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.

Celebrating Trump in a Deep Blue Land

Rick Hess and Checker Finn complain about the schools and  teachers  who are encouraging their students to be fearful and angry at a Trump victory.

I agree, but as long as media outlets are determined to make this about teachers and students, I see two narratives missing. First, somewhere in  Michigan, Ohio, Wisconsin, or Pennsylvania, is a high school in which students are ecstatic over a Trump win. Where they’re saying “Wow, I had lost all hope! The media was so sure he was going to lose!” and consoling the despondent Hillary-voting teacher, “Don’t worry. Trump’s going to be great! This is how my uncle felt when Obama won.”

Also missing are any examples of GOP-voting teachers talking to students about the election, particularly that unicorn Trump-supporting teacher living in a blue state.

Hey. I can help there.  (Note: this this piece gives some additional context to some conversations below.)

Wednesday morning I came into work with maybe two hours sleep and some mild trepidation  layered over the euphoria. It’s one thing to be a cheerful loser while students view you as a curiosity, secretly somewhat impressed that their teacher is a defiant non-comformist. Quite another to be the only Trump voter in the school after Hillary’s catastrophic, wholly unexpected loss.

My ELL class, immigrants all, was buzzing at the results. Charlotte was upset. “Hillary wanted to make life better for us.”

“But so does Trump.”

“No,” Charlotte sighed.

“Yes.  Donald Trump wants to make life better for all Americans, just like Hillary does. ”

“But  Hillary wanted to make it so more people could come in to America.”

“Not everyone wants that,” said Julian. “I think many Americans don’t want that.”

We watched Hillary’s concession speech. CNN reprinted lines on the screen, and I pointed them out, repeating key words.

“Why ‘not lose heart’?” asked Marshall, confused.

“Heart. Passion. Ganas?”

“Ah!”

“She is telling her supporters to not give up. To keep in their hearts their ideas, to continue to working for their beliefs. That’s what you all should do. That’s what America offers, right?”

I was walking from my ELL classroom to my regular class when I ran into Chuy, whose support had remained steadfast despite his activist girlfriend.

“I TOLD YOU!”

Chuy had, in fact, told me Tuesday morning he was sure Trump would win. I had smiled, told him I hoped he was right, secretly thinking I’d be pleased if he kept it close. “I doubted. You called it.” We bumped fists. “I’ll stop by later,” he promised.

In my brief advisory class,  Sasha the drama queen, who the gods have granted me as a student three times, flounced in with a pout.

“I can’t BELIEVE you voted for Trump!” she announced.

“Hey, I’m a Republican. It’s kind of what we do—you know, vote for Republicans.”

“But Trump is EVIL!”

“You were fine with me voting for him yesterday, when you thought he’d lose.”

I suddenly noticed another student, Marjorie, who just saw me in this once-a-week class, realize the import of our conversation–realize that I’d  voted for Trump. What I remember most is the purity of her shock. Maybe later she’d be disgusted or angry, but for now the dominant factor in her reaction was that never once had Marjorie considered, for a single moment, that she might know a otherwise totally normal person–a teacher, no less–who voted for the orange demon.

Devon said, “Remember the first advisory day? You told us that Trump would probably lose, but that it was weird how close it was.”

“Yeah, I told my dad you said that,” said Jesus. “My dad said Hillary was a bad candidate.”

“I don’t think candidate quality matters these days,” I said. ” We only get two choices. Hillary couldn’t openly appeal to Trump voters without risking the loss of media approval. At the same time, she couldn’t do more to appeal to win enthusiasm for young voters by making promises that would lead to criticism.”

“Yeah, but Trump didn’t care about making everyone happy. I guess that’s the difference.”

I laughed, genuinely surprised. “Yes. That’s right. He didn’t care.  OK, you’re right. She was a terrible candidate.”

I ran into Abdul in between second and third block.

“Hey! You still American?”

“God, don’t depress me! But at least I’m a citizen. We’ve been ragging on Omar, nyah, nyah, Trump’s going to deport you!”

You know how they say smiles fade? Mine was wiped clean. “Hey. That’s not even funny. Is Omar worried about that?”

“Well, you know what Trump says about Syrian refugees.”

“Yes, but that’s about reducing future refugees. Omar’s here now. Look, is he worried? Is his family worried?”

“Naw, we’re just giving him sh**. They won’t kick anyone out, right? If they’re legal?”

“Right. Tell Omar to stop by, ok?”

All those stories about students  teasing immigrants–did anyone ask if in some instances, the kids teasing were also immigrants, razzing their friends goodnaturedly? Oh, don’t be silly, Ed. The media wouldn’t distort a story like that. (Omar did stop by, to ask me if I’d write him a letter of recommendation and edit his application essay about the pressures his parents were forced to put on him and how he’d developed a tremendous facility for languages. He seemed fine.)

Many snickers in my algebra 2 class as I explained how to test regions for  systems of inequalities , which took me a while to figure out.

“…so you test. Is (0,0)  on the true side of the border or the false side?”

“Ask Trump,” Eddie snarked.

“Yeah, ask Trump whether I’m born on the right side of the border,” said Elian, more seriously.

“I’m sorry if you’re worried, Elian. And to anyone else worried. But I think things will turn out well. Now, let’s focus.”

“I can’t focus. Trump’s gonna kick me out of the country.”

“You don’t focus, Eddie, I’m gonna kick you out of the classroom.”

“See, already you’re marginalizing me!” Eddie does deadpan hysteria very well.

“It’s true. I’m marginalizing Eddie’s fears that he’s faking because he’s a citizen. SAD! ” Eddie grinned.

After I’d released them to work, Mark ambled up. “So what do you think Trump will do?”

“I hope he appoints a good judge, and rolls back some of the executive orders. Past that, I don’t know.”

Peter came up to me quietly. “You voted for Trump, right?”

“Yep.”

“I think the anti-Trump demonstrations are….idiotic. Totally insane.” I nodded.

“Oh, hell yeah,” said Mark. ” I didn’t want him to win either, but it can’t be that bad. Those people are crazy, wasting time, whiners.”

In pre-calc, class began with an announcement reminding students that a walkout would result in a zero grade.

“Total waste of time,” Antonio said.

“It’s so depressing,” sighed Janelle. “We could have had a female president!”

“Not for me,” said Teng. “I won $500 betting on Trump!”

I commiserated. “I only ever voted for one other president who won.”

“Bush?”

“No. Hillary’s husband.” Pause as they absorbed this.

“And cheer up. You’ve lived through an amazing moment in history. Every powerful institution in this country wanted Trump to lose. The leaders of academia, almost every owner of a media publication, television or print, our political leaders. Business largely rejected Trump. Even most Republicans in the media rejected him. Most politicians kept their distance. He had few advisers. Trump’s supporters were insulted and mocked–or at best presented as….”

“Total losers living in little white towns with meth addicts and hillbillies” finished Morgan.

“Exactly.  Last night I was tired.  I hadn’t voted yet. Trump was going to lose my state anyway. Everyone was saying the exit polls were showing a huge Clinton win. So why bother voting? It wouldn’t make a difference. But I literally…I mean this,  I literally thought ‘I want my vote to count.’  So I went and voted.”

Leah, always imaginative, spoke up. “It’s like….all the other Trump people did that too. Instead of staying home.”

“So Hillary voters didn’t care as much,” Kenny said.

“Trump convinced voters to care. He screwed up in a million ways, he was rude and obnoxious and you can’t really take anything he says literally,  but he never backed down when all the cool people on TV, in the movies, in the media, in the universities were laughing at him, mocking him. It made him angry, he often responded in infantile ways, but their hatred never stopped him from understanding what his voters cared about. He went everywhere and asked for votes.”

“People are treating it like an earthquake. ” argued Inez.

“Yeah, but an earthquake is a natural event. A powerful one. We are living through an epic moment, where ordinary people created an earthquake, defied the will of the media and most of our leaders, simply by showing up and voting. What you should take away most of all from this is that earthquakes are possible in politics. You’ve now seen one.”

“I’ve spent a lot more time than you guys have, feeling sad my candidate lost. You focus on the good where you find it. So Hillary lost. Feel sad about that. But feel good that elections aren’t all about turnout and commercials and interest groups. Sometimes, every so often, an election turns on ideas. No one in the media, in academia, or our businesses really liked Trump’s ideas. They tried to  shut them down. And they failed. Because people came out and voted for Trump’s ideas.”

“So sometimes ideas really do win.” mused Teng.

“Yeah,” Adriana agreed. “It’s really…epic.”

“Epic?” snorted Gita. “Hillary won more votes! Trump’s a racist!”

“Yeah, well, no one said epic was perfect, yknow?  So let’s look at inverse functions and turn our thoughts away from epic wins.”

The Thursday after the election, I was standing in front of my trig class, explaining angular velocity, when I suddenly stopped and said “There he is.” Hustled across the room to the left door, opened it, and hollered.

“DWAYNE!!”

The beefy senior had just strolled past, and turned. “What? I’m not out of class, I have a pass.”

“That’s not the point. It’s TWO DAYS and you don’t stop by to celebrate? I’m pissed.”

He grinned, came back towards me. “My mom called me in sick yesterday because I stayed up all night watching returns! Can you believe it?”

“I really can’t.”

Last weekend, I was in a different, equally blue, state for my grandson’s first birthday party.   A successful salesman in a roofing and windows company, my son has only Trump co-workers and only Clinton friends and family (save me). A colleague showed up in his MAGA cap, and  my son steered him over to me for safety and celebration.

“I’m a gambler, you know? And when Florida’s returns were nearly done , when you could see Michigan and Wisconsin ahead, North Carolina won, it was like a $100,000 hit on 20:1 odds. That’s how good it was.”

Yes. That’s how good it was.

 

This Great Election

This is the first election day since 1992 that I’ve really enjoyed. 1992’s election was exhilarating and in many ways a set up for this one. Bill Clinton back then gave a master class in how far a politician could go if he lacked shame and had a message the voters cared about. In 2000, I thought Gore ran a poor campaign over the summer, and the recount was a little too much evidence that our court system is just a reinforcement of our political system. I was just pleased it was close.

2008 radicalized me. I didn’t mind Hillary much back then (she was against driver’s licenses for illegal aliens, remember that quaint old restriction?), and the media’s anvil on the scale for Obama in both the primaries and the general was just nauseating.

I quit watching or reading about politics from late October 2008 to the Obamacare fights of 2009. And when I came back to it, I stopped trusting any media. Going on Twitter in 2012 further reinforced my understanding that even the ones who write in a seemingly neutral and unbiased style are, in fact, predictably liberal with tremendous disdain for half the electorate. For a news junkie living squarely in the mainstream, this comes as an unhappy shock.  (This time around, Sean Trende and Jack Shafer, two of my favorites, have been the most disappointing re the disconnect between the bias in their tweets and their carefully cleaned up columns, Josh Kraushaaer the one I still have illusions about so dammit Josh, don’t screw it up. Michael Goodwin, Mickey Kaus, and Byron York have, in their various ways, been solid gold treasures.)

Anyway. One thing I did learn from 2008 was that outside of progressives, white voters aren’t very interested in the presidential election issues. It’s been clear to me for a while that the public, particularly the GOP base, was not getting the candidates or the issues they wanted. Two elections in a row, I thought it likely that white voters were staying home, not bothering. Two elections in a row, I thought that the GOP was ignoring its voters in favor of ideas that no one really wanted–from immigration to education to social issues to entitlements. (I never thought of trade, sorry.)

Then came the 2012 autopsy, in which the GOP said hey, we need outreach to Hispanics in order to win back the presidency. Not to blacks. Noooo, the much-vaunted Party of Lincoln didn’t even think of blacks, didn’t think to find the common ground between their base of working class whites and the many blacks (and non-immigrant Hispanics). No notion of using immigration restriction as a uniter. Nope. Their money men wanted cheap labor, and they all figured that the 2012 loss could be used as rationale to argue against the base’s desire for restriction.  “See, we’d love to end H1B visas and implement e-verify, but we gotta do outreach!”  Because that’s how you grow the economy, with lots of businesses making money off of cheap labor. Good for the stock market. Meanwhile, of course, the GOP wanted to double down on blaming schools for failing to educate kids–that’s why they need immigrant labor, because teachers suck!

So I wasn’t excited about 2016, what with all the talk about another Bush, hints of returning to the autopsy plan, even after Rubio got his ears pinned back.

And then came Trump, down that damn escalator.

He never had to win to make me happy.  I wanted the message out there.  I wanted another politician to defy conventional wisdom, to refuse to step down or apologize, to insist that the people be given their choice. I wanted someone to show the popularity of issues the media and elites considered completely unthinkable, to force them into the debate. The Overton window has shifted feet–yards, even–back in the direction of sanity.

But GOP elites are trying to bargain their way out of reality. They  think fondly of a world where Rubio–the GOP’s version of bland, teleprompter-ready Obama–could have won if Kasich and Christie had dropped out because golly, he gave a good speech. Or Cruz–whose voice is so awful I change the channel when he shows up–could somehow win over enough swing voters.  Or they blame the media for giving Trump air time, forgetting that the airtime was devoted to blasting Trump for insensitivity, for “racism”, and demanding the public share their opinion. Instead he won more votes every time he refused to back down.

If you want to rebuild the GOP, start by asking a Trump voter what the key moment in his success was. Most will point to his refusal to apologize for his June 16 announcement. NBC dumped him. Univision fired him. And he didn’t back down. He didn’t play the game. He didn’t apologize, mend fences with the media. That was……well, huuuuge in the world of Trump’s base.  He snarled back, and got more popular.

What we’ve needed in America is someone willing to defy the media and the elite. Someone who had the money and message to succeed despite blasted disapproval. This forced the media and the GOP leadership to realize that all of their power relied on their ability to shut off the microphone. Take that ability away, they got nothing.

I don’t lionize Trump. I think he tried for years to win approval from the same elites who despise him now. I’m glad he chose to run. I’m glad he showed them, through the people, how wrong they were.

Because unless the polls are dramatically wrong in Clinton’s favor, Trump is not going to get destroyed. If he loses, it will be be a margin less than McCain, possibly less than Romney. With few ads and even fewer experts to advise him–the experts being the one class who still needs elite approval.

All he had was a message.

Next steps: win or lose, Trump voters need to see that class, not race, is the way to grow their ranks. This Sheryl Stolberg story on the decimated black working class that see no hope from Hillary but hate Trump–they’re the first step. I believe that African Americans can be convinced that our immigration policies are incredibly harmful to their interests: in jobs, in education, in reducing their political viability. Working class Hispanics, those of long-standing in this country, are also a great opportunity for actual outreach.

I’m not sure where it goes from here, because very few Republicans in media or leadership have any interest in rebuilding. Most of them believe that surgical removal of Trump voters is not only necessary, but simple. Laugh at them.

It’s all the meme these days for the media to talk about how horrible this election has been, how dispiriting it’s been to true believers in democracy and American greatness. That, again, is one reason why we all hate the media and elites, for failing to realize how exciting many of us are by the opportunity to vote our issues.

To all of you out there in Trumpland, I hope you share my sense of joy in this campaign. Watching everyone in power realize they had no power to stop Trump and his message.

If our side loses, it wasn’t because the media won the narrative. Entire publications were dedicated to convincing the public of Trump’s evil nature. They failed. They weren’t able to frame this election, because in their framing, Trump is unthinkable, a fascist racist misongynist who’ll start nuclear wars. But “unthinkable” doesn’t include close to half the country’s support.

If we lose, we’ll lose because we don’t yet have enough votes. Trump’s important qualities are alienating. I believe they were also essential. There was no moderating, no winning approval, that wouldn’t likewise end his ability to sell his message. And the conservative wing of the party has had it their way for so long that they can’t conceive of voting for a candidate they aren’t crazy about. That, too, was a non-negotiable constraint.

But moving forward, I believe this can be fixed. I believe the media  and the GOP will find it impossible to shut down these issues. I believe we’ll get more compelling candidates. I believe we’ll find a way to win more support.

If not, well, at least we had the chance to try.  That’s more chance than I ever expected.

Go Trump!