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.

Citizens, Not Americans

I’ve been pro-Trump from the beginning, a supporter who thinks his rhetoric essential to facing down the unending opposition from the media and the political establishment. He may lose; I don’t make predictions. I’m unflustered by the establishment hysteria and even now, in the face of all the unrelenting Trump condemnation, see little to fuss about.

Trump did make me flinch once, when he said that Judge Curiel was Spanish, or Mexican—fundamentally Not American.

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At a recent department meeting, Benny said: “Look, Honors Algebra 2/Trig has…what, five Americans? Honors Precalc has just four and then Walter and Victor.”

Wing nodded. “Most of the Americans take regular Calculus, and a few Hispanic kids.”

I sighed. “Yo, China boys, do you think you could remember that ‘American’ doesn’t mean ‘white’?”

Benny’s ABC…that’s a weird thing, isn’t it? American Born Chinese. Not Chinese American. Wing’s just plain old Chinese, with either a green card or citizenship, I’m not sure which. Walter is black. Victor is Hispanic. Both are American.

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Immigration romantics usually live in all white enclaves, because white regions don’t have any immigrants.

Immigrants can be whites, of course, and don’t kid yourself into thinking their skin color makes them more popular. Those of us in high immigration areas rank Russians and Eastern Europeans well below Asians as desirable neighbors and I, at least, would pick Hispanics in a heartbeat over anyone from east of Berlin who showed up after the Wall came down. The corruption levels are freakishly high, and they’re often nasty neighbors. Irish immigrants are more popular–cute accents! plus, Western culture.

White immigrants don’t cluster in white American enclaves, though, but in already diverse urban areas: New York, San Francisco, Los Angeles, Boston.

It’s much easier to get all misty-eyed about the immigrant dream if you aren’t experienced enough to categorize them by ethnicity.

Paul Ryan represents a district that’s 91% white, ahead of the state’s 86% white average. Mitch McConnell’s Kentucky is likewise 86% white. Both Wisconsin and Kentucky’s second largest population is African Americans, coming in at around 6%. Naturally, these white men living in white enclaves feel entitled to judge those who don’t live in genteel segregation for questioning the onslaught of diversity imposed on them by all three branches of federal government.

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In addition to summer school, I’m teaching test prep all day Saturday, with populations very like the enrichment classes I taught until last year: 99% 1st or 2nd generation Asian.

“I need a Saturday off over the 4th of July. Is there anyone here who absolutely can’t meet on a Sunday to make up that session?” The kids all seemed fine with it.

“Besides, look at the bright side–you’ll get the whole fourth of July weekend off!”

Yun snorted. “I’m not American. It’s not my holiday.”

I stopped cold. Just looked at him. And waited.

“It’s not me, it’s my parents. They weren’t born here; they don’t care about that.”

“Were you born here?”

“Sure.”

I looked at the class.

“How many of you were born here?”

Most of the hands went up.

“How many of you know what the Fourth of July is, much less celebrate it?”

All of the hands went down.

“Yeah. You all SUCK. Do it different this year.”

They all looked abashed.

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So yeah, I twitched a bit when Trump declared that the judge was Spanish. I would have preferred that Trump simply question the judge’s objectivity, as Byron York outlined.

Is he American? I’d say yes. He was born here. But I know countless children of immigrants who laugh at the idea they might be American just because they were born here. Just one more way in which misty-eyed white people in all-white enclaves are allowed delusions that the rest of us–white or just American–are forced to abandon.

Trump on CNN: "the last thing we need to do is take in people like this" except shooter was born here.

— Zeke Miller (@ZekeJMiller) June 13, 2016

It’s always “he’s born here” or “is a US citizen?”

Jake Tapper, to Trump, “He was born in Indiana.”

Rarely “He’s American”.

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Last fall, Abdul said “Can you believe Trump wants to kick out all Muslims?”

“Pretty sure he wants to ban Muslim immigration.”

“Same thing.”

“Well, no. Even assuming the ban happened, it wouldn’t be for citizens and you’re a citizen, right?”

Abdul spat. “I’m not a citizen of this country. Not if they nominate Trump.”

“Oh, that’s such crap.”

“I’m Palestinian. I can go there after I get my degree.”

“Yeah, because Palestine is just a paradise of tolerance and religious freedom.”

Abdul was shocked at my, er, lack of support for his pain. “You think I should just accept people here hating Muslims and electing Trump?”

“Jesus, Abdul. You want to oppose Trump? Start a voter registration drive. Put a sign in your yard. Go door to door. But oppose him as an American.”

“But why would I want to be an American if Republicans hate Muslims? “

“Republicans don’t hate Muslims. Trump doesn’t even hate Muslims. And America didn’t demand you lived up to any expectations, didn’t make any demands of you to give you citizenship.”

“Wow, go ED!” shouted Al, who demands we pledge every day even if there’s no announcements because “otherwise the Commies win!” “And go TRUMP!”

“Shush. Look, Abdul, you should oppose Trump. You’ll have plenty of company. But you are a shining example of what Islam can mean in America–you work hard, you challenge yourself, you’ve achieved tremendously. But you reject the country that gave your parents a home–well, no, not rejecting it, but making it conditional.”

“It’s conditional on people accepting my religion!”

“They do, but never mind that. If you reject the country of your birth in favor of Islam, you who have done so much and so well, isn’t it logical for Americans–actual Americans, those who don’t set conditions on their country–to wonder if Muslims are right for this country? Shouldn’t we wonder if they’ll be loyal, if they’ll appreciate what the country has to offer? If you make your acceptance conditional, how can you blame the America you want to reject for doing the same?”

Abdul mulled, shifting his shoulders back and forth. “That’s a bunch of good points.”

“Well, we shouldn’t be talking about politics in class. Back to trig.”

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You will find Muslims across the US in business, in medicine, and fighting and dying for their country in military. https://t.co/WSMk5NgqcU

— Joe Scarborough (@JoeNBC) June 15, 2016

Joe Scarborough has been carrying buckets from the “Muslims are productive citizens” well on a regular basis ever since he took Ted Cruz to task last March. But if Joe actually met a Muslim immigrant now and again–which he’s unlikely to do in New Canaan, population 95% white–he’d realize the reality doesn’t quite live up to the ideal. Abdul’s one of many Muslims I know who doesn’t think of himself as American, and Abdul’s a minor glitch compared to the reality of intense Muslim immigration. Just look at Hamtramck filled with recent Muslim immigrants.

Look close at Hamtramck, Joe. That’s not Muslims choosing to be Americans first. That’s Muslims imposing religious tyranny through numbers, not granting Americans what they demand for themselves. And they’ve created a place that no American, a word I use advisedly, would willingly choose to live.

I spent a year recently in a town over half Asian, the vast majority recent immigrants, and whole pockets of the area are….unappealing, because of mores and cultures that simply aren’t anything Westerners find acceptable. Nothing you’d find compelling or convincing, unless you had to live with it. I moved, for reasons not involving my discomfort, to a town that’s 60-40 white/Hispanic, and am much happier. Victor Davis Hanson warns of what happens when illegal, high poverty Hispanics hit critical mass, and that’s not pretty either. I’ve heard similar tales of Armenians and Russian enclaves.

Heavy immigration of any ethnicity and de facto, willing segregation by ethnicity does not lead to immigrants thinking of themselves as Americans, but rather immigrants imposing their ethos (or lack of same, as we see it) on America.

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Dwayne, the closest thing you find to a good ol’ country boy in this area, gave me a note:

I just want to tell you I’m sorry I’m such a jerk who talks too much. I’m just really stupid at math and I hate school. Please give me a C. I’ll paint your car any time you want. Free. And you’re a really cool teacher. Go Trump!

Dwayne and his buddy Paul live and breathe cars. Dwayne likes body work, Paul does engines, and both are highly regarded by the mechanics at our vocational training program.

A couple months ago I asked my mechanic if he was interested in training high school graduates with experience. Hell yes, came the answer, we always need mechanics. I gave Dwayne and Paul the address, they dressed in nice shirts and (on my advice) made single page resumes for their visit. They returned impressed but nervous (“There were FIVE PORSCHES in the garage!! There’s no way he’d want us!”) but said it went well.

My mechanic concurred. “Good kids. You know what’s really nice, although I didn’t say this: they’re white kids. These days, the only young men showing interest in mechanics are Hispanics. That’s fine, don’t get me wrong, but can it be that no white kids want to be mechanics? When did that happen?.”

“Maybe when they needed Spanish to speak to their co-workers?” I suggested. He laughed, but not in a happy way.

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—Christopher Caldwell, The Migrants of Calais

Leave it at that.

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Last February, third block algebra 2: “Hey, Ed, I heard you’re a Republican. You voting Trump?”

Chuy broke in, “Yeah! Go Trump!” and beamed at me approvingly.

Daniel was shocked. (Truth be told, I almost fell out of my chair.) “What? How can you like Trump? He wants to deport Mexicans.”

“So what? I ain’t Mexican.”

“Well, neither am I, but…”

“Then what do you care? They can go back home. Trump’s strong. This country needs someone strong and tough. I like him.”

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“I ain’t Mexican.”

The Chuys in my world are rare. Maybe it’s just 40s movie mythology that I’m thinking of, that there was ever a time when immigrants felt American in their souls, in their hearts, and were overjoyed when they finally became citizens. If such a time ever existed, it’s gone. Today, citizenship is taken for granted but it’s just a technicality, a legal state that gets you low tuition, benefits, shorter lines at the airports. Being American, holding your country in your heart, doesn’t seem to be part of the equation any more.

Why is Chuy so ready to stand for America as his country? I don’t know what made this second generation citizen “American”. So many others treat their citizenship as a business proposition, like the American-born Chinese and Koreans returning to their parents’ homelands, where they are welcome “home” as part of a diaspora, regardless of the trivia involving birthplace. Others, like Abdul, treat their citizenship as a choice: which will be best for my culture?

We must learn how to demand that immigrants think of America not just in economic terms. We must inculcate the understanding that their children aren’t just citizens of convenience, but Americans. Until then, Hillary Clinton is wrong in claiming that neither Trump’s wall nor a Muslim ban would have prevented the Orlando massacre, San Bernardino, Fort Hood, the Boston bombings.

We imported Nidal Hasan, the Tsarnaev brothers, Syed Farook, and Omar Mateen. Not directly, no. We imported their parents, who came here utterly indifferent to the American way. They passed their culture to the next generation intact, creating citizens, but not Americans. Their children, raised as cultural Muslims, found the totalitarian branch of radical Islam appealing.

But radical Islam is just our current threat, the present exposure. Our lax immigration policies, our own indifference to creating Americans, our unthinking donation of citizenship, create the conditions for any immigrant population to turn against the country.

America should not assume that its citizens are Americans. Each year millions of children are raised to see themselves as citizens of convenience. I see the results. Individually, I couldn’t point to any attribute, any character flaw that is companion to this mindset. But I wouldn’t consider it the sign of a healthy cultural polity.

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Chuy’s open embrace of the GOP candidate withstood the outrage of his girlfriend, a citizen who thinks of herself as Mexican. Employed boyfriends aren’t low hanging fruit, and Chuy still has both his political preference and the girlfriend. Last day of school, Chuy stuck his head in and said “Yo, I hope I have you for Trig. Trump!”

Abdul stopped by all year, asking for advice or just to chat. And so once, in mid-May:

“Man, Trump’s killing it. You happy?”

“Yeah. You American?”

He laughed. “I think about what you said. I really do.”

That’ll have to count as a win. For now.

As for my test prep class, I’m assigning 1776 for homework over the 4th.

¹Note: most of these stories happened verbatim, in a couple cases I collapsed or combined events. Nothing that changed import.