Post-Pandemic Update

This is me:

Wednesday morning my phone rang and instead of the usual “scam likely” (thanks, T-Mobile!) it was the medical center. I picked up, apolozing to my class for taking the call but I was having surgery soon.

“You have been assigned a surgery time tomorrow at 11:00 am.”

“Tomorrow? No, Friday.”

Pause. “Uh, no. December 22nd.”

I went to my google calendaar. For Thursday it  says “Surgery.”  For six weeks I have been telling everyone, including myself, that my surgery is on Friday.

“Um. Could we schedule it later in the day?” The scheduler is very nice and promises to call back.

“Your surgery is tomorrow? Will we be able to finish our final?” I don’t know why Ahmad is concerned about anything; I’m the fool who doesn’t know what day my surgery is. My answer is forestalled by the phone again, it’s the scheduler with an offer of 1:00.

“Oh, that’s perfect. Thanks so much.” Thursday was a half day and half days end at 12:30. I’m about 20 minutes away from the medical center. Plus, no covid test! My record is pure.

“OK, guys, it’s good. I’ll be here tomorrow. Just have to jet out at 12:30 when school ends.”

“School ends at 1:30 tomorrow.”

“What?”

“Yeah, it’s some new schedule.” This is the 9th half day this year.

“Fucking fucking fuck.Ignore me.” Just as my calendar said Thursday, the posted bell schedule says school ends tomorrow at 1:30 for the first time in thirty years. I have to get coverage for fourth block.

So I go see the principal’s secretary to confirm procedure. Teachers accrue ten days a year. I’ve been teaching for 14 years and have 120 days. Time off is rare and I usually screw it up so I check with her first.

“Yes, you have to enter an absence record, no sub, get coverage.”

“Why an absence record?”

“You’ll be gone for three hours.”

I look at her. “Tomorrow’s a whole day?”

“No, but you’re contractually obligated to be here until 3:30.”

“This campus will be a ghost town at 1:30. In fact, the only person who would be here tomorrow after 1:30 would be me, if I weren’t going to be gone.” (I’m routinely at school until 6 or 7.)

“Doesn’t matter.”

Fuck that. I enter an absence record of an hour. If she wants to prove I’m not here, she can wait around.

For teachers, there’s a big difference between contractually obligatated hours and actual time worked. You can’t shift the hours. All the extra hours in the world don’t get teachers out of contract hours. I get that. But this is bullshit. Probably the most irritating part is that the secretary knows full well how rare my absences are and how jackassed it is to tell someone who’s never absent that the bill will be for three hours instead of one.

Anyway. I get coverage and in the process notify several other teachers that we’re out at 1:30. Wasn’t just me who didn’t know. That, at least, is comforting.

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You might think that since I’m never absent,  I might be intolerant of kids taking time off. Or you could more accurately conclude that people who mangle their own surgery dates are tolerant of all sorts of hiccups.  But this year has taken a toll on my approach.

The first post-pandemic year the problem was first-bell tardies. We had literally 300 or more kids showing up more than 15 minutes late every day. I’d have fourteen kids out of a class of thirty by the tardy bell, and four kids still missing after 30 minutes.

This year, the tardies are limited to five or six kids, but the absentee rates are stratospheric.I’m not fussy about occasional absences. But I have thirteen students with more than twenty unexcused absences thus far, out of 77 days in school, and most of them had excused absences as well: mental health days (two a month. really.), flu, covid, monthlong trips to India. Three kids didn’t bother to show up for finals. Another kid said “I’m going on vacation next week so I won’t be here. What can I do to make up the work?” Nothing, I said. You’ll probably fail the class.

I told one girl the first week of school to drop precalc when she explained she’d had to leave early every day to go to a college class (I still don’t know how this was allowed). She didn’t. Then she had a crisis with her mother’s health. Then her car got totaled. And in between all these absences, she took her two mental health days every damn month and then, after promising to be in class, spent three days in a row talking to her counselor. Then complained when she got a D, which was a gift. (Great Roger Sweeney joke: Student asks why she got a D. Teacher says because I like you.)

Sing me no sad songs about kids’ emotional health. Most of them are fine. But even if some of them are not, the batshit tolerance is just begging for exploitation, and getting it good and hard.

School’s wonderful. I’ve been so busy I haven’t had time to write, plus the Mounjaro tends to wipe me out (22 pounds! although it’s a lot more expensive now).  But this year has been even more joyful than last, when just being in the classroom again was a jolt of happiness, masks notwithstanding.

Achievement is up. Last year my trig and algebra 2 classes were hideously behind as these were kids who’d had two years of cheating with photo-math. This deficit is reflected in my precalc class, which is just….not fun. The juniors are ready to go, but the seniors are comatose. My algebra 2 class has by far the worst absenteeism–nine of the +20 absences are in this class–but the rest of the kids are good to go. Not quite up to pre-pandemic skills, but well on their way. I’m teaching a freshman algebra class for the first time since my traumatic all-algebra year of terrors and it’s marvellous. Terrific students, hardworking, ready to face the world.  Another thing: last year, most kids wore basically pjs to school. Way more jeans this year.

It was harder to see the division in abilities and mindsets last year, but I’m centering on the view that the middle of the bellcurve kids who spent a couple years of high school in remote and are now seniors were the worst hit. The top kids are still on the ball and kept activities going during the pandemic (I ran five clubs pandemic year to aid in this, plus my usual stuff), but kids who were just planning on community college anyway and had no resumes to worry about found it pretty easy to skate by. I’m seeing mostly the top half of the bellcurve of those who spent middle school in the pandemic, but they seem to be doing very well. Yes, I know what NAEP scores say, but people are stupid about NAEP anyway. I’ll yell about that later.

My worst enemy wouldn’t call me a hardass teacher, but I am not aligned with the current obsession with student mental health. Get your ass back to class, kids. (And don’t say “OK Boomer” to anyone born in the early sixties. It REALLY pisses us off. Early gen boomers got free love, the Beatles, and cheap college. We got disco, HIV, AIDS, stagflation and a far more competitive job market.)

We’re supposed to be doing these SEL lessons every week. I did one. Said “This is stupid.” My advisory class of sophomores agreed so we didn’t do them anymore.

When my TA told me he’s still covid+ after two weeks off, I say “then don’t take the damn test. Lie. What the hell. Get back here.”….and well, not entirely kidding.

But generally, I’m having a blast. It’s good to be back. Most kids are doing great, world. Stop worrying.

Surgery is for arthritis. I’m a bit nervous.

Song Blue

Song spins are fun for English learner classes. So Monday, right after Scott Walker had just died, when I was reminded of the only hit of his I knew (remember, my tastes are relentlessly mainstream), the lyrics suggested a lesson.

“OK, listen up. Listen to this song once through.” While they listened, I printed out the lyrics, although I’m not sure that’s any better than putting them on the board. Huh. Next time.

“Take a look at the lyrics, now.”

Loneliness is the cloak you wear
A deep shade of blue is always there

Chorus:
The sun ain’t gonna shine anymore
The moon ain’t gonna rise in the sky
The tears are always clouding your eyes
When you’re without love, baby

Emptiness is the place you’re in
There’s nothing to lose but no more to win

Chorus:
The sun ain’t gonna shine anymore
The moon ain’t gonna rise in the sky
The tears are always clouding your eyes
When you’re without love

Lonely, without you, baby
Girl, I need you
I can’t go on

Chorus

Hooriyah can’t write an intelligible sentence and her reading skills are shaky, but what English she has is always ready.

“Loneliness is like being alone?”

“What is a cloak?” Matias puzzled.

Pro-tip: Pictures are far more useful than words to give definitions. The kids instantly make the connection without additional words to confuse them. I google “cloak”, click on “Images” and eight kids go “ahhha”.

“But how can clothes be alone?” Hooriyah asked.

I wrote “metaphor”.

“When you listen to songs, read poetry, read literature, you will see different ways that the artists use to communicate feelings. One of them is a metaphor, which is a way of comparing.”

“So…wearing lonely?” Vanessa squinched her face doubtfully.

“Yes. Imagine being so lonely, so without people, that you are surrounded in it.”

“Like a coat! And a coat is a cloak!” Matias again.

“Yes, a cloak is even more of a coat. You know, look at the picture. It covers even more of your body. So everywhere around your body, is loneliness. It’s not just in your head, in your heart. The song saying the loneliness is so painful, it’s like a real thing. The next line uses the color blue.

Taio frowned. “Chorus?”

“It’s a ‘k’ sound, not ch like cheese. Korus. What do you notice about the chorus?”

“Again and again.” Taio has, hands down, the best grammar and best vocabulary of my eight students. Vanessa is next. Both of them simply hate to talk.

I enjoy this activity so much because it carries itself along. Hooriyah was already mulling over the chorus. “So if the sun does not shine, then it’s night time.”

“But the moon isn’t rising,” I observed.

“Then it is not night…and not day?”

“Can it be day without the sun?”

“Yes, but it’s…what’s the word? Clawdy?”

“Cloudy! Good. What is a day like when it’s cloudy?”

Silence, so I google me up some cloudy days for clarification.

Vanessa snapped her fingers. “That’s gray!”

“Yes. No sun and no moon turns the world gray.”

Curiousity finally spurred Taio to venture a query. “The sky is blue when sun shines. But blue means sad, when sun isn’t shining.”

Huh. “That’s a great point, Taio, and I never really” I remember waving my hands shaping my thoughts, “I never had to explain the difference before. We talk about feeling blue, a cloak of blue, singing the blues. But yeah, you’re right. The color blue isn’t really sad. Just the….feeling.”

“Oh,” said Taio, still confused. “Like…song blue?”

“Hey, yeah. It’s a blue for songs. For poems. For words, in English. But now, think about someone you love very much. Your mother, father, grandma, sister, brother–family. Think how you would feel if they died.” I paused for a second to google translate to the newest students, Geovany and Jorge. This lesson was mainly for the more fluent kids; I was just keeping the new boys in the loop.

“Can you think how sad, how horrible you’d feel? What color would you call that?”

“Black,” Matias said, with no hesitation. Vanessa nodded.

Hooriyah, “I would feel very, very, terrible. So black, yes, that makes sense.”

Now Taio was really confused. “White is for death.”

“Oh, that’s right, Chinese wear white for funerals. Arggh.” I tap my chest with my palm. “Black is about feeling. Feeling death. Not what you wear. How you feel.” Taio nodded, getting it.

“Now I get to complicate it more. Bad grief, bad sadness, is black. But the world, when you feel so sad, well,…”

“THE SUN DOESN’T SHINE!” Hooriyah clapped her hands! “I GET IT!”

“And the moon…..” I prompted.

“Doesn’t rise!” Vanessa shouted.

“So blue is feeling sad and black is horrible loss, and the pain means you feel like the sun doesn’t shine and the moon doesn’t rise. And so we have a third color–or lack of color, which is….?”

“Gray!”

“Yes! Now look at the next verse. What does “empty” mean?”

“Nothing inside.” Taio held up his water bottle, turning it upside down.

“So being sad and black and gray is like being in an empty place,” Sanjana spoke up for the first time. She’d been watching carefully, but although she’s been in America twice as long as any of the others, speaking English comes very slowly.

“Think about the next part: nothing to lose and…”

Matias frowned. “But if you can’t lose and you can’t win then what happens?”

“Nothing!” Hooriyah again.

“Sanjana, what do you think? How is having nothing to win and nothing to lose part of the song?”

“I don’t know.”

“Oh, come on. Play along. See, Geovany and Jorge are waiting for your answer.” Silence.

“So you know why you’re not speaking?”

She grinned. “I don’t know the answer!”

“So are you worried you’re going to lose? Lose what?”

“Nothing.”

“Nothing to lose!” Vanessa jumped in.

“In English, ‘nothing to lose’ can sometimes be a good thing because it means taking risks, taking a chance, has no bad side, no down side. So if you want to reach for something, try to get something…”

“…but the song says you can’t win.” Taio pointed out.

“Yes! Exactly. You have nothing good in your life, nothing that you can lose. But there’s no way to win.”

“It’s just like the sun and the moon.” put in Sanjana.

“There you go. The two verses are saying the same thing. They are reinforcing. Reinforcing…to make stronger. To make the meaning stronger. Even the tears. What do tears normally do? Jorge? When you cry, what happens?”

Jorge read the Google translate and looks at me like I’m crazy. “Lagrimas?” He mimed tears falling on his cheeks.

“YES! Perfect! The song doesn’t say tears falling from your eyes, but clouding your eyes. And clouds are….”

“GRAY!”

“More reinforcement, see? When artists use words to build a picture in your minds and hearts, that’s what creates…poetry. Songs. And part of knowing English, of knowing any language, is understanding the deep meanings of words. Of knowing not just blue, but song blue.”

And with that, I put on a second song, one I’ve played each year of my ELD class:

Song sung blue
Everybody knows one
Song sung blue
Every garden grows one
Me and you are subject to the blues now and then
But when you take the blues and make a song
You sing them out again
Sing them out again
Song sung blue
Weeping like a willow
Song sung blue
Sleeping on my pillow
Funny thing, but you can sing it with a cry in your voice
And before you know, it get to feeling good
You simply got no choice
Me and you are subject to the blues now and then
But when you take the blues and make a song
You sing them out again

Song sung blue

We go through the lyrics every year, but while they always love the song (even little Geovany tentatively sang along), this year I had the fun of seeing them realize that “Sun Ain’t Gonna Shine Anymore” was a case of a Song Sung Blue, and take joy in their knowledge.

With fifteen minutes left, I introduced them to “Here Comes the Sun” and watched them grok the symbolism all on their own.

I swear, sometimes I’d do this for free.

 

Not Really Teaching English Once More

Have I ever gone through the steps that led to my teaching ELL?

Year 1

Back in August 2016, the day before school started, my principal walked into my classroom, which he rarely does, and I said, fearfully, “you’re not taking my pre-calc class, are you?” because I rarely get to teach precalc and principals only walk into your room when they’re asking for something you won’t like, and the only thing I wouldn’t like was losing my precalc class. The other upper math teachers complain, which isn’t fair, because the state tests show my kids do as well as theirs on average (their top kids do reeeeeeally well, but the rest of the kids do horribly, while my top kids do well, but everyone else does respectably.)

But no, he wasn’t taking my precalc class, he was taking my prep period. The non-tenured mostly ELL teacher who was on track to for termination in the upcoming year had taken a new job with less notice than is legally allowed. It’s not well known, but teachers aren’t allowed to quit without some degree of notice, usually between 30 and 60 days, unless the district gives permission. Otherwise, they won’t be able to teach anywhere else in the state. My principal and the English department felt the rejected teacher should be allowed to take a new job under the circumstances. English teachers hate taking extra preps, but they scrounged up two volunteers and suddenly, someone remembered I have an English credential.

So with no notice, I started ELL instruction. I had four classes planned for the following semester. But despite an ongoing hiring campaign, no one would accept the job. With a whole bunch of juggling,  Bart was handed one of my trig classes and I taught ELL the entire year. That was year one. Articles: The Things I Teach, Not Really Teaching English,ELL isn’t Language Instruction

I expected it to be an anomaly. I loved the kids, and my first, large, ELL class remains my favorite both for the students and the experience. I’m currently teaching Marshall and Kit (from the Things I Teach), both of them juniors, doing well. Juan, Anj, and Tran are all academic rock stars, with several AP classes (including English) to their credit. But the political and instructional aspects of ELL bothered me tremendously, and I was happy to be out of it.

Year 2

Then almost exactly a year ago right now, an AVP walked into my classroom just a week before the first term ended, just as I had convinced myself I was actually going to get the pay cut of a normal prep period, to ask me if I’d help them out by teaching the ELL Connections class. She didn’t say why. In fact, I’d asked the principal a month earlier to confirm that there’d be no additional class coming my way, and he assured me there were no plans to use me. Which, well, wasn’t true. He couldn’t mention that he was in the process of firing the primary ELL teacher, the 20 year expert, which he achieved in three months from start to finish, including investigation. They collapsed two of her courses into one, and gave me the other.

The second year (2017-2018) wasn’t particularly enjoyable. The kids had hated the fired teacher, and had enjoyed three months of substitutes and movies. None of them had any particular interest in learning English. I had two Chinese boys who wouldn’t (and won’t) stay off their phones, one Afghani girl who liked (and likes) to cause trouble, a German girl who was seriously pissed off at her dad for bringing her to America (I hear she’s forgiven him), a Mexican boy from my previous year who went from being the weakest student to the second strongest simply because all the others moved on, a Salvadoran girl who was friendly and helpful and hardworking unless she wasn’t, and three Guatemalans who chattered constantly in Spanish, generally refusing to even try to speak English.

Simply getting them to enjoy being a class, to tolerate each other, took a long time, although I’m pleased to say that the Disney/dead animals day was the turnaround I took it for. But as far as actual progress in learning English went, there was none. In fact, I didn’t spend much time teaching them English at all, not directly. I taught content in other areas and got them thinking and talking, which trust me was more than enough of an accomplishment. But even if I’d wanted to teach actual English, I had no curriculum.  No books. All the stuff from last year disappeared from my old room (I was using the previous teacher’s room, until the kids asked if we could just stay in mine).

One significant improvement over the year before, though, was Miko.

Miko was a science teacher with an English credential. But he loathes the new state science curriculum. So he volunteered to be the permanent replacement that I’d been temporarily, and now he’s an English teacher with a science credential. He likes running things, he likes after-school activities, teaching drama, cheerleading, stuff like that. You might have noticed that between the two of us, we could open a school. He could be in charge, even.

So after the 20 year expert got fired, Miko was put in charge, and made changes that I’d advised the year before. He reduced the infuriating “three English classes” requirement, arguing as I had (but more successfully) that just two classes would give students needed time to build credits towards graduation. He was considerably more aggressive about moving students up into second year, ending the absurd practice of forcing highly educated students who read English at a 9th grade to learn “cup”, “stand”, “pencil”, and “sit”.  And, as a guy who likes to be in charge, he took a very hands-on approach to the kids’ status, so I had someone to talk to about behavior problems and frustrations.

I apparently impressed him, too, because he asked me to teach two courses of ELL in the next year (2018-19). How to put this politely? I demurred, saying that I’d be happy to help out if another teacher quits (hahahaha! What are the odds?), but that math was my bag, thanks.

I didn’t see much need for me. The other, brand new, ELL teacher was let go. The ELL specialist (non-teacher) was leaving and we’d hired a new one. The principal decided to use experienced English teachers, non-ELL, to take over the classes rather than try to hire new teachers again.  So Miko would teach first year and Connections, Karinna, who taught AP English, would pick up second year, and Joanne, also an honors teacher, would take third year.  A full-fledged ELD department would be created, with Miko and the newly hired ELL specialist, and Karinna and Joanne in both.

So I left for summer thinking I’d be teaching two precalc classes, or even three, which I’d strongly requested, maybe a trig or algebra 2. Four blocks again, definitely–the 33% premium now for nine semesters running!–and no ELL. Alas, no history either. We’ve been hiring up in that area, so I won’t be teaching US History possibly ever again. Sad.

While I wasn’t crazy about the kids, I still felt the year finished productively, given where I started. At the time, I felt it was a good way to leave the topic.

Year 3

We teachers were notified of our schedule for the year by email, from yet another AVP, the week before school started. My note said:

• Trig
• ELL 1
• Algebra 2
• Algebra 2

WHAT THE HELL! I sent off a cranky–too cranky–note to that AVP and hurt her feelings, which wasn’t my intent and I apologized later, saying I wasn’t blaming her. I was just pissed off, having been reassured that my agitation for more pre-calc had been heard.  Why no pre-calc? Well, because one of the pre-calc classes was second block, and they needed me to teach ELL. OK, we’ll get back to that. Why no pre-calc? Well, because Chuck had though the fourth block class was mostly juniors and seniors and better suited to me than Wing, who got the pre-calc class. I gave her a look, and she thought it a good idea to give that pre-calc class to me.

Now, why was I teaching ELL again?

Well, the newly hired specialist had quit. Not quit, but gone…oh, I don’t know, fishing. I don’t remember the details. Off to another school somewhere. But we needed a specialist. So Miko stepped up. Told you, he likes to run things. He is still teaching the Connections class, and his drama class, but being specialist takes time, so he gets a block off. And so, here I am. Out of the tree, but still in the car.

I told Miko I wanted curriculum and a reasonably homogeneous class. I eventually got curriculum. I’ll discuss the class another day. But for all the frustrations, this year has been much more enjoyable. Miko is in charge, and moves over-qualified students out of my first year class at a gratifying pace. I have a curriculum with textbooks and workbooks, as well as an online program. We have an over-arching framework that allows us to focus in on kids who need a particular skill. For example, second and third year students who needed grammar focus get additional time in small group instruction, while others got practice time listening to long, involved stories and answering questions about it, just like they would in “real school”.

We’ve had several days of professional development which has been reasonably useful. And having a small department that allows us all to discuss the craziness that is ELL policy has been most cheering. I’m part of a group, one that considers me valuable as opposed to a dangerous renegade, which is a pleasant change.

Note well that the school hired two ELL teachers and fired both of them, then fired its 20 year expert, then hired and lost a specialist–all in less than two years.

I say again–hiring, not firing, is the pain point.

Here’s irony: As a math teacher, I am longest-standing ELL teacher at my school. I speak no languages other than English, yet I am the designated entry class for students who speak no English at all.

Apparently, I’m pretty good at it.

 

 

 

 

 

Great Moments in Teaching: From Dead Animals to Disney

ESL this year hasn’t been particularly enjoyable, unlike last year, which troubled me ideologically but was a joy to teach. I am primarily challenged by a hard truth: my students simply aren’t interested in learning English. In fairness, they’ve had a tough year, the details of which I won’t share. When I arrived, they weren’t grateful, but rather annoyed that they had a teacher who expected them to speak English rather than watch movies.

Most are eager to learn, having been out of regular school for a year or more. They’re just not  eager to learn English, and they particularly don’t want to speak English. I’ve been having trouble getting any conversation going; my questions are met with either utter silence or a request, in Spanish, that someone give them a one word answer to get me off their backs.

I can focus on any content, anything that sparks their interest while reading or at least hearing English.  I taught them ratios and fractions. We constructed some robots. They enjoy grammar, primarily because they just like completing worksheets instead of talking.  I showed them Zootopia, a clever little movie, and tied it into “prey” and “predators”, which then expanded into “producers”, “consumers”, and “decomposers”, then into “herbivores”, “carnivores” and “omnivores”. This went over pretty well, so I found an ESL science book and reinforced all that with pictures and text.

I’m a teacher tailor-made for covering a wide range of topics, and I’ve improved their compliance and cooperation. But they are still a sullen lot, with no cohesion and they aren’t that crazy about me, which is a hard ego hit for someone who’s quite used to being “favorite teacher”.

So I needed a day like last Friday.

Notably, Reyes was absent. “Behavior problems” and “ESL students” don’t see a lot of overlap; unhappy ESL students act out by passive inaction, in my experience. But Reyes, a junior from Mexico, became a huge behavior problem once the others started showing even minimal compliance and improvement.  He chases girls around the room. He pulls his hood over his head when he’s trying to ignore me. He constantly speaks Spanish, interrupting me and making crude comments  that cause the other Spanish speakers to giggle.  He refuses to speak English, even simply to ask to go to the bathroom. He’s not a bad kid, really, but nonetheless a disruptive force in the room was gone, and that mattered a lot.

We’d left the day before on “food web” and “food chain” and I brought the image of a spider web up again, intent on explaining in some way  that the original meaning of “web” has transformed, to start to get across the notion of metaphor. Then  I googled “web” without spider and bring up one of the results.

You get this sound, in ESL classes–at least you do in mine. It’s a genuine “Aha” of comprehension and connection. It’s a great sound.

“See? We use ‘web’ to describe the connection because it’s many connections to many other connections. It’s not one way up or down. Now look at ‘chain’” and I googled the word and tabbed to images.

Again with the “aha”.

“See the difference? In a chain, every link is directly connected to only two. See this one? In English, we often use the word ‘chain’ to mean one up and one….”

“Down!” they chorused.

“So when we talk about food web, we are talking about many to many.  See the many connections? All these animals exist in a web, with different relationships. Now look at a food chain. See the clear cycle, or circle?”

So far, so good. Then I lost them: “First, we’re going to focus on food chain, which is a basic way of seeing who is eating, and who is being eaten.”

I was quite surprised to hear a big groan from Allie. “I HATE English!!!”

Taio agreed. “Both eating! Why eaten sometimes, sometimes eat?”

Ah. “So when is it eat? When is it being eaten?”

Allie threw up her hands. “They are both the same thing!”

“No, they’re just the same verb root. But…. Huh. Let me think.”

“See? English is stupid!”

“No, no, I get that! And you’re right. English can be insane. But I’m not teaching you verbs right now. I just want to figure out how to make you see the difference. Oh, wait.”

And I quickly googled up “rabbit eating carrot“.

“The rabbit is eating the carrot. The carrot is being eaten by the rabbit.”

Pause, but I could see they were thinking. So I googled up “fox eating rabbit”.

“The fox is eating the rabbit. The rabbit is being eaten by the fox. So if you are eating, you are the one getting food.”

“If you are eaten, you are the food?”

“Exactly!”

Elian stood up and came to the front by the projector. “Who eats fox?”

“Great question. I don’t know? Who would kill and eat foxes?”

“Birds?” Allie again.

“Hey, that’s an idea.” I google “eagles eating foxes“.

“So then someone eats eagles?” Taio asked.

“Maybe. But some predators aren’t eaten. Like humans. We kill other predators, though, because of competition. So we kill foxes because foxes will eat our chickens and rabbits. Or we kill eagles because we like their feathers.” Elian nodded, and leaned against a desk, still up front.

“Let’s try another chain.” I google “mouse eating“.

“Elian, is the mouse eating or being eaten?”

“Eating!”

“Yes! So Taio, what is happening to the blackberry?”

“The blackberry is…eaten?”

“Allie?”

“The blackberry is eaten by the mouse?”

“You got it! So who eats mice?”

“SNAKES!” I had all seven kids playing along as I google snake eating mouse.

“The snake…” I prompted.

“the snake is eating the mouse!” even my non-English speakers, like Chao, was moving his lips, at least.

“THE MOUSE IS EATEN THE SNAKE!” announced Hooriyah, my lone Afghan student.

“No. Eaten BY,” from Elian.

“Yes. The BY is very important. Otherwise, in English, it sounds like you are saying ‘eating’.”

“That’s why I don’t like English. Eaten and eating sound the same!” Allie nodded.

“So remember the ‘by’. That will help.”

“Do snakes eat deer?” Taio asked.

I can’t begin to explain how pumped I was. We’d now kept steady conversation for close to ten minutes, where everyone was chiming in without prompting. So I googled “snake eating” and we paged down looking.

“THERE!” Taoi pointed.

“I have a question,” Allie announced. “What do you call that word that snakes do to….” she paused. Kept pausing and then shrugged. “I don’t know the word.”

“Crushed? Constricted? Squeezed?”

Allie had come up to join Elian, standing by the Promethean, looking at the images for one specific thing. “No. The other way. Before.”

“Poison? Some snakes bite their prey and the poison kills or at least paralyzes–makes the animal not able to move.”

“No, not that. It’s….” and here Allie gave up  in frustration, looking at me, trying to “think” the word at me.

Up to now, I’ve been doing a good job, but it was all ad hoc teaching, taking what comes.  But I don’t think all teachers grasp the essential moments of their job. This was an essential moment and I made it a great one.¹

Nothing is more important to me in that minute than identifying Allie’s word. Writing this a week later,  I have a vivid memory of standing next to the projector, looking intently at Allie, oblivious to everything else, trying to grab the word out of her brain. And best of all, I could see that she knew this. She knew I was absolutely intent on figuring out her word, that I wanted this, that I wanted to be useful because hell, she’s stuck in this class learning a language she hates, can’t the teacher give her information she actually wants? For once?

My second great moment arrived, but I’m not sure it’s a pedagogical moment or just that of a very good and quick thinker. Because instead of trying to prompt more information from her, I started thinking about snakes. What are the ur-Snake things? I’d gotten constriction, gotten poison, what other snake categories are there?

“Cobra?” Allie stared intently at the google results, but shook her head. “No, it’s…” she paused again, giving up.

“What do you call that?” Elian pointed.

“That’s a hood. Cobras have a really distinctive look. That’s why I thought maybe Allie was thinking of them.”

More ur-Snake. What else? I stare at the cobra images, and suddenly, miraculously, I think of Indian snake charmers.

“HYPNOTIZE!” I practically shouted.

“YES! WITH THE EYES!” Allie was overjoyed. “It makes the animals….something.”

“Obedient. Calm.”

“What’s hypnotize?” Hooriyah.

Third great moment, back to teaching. How to show kids what Allie is thinking of, and the meaning of “hypnotize”? I switch over to youtube.

“This is a famous Disney movie. Has anyone seen it?”

“Yes!” Allie was over the moon with excitement. “This is what I was thinking of!”

So as the scene progressed, I showed the students the broadly caricatured meaning of hypnotize.

When this was over, Allie rested content, sitting back down.

“How do snakes hypnotize?” Taio asked, saving me the trouble of raising the issue.

“I don’t think snakes actually do. I think people just think it is true.”

Allie nodded. “My neighbor has a snake. He says they don’t hypnotize.”

So I googled again, and we found a few highly verbal sites that seemed to deny it, but I didn’t dwell on this much.

Final pretty great moment in teaching: I brought it back to food chains!!

“So. Remember where this all started? Eating and…..”

“Being Eaten!”

“Let’s go through some food chains that you might see in a farm.” I wrote on the board.

corn->mouse->owl

“Owl?” asked Hooriyah, and I googled “owl eating mouse”.

“So now we know three bird predators: owl, hawk, eagle.”

Another food chain: wheat->caterpillar->black bird

“What’s wheat?” Taio again. “I don’t know wheat.”

“Every country has a primary grain. In South America, the big grain is corn. Maize.” Elian nodded. “In China, in most of Asia, it’s rice. In Europe and in America, also the Middle East, wheat is big.”

Allie, who has Brazilian parents but was born in Germany, nodded. “Yes. Bread is made from wheat.”

“And the Germans do amazing bread.”

“Bread!” Suddenly Taio is galvanized. “We have bao bread!”

I know a lot of Chinese food, but this one was new, so I googled.

“Oh, like in pork buns! I didn’t know that.”

“Dumplings. I hate dumplings,” Maria, Salvadoran, my best English speaker, had been missing from most of the class and had just arrived.

“No, this isn’t dumplings.” I corrected her. “Dumplings are like shu mei. It’s food wrapped in a pastry.” Chao sat up and chattered excitedly to Taio, who answered in English.

“Yes, that’s dumpling.”

I grinned at Elian, my only repeating student. “This feels like last year,” and he smiled in recognition. Last year, we’d talked about food in class all the time, going around the room talking about various foods just for fun–what they eat in Afghanistan for breakfast, what they eat in Vietnam for dessert, why Westerners make the best desserts (that was my claim, anyway, although my students roundly disputed this assertion).

We finished up with explanations of caterpillars and cocoons, and discussing the difference between blackbirds and crows–“One is just a black bird, the other is a blackbird.”

The bell rang off for once on an animated conversation.

I started this article a week ago, and was originally going to finish it with the hope that my class had turned the corner. My perpetual lagtime in writing allows me to say that it is better. Last week was a distinct improvement on every day that came before the great moments. More conversation, less lag time, and a much improved sense of camaraderie, even Reyes is speaking with a bit less prompting.

Before last Friday, I’d been telling myself regularly that tough classes are good for me. They keep me humble, keep me looking for answers, for methods, for strategies to help my students want to learn.

Besides, I’d tell myself grimly, tough classes make the triumphs all the sweeter.

I love being right.

*********************************************************************************

¹Again, the great moment is mine. I’m standing there going oh, my god, this is a great moment in teaching, in my life. For me! The kids, hey, if they liked it, that’s good.

 

 

 

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.

“OK.”

“YOU DON’T UNDERSTAND HOW TO DO THIS PROBLEM.”

“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.”

“STOP SAYING THAT.”

“I kno…OK.”

“What’s OK?”

“I understand what you’re saying.”

“No, you don’t.”

“OK.”

“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.”

“OK.”

“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.”

“OK.”

“Really?”

“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?”

Pause.

“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

etc.

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?

“Output!”

“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?”

“Circumference!”

“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.”

Pause.

“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?”

Pause.

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.”

“Grandfather?”

“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?”

Silence.

“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.”

“Marshall?”

“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?”

“Water.”

“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?”

“…around?”

“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?”

“Rio”

“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.”

“Oh.”

“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.

“…to…”

“explore America!”

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

“What is ‘amazed’?” asked Charlotte.

“Julian?’

“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 eat..how 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?”

“Maravilloso?”

“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…”

“..prism…”

“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.

“Yeah.”

“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.

“Exactly.”

“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:

x=3y
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.

Epic.

(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.