Category Archives: teaching

Great Moments in Teaching: Or, Browbeating Psychoanalysis

One of my strengths as a test prep instructor was spotting weird mental glitches that was interfering with a student’s success. I miss this part of the job, but every so often I get the chance in classroom teaching.  In this case, summer school trigonometry. I taught first semester in block 1, second semester in block 2, but I taught the same material in both classes.

I had about eighteen kids in the two classes, but eight of them took both classes, meaning they’d failed both semesters. All eight students repeating both semesters were stronger than the three weakest students repeating the second semester, and the weakest student just repeating first semester. Remember what I said about GPAs? Shining example, right here.

So this is a conversation I had with Warren on the next-to-the last day of class. Before you decide I’m a rotten bully, understand that I had raised this issue several times with Warren, but the message had, like everything else, rolled right off his back like whatever water does with a duck.

He was taking the final test, and had asked me to check it over before he turned it in. (This is a normal part of my class routine).

“OK, you’ve got quite a few cases where I’m asking for onions and you’re giving me a Jeep.”

“I know.”

“No, you don’t. Like question 5. You’re using the Pythagorean theorem on a question asking you to understand and evaluate a trigonometric model.”

“I know.”

“No, you don’t.”

“I get it.”

“No, you don’t.”

“Yes. I see now.”

“NO YOU DON’T!” The class was now snorfling quietly, not out of mockery of Warren, but amusement at me. I was playing my aggravation very big.



“I know.”

” Then why are you done with the test? This one is not just mildly wrong. It’s Jupiter and we’re Earth.”

“I know.”


“I kno…OK.”

“What’s OK?”

“I understand what you’re saying.”

“No, you don’t.”


“No, it’s not OK!”

“OK..I mean, I know…Oh, sh**.” Warren is no longer a duck, but a deer, frozen.

“Listen to me.”


“No, see, already you’re not listening. Don’t try to make me happy. Don’t try to give me what I want. You’re trying to figure it how to make me happy and that one task is consuming all your brain cycles. JUST LISTEN.”

“I know…no. OK. I get it.”

Half the class was howling by this point, and I shushed them.

“This problem is incorrect. Not mildly incorrect. Way off. DON’T SAY A WORD. Continue to listen. Cover your mouth if you must. Say nothing until I ask you a direct question.”

Warren stood. Affect way off, smiling nervously.

“You came up here, telling me you were done, asked me to just look through for minor errors. But as I look through the test, I see that you have no idea how to do at least three of the eight problems. SAY NOTHING!” Warren closed his mouth. “In two cases, you came up here earlier and asked me for help. I gave  you guidance, you said ‘I know’, I told you no, you didn’t, tried again, got nowhere. And now you’re up here saying you’re done. We have an hour left of class. You are MANIFESTLY not done. When I point out an error, you say ‘I know’ but you clearly don’t mean it because you are up here saying that you are finished! No–I haven’t asked a question. Stay put.”

Warren stood. But I could see the panic fade a bit. He was starting to actually listen.

“This is a trig modeling question. It’s about temperature in a room. Max and min temp. 24 hours in a day. Yet you are using the Pythagorean theorem. Why, Warren, are you using the Pythagorean theorem? That is a question. You can answer.”

But Warren stood mute. I waited. The class snickered and I ferociously signaled them to stop.

“I….I don’t know what you want me to say.”

“EXACTLY! That’s it. Exactly. Perfect. Now, continue to listen. The reason you are confused, Warren, is because you are uninterested in math at this particular second, and entirely interested in making me happy because you think it will help your grade. But I want you to learn. And that involves asking questions. It involves thinking. It involves furrowing your brow and asking for clarification. Normally, you ask your friends for help, or copy what they’ve done and think you understand the math. Sometimes you do. Mostly, you don’t.You just know how to go through the motions.”

“But I asked for help.”

“No, you didn’t. You asked me to ‘look through’ the test. Earlier, you asked for help and ignored my response. You aren’t asking for feedback. You are going through a self-imposed ritual in the hopes that I will be impressed with your  effort. Then after each conversation,  you return to your work to try another random approach to a problem you don’t understand, as completely clueless as you were before you came up. You only know the math that triggers a routine in your brain. And when I try to fix that, you nod or say ‘I know’ but you have absolutely no conception of the possibility that I might be able to help you! In your learning world, friends are for help. Teachers exist to be placated and grade your work so you can get an A.”

Warren’s eyes widened. Apparently, he thought we teachers weren’t onto his scheme.

“But Warren, talking to me–talking to any teacher–is a conversation. A process. It is your job to communicate your confusion. It’s my job to try and give you clarity and undertanding. Our conversations are not mere rituals mandated by the Chinese American education canon. So let me ask the question a different way: When you were modeling trigonometric equations all this week, what pieces of information were relevant?”

Warren answered readily, “Amplitude. Period.”

“If I gave you the maximum and minimum points, how would you find amplitude and period?”

“I would sketch them and look for the middle.”

“Which is also the…..”

“Vertical shift.”

“OK. Now. Look at this problem. Do you see how this problem fits into that format? It describes the temperature in a corporate office. So what I want you to do now is go back and think about this problem. Think about how you could describe temperature in terms of max and min. Think about relating it to the time of day, hours past midnight. And then see if you can figure out how to work the problem.”

Warren obediently took the test and started to return to his seat, but stopped. “OK, but here’s what I don’t get. You’re asking us to solve an equation. But modeling is just building the equation. How come you’re asking us to solve the equation?”

I looked at the class. “Whoa. Did you hear what I heard?”

“A QUESTION!!!” and we all clapped loudly and genuinely for Warren, who smiled nervously again.

“Warren, I mentioned this over the past couple days: Trig equations don’t just occur in a vacuum. We build the equations to model the world. Then we look to the model to predict outcomes, which we do by solving for outputs given inputs, or vice versa. The problem covers both. It asks you to evaluate and explain the given model,  then it asks you to use the model as a trigonometric equation. In this case, I actually used function notation because I want to see if you understand it, but at other points, I’m using verbal descriptions.”



“Um. No. I don’t know how to start.”

I waited. The class waited.

“Could..could you give me a suggestion on how to start?”

“Is there something you could do to the given equation that might give you some insight?”


“I could…graph it, maybe?”

“There’s a thought. Then look at it, look at the multiple answers, and see how it goes.”

As Warren walked back to his desk, I mimed collapsing in fatigue. “And now, everyone, entertainment’s over. Get back to work.”

Warren worked on the test for another hour. He forgot and said “I know” and “OK” reflexively a few times, but stopped himself before I could, to both of our smiles. He came up each time with a specific question. He listened to my response.  He went back and worked on the problem based on my response and his new understanding.

On the last day of class, after the final bell rang, Warren came up to chat with me.

“Thanks for yelling at me.”

“You know, I was working towards a good cause.”

“You were right. I was coming up to ask you questions because that’s what other kids did, so I figured that’s what you wanted. I never really thought about getting help from you. I just kind of…work through something using whatever I remember, until I’m done.”

“Don’t be a zombie.”

“Okay–wait. What’s a zombie?”

“Don’t just work problems without any sense of what’s going on. That’s why you flunked Trig the first time, I’ll bet.”

“Yeah. I didn’t always understand Algebra 2, but I could follow the procedures. But Trig, I just couldn’t do that.”

“Yeah. Zombie thinking. Don’t do that. I mean–zombie thinking is what you’re doing in math. You get the answers from friends, you don’t care about understanding the math. You just go through the motions. The driving me crazy saying ‘I know’ stuff, that’s different. Plenty of zombies do a better job of asking for help!”

“I understand math a lot more the way you teach it, but I also….I couldn’t always figure out your tests.”

“That’s why you ask for help. And not from your friends. Look–school is about more than getting an A. It’s about more than giving teachers what they want so you’ll get an A. It’s about learning how to learn. You have to start communicating with teachers–good, bad, indifferent–and learn how to figure out what they’re telling you. That starts with asking for what you need. If you can’t communicate with a teacher right away, don’t just ask a friend. Half the time, they’re just doing what you do! Find teachers you can work with. You’re a really bright guy. Don’t let school ruin you.”

And then we talked about his college plans where–no joke–he asked me for advice.

Ten minutes later, as he walked out, he said: “Thanks again. I mean it.”

He knew a lot of math, and worked his way out of being a zombie. I gave him an A-.


Teaching Transformations

One of the most important new concepts in algebra 2 and beyond is the notion of transformation. That is, given the function f(x), we  can change any function’s position and growth by using the same instructions, much like giving directions from a map.

I’ve just introduced functions at this point in the calendar, so I’ve designed this activity to reinforce f(x) as a rule, that once a mapping is created, the mapping holds for all subsequent calls.

So just create a random table, one that’s simpler than anything I’d do in class. (One of the incredibly irritating things about blogging is that it’s insanely time-consuming to create images for publication that take next to no time at all to do on  a smartboard, but I never think of capturing images while on a smartboard.)

x f(x)
-3 2
-1 5
1 6
3 3
5 -1

That looks like this:


So then I ask if this is f(x), what would f(x+2) look like? Someone brave will always say “Two to the right”.

At that point, I always say “This is a totally logical guess and one of the most annoying things in math from this point on is that your guess is wrong.” (I originally developed the concept of a parabola as the product of two lines as another way of explaining this confusing relationship. Confusing to normal people. Mathies think it makes sense, but they’re weird.)

I add a column to the table. “We start with x. Then we add 2. Then we make the function call. Note the function call comes after the addition of the value. This is important. Now, we have three columns, but we are starting with our x and that’s still our input value. We graph it against the outer column, the output value for f(x+2).”

x x+2 f(x+2)
-3 -1 5
-1 1 6
1 3 3
3 5 -1

I’ll ask how we can bring the -3 back in, and after some mulling, they’ll suggest that I add -5 to the table. So I add:

-5 -3 2

to the bottom. But I’ve been plotting points all along, so the kids can see it’s not going as expected.


“Yes, indeed. I’ll be teaching this concept in many ways over the next few months, and I ask you to start wrapping your head around this now. We have many ways of envisioning this. When working with points as opposed to an entire function, it might be helpful to think of it this way: Suppose I’m standing at -3, and I want to add two. This has the effect of me reaching to the right on the number line and pulling the output value back to me–to the left, as it were.”

I go through this several times. Whether or not students remember everything I teach, I always want them to remember that at the time, they understood the concept.

“So if standing on -3 and reaching ahead is addition and move the whole function to the left, how would I move the whole function to the right?”

If I don’t get a ready chorus of “subtract?” I know that I need to try one more addition example, but I usually get a good response.

“Exactly. So let’s try that.”

x x-2 f(x-2)
-3 -5 NS
-1 -3 2
1 -1 5
3 1 6
5 3 3
7 5 -1


One year, I had a doubter who noticed that I’d made up these numbers. How did we know it’d work on any numbers? I told him I’d show him more later, but for now, imagine if I had a table like this:

x f(x)
1 1
2 2
3 3


Then I told him, “Now, imagine I put decimal values in there, fractions, whatever. Imagine that no matter how I change the x, the new value has an entry in the table and thus an output. So imagine I added 50. There’d be a value 50 ahead that I could reach forward or backwards.”

“In fact, we’ll eventually do all this with equations that are functions, instead of randomly generated points. But I start with points so you won’t forget that it works with any series of values that I can commit math on. Which isn’t all functions, of course, but that’s another story.”

“But if adding makes it go left and right, how do we make a function go up and down? Discuss that among yourselves for a minute or so.”

Sometimes a student will see that we’ve been changing x so far. Otherwise I’ll point it out.

“The function call itself is key to understanding this. If you change the value before you make the function call, then you are changing the input to the function. Simpler: you’re changing x before you call the function. But once the value comes out of the function, that is, once it’s no longer the input, it’s the….” I always wait for the class to chime in again–are they paying attention?


“Right. But output is no longer x. Output is”

“f of x!”

At this point, I call on a mid-level student. “So, Sanjana, up to now, we’ve been changing x before making the call to the function. See how the new column is in the middle? What could I do differently?”

And I wait until someone suggests making the column on the right, after the f(x).

x f(x) f(x)- 3
-3 2 -1
-1 5 2
1 6 3
3 3 0
5 -1 -3


I’m giving a skeletal version of this. Often the kids have whiteboards and are calculating all this along with me. I’ll give some quick learning checks in terms of moving to the right and left, up and down.

The primary learning objective for is to grasp the meaning of horizontal and vertical translations–soon to be known as h and k. But as an introduction, I define them in terms of function notation.



We usually end this activity by combining vertical and horizontal shifts.

What would f(x-2)+ 3 look like? Well, you’d need another column.

x x-2 f(x-2) f(x-2)+1
-3 -5 NS
-1 -3 2 5
1 -1 5 8
3 1 6 9
5 3 3 7
7 5 -1 2


I connect them this time just to show that one point is in both the original and the transformation.

Ultimately, this goes to transforming functions, not points. That’s the next unit, transforming parent functions. I have a colleague who teaches transformations entirely by points. I start down that path (not from his example, just because that’s how this works), but the purpose of transformations, pedagogically speaking, is for students to understand that entire equations can be changed at the unit level, without replotting points. At the same time, I want the students to know that the process begins at the point level.

Over time, the students start to understand what I often call inside and outside, or before and after. Changes to the input value affect the x, or the horizontal because they occur before the function is called. Changes to the output value affect the y, or the vertical, because they occur after the function is called. Introducing this on a point by point basis creates a memory for that.

At best, this lesson functions as more than just a graphing exercise, something to introduce vertical and horizontal shift. It should ideally give students an understanding of the algebra behind it. Later on, when they are asked to solve equations like:

Find f(a) = 32 for f(x)=3(x-2)2+5

Weaker students have trouble with understanding order of operations, and a memory of “inside” and “outside” the function can be helpful.

If I were writing algebra 1 curriculum, I’d throw out quadratics, introduce a few parent functions, and teach them function notation and simple transformations. It’s a complicated topic that they’ll see all the way through precalc, at least.

I’ll discuss stretch and its complexities in another post.

Realizing Radians: Teaching as Stagecraft

Teaching Objective: Introduce radian as a unit of angle measure that corresponds to the number of radians in the length of the arc that the angle “subtends” (cuts off? intersects?).  Put another way: One radian is the measure of an angle that subtends an arc the length of the circle’s radius.  Put still another way, with pictures:

How do you  engage understanding and interest, given this rather dry fact?  There’s no one answer. But in this particular case, I use stagecraft and misdirection.

I start by walking around a small circle.

“How far did I walk?”

“360 degrees.”

“Yeah, that won’t work.” I walk around a group of desks. “How far did I walk?”

“360 degrees.”

“Really? I walked the same distance both times?”

“No!” from the class.

“So what’s the difference?”

It takes a minute or so for someone to mention radius.

“Hey, there you go. Why does the radius matter?”

That’s always an interesting pause as the kids take into account something they’ve known forever, but never genuinely thought about before–the distance around a circle is determined by the radius.

“Yeah. Of course, we knew that, right? What’s that word for the distance around a circle?”


“Yes. And how do you find the circumference of a circle?” There’s always a pause, here. “OK, let me tell you for the fiftieth time: know the difference between area and circumference formulas!”

“2Πr” someone offers tentatively.  I put it up:


“So the circumference is the difference between this small circle” and I walk it again “and this biiiiigg circle around these desks here.” Nods. “And the difference in circumference comes down to radius.”


“Look at the equation. 2 Π is 2 Π. So the only difference is radius. The difference in these two circles I walked is that one has a bigger radius.”

“So the real question is, how does the radius play into the circumference?”

“Well,” it’s always one of the better math students, here: “The bigger the radius is, the farther away from the center, right?”

“So then…you have to walk more around…more to walk around,” some other student will finish, or I’ll ask someone to explain what that means.

“Right. But how does that actually work? Can we know exactly how much bigger a circle is if it has a bigger radius?”

“A circle with a radius of 2 has a circumference of  4Π. A circle with a radius of 4 has a radius of 8 Π. So it’s bigger.” again, I can prompt if needed, but my class is such that the stronger students will speak their thoughts aloud. I allow it here, because they can never see where I’m going. See below for what happens if they start with spoiler alerts.

“Sure. But what’s that mean?”


I pass out pairs of circles, cut from simple construction paper, of varying sizes, although each pair has the same radius.

“You’re going to find out exactly how many radius lengths are in a circle’s circumference using the two circles. Don’t mix and match. Don’t write annoyingly obscene things on the circles.”

“How about obscene things that aren’t annoying?”

“If you can think of charmingly obscene comments, imagine yourself repeating them to the principal or your parents, and refrain from writing them, too. Now. You will use one of these circles as a ruler. All you have to do is create a radius ruler. Then you’ll use that ruler to tell me how many times the radius goes around the circumference.”

“Use one of the circles as a ruler?”

“You figure it out.”

And they do. Most of them figure it out independently; a few covertly imitate a nearby group that got it. Folding up one of the circles into fourths (or 8ths) exposes the radius.


Folding up one circle exposes the radius.

It takes most of them a bit more time to figure out how to use the radius as a ruler, and sometimes I noodge them. It’s so low-tech!


Curl the folded circle around the edge of the measured circle. 

But within ten to fifteen minutes everyone has painstakingly used the “radius ruler” to mark off the number of radius lengths around the circumference, and then I go back up front.


“Okay. So how many times did the radius fit into the circumference?”

Various choruses of “Over six” come back, but invariably, someone says something like “Six with and a little bit left over.”

“Hey, I like that. Six and a little bit. Everyone agreed?” Yesses come back. “So did everyone get something that looks like this?”


“Huh. And did it matter what size the circle was? Jody, you had the big two, right? Samir, the tiny ones? Same difference? Six and a little bit?”

“So no matter the circle size, it appears, the radius goes into the circumference six times, with a little bit left over.”

No one has any clue where I’m going, usually, but they’re interested.

“‘Goes into’ is a familiar term, isn’t it? I mean, if I say I wonder how many times 2 goes into 6, what am I actually asking?”

Pause, as the import registers, then “Six divided by two.”

“Yeah, it’s a division question! So when I ask how many times the radius goes into the circumference, I’m actually asking…..” The pause is a fun thing. Most beginning teachers dream of using it, but then get fearful when no one answers. No. Be fearless. Wait longer. And, if you need it:

“Oh, come on. You all just said it. How many times does 2 go into 6 is 6 divided by 2. So how many times the radius goes into the circumference is…”

and this time you’ll get it: “Circumference divided by the radius.”

“Yeah–and that’s interesting, isn’t it? It applies to the original formula, too.”


“Cancel  out the radius.” the class is still mystified, usually, but they see the math.

“Right. The radius is a factor in both the numerator and denominator, so they can be eliminated. This leaves an equation that looks like this.”


“The circumference divided by the radius is 2Π. Well. That’s good to know. Does everyone follow the math? Everyone get what we did? You all manually measured the circumference in terms of radius length–which is the same as division–and learned that the radius goes into the circumference a little bit over six times. Meanwhile, we’re looking at the algebra, where it appears that the circumference divided by the radius is 2Π.”

(Note: I have never had the experience where a bright kid figures it out at this point. If I did, I would kill him daid, visually speaking, with a look of daggers. YOU DO NOT SPOIL MY APPLAUSE LINE. It’s important. Then go to him or her later and say, “thanks for keeping it secret.” Or give kudos after the fact, “Aman figured it out early, just two seconds before figuring out I’d kill him if he spoke up.” Bright kids learn early, in my class, to speak to me personally about their great observations and not interrupt my stagecraft.)

And then, almost as an aside: “What is Π, again?” I always ask it that way, never “what’s the value of Π” because the stronger kids, again, will answer reflexively with the correct value and they aren’t the main audience yet. So the stronger kids will start talking yap about circles, and I will always call then on a weaker kid, up front.

“So, Alberto, you know those insane posters going around all the math teachers’ walls? With all the numbers?”

“Oh, yeah. That’s Π, right? 3.14.”

“Right. So Π is 3.14 blah blah blah. And we multiply it by two.”


That’s when I start to get the gasps and “Oh, MAN!” “You’re kidding!”

“….so 3.14 blah blah times 2 is 6.28 or…..”

“SIX AND A LITTLE BIT!” the class always shouts with joy and comprehension. And on good days, I get applause, too, from the stronger kids who realized I misdirected them long enough to get a deeper appreciation of the math, not just “the answer”.


So a traditionalist would just explain it, maybe with power point. I don’t want to fault that, but I have a bunch of students who would simply not pay any attention. They’ll take the F. I either have to figure out a way to feed them the math in a way they’ll remember, or fail more kids than I’m comfortable failing.

A discovery-oriented teacher would probably turn it into a crafts project, complete with pipe cleaners and magic markers. I don’t want to fault that, but you always get the obsessive artists who focus on making a beautiful picture and don’t care about the math. Besides, it takes forever. This little activity has to be 15-20 minutes, tops. Remember, there’s still a lot to explain. Radians are the unit measure that allow us to talk about circles in terms akin to similarity in polygons–and that’s just the start, of course. We have to talk about conversion, about the power that radians gives us in terms of thinking of percentage of the entire circle–and then actual practice. I don’t have time for a damn pipe-cleaning activity.

As I’ve written before somewhere between open-ended, squishy discovery and straight discussion lecture lies a lot of ground for productive, memorable teaching. In my  opinion, good teachers don’t just transmit information, but create learning events, moments that all students remember and can use as hooks for further memories of learning. In this case, I want them to sneak around the back end to realize that  Π is a concrete reality, something that can actually be counted, if not exactly.


Teaching as stagecraft. All the best teachers use it–even pure lecture artists who do it with the power of their words (and an appropriate audience).  Many idealistic teachers begin with fond delusions of an enthralled class listening as they explain math in terms that their other soulless, uncaring teachers just listlessly put up on the board. When those fantasies are ruthlessly dashed, they often have no plan B. My god, it turns out that the kids really don’t find math interesting! Who do I blame, myself or them?

I never had the delusions. I always ask my kids one simple question: is your life better off if you pass math, or if you fail?  Stick with me, and you’ll pass. For many, that’s a soulless promise. To me, that’s where the fun starts. How do you get them interested? How do you create those moments? How do you engage kids who don’t care?

It’s not enough. It’s never enough.

But it’s a good way to start.

The Things I Teach

“OK, today in focus we’re going to read  Grandfather’s Journey together. We will find new words on each page, talk about vocabulary and meaning.”


“Me! I know!” Marshall waved his hands. “It is….the father of your father.”

“Also the father of your mother, right?” Charlotte asked.

“Abuelo?” Kit looked to Marshall.

“Yes, abuelo,” I nodded. “But what about journey?”


“I think it means hat,” offered Julian.

“Sombrero?” Kit was surprised.

“No,” I shook my head. “Journey means ‘trip’. It means…to travel. To go somewhere else.” Blank looks. I grabbed a white board and drew–badly–what I call in my history classes the Great American Porkchop with an airplane, also rendered poorly.


“Ahhh!!” Comprehension. They didn’t laugh. So don’t you mock my artwork.

Charlotte said, “So I took a….journey from the Congo?”

“I took a journey to India?” asked Amit.

“No. From.”

I pointed to “Here” on my sketch. “In a journey, your beginning point is from. Your end point is to.”

“So I came from China to America?” asked John.

“Use journey.”

“OK. I took a journey from China to here.”


“I…journey from Mexico to America.”

took a journey,” said Charlotte.

“Either. I journeyed from Mexico to America is good, or I took, or I made, a journey” is good. Kit?”

“I….took journey from Mexico to America.”

“Good! Sebastian.”

Long pause.

“Sebastian, put the phone away or you’ll lose it.”

“I journey from China to…here.”

Fun, clear learning, but five minutes had gotten me through two words.

“My grandfather was a young man when he left his home in Japan and went to see the world. He wore European clothes for the first time and began his journey on a steamship.”

“Look at the difference between Grandfather in the first picture and then on the steamship.”

“He is not wearing…same clothes.” from Amit.

“Oh! He is dressed like he is from Japan!” said Julian, “and now he is dressed like an American. Why is that European?”

“So does everyone see what Julian means? He is dressed in what we call traditional clothes. This story is about the past, yes? About a long ago time?” Nods. “Well, in this long ago time, Europe was more well-known than America. Today, Julian thinks of America before Europe. Today, probably the best word to use for this sort of difference is ‘Western’. Why would he want to dress in different clothes, Kit?”

Kit is quiet, particularly compared to Marshall, whose American aunt is really helping him develop skills. He paused. “He…belong?”

“He won’t be strange,” offered Charlotte.

“Yes, he wants to fit in, or assimilate. Good! Back to the book. The Pacific Ocean surrounded him.

“Océano Pacífico!” Marshall beamed. “That’s here.”

“Yes, and now we know the first part of his journey,” I walk over to the large wall map. “He left from Japan” (points) “and traveled across the Pacific Ocean. Where will he end up?”

“AMERICA!” chorused from all six.

“What does surround mean? Sebastian?” Sebastian tried to check with Julian in Chinese, but I stopped him. “He is on a boat, yes? In the Pacific Ocean? What would he see?”


“Amit, would he see land?” Amit was puzzled. I went back to the map, showing the trip. “He would be here. Would he see land?’

“No. Only water.”

“Yes. Surround means that everywhere you look, you see only one thing. It could be water. It could be people.”

“So what does ‘surround’ mean, Kit?”


“All around.” Sebastian.

For three weeks he did not see land. When land finally appeared, it was the New World.


“Kit, we just talked about days of the week. How many days in the week?”

“Seven,” jumped in Amit.

“Is that right, Kit?” Kit nodded. “So if the grandfather traveled for three weeks, and each week is seven days–and this is only for Kit–how many days did he travel?”

Kit clearly knew the answer, but needed time to put it in English. I held back everyone else with my hand, giving him time. “Vienti…no. Twenty. Twenty one.”

“Twenty one days on a boat?” Charlotte was skeptical.

“It was a steamship, which would be faster than sailing.” I googled up an image on my cell phone and held it up and walked around to give kids a look.

“Oh, so he didn’t fly on a plane,” Julian. “Twenty one days is a long time.”

“Yes. We can travel more quickly these days. That changes everything. Think about how different you would feel if you had to travel for twenty one days.”

“Please–I would travel more, yes?”

“Longer, not more. Yes, it is a longer journey from India.”

Sebastian was puzzling over the second sentence. “What is New World?”

“America,” Marshall offered.

“Yes, all America. North and South. Mexico is part of the New World. So is Canada.” Back to the map. “All of this.”


He explored North America by train and riverboat and often walked for days on end. So a riverboat is a boat that travels on a river, yes? Who can tell me what a river is? Kit?”


“Yes. Like the Mississippi, here on the map. It’s a…long.. you know? It’s long, but much skinnier than an ocean. Also, ocean is salt water. Rivers are in countries and are not salty.”

“Rio Grande!” from Marshall.

Amit looked confused. I googled “Punjab rivers” and then brought up an image of the Chenab to show him.”

“Oh! Yes. Rivers. Big. Punjab has many rivers. Five.”

“Charlotte is from Africa, which has the Nile,” said Julian.

Charlotte snorted. “The Nile is in Egypt. We have the Congo River.”


“What does explore mean?”

“Aagh!” Marshall smacked his head. “No sé cómo decirlo en Inglés (at least, that’s what Google says he said.) Uh, he looks at. No. Looks…deep.”

“Explore means to learn about…to study. No…is that it?” said Charlotte.

“Yes, Marshall and Charlotte have it right. Explore means to learn about a new place, a new idea–or maybe something you already know a little bit about. Marshall says ‘deep’, to go deep into a subject. Good work! Now, think about that with journey.”

Julian said, “So you go on a journey to explore.”

“Outstanding. Let’s put it in the story terms. We are reading a story about the author’s grandfather, who has crossed the….”

“Pacific Ocean” they chorused.


“explore America!”

“Good! Deserts with rocks like enormous sculptures amazed him.”


“What is ‘amazed’?” asked Charlotte.


“I don’t know. What is a sculpture?”

“it’s art formed out of a hard material–rock, or metal.” I googled “rock formations America” and held up the results one by one. To a kid, they all gasped in…

“Yes. You see that feeling? That is amazed. See how you are all thinking oh, how beautiful. How you didn’t know about such beauty. It’s when you see something good…or bad..or just different. But something you didn’t expect. So when you came to America, what amazed you?”

“The food,” offered Charlotte instantly. “I was..amazed at how much food. How much you could much you could have. It is wonderful.”

“I was amazed that you can take cellphones to class. But mostly that you can ride bikes on the road, with cars,” from Julian.

I chuckled. “Yeah, that’s a quick way to die in China, huh?”

“Here the cars have to stop!”

“See Julian’s behavior, guys? He is acting amazed. Sebastian, what amazed you about America?” Sebastian clearly understood the question, but said something in Chinese to Julian.

“Oh, that’s true,” Julian turned to me. “He said..oxygen. You can’t see it here.”

“The air! Yes, the air in America is so much cleaner, so much clearer, is that it?” Sebastian nodded. “So can you put that in a sentence?”

“I was amazed at the clean air in America.”

“Good! Back to the book. The endless farm fields reminded him of the ocean he crossed. Endless? Kit?”

“No stop?”

“Keeps on going.” said Marshall. “But what is field?”

“A field is an open space, a big one. A farm field is an open space used to grow food.” I googled corn fields and wheat fields .  We determined that the grandfather was seeing wheat fields in this picture.

“So the author is making a comparison. Just as he traveled across the Pacific for twenty-one days, surrounded on all sides by water, so too did these fields seem to go on forever.”

“Like an ocean,” said Max.

“Yes. See how the author drew the fields to look like an ocean, surrounding the grandfather? Huge cities of factories and tall buildings bewildered and excited him.


“Who can tell me what bewildered means?”

Amit, galvanized, pulled out his phone, looked at me for permission. I nodded, and he handed me the results.

“Oh, perfect!”

“Ah!” the class chorused. They all got it at once.

“So bewilder means to confuse you, to see or experience something that fills you with questions. Nice job, Amit.”

“I…feel bewildered a lot.” Amit replied, and everyone nodded.

“Welcome to America!” laughed Chancelle.

He marveled at the towering mountains and rivers as clear as the sky.


“But ‘tower’ is like a building,” puzzled John.

“Maybe the mountains are big, like tower,” offered Max.

“Yes, that’s it. Like a tower. He’s comparing the mountains to a tower, like this.” and I googled some towering buildings. “See? What does marveled mean?

Kit muttered something.

“What? Could you say it again?”


“Ah, yes, like…” Max, like me, uses his hands to fill in blank spaces.

“Would you say marvelous is like amazed?”

“Yes!” Charlotte beamed. “They mean the same thing!”

“Close to it. So notice, let’s page back. The author said his grandfather is amazed, excited, and that he marveled. All of these words have similar meanings. So the author is creating…making a mental image for you.”

“The grandfather is seeing many things that surprise him but…they are good things,” Julian nodded.

“Please–the words mean the same?”

“Not every word, Amit–but amazed…do you see, go back? Amazed and now to the cities page. Excited and now the mountains page…marveled. Everyone see those words? They all have very similar…very close meanings.”

“But not ‘bewildered’.”

“Good! Bewildered is something different. That’s why the author writes yet .See that small word? Yet means that he was confused but still feeling…”

“He is confused but happy he is seeing all this.”

“Exactly! Going on: He met many people along the way. He shook hands with white men and black men, with yellow men and red men. In Japan, would he have seen only other Japanese people. Julian, Sebastian, did you see people who weren’t Chinese before you came to America?”

“No,” Sebastian shook his head. “Only…movies.”

“Only in movies. Charlotte, Congo is mostly black people, but there are some white people there, too, right?”

“Yes, also Chinese. Not…many. But some Chinese.”

“Chinese people in Africa?” John couldn’t believe it.

“Yes, Chinese people are starting to build businesses in Africa.  Asia and Africa are less diverse–they are mostly one race. Well, not North Africa.”

“Yeah,” Charlotte nodded emphatically. “Egypt, Libya, they… have more types. More races. More…mix.”

“Mexico, too,” said Max, and Kit nodded.

“Yes. North and South America have had more than one race for many years–because we’re the New World. Many people from different places came here. Mostly white in North America at first, but still blacks and Hispanics, and even some Asians. But Asia, particularly East Asia, doesn’t see many differences.”

“India has many types,” said Amit.

And the bell rang. Nine pages.

Debrief and other thoughts soon.

Great Moments In Teaching: The Third Dimension (part II)

In our last episode, the class was engaged in sense-making, thinking aloud, arguing aloud, just plain being loud, at the math behind this sketch:

“So up to now we’ve spent a lot of time in the coordinate plane thinking about lines. In the two-dimensional plane, x is an input and y is an output. A line can be formed by any two points on the coordinate plane. We’ve been working with systems of equations, which you think of as algebraic representations of the intersections of two lines. We can also define distance in the coordinate plane, using the Pythagorean theorem. All in two dimensions. So now we’re seeing how this plays out in three dimensions.”

I drew another point, showing them how the prisms were formed, how you could see a negative or positive value:


More students began to see how it worked, as I’d call on a kid at random to take me to the next step. When I finished a second point, the chaos was manageable, but still loud.

“Yo, you want me to be honest with you?” Dwayne shouted over everyone.

NO!” I bellowed. “I want you to be QUIET!” Dwayne subsided, a little hurt, as I go on, “Look, this is a great discussion! I love watching you all argue about whether or not I’m making sense. But let’s stay on point! I got Wendy questioning whether or not I know what I’m doing, Dwayne howling every time he loses attention for a nanosecond….”

Teddy jumped in.

“Here’s what I don’t understand. How come you have to draw that whole diagram? We don’t have to do that with the usual graph….in, what, two dimensions? So why do we have to do it with three-D?”

“Great question. Here’s why. Go back to my classroom representation. According to this, the Promethean is a quarter inch from Josh, Hillary and Talika in the front row. Was anyone thinking that I’d drawn it wrong?”

“I learned this in art!” Pam, also up front and up to now watching silently, said, while comments around nearly drowned her out. I hushed everyone and told her to say it again. “It’s like…we need to draw it in a way to make our brains see it right.”

“That’s it.”

“My brain hurts to much to see anything!” Dwayne moaned.

“But we don’t have to do it with, you know, x-y points.” Natasha.

“Good! Let’s go back to two dimensions. If I want to plot the point (2,3), I’m actually plotting the lines x=2 and y=3, like this:

Alex said “Oh, hey. There’s a rectangle. I never saw that before.”

“You never seen a rectangle before?” Dylan. I ignored him.

“Right. The point is actually the intersection of the two lines, forming a rectangle with the origin.”

Alex again: “Just like this one makes a cube…”


“a prism with the intersection of the three points. But how come you have to draw it? I don’t have to draw a rectangle every time I plot a point.”

“But three dimensions make a single point much more ambiguous.”

Dwayne sighed loudly and held his head. “This ain’t English class. I can’t handle the words.”

“Ambiguous–unclear, able to be interpreted multiple ways. Let’s start with this point:


“What are the coordinates of this point?”

“Easy,” said Wendell. “Just count along the lines.”

“Okay. How about (-1, -1, 2)?” I count along the axes to that point.


“How about (8, -10, 11)?

“They can’t be both!” protested Wendy.

“How about (2, -4, 5)?”

While I listed these points, I followed along the axes, just as Wendell suggested.

“How can you have three different descriptions of one point?” Josh asked.


“But that means there would be three different cubes…prisms?” Manuel.

“Yep. Let’s draw them.”

Something between controlled chaos and pandemonium dominanted as I drew–with class participation–three paths to that point, and the cubes. With each point, I could see again that increasing numbers were figuring out the process–start with the intercepts, create the two-dimensional planes, join up the planes.

When all three were finished, I put them on screen one after the other.

Sophie, ever the skeptic, “Those can’t be the same point.”

Wendy: “I’ve been saying that.”

Arthur stood up. “No, you can see!” He came over to the Promethean and I gave him the pen. “Look. Here’s the original. Start at the origin, go two to the left, and one up. Each one of the pictures” and he shows it “you get to the point that way. So the point is the same on all the pictures.”

Sophie was convinced.

Dylan: “So how come it’s not just (-2, 1)?”

Arthur looked at me. “It’s not (-2,1) but it is (-2, 0, 1).” I replied.

“Oh, I see it!” Wendell came up. “See, you go along the x as negative 2. Then you don’t go along the y. Then you go up 1 on z?”

“You got it.”

“Can you do it for x and y, without z?” Josh.

“Take a look. Let’s see.” And with many shouts and much pandemonium, the class decided that the point could be plotted as (-3,1,0) or (-3,1) on the two dimensional plane.

“Whoa, there’s lots of possible ways to get to the same point.”

“Can you figure out a way to count how many different ways there are to plot the point?” asked Manuel.

“That’s a great question, and I don’t know the answer. My gut says yes, but it’d be a matter of combinatorics. Outside the scope of this class.”

“Thank god,” said Wendy, and I shot her a look.

In other classes, I let them work independently on the handout at this point, but 4th block is crazy loud and easily distracted, so I brought them back “up front” to check their progress every 10 minutes. The whole time I was thinking, man, I’ve controlled the chaos and the skeptics this far, do I dare do the last step? Or should I keep it for tomorrow?

Here’s where the performance aspect kicked in: I wanted a Big Close. I wanted to bring it all together. Even if the risk was losing the class, losing the tenuous sense of understanding that the weaker kids had.

I wanted the win. After they’d all sketched three prisms, I started up again.

“I began this lesson with a reminder of two dimensional planes. The only thing we have left is distance.”


“How do you find the distance of a prism?” said Francisco.

“What would the distance be?”

I drew a prism:


“Oh, okay,” said Sanjana. “So the distance would be to the corners.”

“Yeah, the lower left is the origin, right? And the top right is the point,” said Sophie.


“Now, you learned this formula in geometry. Anyone remember?” I look around, and sigh. “Really, geometry is a wasted year. Does anyone see the right triangle that the distance is the hypotenuse to?”

Dwayne sighed. “God, Hypotenuse! I don’t…”

“HUSH. Tanya?”

Tanya frowned. “Would one of the legs be the….the height?”

“But where’s the other leg?” asked Jenny. “It can’t be the length or the width.”

“No. It’s across the middle,” and I drew the second hypotenuse and labeled everything.


“So now you can see that the x coordinate is the length, the y is the width, and z is the height. I just labeled the other distance w. So how do I find a hypotenuse length?”

Relative silence. I growled.

“Come on, it’s the mother ship of geometry. In fact, I mentioned it earlier.”

Still silence.

“Keerist. You all bring shame to your families.”

“Wait, do you mean the…the thing. a squared plus b squared?”

“The thing?”

“Pythagorean theorem!” half the class chorused.

“Oh, NOW you know it. So how can I use that here?”

“x2 plus y2 equals w2,” offered Patty.

“Yes, and the other one would be “w2 plus z2 is distance, squared,” said Dylan, who’d decided he couldn’t disrupt class, so he may as well participate.

“Great. We have two equations, right? Kind of like a system.”

“But there’s way too many variables. We only have two, right?” Sophie frowned.

“Great question again. To solve a system, we need as many equations as we have variables. But since we aren’t using any specific values at all in this discussion, we aren’t looking for a full solution.”


“So what are we looking for?”

“The EXIT!” shouted Dwayne.

I howled back, “Exit is in 15 minutes. BE QUIET!–Again, a good question. What we do in math sometimes is look for meaningful algorithms that can be formalized into useful tools. Like in this case. Right now, I’d need to go through several steps in order to find the distance of a prism. I want it to be simpler. How can I simplify or restate a system? Natasha?”

Natasha, tentatively, “We can add them up.”

“OK, like combination. Anything else? (SHUT UP DWAYNE!) Natasha? No? Fine. Dwayne?”

“Who, me? I don’t know anything!”

“Bull crap. I scrawled on a new page:

2x + y = 14

“What do you do?”

“Wait. You mean where you…oh, ok, you put the 3y in for the x and then multiply by 2.”

“Try not to be shocked by your own comprehension. That’s correct.”

“Yes,” Dylan said. “You can substitute. How come you never have a substitute?”

“Excluding his extraneous crap, Dylan has also stumbled onto truth. Substitution. And looking at the equation, I see an opportunity. An isolated opportunity, even. We have two variables that aren’t length, width, and height, right?”

“Yeah. w and distance,” Alex offered. “But we’re trying to find the distance.”

“Exactly. What I’d like to get rid of, really, is that pesky w. And oh. Hey.”

I drew a circle around x2 plus y2 and pointed it at w2 in the second equation


Manuel got it first. “Holy SH**!”

Teddy, Alex, and Sophie were right behind him verbally, although Prabh and Sanjana had already figured it out and were rapidly taking notes, working ahead of me.

As I wrote down the final equation, they were shouting it along, and eventually all the class joined in: “X2 PLUS Y2 PLUS Z2 = DISTANCE SQUARED!” and as I finished it up, hand to god about a third of the class clapped madly (while the rest looked on in bemusement).

I bowed. I don’t, usually.

“But where’d the w go?” asked Josh in bewilderment.

“That’s the point, the system substitutes x and y so you only have to use the length and width and height!” shouted Manuel.

“So you never have to find w!” added Sophie.


“Pretty cool, huh?”

And the bell rang.

How to explain the adrenaline rush all this gave me? It took me a good hour to return to earth. They clapped! Not all of them, but so what?

So much of my time is spent slowing down math to be sure everyone gets it. I rarely can really engage and challenge the top kids “up front”. I give them challenges at other times, and they usually like my lectures, but I don’t often have the opportunity teach something in a way that makes sense to the less advanced but still captures intellects at the high end.

Here, it happened. The top kids understood that I’d piece by piece revealed that 3-dimensions are just an extension of the two dimensional system they all knew, but had never thought of that way. Not only did I reveal it, but I did so while using systems, something they’d just been working with. They were admiring the artistry. They got it.

And I did it all with Dwayne and Dylan yipping at my heels.


(Here’s an actual promethean shot from that day. The rest of them I rebuilt.)


Not Negatives–Subtraction

In summer school, I’m teaching what used to be known as pre-algebra and happily, my colleagues had a whole bunch of worksheets that I got on a data stick. Very nice, and the curriculum was very good, leaving me time to tweak but not spend all my time inventing.

It’s not like the curriculum was a surprise: integer operations and fractions played a big part.

Of course, when we math teachers say “integer operations”, we mean “operations with negative integers” because while we don’t really care all that much if they’ve memorized their plus nines and times sevens (sorry, Tom!), kids that don’t fundamentally understand the process of addition are usually un-included by high school.

But negative numbers are one of those “Christ, they’ll never get it” topics. I don’t reliably have an entire class of kids who answer 9-11 with -2 until pre-calculus. I’m not kidding. They say 2, of course. But not negative 2. And if you give them -3-9, they will decide it’s 12 or -6 or, god forbid, 6. But not -12. They’re actually not terrible at subtracting negatives, provided that it’s subtracted from a positive. So they know 9-(-12) is 21, but have no idea what -9-(-12) is, and wildly guess -21.

I’ve suddenly realized that negative numbers aren’t really the problem. Subtraction causes the disconnect, as a result of the tremendous bait and switch we pull when moving from basic math to the abstractions needed for advanced math.

In elementary school, kids learn addition and subtraction. They are not told that they are learning addition and subtraction of positive integers. Nor are they told that they are only learning subtraction when the subtrahend is less than the minuend and, by the way, we need new terms. Those are horrible. In fact, kids are told that they can’t subtract in these cases.


At no point are kids told that everything they’ve been taught is temporary, and that much of it will become irrelevant if they move into advanced math. Consider the big fuss over Common Core subtraction, which is all about an operation that has next to no meaning in advanced math other than grab your calculator. (No, this isn’t an argument pro or con calculators, put your hackles down.) Or consider the ongoing drama over the aforementioned “math facts memorization” which, frankly, gets turned ass over tincups with negatives and subtraction.

Common Core requires that sixth grade math introduce negatives. Along with ratios, rates, fraction operations, and statistical analysis, all tremendously complicated concepts. In seventh grade, things get serious:


Never mind that most non-mathies would clutch their pearls at the very thought of parsing these demands, or that these comprise one of nearly twenty standards that have to be covered in seventh grade. Leave that aside.

Focus solely on NSA1B and NSA1C which, stripped of the verbiage, define the way we math teachers reveal the bait and switch.

So first, you teach the kids about these negative numbers and how they work. Then you show them that okay, we kind of lied before when we taught you that addition always increases. Actually, the direction depends on whether the added value is positive or negative.

But that’s it! That’s all you have to know! Just this one little thing. So negative numbers allow us to move in both directions on the number line.

And subtraction? Piffle. Because it turns out that (all together now!) Subtraction is addition of the opposite. Repeat it. Embrace it. Know it. Then everything makes sense.

So we teach them these two things. Yeah, we lied about adding because we had to wait to introduce negative numbers. But there’s this one little change. That’s all you have to know! because subtraction is a non-issue. Just turn subtraction into addition and funnel it all through the same eye of the same needle. Dust your hands. Done, baby.

Well, not done. As I said, we all know that negative numbers are brutal. We build worksheets. We support the confusion. We do what we can to strengthen the understanding.

But over the years, as I started teaching more advanced math, I realized that subtraction doesn’t go away. Subtraction is essential. It’s the foundation of distance, for starters.

And what the standards don’t mention is that introducing negative numbers changes subtraction beyond all recognition. The people who “get” it are those who reorder the integer universe spatially. Everyone else just stumbles along.

Until this summer, I never addressed this issue. I’m pretty sure most math teachers don’t, but I welcome feedback.

How do we change subtraction?

For starters, we violate the rule they’ve been taught since kindergarten. Turns out you can subtract a bigger number from a smaller number. (And, when a kid asks, “Well, in that case, how come we have to borrow in subtraction?” we teachers say…..what, exactly?)

But that’s just for starters. Take a look at the integer operations, broken down by sum and difference. (Much time is spent on teaching students “sum” and “difference”. More on that in a minute.)


So first, a row of numbers like this brings home an important fact: the Commutative Property ain’t just for mathbooks. This provides a great opportunity to show students the relevance of seemingly abstract theory to the real world of math.

But notice how much simpler the addition side is. I color-coded the results to show how discombobulated the subtraction pairs are:


Middle school math teachers spend much time on words like sum and difference, but I’m not entirely sure it helps.

For example, consider the “difference” between -9 and -5, which is -4. First, -5 is greater than -9, a complicated concept to begin with–and -4 is greater than both. And–even more confusing to kids taught to limit subtraction–none of those relationships matter to the result.

So -9 – (-5) = -4. Which is the same as adding a positive 5 to -9. So the difference of -9 and -5 is the same as the sum of -9 and 5.

Meanwhile, -9 – (-5) is subtraction of a negative, which we have hardwired kids to think of as “adding”–which it is, of course, but adding in negative-land is subtracting. So what we have to do is first get kids to change it to addition, then realize that in this case, the addition is a difference.

It’s not illogical, if you follow the rules and don’t think too much. But “follow the rules and don’t think too much” works for little kid math. As we move into algebra, not so much–we discourage zombies. Math teachers are always asking students, “Does your answer make sense?” and how can a student answer if subtraction makes no sense?

One of the things I’m wondering about is the end result of the operation. Any two numbers have a difference and a sum, all expressed in absolute values. 9 and 5 have a difference of 4 and a sum of 14, and no matter what combination of sign and operation used, the answer is the positive or negative of one of these two. So I ordered them by the end result.


Notice that P+P, N+N, P-N, N-P are ultimately collective sums. No matter the relative size, P+P and P-N move to the right, N+N, N-P move to the left, and result in a positive or negative sum of the two terms.

That looks promising, but I’m not sure how to work with it yet, particularly given the confusion of the actual meanings of sum and difference.

Here’s what I’ve got so far, and how I’m teaching it:

  1. What students think of as “normal” subtraction is actually “subtraction of a positive number”, where the subtracted number is smaller. Subtraction of a positive number always involves a move to the left on the numberline.
  2. In subtraction, the starting value does not change the direction of the operation–that is, -9 – 5 and 9 – 5 will both go to the left.
  3. The starting value must not change. This is a big deal. Kids see -9-5 and think oh, this is subtracting a negative so they change the -9 to 9. No. It’s subtracting a positive.
  4. Please, please PLEASE sketch it out on a numberline. Please? Pretty please?

Hey, it’s a start. I also use a handout I built six years ago, during my All Algebra All The Time year (pause for flashback) and has proved surprisingly useful, particularly this part:

I am constantly reminding kids that subtraction is complicated, that the rules changed dramatically. Confusion is normal and expected. Take your time. I am seeing “success”, with “success” defined as more right answers, less random guessing, more consistent mistakes in conception that can be addressed one by one.

I don’t know enough about elementary and middle school math to argue for change, except to observe that much more time is needed than is given. I once took a professional development class in which a math professor covered an abstruse explanation of negatives and finished up by saying “See? Explain it logically and beautifully. They’ll never forget it again.” We laughed! Such a kneeslapper, that guy.

But I’m excited to get a better sense of why kids struggle with this. It’s not the negatives. It’s subtraction.

Great Moments in Teaching: When Worlds Collide

I’m on vacation! I actually took a whole half day off to add to my spring break, spent a couple days with my grandkids (keep saying the phrase, it will get more real in a decade or three), then embarked on an epic road trip through the northwest. My goal to write more posts is much on my mind–despite my pledge, I’ve only written 10 posts this year. But I’ve gotten better at chunking–in years past, I would have written one “teaching oddness” post, rather than three.

So this new semester, new year, has already seen some teaching moments that are best thought of as crack cocaine, a hit of adrenaline that explodes in the psyche in that moment and every subsequent memory of it, the moments you know that all those feel-good movies about teaching aren’t a complete lie. Not all moments are big; this one would barely be noticed by an outsider.

I was explaining slope to one of my three huge algebra 2 classes, the most boisterous of them. Algebra 2 is tough when half your kids don’t remember or never learned Algebra 1, while the rest think they know all there is to know, which is y=mx+b and the quadratic formula (no understanding of what it means or how to factor). Meanwhile, my recent adventures in tutoring calculus (be sure to check out Ben Orlin’s comment) has increased my determination to improve conceptual understanding among my stronger students, even if my weaker ones get a tad bored.

“I want you to stop just thinking of slope as a number, something you can only get by looking at two points, subtracting y1 from y2, then x1 from x2. The simplest way to start this process is to consider the slope triangle, which I know a lot of you use to find the slope, but don’t really think about.”

“But think of slope as represented by an actual right triangle. The legs represent the relative change rates of the horizontal and vertical (the x and the y). The hypotenuse is the slope. You can see the rate of change. It’s not just a number. Evaluate slopes by their triangles and you can see the ratio in action.”

I’m skipping over some discussion, some give and take. As I drew pictures, I “activated prior knowledge“, elicited responses as to what slope was, what the slope-intercept form represented, etc. But this was pretty close to pure lecture. I can read the audience–they’re not hanging on every word, but they get it, I’m not preaching to snoozers.

“How many of you remember right triangle trigonometry last year, in geometry?” A few hands, mostly my top kids.

“Come on, SOHCATOA?”

“Oh, yeah, that stuff” and most hands go up.

“So when I teach right triangle trig, I do my best to beat into your heads that the trig identities are ratios. Trigonometry is, in fact, the study of the relationship between the ratios of triangle legs and the triangle’s angles.”

“And that means you can think of the slope of a line in terms of its trigonometric ratio. Take a look at the triangle again, but now use your geometry lens instead of algebra.”


“The slope of a line is rise over run in algebra. But in geometry, it’s opposite over adjacent. The slope of a line is identical to the slope triangle’s tangent ratio.”

“Holy SHIT.” Every head turned around to the back of the room (where the top kids sit), where Manuel, a big, rumpled, exceptionally bright sophomore was staring at my board work.

I smiled. Walked all the way to the back of the room, to Manuel’s desk, tapped it lightly. “Thanks. That means a lot.” Walked back all the way to the front.

Remy smiled knowingly. “That was like some sort of smart-people’s joke, right?”

“Naw,” I said. “His worlds just collided.”

I could do a bit more, explain how I followed up, but no. You either get why it’s great, or you don’t.

<mic drop>

Handling Teacher Preps

I was initially horrified at my schedule when I first saw it last June. Having since conceded the possibility–just the possibility, mind you–that I might have overreacted, I thought I’d discuss teacher preps.

Preps is a flexible word. A teacher’s “prep period” describes the free period the teacher gets during the day, ostensibly to “prep”are. “I’ll do that during my prep” or “I go get coffee during “prep”. But if a teacher asks “How many preps do you have?”, the query involves the number of separate courses the teacher is responsible for. So a teacher could say “I have no prep, but I’m only teaching one prep–geometry” or “I’ve got three preps and it’s brutal” without explaining which prep is which.

Non-teachers can’t really understand preps properly without realizing something I’ve mentioned frequently: teachers, particularly high school teachers, develop their own curriculum.

Odd that I’m mentioning Grant Wiggins again, but a little over a year ago, he said that too many teachers are “marching page by page through a textbook”. I’m sure that’s true, but said even teachers who march through a textbook using nothing but publisher generated material, make decisions about which problems to work, which test questions to use, and, unless they are literally walking through the textbook as is, which sections to cover. And those are extreme cases. Most teachers that I would describe as “textbook users” still make considerable decisions about their curriculum, including going “off-book”.

So preps are a proxy for workload. A teacher with four preps has a much greater workload than a teacher with one prep.

I’ve taught at 4 high schools (including my student teaching) and observed how many others operate. So this next description is typical of many schools, but variations on the theme occur.

At both the middle and high school level, math teachers are kind of like the swimmers in Olympic sports—we’ve got the most events.

English has many courses, but more of them are electives (journalism, creative writing) and then there’s the “ELL” split that few teachers cross. Most students take a four year sequence by grade, either honors, AP, or regular. Science and history courses add up because unlike math, each course has an AP version. Science has a 3-year sequence that lower ability students take four years to get through; the rest take an AP course in one of the same subjects, or an elective. History has a four-course sequence over three years, and can’t take an AP course again, which is too bad.

High school math has a six-course sequence that students enter at different points–five course if you count algebra 2/trig as one. From geometry on, each course has an honors version. Calculus is generally offered in both general and AP versions AB and BC. Algebra often has a support course. Then there’s statistics and AP Stats, and usually Business Math. Toss in Discovery Geometry. What is that, 17? And unlike ELL vs. regular English, we math teachers cover it all.

English and history high school teachers rarely have more than two preps, often a primary and secondary. I won’t say never. Science teachers are the most likely to have single preps, or general and honors in the same subject, because they have specialized credentials.

Math teachers often have three preps. Larger high schools may have more specialization. Maybe in big schools you’ll hear someone described as a geometry teacher, or a calculus teacher. But that’s just never been the case in any school I’ve seen.

To the degree math teachers do specialize, it’s a range of the 6 year sequence. The most common is the algebra specialist, a gruesome job that others are welcome to. (It’s only been four years since algebra terrors, my all-algebra-all-the-time year, can you tell? I still get flashbacks.) Some algebra specialists have limited credentials and unlimited patience. Others are genuine idealists, determined to create a strong math program from the bottom up. All of them can go with god, so long as I don’t go with them.

Sometimes you find the high-end experts, the ones that teach AP Calc, honors pre-calc, AP Stats, or some combination of. Sometimes these folk are the prima donnas with the math chops. Other times, they just aren’t very good with kids so they get stuck with the most motivated ones—they also teach the honors algebra 2 and geometry courses sometimes, because they just can’t deal with kids who aren’t as prepared or motivated. (No, I’m not bitter. Why would you think that?) And while we don’t have a name for what I do, it’s not uncommon for a math teacher to focus on “the middles”, the courses from geometry to pre-calc.

But not all schools go the category route. Others require all math teachers to cover a low, mid, and high level course in the sequence to be sure that no one gets cocky.

So now, after that explanation of preps, go back to the beginning, when I mention my hyperventilation over easy, familiar preps that I thought would be boring. Many teachers would agree—quite a few colleagues in all subjects commiserated with my dismay. Other teachers consider it rank abuse of power when admins assign them two preps, much less three.

Why? Because some teachers love the additional workload, love building and developing curriculum, mulling over the best way to introduce a new topic. For teachers like me, that’s an essential element of teaching—and repetition, teaching the same content three or four times a day, is so not essential, but rather Groundhog Day tedious. Others see curriculum as something they want handed to them or will do, reluctantly, once. Or, something they’ve honed after umpty-ump years and it’s perfect so they aren’t changing a thing. To these teachers, curriculum is a distraction from their primary job of teaching, the delivery of that curriculum–the job they actually get paid for. Give them the day of the school year, they know what they’re teaching.

If you’ve never really considered teacher preps before, certain questions might come to mind. Does teacher effectiveness (however measured) vary with the number of preps? Does teacher effectiveness vary by subject? (I’ve wondered before if I’m just better at geometry than algebra, for example.) Could we improve academic outcomes by giving weak teachers one prep in a limited subject, and strong teachers multiple preps (assuming we know what that is)? Do teacher contracts negotiate the maximum number of preps that can be assigned? While Ed’s informed assertions are interesting, surely there’s better data that gives a better idea of how many preps high school academic teachers have, on average? Or middle school teachers?

What terrific questions. They all occurred to me, too. And while I’m a pretty good googler, I began to wonder if I wasn’t using the right terms, because I could find no research on teacher preps, no union contracts restricting preps.

Let’s assume that some research has been done, that some contracts exist but escaped my eagle Google. Teacher preps still are clearly not on the horizon. I can’t remember ever hearing or reading a reformer mention them. When I was in ed school, the subject never came up—how to identify the best combination of preps, what number was optimal, and so on. Given how little control teachers have over preps, ed schools may just count it as one more of the nitty-gritty elements of the job we’ll discover later.

Education reformers simply don’t understand the degree to which teachers develop or influence curriculum and the resources it takes. They don’t understand the tremendous range of curriculum development that takes within a school. Moreover, most reformers don’t even understand that preps exist or have any impact on teacher workload. Few of them ever taught at all. So they don’t really know what a “prep” is, and then assume that most teachers rely largely on a textbook. That doesn’t leave them much room to mull.

Researchers don’t discuss preps much, either. I’m not even sure Larry Cuban, who describes teacher practice better than almost anyone, describing here the multi-layered curriculum which explicitly describes teacher-designed curriculum, has never written about preps. Many researchers also tend to confuse textbooks with curriculum.

I wonder if researchers are prone to ignoring high school preps because they would have to acknowledge how questionable their conclusions are without taking preps into consideration. If a researcher compares two high school teachers using a new curriculum, does it matter if one teacher has one prep and is teaching the same topic all day? This may give that teacher more time to adjust, notice patterns, change instruction. Meanwhile, the busy teacher with three preps who is just teaching one class with the new curriculum may just be doing it as an afterthought. Alternatively, teaching one class all day may also bore the teacher to the point of rote delivery, while the teacher with one class jumps in with enthusiasm.

Once I really started thinking about preps from a policy perspective, I became really flummoxed at the lack of play it gets. I may be missing a whole field of research, that’s how odd it is.

Administrators keep preps firmly in mind; whether contracts require it or not, they rarely give high school teachers more than whatever a commonly agreed amount is (usually three). Ideally, they will limit new teacher preps, although my mentee from last year had three preps each semester. Now that I think on it, I had three preps, too. Never mind—they pile it on newbies, too.

If VAM ever gets taken seriously at the high school level (which I find very unlikely), preps are likely to become a contract issue. Teachers being judged on test scores will probably demand a large sample size, which means fewer preps.

Fewer preps for teachers, of course, means far less flexibility for administrators putting together the dreaded master schedule. Ultimately, it means more teachers on the pay roll or fewer courses offered, because fewer preps and less flexibility must be compensated for somehow.

And hey. I just realized that Integrated Math (bleargh) schools have fewer preps. Maybe this is another foul plot of Common Core.

For myself, I do not want limited preps, even if my feet are forced to the fire on the point that hey, I’m really enjoying this easier year. But honesty compels me to point out that preps should be explored for their impact on teacher satisfaction, teacher productivity and–to the extent possible–academic outcomes.

I have no real ideas here. Only thoughts to offer up and see what others have on tap.

However, there’s another issue never far from my mind that perhaps the above mullings cast some light on: that of teacher intellectual property. Stephen Sawchuk just wrote a great piece on various issues in the related arena of teacher-curriculum sharing, and mentioned IP and copyright. I have huge issues with the absurd notion that districts own teacher-developed curriculum, which I’ll save for another post.

But surely this post makes it obvious that if teacher preps vary, then one of two things must be true. Either teachers in the same subject are getting paid the same salary for doing dramatically different jobs–and I don’t mean quality here, just work expectations.

Or teachers are paid to teach, in which case the actual delivery is the same no matter how many preps we have. Teachers then have the choice–the choice–to use the book and supplied materials extensively, or develop their own, to do the job as they determine it should be done. This seems to me to be the obviously correct interpretation of teacher expectations and the “work” they are “hired” for.

And in my world view, teachers are not paid to develop the curriculum, and therefore the district can keep its damn paws off my lessons.


Troubling Students

My classes are easy to pass, hard to do really well in. I’m a pushover for a D, but think three or four times about giving out an A. I didn’t fail a single kid last year. Save for Year Two, All Algebra All the Time, I’ve failed fewer than six kids a year, and even Year Two I had the second lowest fail rate of the math teachers.

I teach mostly math at a comprehensive high school, and the previous paragraph is very near heresy. Some math teachers cheer me on as a brave, admirable soul, but I spot them making the Mano Pantea while they walk away, just in case the Overlord is Watching. Others think I’m What’s Wrong With Education Today. These teachers hold as gospel that math standards could be upheld if we teachers were just willing to fail 60-70% of our students. In contrast to Checker Finn, who thinks teachers like me are spreading out two years of math content over three years of instruction because we can’t be bothered, these folks don’t think I’m lazy. They think I’m soft. They think I’m damaging their ability to cover all the course content they could get through if there weren’t all these kids who shouldn’t be there.

I became a lot less conflicted about my high pass rate–not that I ever lost sleep over it–after teaching precalc and discovering that a third of the kids had forgotten how to graph a linear equation and half couldn’t graph a parabola. These were kids that those other teachers had, teachers who had covered everything. Meanwhile, my kids do well in subsequent classes, so I’m not doing any harm.

But I digress. The students who trouble me aren’t the strugglers. I can take a kid who hates math, doesn’t want to be in class, and get him (it’s usually a him) to try. I can get that kid to attack a projectile motion problem and, even while making multiple small mistakes, beam with pride because by god, he kind of gets this and who ever would have thought? Kids like that, I can pass with nary a qualm.

The worrisome ones pretend they understand, but don’t have a clue. They cheat whenever they can, and not just on tests. They copy classwork in the guise of “working together” or “getting help”, and do their best to sit next to strong students. I group students by ability and, unless they can cheat on my assessment test, they are outed and placed up front, where I can keep an eye on then. They will then ask if they can sit next to John, or Sally, or Patel, their friend, because “they explain it so well”. I say no.

But if they cheated on the test, they can sometimes escape notice for a while. I circle constantly, watching kids work, changing seating when I see too much “consulting” with little discussion. Still others are more clever, and it takes a while before I realize they’ve been cheating not only in classwork, but on the tests–even when I create multiple tests. As a new teacher, I would sometimes miss these kids through the first semester. My success rate at pegging them early has improved.

This isn’t a big group, thank god. I might run into one or two a year. They have a telltale bipolar profile: for example, failing English entirely one year, and passing it the next year with Bs. Passing algebra with straight As, failing geometry completely–and failing the mostly pre-algebra and algebra state graduation test with a spectacularly low score. They aren’t fooling all of the teachers all of the time.

These kids are not your Stuyvesant cheaters, conspiring with others to satisfy demanding parents and create a fraudulent resume to get into a good school. Nor are these the low achievers who just want to get a passing grade in these time units called classes organized into a larger time period called school that others apparently view as a place of learning but they see as little more than a community network in which they have invested considerable social capital.

In fact, they’re almost worse than identified low incentive low achievers, cheating or otherwise. These kids almost seem incapable of learning. I can’t get them to slow down. They often resist help from me. Typical conversation:

Me, stopping by: “Okay, let’s start this again. You’ve plotted these points….”

Student: “Oh, yeah, I see.” Frantically erases.

Me: “Well, hang on, I want to be sure…”

Student: “I got it I got it I got it.” Starts to plot a point, then pauses.

I realize the student is waiting for me to say where to plot it in order to say “Yes, I know, I know.” So I wait. The student takes a deep breath and plots the point then lifts his pencil. “No, that’s not right, duh…”

Me: “You aren’t sure how to plot points.”

Student: “Yes, I am.”

Me: “Great. Plot (7,-7).”

Student plots (-7, -7).

Me: “Stop there.” I go grab a handout I have specifically for these situations, a simple handout that explains plotting points with some amusing activities to drive the point home.

Student: “I don’t need this. I know how to do it!”

Me: “Great. Then it should just take you a few minutes.”

At this point, I get a variety of reactions. Some students become furious. Others get sulky. Still others do the handout, making many mistakes, all the while assuring me that this is easy. I obligingly correct the mistakes, make them do it correctly. The ones that get furious, I shrug and let them continue.

Regardless, within a day, they are making the same mistakes. Nothing sinks in. Don’t get overly focused on plotting points; the problem could be anything–factoring, solving multi-step equations, working with negatives, exponential properties, fractions, whatever. Or a new concept. They have absolutely no clue, and can’t do much of anything.

Yet they don’t have the profile of a low ability student. Test scores, yes. Profile, no. They often have As, win praise from teachers for their teamwork and effort. They are heavily invested in appearing “normal”. Serious control freaks. Sometimes, but not always, with parents who expect success. More often, but not always, Asian. All races. Both genders.

I haven’t taught freshmen since oh, lord, fall 2012.1 I teach relatively few sophomores these days, running into them only in Algebra 2.

That matters because when I taught freshmen and sophomores, I would go full-scale intervention. I might talk to a counselor to see if they should be assessed for a learning disability. I would insist that they stop lying to me and themselves. I had no small success at getting some of them to acknowledge their desperate attempts at fraud, get them to work at their actual level, deal with the discomfort. They didn’t make much progress, but it was real progress, and they had skills to move forward. I ran into some of them again the next year, and we could start on an honest basis and make additional progress. Those who didn’t acknowledge their issues were among the few students I failed.

But that’s a lot harder to do when dealing with juniors taking trigonometry or, god forbid, precalc. Should I fail them? They will probably do better in a class with teachers who give “practice tests”, study guides that have exactly the same questions as the eventual real test but with different numbers. They will definitely do better with teachers who actually grade homework and count it as 25% of the overall.

A small problem. This approach turns my grading policy into: work hard and honestly acknowledge your ignorance and I’ll pass you. Lie and do your best to cheat with similar ignorance and I’ll fail you. I’m comfortable with holistic grading at the bottom of the scale, but I don’t like morality plays.

Then I remember that kids who honestly acknowledge their inability in a trig or pre-calc class are usually seniors, off to junior college and a placement test that will accurately put them in remedial math. I’m only ensuring they are learning as much as possible for free before paying. If they are juniors, I always have a talk with them about their next steps, telling them not to take the next course in the sequence but maybe stats or something else that will keep them working math, but not out of their league.

The kids who cheat and fake it in trig and precalc are usually juniors, and they will be going onto another course. They will not listen to me when I tell them under no circumstances should they continue into pre-calc or, god forbid, calculus. I might be teaching that course, which just gives me the same problem again. Or they’ll be cheating their way through with another teacher–or, that teacher will do what I should have done and flunked them.

This quandary doesn’t make any sense unless you realize that in my view, these kids are pathologically terrified of facing reality, the sort of thing that some of them, forced to face up, might not survive in good form. These aren’t blithe liars gaming the system to look good. Then I remind myself that they’ve been caught before, they’ve flunked other classes, they’ll survive. But I still don’t like the quandary, because these are kids who literally can’t learn. (And remember, I’ve seen them in my non-math classes, too). By junior year, given their denial and fear, does it do any good to make them aware of this? They’re going to be able to point to any number of teachers who disagree with my assessment, and have all sorts of excuses for why they got those Fs. Besides, they just don’t test well. It’s always been a problem.

At times like this, I envy my colleagues who never notice the cheating, or who focus purely on achievement and aren’t interested in the distinctions I’m making.

But these are the students who trouble me.

1Holy Crap. That’s an amazing realization. New math teachers doing your time in the algebra/geometry trenches, take heed. If you want variety, it will come.

Teaching: The Movie

Another entry in “teacher as entertainer”:

Dave of Math Equality writes that Taylor Mali captures his zeal for teaching. Eh. I get vaguely embarrassed when they play Taylor Mali at PD sessions; he’s like teacher martyr porn or something. I naturally have all sorts of teaching miracle stories. But I don’t tell them to inspire you, dear readers, to convince you that here’s another wonderful, self-sacrificing teacher slaving away unappreciated and exploited, yet nobly giving every drop of sweat and blood to to help navigate self and soul to adulthood or sanity, whichever is needed more.

I’m saying “Look, another day at work turns out to be a F***ING MOVIE!” I made more money in tech, sure, but I didn’t ever experience moments where I thought jesus, people would pay money to watch this on screen and not feel ripped off.

Make no mistake: I am the STAR of this movie. I have a contract giving me a guaranteed audience of thirty for 90 minutes, three times a day. They are to be attentive, listen, watch, and if they learn too, well, cool.

Anyway, I had a moment today that many other teachers have had, and for me it was like, I’d have kicked back $20 to the district for the sheer joy of the experience.

It was fourth block, my prep, and I was just about to leave for Starbucks, as is my routine, when Steve, from third block, knocked on the door.

“Hey, why aren’t you in class?”

Steve, white, tall, skinny, glasses, shook his head. “Can’t handle it. It’s insane in there.” He pointed to the class next door.

The class next door is taught by a long-term sub, because we haven’t been able to find a math teacher. But of course, the big pain point for principals is firing bad teachers. (The AVP offered the job to first one, then the other of my interviews, both took other jobs.) This sub is a qualified physics teacher, new to teaching, just got work permit, teaching a brutal schedule (two Discovery Geometry classes. Shoot. me. now.) I’ve talked to her a couple times, given her some advice.

I got up. “Come on.”

Steve shook his head, “No, they’ll know I brought you over. Can I stay here?” I gave him a withering look–sissy!–and as I walked next door I have to admit I envisioned myself pushing open the saloon doors as the sheriff, come to beat this brawl down.

The sub opened the door and gasped, “Thank you for coming!” The room was….not quite a barroom brawl, but kids were talking and chatting and eating, purses and backpacks on their desk covering the handout. They were manifestly not doing math. One big guy with cornrows (and no, not black) in the back of the room was leaning back in his chair, texting. I took his phone and gave it to the sub.

“What are they supposed to be doing?” I asked, softly.

“They are taking a test.”

“A TEST?” Cue Ennio Morricone.

Heads swiveled. I walked to the front of the room, slowly, looking at students. At least ten of them are in my third block class (not math), and they quieted down immediately. Some of the others were still talking. Discovery Geometry is a tough crowd.


“Who are you?”

I just look at him, a big guy, Asperger’s, not malicious. He picked up on a facial cue (hey!) and didn’t demand an answer. The room got quiet in a hurry. Another, smaller guy (this one is black) is perched at the door, half open.

“Are you in this class?”

“Yeah, I have to go the bathroom. Waiting to see what you said.”

“Good plan. You can go. Be back in under two minutes.” To the class, which had briefly started to rustle: “I said QUIET.” Quiet.

“Purses and backpacks on the floor. Now.”

Instant obedience.

“You three are way too close together. You, in red, move to that desk. Then you two spread out. Girls, you in pink sit at the end of the table, other two spread out.” Again, obedience.

“You work the test in silence. I don’t want to hear about any problems. Next time I come here, it’s with an administrator. Is that clear?”


“Get to work.” They all instantly bend over their tests, except Texting Kid, who raised his hand.


“Could I have a pencil?” (Keep in mind, he’s had the test for 20 minutes.) He got a pencil, and got to work.

I left as Bathroom Guy comes back, well under two minutes.

Steve hustled back to the test, gratefully, after taking my cell and room phone number so he could text or the sub could call me in the event of future disaster.

I never did get to Starbucks, so did some copying. On the way back to my room, who should I run into but Bathroom Boy.

“Hey. What are you doing out?”

“Had to go to the bathroom.”

“You already did that.”

“Had to go again.”

“Yeah, no.” Walked him back to the room. He didn’t even protest. I told the sub no one, but no one without health issues, goes to the bathroom twice in one day. They’d finished the test, and with fifteen minutes left in class, they were talking loudly with nothing to do. I told her no to that, too, in the future. But they’d worked harder and more quietly than ever before, she told me.

I remember an actor saying that in a performance if you have to cry, you can either dredge up a horrible memory or just use an onion. This was all onion. And yet it was also a good fifteen minute’s work. Kids learned someone was watching; they know it’s not free beatdown on sub week. But the whole time I was thinking “Oh, my god, this is SO COOL. I’m CLINT. Or at least the badass principal in The Wire.” Self-absorbed puppy that I am, there is my takeaway.

I am teaching two brand new classes, and an Algebra 2/Trig class I’m struggling to keep somewhat true to its name. It’s not an easy year, I’m not brimming with confidence—although I’m having a great time. So getting to be Clint or the badass principal was just a great moment, a reminder I still have teacher mojo.

Right about now, I realize son of a bitch, I’m a lot more like Taylor Mali than I’d like to think. Yes, I’m more Movie Star than Teacher Martyr, more audience participation than individual redemptions. But ultimately, I’m one of those teachers who can walk into a room of adolescents and command them—-to learn, to think, and sometimes just to obey. And just like Taylor Mali and the people clapping him on, I like what that says about me.

And hell, if you think it’s easy, you try it.