First block was on point, even the cranky ones. They loved the test, wrote detailed responses, discussed it thoroughly in group, and were wildly participatory in the open discussion. Easily 90% of them came up with the correct response to imaginary numbers (and the ones from my algebra 2 class identified multiplying by i as 90 degree rotations in the complex plane, which was quite gratifying, thanks so much). Second block, the amiable, mildly uninterested ones pulled things down slightly, goofing around and making jokes while the stronger kids would have preferred more time to explore things. The conversation was still great, the students learned a lot and enjoyed the discussion, but I had the enthusiasm levels backwards.

My algebra 2 classes, I nailed in terms of expectations. Block three is a fairly typical profile, except I have a lot more sophomores than usual (which is due to our school successfully pushing more kids through geometry as freshmen). But still a good number of seniors who barely understood algebra I, a lot of whom are just hoping to mark time til graduation without ending up in summer school. (One of my specialty demographics.) And in between, juniors and seniors who are often thrilled to find themselves actually understanding math and succeeding beyond anything they’d ever hoped (another specialty of mine). Typically, many of the seniors were in class, as they lacked the the behavior or grade profile (and sadly, in some cases, the money) to go to the water park. So I expected conversation here to be a bit lower level, with less interest. Happily, everyone engaged to the best of their ability and many told me later how much they loved just “talking about math”. I spent much more time on questions 4, 6, and 8, and could see them all really registering why they’d made the mistakes they did. But they still were enthralled by questions 1, 3, and 5, which is great because it’s going to give them some memories when we review percentages in preparation for exponential functions.

Last up was block 4 algebra 2, a ridiculously strong class; only five students are of the usual caliber I expect. The seniors are all well above average ability level. Two of the kids are so skilled that I’ve already introduced three dimensional planes and the matrix, while still forcing them and the other really strong kids to deal with complex linear word problems (mixture questions! I usually skip them, so it’s a trip). They stomped all over the test, writing at great length, discussing it with their teams and then shouting out to other groups to see what they’d answered for multiplication. The class discussion took so long that I actually allowed it to continue for 20 minutes into the next day, when I invited one of my mentees to watch. He came away determined to try the test in his honors geometry class.

Look, the whole day was teacher crack. Take a day. Try the test. I’ll be discussing individual questions and my explanations in future posts, but this introduction is offered up as invitation. High school teachers working in algebra 2 or higher would be a good starting point. Honors classes in algebra and geometry would also benefit. Every math teacher can find links from this test to their math class—but then, that’s not the point.

As for me, I started out the day with hope, but also a determination to see it through as part of a way to honor Grant Wiggins, who felt very strongly that students needed to do more than just march through curriculum. I promised myself I wouldn’t abandon the effort even if it went wrong. It didn’t go wrong. Quite the contrary, the test sparked delighted interest and intellectual curiosity among students who are often hard to push into exploring mathematics in depth. So hey, Grant, thanks for the idea–and the inspiration.