What I Learned: Years 4-7

I was going to continue my year by  year  (by year) retrospective, but I decided that the last four years can be considered as a block. Which is good, because if I did a post per year I’d never catch up.

tl;dr Years 4-7 were all about happiness.

I began at this school in late August of Year 4, just a week or so before school began.  Utterly desperate for work, I would have accepted any offer, anywhere in a 35 mile radius and really, by late August, 50 mile drives weren’t out of the question.  That I got my first choice, a school that had actually offered for me the year before, seemed almost a miracle.

And really, I’ve been happy ever since. Teaching has always been a joy. The previous schools, with all the challenges, never dented my belief in my own abilities or the faith that kids were benefiting from my teaching. But at the other schools, the administrators didn’t agree. It’s not that they thought I was a bad teacher. I just wasn’t what they wanted–someone younger, ideally.  Here, for the first time since student teaching, my bosses also thought I was a darn good teacher. May this bliss last at least eight years more.

So in those four years, what changes and accomplishments can I point to?

Teaching Persona

As mentioned in Year 3 retrospective,  I’d begun to establish the same ambiance in my school classrooms that existed in my test prep and enrichment instruction classes. At this school, the process was complete and I never looked back. From day one, I was unpredictable, flexible, friendly, ruthlessly sarcastic, and damn funny, which is where I live the rest of the time. The second year there, I introduced a meme: I am the star of my classroom.  I get a guaranteed audience three or four times a day. It is in my contract. Students are the audience. Their job is to attend. If they’re lucky, they might get some lines. A walk on part. But mostly students are to watch. To listen. Eh…learning would be nice, but that’s up to them.

Someone somewhere is going to take this as a serious statement of priorities, rather than a mindset. Remember that I spend very little upfront time teaching. It’s more of an attitude. It allows me to be big, overblown, demanding attention, dammit, whether you learn or not. Students enjoy the spin on the usual pay attention because education is good for you. Hell with that, kids, your attention is good for me.

I count it as a good sign that I’m regularly in the Teacher Awards section of the yearbook, and for fun things: Storyteller, Unpredictable, Mostly Likely to Lose Whiteboard Eraser. May that, too, extend through the next eight years. I’m a geek; popularity is a nice change.

Building Curriculum–Never Be Satisfied

I vividly remember in the spring of year 4, my first year at this school, when I was looking ahead to linear inequalities. I was just about done with my new method of modeling linear equations which had now gone well twice in a row (remember, this school does a year in a semester and then repeats).

But at the time, I did little more than go through the procedures on linear inequalities, and felt a twinge of shame the first semester, as we moved from a unified modeling approach to…here’s how you test a value.  And suddenly, out of the blue, I remember my ed school professor saying “You should never be satisfied. You can always do better.”

At the time, I rolled my eyes. She was saying this in the context of our first year of teaching, to never feel satisfied. I think this is absurd. “Good enough” is fine a lot of the time. But at this moment, I realized it could apply to an entire career (and in fairness to the professor, that’s probably what she meant.)

So I challenged myself in that moment to come up with something different. How could I introduce  linear inequalities in such a way that would build on the linear equations, while showing their differences? I still use the methods I built that day, although I’ve developed them somewhat.

But from that point on, I always take that moment, that wince away from a piece of curriculum I don’t like. What can I rebuild? How can I make it better? I’m not a perfectionist, not a driving careerist, definitely not hard-charging in approach. (my affect and opinions, whole different deal.) Just one of many ways in which my pricey ed school degree has transformed me well after the fact.

I’ve written about many other curriculum improvements over time. All of these were done with that same spirit of yeah, the old way wasn’t working, let’s try this:

I’ve completely reworked quadratics and exponents as well, but haven’t written them up. It’s been fun.

I don’t have one approach to curriculum, but if I have a go-to process, it’s the “illustrating activity” or problem, which can be seen here in the Projectile Motion writeup, or this lesson on proving the pythagorean theorem and geometric mean activity. I began to write it up as part of this entry, but decided no, do a separate post. (I do apologize for my scarce blogging lately.)

Classroom Ambiance


This is the first time I had my students “work in the round”, which is how you’ll find my class at least 12-15 days a month since then. My current classroom has whiteboards all the way around. The walls have a 5 x wall-length strip of white board paint (which is really cool). I have small white boards with coordinate planes etched in. I also have a wonderful donor who sends me $100 worth of whiteboard pens every year, so the kids can always be working on a big surface, with plenty of room for mistakes.

There must be whiteboards.  Working constantly in class, moving around, reduces the risk of math zombies. Earphones are allowed to shut out the noise, provided I don’t see the student enjoying the music more than the work.

I sit my kids in groups, which has been true since my first year. I don’t do homework, which has been true for two years, but I never counted failure to do homework.

There must also be movies. Twice a semester in the fall, because Christmas means “It’s a Wonderful Life”. Always at the end of the semester.


The December of Year 5, I got a look at the new Common Core tests and laughed, again, at the ludicrous notion that the new mean achievement would be centered around these ridiculously difficult standards. Except…..

I noticed that many questions required more than one answer. They weren’t simple multiple choice questions. “Identify all the solutions.” “Select all the equivalent expressions.”

Hey, now. That’s interesting. And my first multiple assessment test was born a week later.

Thanks, Common Core! sez I. My kids, not so much.The tests are really helpful for lower ability kids to show what they know, but the strongest kids have to be on their game to do well.

Here’s my first post on the topic, but I’ve revisited it often. They allow flexibility way beyond the usual multiple choice–I can mix and match between freeform and formatted response, or include both in the same question. I can create one question with varied procedural tasks, or one question that dives deep into one situation. They also allow me to greater access to student thinking.

I’ve also had fun redoing my quizzes in the past year. Typically, my quizzes have been straightforward affairs that contain no surprises. But I’ve started to mix it up. helicopterquest

These are nice stylistic changes, even if the underlying question is still straightforward.

Meta Teaching
While I mentioned that my third year saw fundamental .changes in my approach to teaching, I completely forgot to mention that the year also gave  birth to this blog.  While I began blogging at my last school, all but eight months of it have been here. The blog itself is a constant insight into my teaching practice–among other things. I’ve kept it primarily on education, whether it be policy, practice, law, or the reality, which often violates all the others. And occasionally Trump, of course. But then, a good chunk of my Trump support is also related to teaching.

I was not terribly popular with the head of my ed school, but on at least two occasions, she mentioned my gift for writing about classroom experiences. My second year out, I was telling one of my ed school professors about my administration woes, and he told me that he wanted me to keep teaching so I could write about it.

“You’re a very good new teacher. But writing about teaching will be your unique contribution to the field.” I was immensely complimented, and said so–wondering how I could possibly get someone to ever be interested in publishing my thoughts, and how I could get my thoughts down to 750 word chunks.

Turned out I could do first part myself, making the second (ahem) unnecessary.

I try to avoid doing too much with clubs or other school activities that involve stipends.Mentoring credentialed and student teachers1on the other hand, fits in well with my temperament. I’ve spent most of my life being paid for opinions. Consulting new teachers carries on that piece of my past. I’ll be doing induction this year, and hope to find another student teacher soon.

And so, I move onto years 8 and beyond. Looking forward to it.


1My student teacher got at least two job offers from the district; I’m assuming he took one of them but haven’t talked to

Writing a Tweet Storm Chain

Here I offer a practice that will bring all of us Tweeters together as one. Discovery zealots or zombie-denying traditionalists. Content knowledge worshippers or skeptics. Math, English, or history teachers–or those of you who, you know, do that other topic. Immigration restrictionists or citizens, not Americans.

Twitter will be a happier place if its users learn that tweet storm are not as effective as tweet chains.

Others have come before me, but they were writing for early adopters, the ten percenters. I wasn’t an early adopter.

Like other innovators of the obvious, I began with a question: How the hell can I write more than one tweet without forcing everyone to read backwards? And irritants: numbering my multiple thoughts. I could never remember what number I was on–or worse, not bothering to number at all. Yes, I know there’s an app somewhere, but since I didn’t like making everyone reading backwards, I didn’t want that solution anyway.

Lately, I’ve seen the nested retweet, as used here by Megan McArdle and Mickey Kaus, but while this approach does link the tweets, they are still presented in backwards order and also gives readers the feeling they’re spiraling in an endless loop. I recommend against.

At some point I noticed elegant chains of comments, such as these put together by Ed Asante and Spotted Toad and wondered hey, how can I get in on that?

And the answer is: Reply.

Just hit reply on your own tweet. Remove the moniker. Twitter still treats the tweet (try saying that three times fast!) as a reply, and chains it to the original, which also appears as a reference point to show that the new tweet is part of an ongoing series. Better yet, click on any tweet in the series, and they all appear, in order, going back to the first.

Bam. No need to number, no need to use some sort of tweet deck to organize. It’s all kept track for you. Twitter isn’t the easiest interface, and certainly not designed for archival, but if you want to dig up an old series, you can just “Reply” to the last tweet and it chains perfectly. Then, to draw attention to the whole series, use retweet.

Examples: Here’s one of my earliest tweet chains, just to show how late to the game I was.

David Frum, who I linked in an image of above to show how NOT to do it, at very nearly the same time used the more elegant chaining method, so I’m not sure why he’s still stuck in the old ways (perhaps it’s an app).

If people reply to a tweet chain with another tweet chain, you get a nice elegant conversation, like this one between me and Billare, on whether the canonization of the Khans and their appeal to emotion is unseemly. And here’s Dan Meyer not chaining, but showing how to reply to tweets in the chain fashion, so you can easily follow the conversation.

I usually stay out of technology issues. In my former techie life, I was unusual in resolutely avoiding power user tricks. I value flexibility over speed, and since I was always entering new environments with new rules, I wanted to get functional as quickly as possible, not whine about how this new program or operating system wasn’t as cool or powerful as my way better one.

But tweet chains have really enhanced my use of the platform. Furthermore, I’ve now twice written essays after organizing my initial response on Twitter–and given how hard it is for me to start pieces, that’s no small thing.

In any event, I needed to prove again I can keep a piece under 1000 words if I try, and wanted another July piece to keep my count to three. Hopefully, another one comes tomorrow.

So if you see someone laboriously numbering their tweet deck or retweeting a chain, send this along.

Note: It occurred to me that while this is well under 1000 words, the advice itself is about 50. Only I could use 500 words when 50 would do. So here’s an image to pass on: twitterchaininstructions

Happy Saturday.

End of Education Reform?

Four years ago, I first described the parallels between cops and teachers. A year after the election, I wrote about unions and asked, again, why the GOP was so intent on attacking teacher protections when cops and other government workers get the same advantages. I mean, even the bitching about gender imbalance is ridiculous, since law ennforcement is far more male than teaching is female.

Then came Ferguson and the start of a bizarre microtrend. Conservatives began this absurd habit of blaming teachers and crappy schools for black kids getting shot by white police officers and ensuing riots. “Choice would end this chaos!” they’d thunder. I’m paraphrasing, but as the sources  show, I’m not exaggerating.

So I’ve been writing about the parallels* between these two jobs since the early days of this blog. But I also—rather presciently, I must say—observed that “acceptable targets change over time” and that maybe we teachers should hunker down and wait for cops to take their turn in the hot seat again.

Lately, I’ve been wondering if the pendulum has swung back, if teachers are getting a breather while the cops take the bulk of the scrutiny.

Just four years ago when I wrote my first essay, cops were politically beyond reproach by either party. Since Ferguson, our police forces are increasingly under rhetorical attack, and the Democrats are “balancing” their comments less often. Those on the right are starting to make noises about police unions. Moreover, while the  attempts to prosecute the police officers for high profile shootings have failed, the pressure to bring these efforts has increased.The brutal murders in Dallas, Baton Rouge of course add to this horrible climate.

Meanwhile, the new K-12 education law replacing the reform-designed No Child Left Behind, has utterly dismayed reformers on both right and left by stripping away a lot of federal control and leaving education back to the states. Conservatives, who gave birth to the reform movement, are now unhappy because social-justice warriors have taken over education reform.

Let’s take a look at the three legs of education reform:

Testing? Extremely unpopular, particularly with suburban whites–and if suburban whites aren’t testing, then there’s no benchmark to beat teachers up for when the black and Hispanic students don’t meet it. Kidding. Kind of.

Teacher value add measurements? Reformers are forced to argue that the American Statisticians Association supported VAM because it says that “teachers account for about 1 percent to 14 percent of the variability in test scores”. As I wrote earlier, I don’t think VAM will last much longer. Teachers are being judged by test scores in some states, but the energy is on rolling back those laws, not adding more states to the list.

Student achievement gap? Jerry Brown actually said hey, someone’s got to be a waiter. Stop waiting for me to close the achievement gap. Ain’t going to happen. The man went unscathed after this heresy. I’m still shocked. But the thing is, once people start rejecting standardized tests, demanding other solutions to “the gap” is sure to follow.

Or, as this paper asked: Can High Standards and Accountability Exist? Their answer: Not easily. My answer: No.


I’m not rehashing the Common Core wars. I will remind you, however, that the governors and education reformers never really cared about the curriculum unless it would drive accountability. As of today, just 20 states are using the Common Core tests. The rest have opted for less stringent metrics.


Choice lives! Well, kind of. Barack to Hillary is a huge step back for reformers. Barack, Arne, and John King were all “neo-Democrats” on education, which means teachers didn’t like them much. Hillary is very popular with teacher’s unions, even if the teachers themselves wanted Bernie. But neither Bernie nor Hillary are big on choice.

The Donald? The most attention an education policy got at the RNC convention was Donald Trump Jr’s line comparing teacher tenure to Soviet-era stores and then only because his speechwriter had used it in an earlier column. Kind of like Carol Burnett: “Don’t pollute, folks!” Puppy chow for conservatives. It’s not a random happenstance that the presidential candidate most dedicated to traditional education reform barely finishied in the top five and is   back pitching the same old ideas that the GOP voters didn’t even bother to consider before rejecting.

Choice will stay around, but I don’t see it having a strong supporter in the White House.

The philanthropy may be shifting, too. Bill Gates admits he’s spent millions on schools to little effect. Mark Zuckerberg wants to convince us that his $100 million in Newark wasn’t wasted, but most of the world thinks he got schooled. So the “billionaire philanthropists” are backing off of education.

But Michael Jordan has just donated $2 million to non-profits in what is clearly a thoughtful and hopeful effort to support community policing.  Perhaps his act is a one-off–or perhaps we’ll see more wealthy African Americans funding ideas and programs that benefit both urban youth and the police serving their communities. I wish them more success than the billionaires had with schools.

Education reform, the era that began with Nation at Risk and traveled through the explosion of choice, the testing era of No Child Left Behind, the imposition of Common Core–well, it may be over. We’ll still have choice in urban areas where many desperate parents are willing to submit to absurd behavior standards in order to get some semblance of peer selection. Voucher programs will have periodic disruptions. I suspect, though, that ongoing regional teacher shortages  will limit charter expansion (same amount of kids, more teachers). I wonder if the public will ever notice that private schools get created simply to grab the voucher money, and whether they will find it unseemly. Or maybe vouchers will continue to exist as a way for parents who can afford tuition to get a discount. Ed tech will continue to disappoint. But I see more of a whimpering out over years, not a sudden bang, if I’m not nuts about this.

And if I’m nuts, well, at least one of the granddaddies of education reform, Checker Finn, agrees with me.

I’m not gloating, not about the potential end of reform and certainly not about the increased scrutiny and pressure that’s being placed on our police forces. I just sense a shift. We’ll see.


*I don’t overstate the parallels.The police are tasked with public safety with all the demands that entails.  We teachers are charged with education and student safety while they’re in our purview. Those are non-trivial differences; the police are compensated with higher pay, overtime, easier access to disability, and better pensions. I’m not complaining.


**I’m in a new phase, apparently, where my new essay ideas come from my tweet storms.

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

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

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

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


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

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

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

Teddy jumped in.

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

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

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

“That’s it.”

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

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

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

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

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

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

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


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

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

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

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


“What are the coordinates of this point?”

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

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


“How about (8, -10, 11)?

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

“How about (2, -4, 5)?”

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

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


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

“Yep. Let’s draw them.”

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

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

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

Wendy: “I’ve been saying that.”

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

Sophie was convinced.

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

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

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

“You got it.”

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

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

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

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

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

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

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

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

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

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


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

“What would the distance be?”

I drew a prism:


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

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


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

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

“HUSH. Tanya?”

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

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

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


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

Relative silence. I growled.

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

Still silence.

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

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

“The thing?”

“Pythagorean theorem!” half the class chorused.

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

“x2 plus y2 equals w2,” offered Patty.

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

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

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

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


“So what are we looking for?”

“The EXIT!” shouted Dwayne.

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

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

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

“Who, me? I don’t know anything!”

“Bull crap. I scrawled on a new page:

2x + y = 14

“What do you do?”

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

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

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

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

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

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

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


Manuel got it first. “Holy SH**!”

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

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

I bowed. I don’t, usually.

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

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

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


“Pretty cool, huh?”

And the bell rang.

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

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

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

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


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


Not Negatives–Subtraction

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

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

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

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

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

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

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

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


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

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

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

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

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

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

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

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

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

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

How do we change subtraction?

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

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


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

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


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

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

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

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

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

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


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

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

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

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

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

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

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

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

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.


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.

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.

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?”


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.

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.

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”.


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.”


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.


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.


caldwellcalaisChristopher Caldwell, The Migrants of Calais

Leave it at that.

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.”

“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.


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.

Vocational Ed: Advancing the Debate

(note: I’m calling it vocational ed in the title, but throughout will use career/tech, career ed, whatever.)

In my last post I discussed the reason that skilled career/vocational education training has declined–which is not because schools themselves sneer at anything less than college. For those who keep griping about the “disdain” America shows “the trades”, I hope the last essay provides a better understanding of the tensions involved. Increasing investment in skilled non-college education will require addressing the concerns raised there.

So I thought I’d give it a shot. Here’s what I won’t take on:

  • Whether career/tech ed is worth it–This Post story shows that getting a certificate from community college only increases income by $1500/year, which is far better than getting the credential from a for-profit school, which loses people $900/year. Wait, what? For other minds than mine, though.
  • Business involvement—certainly, businesses will benefit and thus should fund part of this effort. I’m not sure how, though, and again, turn over to other minds.
  • Currency–how do you keep the offerings current? When does it make sense to put resources into training people for a new field? How are winners and losers declared?

All outside my ken. Designing these features hold no interest for me. Because if the system doesn’t address the disparate impact issue I discussed here, then it will ultimately fail either through lawsuit or lack of interest.

A couple years ago, I offered educational solutions, or at least options, for students of middle to low cognitive ability, in Just a Job. The programs I sketch out could easily be included in this program.

I envision “career” or “skills” training, as being designed for cognitive ability levels a bit above and a bit more below average. If using IQs, from 85 to 105.

Ban college-level remediation

Right now, we spend millions, at least, on students who are incapable of doing genuine college level work. We loan them money to take courses for which they will get no credit—loans which they will often be incapable of paying back. We spend money on instructors, on space, on curriculum. We deny increased opportunities to qualified students in the form of reduced schedules, thanks to the increased costs of remediation.

Increasing the opportunities for vocational/career training requires setting a standard for college access. We have to stop spending somewhere in order to pay for expanding vocational options. Freeing up the wasted money on wasted college spending is a good place to start. I lay out the case in the attached link.

Increasingly people push to do away with remedial education, which would mean colleges wouldn’t be forcing students to take remedial courses. Marc Tucker, for example, thinks it’s unfair that community colleges require their students to know any advanced math, since the jobs they’ll take on won’t need that math.

But if advanced math isn’t needed for the job, then why is the job requiring college? As I said at the end of the original essay: If no one is too incapable for college, then no education is remedial. So give the students credit for remedial courses, let barely functional students get college degrees after 120 credits of middle school work. No?

(OK, you have to be wondering if I’ve noticed that banning remediation is the Mother Of All Disparate Impact Lawsuits waiting to happen. I did. The rest of the proposal is designed to withstand all but the most illogical objections.)

Limit the scope of community college.

Most states offer three levels of vocational education: high school, community college and regional career centers. The last two are far more extensive than high school programs. Community colleges have appropriated many trade credentials (cosmetology, dental hygiene, auto-mechanics, etc). Low cost, but with often absurd waiting lists, community colleges attach many general education requirements to maintain the illusion of college and an AA degree.

That’s in addition to community college’s original mission: to provide an affordable method of taking general ed courses that will transfer to a four year college or provide stand alone associate degrees. Moreover, community colleges own the bulk of remedial education.

So in this new world, community colleges should be required to dump remedial education, which should be returned to adult education (more on that in a minute). Community colleges should only accept college ready students, as defined in the first step.

I can’t say conclusively that all career training should be removed from community college, which Wikipedia says handles 30% of all CTE. Some career training may have a significant academic requirement that warrant an associate’s degree. But if we are to develop meaningful non-college career training, it can’t be in college, which has a wide range of priorities and is also motivated to devote resources to immigrants.

Increase adult education funding.

Banning college remediation would be incredibly controversial. Many will correctly point out the disparate impact of this ban, that it’s funding higher education for predominantly white and Asian kids. What they wouldn’t mention is that restricting college for the qualified, regardless of race, would improve access and resources for the qualified low income students, again regardless of race.

However, everyone should have the opportunity to become college ready. Not everyone “gets it” the first time, and others finally decide to get serious about the brains they were given. All remediation, K-12 education that simply improves abilities or helps prepare for college, must be shifted to the adult education category, currently funded out of the K-12 budget but now (I hope) would have its own category.

Centralize training programs at the regional level

Someone’s going to ask why not private training programs. Go away and leave me be.

So community colleges should retain their original academic mission for college ready students. Adult education should include both adult remedial education institutions (which would be repurposed community colleges) and the regional career training centers. As I mentioned, many states offer these regional training centers already. These centers offer the best solution to skills education that crosses boundaries from high school to early college—and beyond, for those returning to the trades. All those in the area could attend their local center or apply for a center in another region, much like applying to local vs out of state public colleges today. Students can attend a combination of high school/college and regional trade school as needed.

Critically, students opt in to career training. If the students choose to enter the trades, high schools can’t be held responsible for sending them there. This reduces the potential for racism charges.

Increase resources, reduce selectivity

Today, thanks to the scarcity of career tech vocational programs, at least half the kids accepted are more than smart enough for college and would easily pass a placement test. But if step 1 is implemented, college is going to be much more challenging and college readiness will be a much bigger deal, particularly for those who can achieve it easily.

The “lower half” or more of the kids accepted to these selective programs are generally ambitious, hardworking dedicated kids who know they aren’t academic, but understand they need to have a good GPA. They stay focused. They work hard. They get internships. They get accepted into journeyman positions or move onto the next level of trade schools. They do this with the blessing and support, the letters, the advice, the guidance of their teachers and administration who—please know this—do not sneer at their career choices.

But if these rigorous, selective career and technical programs only accept the kids dedicated and focused enough to avoid all manner of screw-ups, the kids who schlep around, get suspended a few times, flunk math because they think the teacher’s a jerk, have shut themselves out of these options, often before they’re old enough to take advantage of them. Others start out motivated, then lose focus and can never get back on that path.

So “career-technical training”, voc-ed, or whatever you call it, is an excellent option for a lot of low-to-mid cognitive ability kids, but we have to recognize certain realities. These kids will be disproportionately, but not exclusively, black and Hispanic. These kids won’t always be perfectly motivated with spotless resumes.

Keep standards realistic, but challenging. Give kids a reason to work hard to qualify for a program that interests them. Then, be absolutely sure there are seats available for the qualified kids.

Qualify for skilled training using the ASVAB

Elites tend to think anyone not as smart as they are exist in a vast undifferentiated blob. Firefighter, plumber, truck driver, fast food worker–it’s all the same. But in fact, “the trades” encompasses a wide range of cognitive ability levels, and creating a fair means of ensuring rigor in the programs. The military’s ASVAB would be well-suited for this.

And–key plot point–students who need to work and study to qualify for a higher score can use the adult education program to get the support they need.

Build economies around the career centers

Remember, Brooks wants high schools to provide career and technical training so that their students can train and then leave the area. Given the expense of career tech, what states would willingly fund programs to train kids to leave the state?

But if the career centers themselves can rejuvenate–or at least bolster–a weak economy, it might not seem so pointless.

For example, West Virginia could place specialized regional centers in its ten poorest counties, each one focusing on a rigorous technology. Maybe not all–or even some–of the locals can qualify for the technology, but the center itself would provide jobs. The trainers and teachers relocated to the area would spur some growth as well.

Invest in the students

I don’t know if it makes more sense to subsidize career training through low-cost tuition or low-cost loans. Probably some of both. Certainly remedial education should be inexpensive and readily available (although we should not fund living expenses for remedial ed). Essential, though, is the understanding that the investment spigot isn’t shut off for kids who aren’t eligible for college. If, as I suggest, we close off college for many, the money saved must be spent not only on increasing resources for low-income college-ready students, but also on training and investment for students who won’t be going to college.

So there’s the broad outline.

Mentioned briefly here, but the main point of my tweet storm and Just a Job: restricting immigration is essential to invigorating the job and training market for the low skilled. (Note to Brooks: Rick Hess, AEI point man on education, pointedly observed a federal role for education that you might want to write about next time.) Until such time as we finally dramatically restrict immigration, we should at least limit our investment to citizens. Not just the loans, either. (To reiterate, I don’t hate immigrants. But they have no place in our educational system these days. Too expensive.)

We need to advance the debate on skills training, from say age 16 and older. We need to get beyond the tired tripe of “education needs to stop demonizing blue collar work” and start understanding why so few options exist. The only people glorifying college are the progressive and conservative elites. Us proles in the middle are just fine with jobs.

But if we are to start dealing with the difficult challenges that come with a diverse society with wide ranges in cognitive ability, motivation, and needs, then we need to stop the combination of “everyone can succeed” happy talk and “schools SUCK!” condemnation that consumes the discourse today.

Vocational Ed and the Elephant

I thought I’d expand my tweet storm on Arthur C. Brooks directive on American relocation, on one point at least. The one involving the Voldemort View, which must not be spoken. Here referred to as the elephant, because it scanned better.


Rod Dreher and his commenters go to this well all the time, about the so-called snobs who sneer at vocational education. Mike Rowe has built a career on it.

But these calls for a friendlier approach to vocational ed, aka CTE, aka career tech, completely misunderstand the reasons for its relative scarcity, which mostly have to do with the elephant in the room.

Keep in mind that the US has never experienced a halcyon period when committed, focused students were provided with meaningful careers through a helpful high school career training program. The term “dumping” has been around for a long time. A 1985 review of California’s vocational ed program showed that high school courses resulted in no improvement in employment or graduation rates, and even regional training centers had little impact on employment. The country’s support for any sort of vocational ed has always been tepid and cyclical. So it’s not as if we had a fantastic functioning vocational education system before the modern era.

The latest cycle began when 1983’s Nation at Risk forced radical changes in high school education in a failed attempt to raise standards. Nation badly damaged what successful vocational ed we had by arguing we needed rigorous preparation and high expectations to get more high school students ready for college. Of course, not everyone could meet the higher standards, because otherwise there’d be no point to the higher standards. The authors expected that students who weren’t ready for college would be well-trained by rigorous vocational education; they just didn’t think about the elephant.

See, Nation‘s call for high standards, joined five years later by Bill Bennett’s report update , dismissed any notion of an achievement gap. The achievement gap, according to these Ur-reformers, owed its origins not to poverty and ability, but unprepared teachers with low expectations and parents who didn’t care as much. Over time, education reformers stopped blaming parents.

But really, blame is irrelevant. Everything is irrelevant, there sits the elephant firmly in the center of unspoken space: large, cranky, completely ummovable. The kids who couldn’t, and still can’t, manage college prep curriculum are disproportionately black and Hispanic and, (often separately, alas) poor. So the insistence that “everyone could succeed”, with “succeed” meaning “go to college” led to that form of accountability otherwise known as lawsuits, which found that tracking resulted in disparate impact, which meant that tracking ended. Everyone took or tried to take college prep, and high school standards declined. Since everyone was taking college prep, no need for vocational ed, which became more of a dumping ground than usual. The low quality and already weak statistics eventually killed funding for the highest quality career training of the 80s and early 90s. (“Nation at Risk Killed Voc-Ed is mine own opinion, but this 2000 NCES report shares it, pg 49).

While many ambitious vocational ed programs were often killed in the Nation era, the next conservative reform movement, “No Child Left Behind”, resulted in an unexpected rebirth of excellence. Forced to prove themselves in order to avoid closure, the remaining voc-ed programs had to keep test scores high. So many career-oriented programs basically re-emerged as rigorous, but incredibly expensive and hard to staff. No longer a dumping ground, career-tech ed (CTE) supply is now outstripped by demand. The programs can pick and choose; the cognitive ability levels required are quite high. Today, career technical training is outstanding, demanding, and extremely selective. At least half the students strong enough for career training programs can easily place into college. The kids who can’t pass Algebra aren’t qualifying for career programs.

So “more technical training” in high school isn’t a magic bullet. Brooks’ AEI stable includes probably the best conservative reform policy guru, Rick Hess. If Brooks asked Rick about vocational education, the answer might have looked something like this:


Comparing Hess’s response to Brooks’, I’m figuring Hess wasn’t asked.

Or Brooks could have read up on Michael Petrilli’s push for moving more kids to career training. Petrilli, president of Fordham Foundation’s education reform think tank, published a harsh message for low ability kids in 2014: Sorry, Kid, You’re Just Not College Material, proposing that kids who can’t cut it in academic courses be rerouted into career and tech ed.

And Petrilli got schooled and schooled hard, as dozens of experts handed him his ass, explaining the history of vocational education, calling him a racist for writing off poor kids of color, pointing out the racial disparities, and basically calling him an uneducated yutz for blindly suggesting solutions that he didn’t understand. Anyone thinking of suggesting changes to vocational/career ed has no better starting point than Petrilli’s chagrined follow up acknowledging the error of his ways, and sounding a bit depressed about the cognitive demands of career training.

Yet here Brooks is, pushing career training again, ignoring the very recent experience of someone on his own team, blandly suggesting vocational education, continuing to avoid the Unspeakable. Twas ever thus. It’s always this vague notion that schools sneer at anything but college degrees, Brooks’ idee fixe. No one ever goes past this reason to wonder why high schools don’t track anymore.

I’m not sure anyone really understands why, until they have their noses shoved into it like Petrilli did. People just don’t understand the degree to which many high schools are forced to choose between failing most of their students year after year, with no hope of ever achieving three years of advanced math or English—that it’s not a matter of trying harder, or teaching better, or that the kids weren’t taught. They lack any real understanding of the layers of cognitive ability. They don’t realize there are perfectly normal folks who aren’t smart enough to be plumbers, welders, or dental hygienists.

But those who do understand often sound callous or dismissive of people with low IQs. Maybe it’s because my father cooks a great meal, fixes a great plane, and has a sub-100 IQ, or maybe it’s just because I was raised working class. Maybe it’s my work as a teacher. But I don’t think “low IQ” is an insult or a dismissal. And so, I’m angry at those who make basically ignorant proposals–move more! create more plumbers!–without even the slightest understanding of the political and social tensions that stop us from tracking kids by ability to the extent that, perhaps, we should.

I have never seen the cause of those tensions more eloquently expressed than in this panel on Education for Upward Mobility, by Howard Fuller. After an early life as a black activist (or maybe “after” is the wrong word), Fuller went on to become superintendent of Milwaukee Public Schools. Pro-charters, pro-choice, the embodiment of neo-progressive education reform and in every way imaginable a partner with Petrilli, the panel moderator, who asked him his thoughts on how best to shake off the ugly history of tracking and use it to help kids succeed. It’s best to listen to him say this, around minute 12, but for those who won’t bother, here’s what Fuller had to say:

“You know Mike, my thing, starting with the whole ‘who goes to high school'[think he means college]….most of the people who talk about ‘kids don’t need to go to college”, hell, they went to college. And so that’s where my problem starts right there. Why is it okay for you, but for these low income kids, “aw, y’all can’t go to college.” ….What do rich people do for their kids?….When I hear some of y’all talk about [vocational education], just know that I’m gonna always be suspicious. It brings up to me…somehow we’re trying to figure out a way…it’s almost like a Booker T./Du Bois argument brought up to this century. Whenever I hear the Booker T. part of that argument, it’s that we’re going to accept that a certain group of people are going to have to be in the lowest level, because that’s the way our economy is set up and so some of these kids, it’s okay for them to be there….And when people say tracking….the issue of power and whose kids get tracked in what ways and where they end up…I can’t get it out of my head…..I’m afraid of whose going to make what choices for what kids.”

This is what’s known as a facer. I have two simultaneous reactions. First, I’m impatient, because Fuller’s response just kills all rational conversation dead. There’s really no way past that. It’s brilliant, effective, and utterly deadening. Why here, I’ll just point out the elephant in the room, shall I? And because everyone’s busy pretending the elephant doesn’t exist, their scrotums will retract up into their livers. We’ll just change the subject, shall we?

But my second reaction, coming right afterwards, is doubt. Brooks’s op-ed is one of many sneering at the working class these days. The GOP head of Congress is wondering if he can talk Trump out of immigration restriction, since his own position is amnesty and more immigration for skilled workers , while Clinton wants amnesty and more immigration of every sort.

So I’m not entirely convinced anymore that Howard Fuller is entirely wrong to doubt the intentions of the elites who want so desperately to make decisions for all the little people.

But that won’t stop me from suggesting a system for career/tech training, of course. Stay tuned.

The Many Failings of Value-Added Modeling

Scott Alexander reviews the research on value-added models measuring teacher quality1. While Scott’s overview is perfectly fine, any such effort is akin to a circa 1692 overview of the research literature on alchemy. Quantifying teacher quality will, I believe, be understood in those terms soon enough.

High School VAM is Impossible

I have many objections to the whole notion of modeling what value a teacher adds, but top of the idiocy heap is how little attention is paid to the fact that VAM is only even possible with elementary school teachers. First, reading and basic math are the primary learning objectives of years 1-5. Second, elementary schools think of reading and math ability in terms of grade level. Finally, elementary teachers or their schools have considerable leeway in allocating instruction time by subject.

Now, go to high school (of which middle school is, as always, a pale imitation with similar issues). We don’t evaluate student reading skills by grade level, but rather “proficiency”. We don’t say “this 12th grader reads at the 10th grade level”. We have 12th graders who read at the 8th grade level, of course. We have 12th graders who read at the third grade level. But we don’t acknowledge this in our test scores, and so high school tests can’t measure reading progress. Which is good, because high school teachers aren’t tasked with reading instruction, so we wouldn’t expect students to make much progress. What’s that? Why don’t we teach reading instruction in high school, if kids can’t read at high school level, you ask? Because we aren’t allowed to. High school students with remedial level skills have to wait until college acknowledges their lack of skills.

And that’s reading, where at least we have a fighting shot of measuring progress, even though the tests don’t currently measure it–if we had yearly tests, which of course we don’t. Common Core ended yearly high school tests in most states. Math, it’s impossible because we pass most kids (regardless of ability) into the next class the next year, so there’s no “progress”, unless we measure kids at the beginning and end of the year, which introduces more tests and, of course, would show that the vast majority of students entering, say, algebra 2 don’t in fact understand algebra 1. Would the end of year tests measure whether or not the students had learned algebra 1, or algebra 2?

Nor can high school legally just allocate more time to reading and math instruction, although they can put low-scoring kids in double block instruction, which is a bad, bad thing.

Scope Creep

Most teachers at all levels don’t teach tested subjects and frankly, no one really cares about teacher quality and test scores in anything other than math or reading, but just pretend on everything else. Which leads to a question that proponents answer implicitly by picking one and ignoring the other: do we measure teacher quality to improve student outcomes or to spend government dollars effectively?

If the first, then what research do we have that art teachers, music teachers, gym teachers, or, god save us, special education teachers improve student outcomes? (answer: none.) If the second, then what evidence do we have that the additional cost of testing in all these additional topics, as well as the additional cost of defending the additional lawsuits that will inevitably arise as these teachers attack the tests as invalid, will be less strain on the government coffers than the cost of the purportedly inadequate teachers? What research do we have that any such tests on non-academic subjects are valid even as measures of knowledge, much less evidence of teacher validity?

None, of course. Which is why you see lawsuits by elective teachers pointing out it’s a tad unfair to be judged on the progress of students they’ve never actually met, much less taught. While many of those lawsuits get overturned as unfair but not constitutional, the idiocy of these efforts played no small part in the newest version of the federal ESEA, the ESSA, killed the student growth measure (SGM) requirement.

So while proponents might argue that math and English score growth have some relationship to teacher quality in those subjects, they can’t really argue for testing all subjects. Sure, people can pretend (a la Common Core) that history and science teachers have an impact on reading skills, but we have no mechanism to, and are years away from, changing instruction and testing in these topics to require reading content and measuring the impact of that specific instruction in that specific topic. And again, that’s just reading. Not math, where it’s easy enough to test students on their understanding of math in science and history, but very difficult to tangle out where that instruction came from. Of course, this is only an issue after elementary school. See point one.

Abandoning false gods

For the past 20 years or so, school policy has been about addressing “preparation”, which explains the obsession with elementary school. Originally, the push for school improvement began in high school. Few people realize or acknowledge these days that the Nation at Risk, that polemic seen as groundbreaking by education reformers but kind of, um, duh? by any regular people who take the time to read it, was entirely focused on high school, as can be ascertained by a simple perusal of its findings and recommendations. Stop coddling kids with easy classes, make them take college prep courses! That’s the ticket. It’s the easy courses, the low high school standards that cause the problem. Put all kids in harder classes. And so we did, with pretty disastrous results through the 80s. Many schools began tracking, but Jeannie Oakes and disparate impact lawsuits put an end to that.

I’m not sure when the obsession with elementary school began because I wasn’t paying close attention to ed policy during the 90s. But at some point in the early 90s, it began to register that putting low-skilled kids in advanced high school classes was perhaps not the best idea, leading to either fraud or a lot of failing grades, depending on school demographics. And so, it finally dawned on education reformers that many high school students weren’t “academically prepared” to manage the challenging courses that they had in mind. Thus the dialogue turned to preparing “underserved” students for high school. Enter KIPP and all the other “no excuses” charters which, as I’ve mentioned many times, focus almost entirely on elementary school students.

In the early days of KIPP, the scores seemed miraculous. People were bragging that KIPP completely closed the achievement gap back then, rather than the more measured “slight improvement controlling for race and SES” that you hear today. Ed reformers began pushing for all kids to be academically prepared, that is hey! Let’s make sure no child is left behind! And so the law, which led to an ever increasing push for earlier reading and math instruction, because hey, if we can just be sure that all kids are academically prepared for challenging work by high school, all our problems will be fixed.

Except, alas, they weren’t. I believe that the country is nearing the end of its faith in the false god of elementary school test scores, the belief that the achievement gap in high school is caused simply by not sufficiently challenging black and Hispanic kids in elementary school. Two decades of increasing elementary scores to the point that they appear to have topped out, with nary a budge in high school scores has given pause. Likewise, Rocketship, KIPP, and Success Academy have all faced questions about how their high-scoring students do in high school and college.

As I’ve said many times, high school is brutally hard compared to elementary school. The recent attempt to genuinely shove difficulty down earlier in the curriculum went over so well that the new federal law gave a whole bunch of education rights back to the states as an apology. Kidding. Kind of.

And so, back to VAM….Remember VAM? This is an essay about VAM. Well, all the objections I pointed out above–the problems with high school, the problems with specific subject teachers–were mostly waved away early on, because come on, folks, if we fix elementary school and improve instruction there, everything will fall into place! Miracles will happen. Cats will sleep with dogs. Just like the NCLB problem with 100% above average was waved away because hey, by them, the improvements will be sooooo wonderful that we won’t have to worry about the pesky statistical impossibilities.

I am not sure, but it seems likely that the fed’s relaxed attitude towards test scores has something to do with the abandonment of this false idol, which leads inevitably to the reluctant realization that perhaps The Nation At Risk was wrong, perhaps something else is involved with academic achievement besides simply plopping kids in the right classes. I offer in support the fact that Jerry Brown, governor of California, has remained almost entirely unscathed for shrugging off the achievement gap, saying hey, life’s a meritocracy. Who’s going to be a waiter if everyone’s “elevated” into some important job? Which makes me wonder if Jerry reads my blog.

So if teacher’s don’t make any difference and VAM is pointless, how come any yutz can’t become a teacher?

No one, ever, has argued that teachers don’t make any difference. What they do say is that individual teacher qualities make very little difference in student test scores and/or student academic outcomes, and the differences aren’t predictable or measurable.

If I may quote myself:

Teaching, like math, isn’t aspirin. It’s not medicine. It’s not a cure. It is an art enhanced by skills appropriate to the situation and medium, that will achieve all outcomes including success and failure based on complex interactions between the teachers and their audience. Treat it as a medicine, mandate a particular course of treatment, and hundreds of thousands of teachers will simply refuse to comply because it won’t cure the challenges and opportunities they face.

And like any art, teaching is not a profession that yields to market justice. Van Gogh died penniless. Bruces Dern and Davison are better actors than Chrisses Hemsworth and Evans, although their paychecks would never know it. Teaching, like art and acting, runs the range from velvet Elvis paint by numbers to Renoir, from Fast and Furious to Short Cuts. There are teaching superstars, and journeyman teachers, and the occasional lousy teacher who keeps working despite this–just as Rob Scheider still finds work, despite being so bad that Roger Ebert wrote a book about it.

Unlike art and acting, teaching is a government job. So while actors will get paid lots of money to pretend to be teachers, the job itself will never lead to the upside achieved by the private sector, despite the many stories about famous Korean tutors. Upside, practicing our craft won’t usually lead to poverty, except perhaps in North Carolina.

Most teachers understand this. It’s the outside world and the occasional short-termers who want teachers to be rewarded for excellence. Most teachers don’t support merit pay and vehemently oppose “student growth measures”.

The country appears to be moving towards a teacher shortage. I anticipate all talk of VAM to vanish. But if you want to improve teacher quality beyond its current much-better-than-it’s-credited condition, I suggest we consider limiting the scope of public education. Four of these five education policy proposals will do just that.

1 I was writing this up in the comments section of Scott Alexander’s commentary on teacher VAM research, when I remembered I was behind on my post quota. What the heck. I’m turning this into a post. It’s a long answer, but not as long-winded as Scott Alexander, the one blogger who makes me feel brusque.

Great Moments in Teaching: The Third Dimension (part I)

“How many other dimensions are there?”

“Well, four, according to Einstein, and five according to Madeline L’Engle, if you’ve read A Wrinkle in Time.”

“I have!” Priya’s hand shot up. “It’s a tesseract!”

I was impressed. Not many girls read that classic anymore. “But we’re going to stick to three dimensions.”

“Isn’t real life three dimensions?” asked Tess.

“Yes. But if you think of it, up to now, we’ve only been working in two. We’ve spent a lot of time in the coordinate plane thinking about lines. In two dimensions, 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.”

Now, no mocking my terrible art skills” and I put up this sketch, the drawing of which occurred to me the night before, and was the impetus for the lesson.


Everyone gasped, as they had in the previous two classes. My instincts about that clunky little sketch proved out, beautifully. No clue why.

“Holy sh**,” groaned Dwayne, the good ol’ country boy who offered to paint my ancient Honda if I gave him a passing grade. He doesn’t like math. He’s loud and foul and annoying and never shuts up. That last sentence is a pretty good description of me, so I’m very fond of him. “What the hell is that? Get it off the screen, it hurts my eyes.”

“That is awesome,” offered Talika, a senior I had last year for history. “How long did it take you to draw that?”

“What is that white stuff spread everywhere? Did someone get all excited?” asked Dylan, a sophomore whose mother once emailed me about his grade, giving me the pleasure of embarrassing him greatly by describing his behavior.

“Ask your mother,” I replied, to a gratifying “ooooo, BURN!” from the class, who knew very well what had happened.

“How did you draw that?” asked Teddy, curious. “It’s not ordinary graph paper, right?”

“No, it’s isometric paper, which allows you to draw three dimensional images. So…”

“This is really stupid,” said Dwayne. “I’ve taken algebra 2 three times and no one’s ever taught me this.”

“Best I can tell, no one’s ever taught you anything , and not just not in algebra 2,” I replied, earning another “Oooooo” from the class and an appreciative chuckle from Dwayne.

“It’s weird, though, because in two dimensions, you start in the middle,” offered Manual, who was consulting with Prabh, another bright kid who rarely speaks.

“That’s a good point! For example, if we were going to plot seating positions in this room in two dimensions, we’d start with Tanya,” I said, moving to the class center and indicating Tanya, who looked a bit confused. “So Tanya would be the origin, and Wendell would be (1,0), while Dylan would be (1,-1).”

“I’m not negative!” Dylan said instantly, talking over my attempt to continue. “You’re saying I’m negative. You don’t like me.”

“Hard to blame anyone for that,” said Wendell who is considerably more, er, urban than Teddy, with pants down to his knees and a pick that spends some time in his hair. Despite his occasional class naps, he maintains a solid C+, and could effortlessly manage a B if I could just keep him awake. “S’easy, dude. It’s like one of those x y things, like we’re all dots on the graph.”

“You’re one down and one to the right of me,” pointed out Tanya.

Dylan was interested in spite of himself. “So Talika’s, like, (0, 4)?”

“Yes,” several students chorused.

“Then I’m negative 8.” said Dwayne, unhappy with any conversation that doesn’t have him at the center.

“More like….(-2,2), yeah,” says Cal.

Ben speaks up, “But how come Tanya’s at the center for mapping the room’s people, but your sketch is, like, from the left?”

“Or right?” Sophie, from the back.

“Or is it….outside?” asks Manuel.

“Yes, it’s kind of like you’re standing on a desk in Ms. Chan’s room and the walls are transparent,” says Ben, more certainly. Ben is repeating Algebra 2 after having taken it with me last semester. Very bright kid who clowned incessantly, confident in his ability to learn without really trying, only to learn that Algebra 2 was different from other nights, and he wasn’t finding the afikoman. I advised him to repeat. The big sophomore not only agreed, but specifically asked to repeat with me. His attitude and behavior is much improved. I ran into him while walking across the courtyard a few weeks earlier, and he said “I just realized I was Dwayne and Dylan combined last semester, and it’s so embarrassing. I’m really sorry.”

“I’m not enough of an artist to know if I could have drawn this any other way. It just seemed intuitive to me last night, when I came up with the idea.”

“See, I knew it,” trumpeted Dwayne. “You’re making this up!”

“Yeah, I know this isn’t in any algebra book” said Wendy, a sophomore whose excellence in math is often hard to discern beneath her complaints. “This is just some weird thing you’re doing to make us think about math.”

I picked up at random one of the four algebra 2 books sitting on my desk (I’m on the textbook committee) and walked over to Wendy’s desk, opening it to the “Three Dimensional Systems” chapter. She looked, and said “Ok, maybe not.”

“So just as we can plot points in two dimensions, we can plot points in three. Take Aditya here,” my TA, who was watching the circus in amusement. “How could we represent him as a point on my graph?”

Teddy said instantly, “Yeah, I’ve been working that out. I can’t figure out which the new one is, and what do we call it? Where’s x, where’s y?”

Sanjaya said, “I think the part along the ground is x. Like if you go along the bookshelf?”

“Like this?”


“Yes,” Sanjaya said, confidently. “That has to be x. So you could count to Aditya, right?”

“Count which way?”

“The bottom!” “The bottom line!” “the bottom..axis, thing. The X!” comes a chorus of voices.

I start counting, and while I do, Sophie objected. “But hang on. I still don’t see what the new thing, direction, is. What’s the third?”

“Up! said Calvin, who rarely participates and often tunes out so far he can’t keep up. But he was watching this with interest. “You know how the class map with Tanya was going north and south and east and west. But it’s all flat, like. This picture has an up.”

“Yeah!” Ben got it. “Cal’s right.”

Dwayne has begun to grasp this. “So you can’t just draw a line? You have to follow along the…things?”

“The axes.” I finish counting along the “bottom” axis and go over to my bookshelf in the furthest corner of my room. “So the sketch starts here…QUIET! One conversation at a time, and I’m the STAR here. The origin starts at this bookshelf. I am walking along the wall, hugging it, on my way to Aditya. Does everyone see how they could track my progress on the axis?”

“YES!” from various points of the room.

“What the hell are you doing?” Dwayne is watching me carefully hug the wall.

“Everyone except Dwayne?”

“YES!” much louder.

I walked along the wall to the table where Aditya sat (fourth along the wall) and stop.

“So now what?”

“You’re there.”

“No, not yet.” countered Nadine. You have to go out….” she waved me towards her. “this way.”

“Yeah, towards Aditya,” this from Talika.

I stepped out 2 steps or so. “That all?”

Josh frowned. “Yeah. You’re there. Except…”

“UP!” Sophie shouted from the back. “That’s the third axis!” General approval reigns loudly, until I wave them all quiet, or try to.

“You go up 4!” Teddy shouted.

“OK. So Aditya is about 40 units out along the wall, 2 units out towards…the door, and 4 units up. Yes?”


“So let’s draw that.”

Of course, while I’m drew this, general mayhem is ongoing with my back turned. I shouted “QUIET! or “Could someone stick a sock in Dwayne?” a few times.

“Wow, so it’s a…cube?”

“A prism, yes. So here’s what we’ve done. We’ve taken the two dimensional x-y coordinate plane and extended it.”

“We extended it up,” from Sophie.

“Yes. And now, instead of a rectangle, we have a three-dimensional rectangular prism. And we can describe things now in three dimensions. But we can do more than that. So let’s step away from my classroom sketch….”


“Whoa. What’s that?” Dylan.

“Man, that’s f***ed up. I just started to get this, and now you’re….” Dwayne, of course.

“No, it’s fine,” Manuel said. “It’s just like the whole thing moved to the center.”

“Oh, I see. It’s like there’s four rooms, all cornered.” Wendell.

“Yes, exactly. Except now, you want to stop thinking about it as a room and think of it as a coordinate plane. As Sophie says, the new plane is the up/down one. So the old x is now here. The old y is now here. The z is the straight up and down one. I think of it as taking the 2 dimensional plane and kind of stepping back and looking down on it.”

“That’s just….”

“DWAYNE BE QUIET. One thing to remember: when you see a 3-dimensional plane, they may be ordered differently. There’s a whole bunch of rules about it that make potentially obscene finger orientations, but I promise I won’t test you on that.”

“So let’s say we’re plotting the point (8,4,5). I’m going to show you how to do it first. Then I’ll go through why. Start by plotting the intercept along each of the planes.”

“Man, does anyone else get this?”

“YES. Shut up, Dylan,” says Natasha.


“The trick to remember when you’re graphing in 3-d is to stay parallel to the axis you’re drawing along. So never cross over the lines when plotting points. Now let’s add the yz and xz planes.

“What? This is weird. Why are you drawing so many rectangles?” Patty, frowning.


“What you have to visualize is that it’s like we’re drawing sides. So far, I’ve drawn,” I look around and grab three of my small whiteboards, “the bottom and two of the sides. Hold this, Natasha, Talika.” and I build the walls. The kids in the back stand up and look over.

“Oh, I see,” Teddy again. “You’re drawing the prism again.”

“Right. It’s just looking different because the axis is in the center.”

“You do all this just to plot one point?” Sophie, ever the skeptic.

“Yes, but remember this is more just to illustrate, to see how you can extend the dimensions. So after you draw the three sides, joining the intercepts for xy, yz, and xz intersections, you extend those out–again, along the lines.”


“So the point we’re graphing is going to be at the vertex, the intersection of the three planes, the furthest point from the origin–just like in two dimensions, the point is at the intersection of the two lines.”

“That’s really complicated.” Wendy sighed.

“No, it’s not” “Don’t you see the…” Ben, Manuel, Teddy, Wendell and others jump in at the same time, while Dwayne bellowed, Wendy and Tess were asking questions of the room, and, as the writer says, pandemonium ensued. It was a shouting match, yes, but they were shouting about math. The Naysayers, the Doubters, and the Apostles were all marking their territory and this was no genteel, elegant, “turn and discuss this with your partner”, no think-pair-share nonsense. This was a scrum, a brawl, a melee conducted across the room with the volume up at 11—but just like any good fight, there was order beneath the chaos, a give and a take at the group level.

And for you gentle souls wondering about the quiet kids, the introverts, the shy ones who need time to think, they were enthralled, watching the game and making up their mind. It may not look like everyone gets time to talk, but pretty much every time you read me call on a kid, it’s a quiet one. And I shush the room. Then the quiet kid sits there in shock as he or she realizes oh god, I’ve got the mike and I can’t be a spectator anymore.

Anyway, the story goes on with a second great moment, but I’m getting better at chunking and this half had too many details I didn’t want to give up. I’ll stop here for dramatic effect. Because oh, lord, I was high as a kite in this moment, watching the room, realizing I was riding a tremendous wave of energy and excitement. Yeah. ME. On Stage. Making Drama.1

Now I just had to come up with a good ending.

1I’m not congratulating myself, saying I’m proving kids with the great moment. No, 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.