Modeling Rational Expressions

As part of our Teacher Federalism agreement, I now include the reciprocal function as one of my parent functions in Algebra 2. But time constraints don’t allow me to really dig into the function–plus, the kids are on overload by the end of the term, what with exponential functions, logarithms, and inverses. I don’t really have time to switch gears. Besides, they’ll be exploring rational expressions in depth during pre-calc.

But then I noticed, during Trig, that my students really weren’t completely understanding that four of the major trig functions are rational expressions and how they differed from sine and cosine.  Meanwhile, I’m always doing a bit of algebra review…and so I decided to kick off my trig class with a rational expressions unit. A brief one, that enabled a review of quadratics and rational expression operations (aka adding and subtracting fractions with variables).

Starting Activity


Task 1 is a straightforward linear function, so almost every kid who has made it to trig, no matter how weak, is able to quickly build the function.

Task 2, of course, is the introduction of division. In function terms, we’re dividing a line by a line, while I will eventually make clear. But practically, the big hop occurs when students realize that cost per hoodie is not constant. Usually students do this incorrectly, graphing either a constant line of 20 or confusedly dividing total cost by 20. So I’ll meander by and ask:

“According to your table for L(h), we spent $520 and got one hoodie. What was the cost of that hoodie.”

“Twenty dollars.”

Silence. I wait. And inevitably, a student will gasp, “No! One hoodie costs $520!!”

It usually takes about 45 minutes for the kids to work through both tasks, including graphing the unfamiliar rational expression. Then I call them back up front for explanation and notes.

After putting the two equations on the board (linear and rational), I point out that our cost per hoodie equation is basically a line divided by a line. I point out the two asymptotes , vertical and horizontal. Why do they exist? Most students, by trig, know that you can’t divide by zero, but why doesn’t the vertical asymptote intersect y=20?

This usually prompts interesting discussions. I usually have a couple students to correct when they build the graph, as they make it linear. So now I redraw it, making clear that the drop is sudden and sharp, followed by a leveling. Why is it leveling?

Usually, a student will suggest the correct answer. If not, I ask, idly, “Can anyone tell me why the cost per hoodie isn’t $20? After all that’s the price.”

“Because you have to pay the $500, too.” and this almost always leads to a big “aha” as the students realize that the $500 is “spread out”, as many students call it, among the hoodies. The more hoodies purchased, the higher the total cost–but less of the $500 carried on each one.

At some point, I observe that certain forms of equations are much easier for modeling than for graphing. For example, when modeling linear functions, we use standard form and slope intercept form all the time–many real-life (or close to real life!) applications fall naturally into these formats. John has twice as much money as Jane. Tacos are $3, burritos are $5, Sam has $45.  But you’d never deliberately model an application in point-slope form. You might use it, given two points, to find the equation. But it’d be an operation, not a model.

So take a look at TL(h) and what does it look like? Usually, there’s a pause until I remind them that we could have negative hoodies, and we graph that in. Then the kids recognize the reciprocal function.

“If  we take a look at the graph and think of it as a transformation of a parent function, what’s the vertical shift?”


I draw the parent reciprocal function . “Remember this? Where are the original asymptotes?” and eventually the kids remember y=0 and x=0.

“Right, so the original parent function, the horizontal asymptote is y=0. Where is it in this function? y=20. So what’s the vertical shift?”

Now they get it, and I hear “20” from all corners.

“Right. Is there a horizontal shift?”


“So we know that h=0, k=20….what’s a=? What’s the vertical stretch?”

Someone always remembers that it’s the vertical distance between (1,1) and the actual output value for x=1, which is….

“Right. a=500. So TL(h) could also be written as 20+500(1⁄h). Notice that if we split the numerator into two terms and simplify, we get the same thing. But we’d never model it that way. Much more intuitive to create the linear equation for total cost and divide it by the line.”

In other words, I point out, the hoodie activity is actually the same function that they learned about last year, but instead of just graphing or solving transformed functions, they’re modeling with it.

The second part of this activity is about 20 minutes, and comes at the end of my 90-minute block. In between, I do a lecture on  the meaning of rational expressions, vertical and horizontal asymptotes, usually bringing up something like this:


But while it works in the context of the lesson, it just pulls the focus of this post so I’ll write about that some other time. Suffice it here to say that yes, I discuss what the defining criteria of rational expressions are, what asymptotes are, and so on. One of the main reasons I teach this now is so the kids will understand both as they bump into them.


Part  three comes quickly because the students see the pattern, and that alone is enough to please a lot of them.  Suddenly, they’ve added an entire model type to their repertoire.

Part 4 is where so much gets tied together. Most students get all the way to part 3 without anything more than nudges. At that point, I usually bring it up front.

Using Desmos, we graph the same system. At this point, I’m obscuring the solution value.

ModRatExpGraph1This sets off discussion about the shift, how one starts out cheaper but stabilizes at a more expensive base cost. And then, look, the lines intersect? What do intersections mean, again? SOLUTION!

And in this first equation, the solution is quite simple because the equations have the same denominator. (note: I’m using x instead of h because I quickly copied these from Desmos):


So it’s a quick matter to solve the system, but again, good reminder. At 40 hoodies, they are $32.50 apiece.

Right around here, I point out that it would be convenient if one equation could show us the information we needed.  How could we show the difference between the two functions?

Fortunately, a number of the kids have lived through my algebra 2 class, and call out “subtract!”. I briefly explain that functions have operations and can be combined, for the rest. So we can subtract one equation from the other. Since ultimately the Hawk’s function will be more, we perform TW(h) – LW(h) and graph it as D(h).


Note first that it’s still a rational expression, although not the same type we’re working with. See how nice and clean the break even point is displayed!

So what we want to do is move this break even point further to the right. Luvs is ahead and has no reason to bargain. Obviously, we need to talk to Hawk’s Hoodies.

The administrators want to buy the better hoodie, but 40 isn’t enough to have a fundraiser–they want over 100. How can we get a better deal?


Suggestions? The ideas come fast.

First up is always “Hawk’s should sell the hoodies for cheaper.”

“Right. Hawk’s could lower its asymptote and slow the rate of increase in total cost. What would be a good price?”

We try $23/hoodie:


$22 is even better, putting the break even point at 100 hoodies.

“But here’s the thing–Hawk’s has real pride in their hoodies. They know they’re charging more, but their hoodies are worth it. That’s why we want those hoodies to begin with! They’re softer, better colors, hold up to wear and tear, whatever. So cutting their price by 12% sets a precedent. There’s a whole bunch of marketing research showing that customers don’t value luxury items if they sense the vendor will cut prices at a later date. So while we might like the price of the hoodies, Hawks could be hurting its brand if it cuts the retail price per hoodie. Take this as a given, for the moment. Is there any other way we could cut the price per hoodie that still maintains the same retail price?”

This always leads to good feedback: give a school discount, cut the price of the logo conversion, and so on.

At some point, I break in (unless the solution I’m looking for has been mentioned):

“All of you are coming up with great suggestions that involve reducing the value of the numerator. How about the denominator?”

Puzzled looks.

“If I have a division problem stated as a fraction, reducing the numerator (the dividend, if you must) will reduce the result, or the quotient. But is that the only way I can reduce the quotient?”

Pause. New teachers, let the pause hang. If it still gets no response, say “What else can I change?” because that will lead to someone saying…

“The denominator. But why would reduce the denominator?”

“Why indeed. 6 divided by 3 is 2. If I reduce the numerator to 3, my answer is 1. But….”

“Oh, I get it! Increase the denominator?”

“What would that do? Or put it this way: what would Hawks have to do to increase the denominator?”

And eventually, everyone figures out that Hawks could throw in some hoodies for free, which would also let them maintain their higher prices while still getting the sale.

“So go figure out the equation if Hawks includes 10 hoodies for free.”

Someone will always realize that this means we could get negative hoodies. So I tell them to test some negative values and remind them to think about what this might do to the asymptotes.


When they’re done, we put the whole thing on Desmos, showing that the vertical asymptote has changed, but not the horizontal.

“See, this way, Hawks is decreasing the time it takes for our purchase to get to the lower prices, getting us to just a little over $25 per hoodie with far fewer purchased, because we’re getting $10 for free.”

Now, take a look at the new equation to find the breakeven point:


“So how many of you remember being assigned these ridiculous equations with variables and fractions and thinking oh my god, none of us will ever use this? Who would ever have to add or multiple or subtract fractions? And yet, here we are. This one has them set equal to each other, but as we said above, function D(h) is the difference between the two :


…look at that! Your math homework in real life!!!

So we discuss what d(h) is doing. I point out that “solving the system” of TL(h)  and TW(h) is nothing more than “finding the zeros” for D(h).

From a curriculum standpoint, I transition pretty quickly from rational expressions to a review of binomial multiplication and factoring. So the D(h) subtraction equation gives me a great opportunity to review the procedures before I set them on their way. I work the problem–which requires the quadratic formula at the last step, ironically, but still gives us a chance to review the steps to determine whether or not a quadratic can factor. Then I show again how Desmos takes the equation and shows us how far we’ve “moved to the right”:


Then they all work out the comparison between a $3 reduction in price and ten hoodies thrown in for free. I take a moment to point out that math drives business analysis. Today, we have technology to do the work for us, but the best analysts have an understanding of the rational expressions driving the graphs.

If I had time, I’d do this in Algebra 2, but from a time perspective, I have a choice between introducing exponential equations and logs or go deep on rational expressions. That’s a nobrainer. They need to at least be introduced to logs, and there’s no opportunity in trig to bring that topic up. Rational expressions, on the other hand, forge a connection that makes sense when we get to the graphs.

And yes, it’s made a difference. I’ve been using this activity for two years, and have seen a noticeable improvement in their understanding of the four rational expression graphs. Remember, I’m not just teaching my kids, so even those who got a full dose of the rationals with other teachers are showing increased understanding. I would like to do this and more in Precalc, and will report back.

Sorry I’ve been so long without writing. We had a ridiculous heat wave and I responded by sticking to Twitter and playing Fallout Shelter, which is kind of cool.


Three VIPs for New Teachers

You’re a new teacher, worried about how to start? Let me tell you about the three most essential contacts to make in your earliest days. Notice that none of these people are, technically speaking, colleagues. If you can find teachers who want to help you, great. I always make sure new teachers have a mentor or at least my help if they need it.  But this is about getting the support you need to do your job and other teachers aren’t really the first line of defense.

The Tech Guy

It’s usually a guy, so I will call him a “he”. Districts usually centralize technology, but each school site usually has a dedicated support guy. The first person you’ll meet is the principal’s secretary (more about her in a minute) but your first real friend must be the tech guy.

Few teachers recognize the advantages to being on first name terms with the guy with keys to the computer room, so they often won’t think to mention him. “Who’s the tech guy?” is a question that leads to other questions. Try “Is my email set up already, or will I need to request it?” or “Do you know if I have an account on the district server?” are excellent questions to elicit the tech guy’s name quickly.

I’ve been at three considerably different schools and found good tech support. But even if the guy is a lazy loafer with no real redeeming qualities, cultivate his acquaintance.

Take printers, for example. You can buy your own printer and request to set it up. This often violates several district policies, needs approval, and in some cases can’t be done at all. Alternatively, you can casually mention to the tech guy that if he has any spare printers, you’d be happy to set it up yourself, keep his workload low….and leave it dangling. Three schools, three printers set up for the asking in under three days. Old ones, sure. But they all worked, and I had them day one. And got replacements when needed.

I’ve seen teachers go two weeks without email, been forced to take attendance (shudder) manually, have no idea how to print to the main copy machine, all because they didn’t take twenty minutes to meet the tech guy. Meanwhile, I’ve gotten ten minute turnaround time when my DVD player doesn’t work on Movie Day, even though I make it clear my problem is non-critical.

Our school paid for our own tech guy for several years by giving up two class sections. He was worth every penny, and we’d still have him except the district technology director didn’t like him and reinstituted centralized control. Our current tech guy, supplied by the district, is also terrific. He likes green beans. I give him two or three bags of freshly picked beans from my garden, every year.

The Principal’s Secretary

Some schools separate actual secretarial support from the administrative tasks of running the school, but in my experience the job is usually centralized. Simply put: who does keys and subs? Who manages the missed prep list? Who runs work orders and facilities requests? If it’s not one person, you don’t need to worry. But it’s usually one person, and it’s usually the principal’s secretary. It’s almost exclusively a woman, so I will call her “she”.

She is actually the VIPest of the VIPs. You will meet her first when you start the year, but that’s the time to get out of her way. She  will be tremendously busy  and ferociously focused, particularly in the days leading up to the start of school.  Get your early business down quickly, smile, and begone.

In my experience, the principal’s secretary has an undocumented but strictly followed communications regimen. I’ll share the one consistent to the three I’ve known; yours may be different.

  • Email–only for work orders or other action items that go to someone else, something she can put in a folder for documentation.
  • Phone–only used for the immediate action of Send Someone Now. There’s a wasp in your classroom. There’s a fight in your classroom. There’s someone injured in your classroom.  You are about to vomit and need someone to babysit while you run to the john. Etc. You don’t call 911. You call her.
  • In person–the best way to handle three or four questions at once. Stop by during prep, or 15-20 minutes after the last bell.

In person, the five most important words to start all conversations with the principal’s secretary are “I’m sorry to bother you….”  Possibly add in “and it’s probably not your job, but I thought I’d check with you first.” Because in most cases, she will have sent out a document giving you the correct procedure, and in most cases you will not have bothered to read it. That’s fine, just slap your head and look apologetic, and try not to ask her two or three times in as many days for the same instructions.

Carefully restrict these in-person visits with questions in the first weeks of school. Don’t be a nag. Whatever other mistakes you might make, never ever think that your needs outweigh the importance of her job. You’re one of, what, 50? 80? If you don’t show up, a sub’s just a few minutes away. If she’s pulled away, a non-trivial chunk of school business gets put off until she gets back.

Eventually, she shares her observations with the principal. You want her report to be positive.

The All Powerful One at my second school had clearly decided long ago that most teachers were trying to make work for her. So outraged was she at the most innocuous query that I resorted to pure groveling.  “I know this isn’t your responsibility, and I swear I wouldn’t ask you except I’ve tried everyone else and you always seem to know everything that’s going on. Do you know where the purchase orders are kept?”


“Oh, ok, it’s not your job to tell people where the purchase orders are kept. Could you tell me whose job it is to tell me where they are? I’m sorry again for bothering you.”

This, she found amusing and deigned to respond with reasonably useful information. After I left, an ex-colleague got in trouble when, irritated at her reflexive outrage, he snapped at her, “I’m helping kids. Your job is to help me.” This earned him a reprimand that went into his permanent file. I advise grovelling.

My current Principal’s Secretary is excellent, properly inspiring fear, respect, and rapid learning curves for all things administrivia. We’d gotten along well for three years until I didn’t call in a sub in a timely manner. No points were granted for my heroic attempt to avoid taking a day off.  I was originally somewhat nonplussed that she didn’t give a rat’s ass about my almost non-existent absentee rate. Then I realized that her job is to get coverage, which meant healthy, noble me was far more hassle than the teacher taking thirteen days a year with a properly notified sub. Humbling.

But she forgave me after a few days of grovelling. I bring her squash and cucumbers every year. Plus, she thinks I’m a pretty good teacher–she’s the mom in this story.

The Attendance Clerk

This will be less focused than the other two because in order to properly value the attendance clerk, you need to understand the importance of attendance.

On the first day at my second school, the union rep reminded us all of the two Do’s and one Don’t: do be on time, do take attendance, don’t touch the kids.  These, she stressed, were the essentials of the job. We all laughed at the truth so brutally expressed: actual teaching is a secondary consideration.

I got a call from my attendance clerk one time, “Why is Darby skipping your class every day?”

I was confused. “He’s at basic training.”

“What? No, he’s not.”

“He said he was accepted to the military and had all the credits he needed to graduate, so he was starting basic training early….this sounds really stupid now that I say it out loud.”

“Yeah, he’s lying. And he’s in all his other classes.”

“Um. No. He’s not. He’s out of town. I know this because he texted another student to ask me not to mark him absent, but I told him…”

“#(S&U#*(&*QT!” and the clerk hung up the phone.

Darby was in an entirely different time zone. His parents were out of town and thought he was in school. When his parents got automatic notifications of his first block absence, he told them he was sleeping in and showing up late. I was the only one of his four teachers marking him absent. The other three thought he was in basic training, too.

At best, that’s embarrassing. At worst, it’s a lawsuit. At really worst, it’s a lawsuit and millions in settlement.

Schools are legal custodians of the children (in loco parentis) while they are in school. Taking attendance creates a legal document, one that is audited and cross-checked, establishing that the student was in the school’s custody. (Note: Many high schools, like mine, have open campuses, allowing students to leave and return. I have never known how that squares with our legal custodial responsibilities.)  That’s not even getting into the fact that schools often get paid for each student in attendance, and the government likes schools to be able to prove in regular audits that they got paid for actual butts in chairs.

All sorts of  caselaw abounds defining school responsibilities, where they exceed parents, what a “reasonably prudent parent” would do, but we’re all just one nasty case and a cranky judge away from utterly ridiculous strictures. Fortunately most of it is out of your purview. Except attendance.  Most of the admins who’ve evaluated me have also checked with the attendance clerks to see how I’ve done. New teachers in particular want that report to be good.

But that’s all just about taking attendance on time, which you should do anyway. Why is it a good idea to be buddies with an attendance clerk?  As you’ll soon observe, these ladies are at best mildly friendly, at worst complete grouches. Their job requires a great deal of nagging teachers, apprehending students in the act of cutting,  and placating parents when teachers (raises hand) accidentally mark a present student absent.  Never mind the daily duties of nagging teachers to take attendance, sign off on their weekly audits, and so on.

But all of this is why it behooves any new teacher to seek them out and befriend at least one clerk. You’ll screw up occasionally. Or a lot, if you’re me. Don’t hide your mistakes. Don’t hope they won’t be noticed, because they will. Acknowledging your errors and emailing them will not irritate the clerks, but win their appreciation. I once apologized to my favorite clerk for being such a screwup–on more than one occasion I’ve somehow missed taking attendance for an entire day and had to email with a deep grovel and my best recollection of who wasn’t there. She laughed. “You’re in the top 15% of all teachers here. Twenty three percent of our teachers don’t ever take attendance.”  I bring them all a bag of heirloom tomatoes to great acclaim.

Pick an attendance clerk to be your “buddy”–she’ll call you up with questions instead of assuming the worst, allowing you to correct minor errors. She’ll send reminders. She won’t nag. She wants teachers to value her work, not despise her picayune corrections. Let her help you. If it ever comes to a lawsuit, you want to feel good about your attendance record.

What about…..?

If you teach K-5, custodial staff replaces attendance clerks. Custodial staff almost makes the cut, but honestly, you won’t need reminders to be nice to them. These are the first folks to enter your room after the last bell, when they get the trash, take a quick look around the room to plan for later. They’re often the first adult you’ll have seen in hours, so smile and take the time to talk.

Leaving administrators. Shouldn’t new teachers cultivate administrators?

Yes, but this is outside of your control. Administrators make their own choices.  I’ve been at two jobs where the teachers loved me and the administrators looked through me, and one job (here) where administrators loved me from the first day, while  three senior math teachers considered me a dangerous radical, best purged.

It sucks to be unpopular with your colleagues. But if you want the time to build relationships, then it sucks more to be unpopular with your administrators. I wish it were a choice. But schools are an ecosystem, and fitting in is outside simple behavior changes.

Of course, that might just be me.

In any event, you don’t need me to tell you to make nice with the boss.

Here’s to a new year.





Great Moments in Teaching: Or, Browbeating Psychoanalysis

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

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

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

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

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

“I know.”

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

“I know.”

“No, you don’t.”

“I get it.”

“No, you don’t.”

“Yes. I see now.”

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



“I know.”

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

“I know.”


“I kno…OK.”

“What’s OK?”

“I understand what you’re saying.”

“No, you don’t.”


“No, it’s not OK!”

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

“Listen to me.”


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

“I know…no. OK. I get it.”

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

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

Warren stood. Affect way off, smiling nervously.

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

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

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

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

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

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

“But I asked for help.”

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

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

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

Warren answered readily, “Amplitude. Period.”

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

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

“Which is also the…..”

“Vertical shift.”

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

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

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

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

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



“Um. No. I don’t know how to start.”

I waited. The class waited.

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

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


“I could…graph it, maybe?”

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

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

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

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

“Thanks for yelling at me.”

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

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

“Don’t be a zombie.”

“Okay–wait. What’s a zombie?”

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

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

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

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

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

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

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

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


GPA and the Ironies of Integration

Grade inflation, score stagnation reports USA Today.  47% of students are graduating with an A- or higher average (A- undefined, but presumably 3.7 or higher). Back in 1998, just 37% were graduating with similar marks. Meanwhile SAT scores have dropped. Inside Higher Education’s take was more skeptical of the SAT connection but covers a lot of the same bases.

Moreover, the SAT scores are stagnant, so these higher grades aren’t evidence of greater learning!  OK, yeah, the SAT isn’t the only college admissions test and it’s changed twice in 20 years. What’s happened to the other college admissions test, which has a larger test base and which has changed very little? Well, one of the researchers works for the College Board, see.


Yes, GPAs are going up. I suspect this is caused by several states banning affirmative action.

Pause. I’ll wait.

[Reader: wait, what What do high school grades have to do with affirmative action?  Affirmative action usually involves college admissions, not high school…oh, well, high school grades are used for college admissions. In fact, now that I think about it,  high school grades don’t really have any purpose save their use in  college applications. ]

Good, you’re caught up.

It appears that voters have given up banning affirmative action not because they approve of it, but because universities have made it clear they have no intention of abandoning their “pursuit of diversity” and the courts have said yeah, okay, we’ll let you And as this how-to guide for avoiding lawsuits makes clear, top of the “diversity strategies” that allow colleges to ignore the will of the voters is the “percent plan”, or taking in students based on their class ranking. Class ranking is set by GPA.

Texas, California, and Florida all created programs to guarantee admission to public colleges for top graduates from each high school in the state. At their most basic level, these programs generate geographic diversity. But since high schools are frequently segregated by class and racepercent plans also create socioeconomic and racial diversity by opening the door to graduates from under-resourced high schools. These are students who may never before have considered attending a major research university. (emphasis mine)

I don’t have any proof that AA is one reason why GPAs are increasing, and I got a bit distracted because frankly, I don’t care about GPA. No, that’s a lie. I care a lot about GPAs. I think they’re fricking evil, and I get a bit nauseous when someone bleats about how they reflect the virtue of hard work. Look, GPAs are worthless information. Grades aren’t even consistent from teacher to teacher, much less school to school, much less aggregated into one big nationwide chunk. Many teachers grade participation and homework on the same basis as tests–some are even required to boost or reduce demonstrated ability with effort or citizenship grades.  Tests are usually the teachers’ own creations. Some are terribly unfair, some are just terrible. And some are very good–so good, in fact, that the teachers reuse those tests year after year, and the students sell images of them to “tutoring services” and each other, thus rendering their goodness inert.

But I don’t really care why GPAs are rising. The italicized part of the paragraph–since high schools are frequently segregated by class and race–operated like a bright shiny object to distract me from an unpleasant subject.

Yes. Since most blacks and Hispanics go to majority black and Hispanic schools, the students with the highest GPAs will be black and Hispanic. Left unmentioned:  the standards will be lower than they are at majority white or majority Asian schools. Unmentioned but not unnoticed, obviously. If blacks and Hispanics were achieving at the same level, then no one would bother with affirmative action, much less banning it.

Evidence of the lower standards are a time-honored journalism time-killer; I wrote about the  Kashawn Campbell saga a few years ago as an example. But sob stories usually involve kids in the deepest of high poverty cases. Often the top 10% of an all URM low-performing high school will go on to decent colleges and do adequately. They might be the ones we read about who abandon STEM and go into an identity major, but a decent chunk of them are getting through the system that was rigged for them just as anticipated.

Still, these kids represent a  chilling inequity. The  de facto segregation that enable this faux meritocracy mean that the B and even C kids at almost any other type of school is more accomplished, on average.

Just recently I looked at African American participation in AP classes over the past 20 years. Mean scores dropped in almost every test, and scores of 1 saw the most growth.  Hispanics have similar stats. Beware any time someone brags about Hispanic AP pass rates–they have the Spanish Literature and Language tests boosting their scores. Whites and Asians…don’t.

Many black and Hispanic students are prepared and can pass the tests.  An open question, though, is whether the qualified kids are going to the schools that offer up the top 10%. I have my doubts.

But urban schools aren’t really playing GPA games–not consciously, anyway. They don’t have time. Other schools are a different story.

Majority URM charters, for example, have the same incentives as urban public schools–more, even, since what’s the point of charters if there’s no bragging to be done? Charters can be very subjective about grades. Other, more diverse (at least at first)  charters are progressive, designed for suburban parents in racially diverse school districts who aren’t quite wealthy enough for private school or houses in less racially diverse districts.

These suburban charters have another advantage. Remember Emily in Waiting for Superman? Emily’s public high school is in Woodside, California, one of the richest communities in the country. Woodside is considered a very strong school for those in the top track, offering a number of high performance classes that aren’t just open to anyone. Emily wasn’t considered strong enough for these classes, so she went to Summit, a school that’s very grateful for any donations. Think Emily got better grades at Summit?

I’ve written much about “Asian” schools (more than 50% Asian), as well as their selection of Advanced Placement class preferences, as well as the fact that their grades and test scores often seem acquired with no retention (and perhaps not acquired). Most of the students take 11 or 12 AP courses in a high school career, valedictorians have GPAs above 4.4, and they’re ten-way ties. Taking geometry freshman year is considered remedial.

But as both Toppo and Jaschik report, it’s predominantly wealthy and white schools, public and private, that have seen the most inflation.  I suspect that these schools have increased GPAs the most because grades were lower to begin with. These kids were once considered in an entirely different context from affirmative action admits. They had better course offerings, better teachers, stricter grades, but of course much higher test scores. Twenty years ago, affirmative action bans kicked in and Asian immigration skyrocketed. These parents began to realize the competitive disadvantage their children faced and I suspect started demanding more. Class rankings probably disappeared for similar reasons–their 40th percentile student achieves far more than the best students from urban schools. Don’t feel too bad for the students–remember, given a choice between a casually high-achieving rich white and an endlessly studying, grade-obsessed Bangladeshi immigrant who has been attending test prep since second grade, the white kid wins every time. Their parents write checks. Plus, legacy.

I know next to nothing about poor white rural schools. Reporters and colleges don’t care about them, and I don’t have any nearby to study.

So that’s all the “racially isolated” cases, be they URM, white, or Asian. What’s left? The Woodside Highs that Emily wanted to escape, at the high end, and schools like mine at the low end. The integrated schools.

Integrated high performing schools, in rich areas that can’t quite shut out the low income and middle class kids, are tracked without fear of lawsuits. Usually three tracks: high (mostly whites and Asians), medium (white boys and  strong URMs, but a mix of everything), low (almost entirely URM).  The rich parents will take their kids, and their money, elsewhere if they can’t be assured of high standards. There will be no talk of insufficient black and Hispanic students in the advanced classes, but nor will there be complaints  if the students are qualified.

Integrated low performing schools, like mine, can’t track and can’t assure high standards. There will be talk of insufficient black and Hispanic students in the advanced classes, and wholly unqualified kids are often plunked in despite loud protests from both teacher and students.

In lower performing integrated schools–stop, for a minute. I don’t mean these schools are terrible or that kids graduate incompetent. But these are schools that can’t really push high achievers hard, because of the racial imbalances that result and get them into  trouble. Asians dominate the top track. Their parents demand that their kids be put into advanced classes early, often look for ways they can test out of requirements. White parents in these schools are usually middle or lower class. While they’re often concerned about school, they aren’t planning on stressing the next four years. They’ve realized that their kids are probably going to spend two years at community college and hey, why fight about it? They know competing with the Asians is out–white kids rarely want academic achievement that badly, and their parents don’t blame them. White parents’ biggest fear is the contagion of low grades. Not only are there many other kids around failing classes, making summer school or repeating classes seem normal, but the teachers are used to giving Fs–in fact, sometimes they get in trouble if their Fs aren’t racially balanced. My guess:  white kids at integrated schools have seen relatively little GPA boost in the last 20 years.

Demographic footprints being what they are, Asians and white kids will still fill the top ten percent plans, leaving room only for really bright, accomplished black and Hispanic kids. Average black and Hispanic kids, who would shine at a majority URM school, are often getting Bs and Cs despite far better skills. This is a point I can speak to personally, having seen it often in test prep.  Black or Hispanic kids with low test scores and 3.9 GPAs from weak progressive charters, while those going to the local public schools have 2.5 or lower GPAs and much higher test scores.

So grades at integrated schools, whether high oer low performing, are a drag. At high performing schools, grades are intensely competitive. At lower performing schools ( these integrated low performing schools are a drag for everyone except Asian immigrant kids.  If Asian parents would stop cocooning, they could probably get much better results by spreading out around the country, ten to twenty a school. Enough to tie for valedictorian. But most of them appear to be doing their best to force racial isolation. Asian immigrants, at least, have little interest in attending integrated schools.

Of course, not all Asian kids fit this profile, just as many blacks and Hispanics pass AP tests in Calculus, US History, and Biology.

If I had to rank my personal preference, the rich white kid schools do some fine educating. All Asian schools and high performing integrated schools are joyless places, although the latter have some stupendous sports.

What the integration advocates want, I think, are what they see in progressive charters. Children of all abilities, working and playing together, learning at the same pace, earnest, hardworking, and virtuous. But charters are artificial environments. True integration would probably look something like my school. Poor black and Hispanic kids would get better educations, but worse grades. Colleges wouldn’t be able to get around affirmative action bans. High standards would be impossible unless we were allowed to track.

I do believe they call this a collective action problem.

Anyway. Grades are increasing because colleges are de-emphasizing test scores. Yes, this means they should be required to return to testing, but perhaps in such a way that Asians couldn’t game it? And as Saul Geiser suggests, perhaps criterion referenced tests would be better.

See why I loathe grades?

This is a bit disjointed; I’ve been having trouble focusing lately. I may rewrite it later.



Teaching Transformations

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

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

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

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

That looks like this:


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

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

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

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

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

-5 -3 2

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


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

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

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

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

“Exactly. So let’s try that.”

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


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

x f(x)
1 1
2 2
3 3


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

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

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

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

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


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

“f of x!”

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

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

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


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

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



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

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

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


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

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

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

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

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

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

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

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

Glenn, John, and Philip K. Dick

In the last segment of their recent bloggingheads discussion, John McWhorter told Glenn Loury he was writing a piece suggesting that discussions of an IQ gap be deemed unacceptable for public discourse1:

I don’t understand what possible benefit there could be. I think it should stay in the journals.

Glenn agreed, and mentioned his famous response to the conservative welcome of The Bell Curve, in which he angrily rejected Murray’s assertions.

My first, instantaneous reaction was hey, great idea.  In fact, it will save us billions. Too bad no one proposed it twenty years ago, before we wasted so much time and energy on No Child Left Behind, forcing states to report by racial demographics. Think of all the schools struggling to meet adequate yearly progress and failing because they couldn’t teach students to perform in perfectly average racial unity. Let’s be sure to tell the College Board; they’ll be happy to stop breaking out test results by race; it’ll save them so much criticism.  This will put NAEP out of business of course, so….

What’s that? You don’t think McWhorter would link banning discourse on race and IQ to ending all scrutiny of race-based academic achievement?

Ya think?

Of course John McWhorter knows that race-based academic achievement is at least tangentially related to discourse on race and IQ.  I also think he understands that race and IQ discussions have been The View That Must Not Be Spoken for forty years. In fact, he even mentions that the only sites engaging in this discusssion are “right wing chat sites” and “some blogs”.

So sarcasm aside, I was a bit puzzled by the proposal, as well as Loury’s endorsement. These are two intellectually honest academics, who are generally fearless on racial topics. Here they are declaring certain topics Voldemortean–and doing so in the event that the link between race and IQ is proven out.

Without even going into the suggestion itself, consider that McWhorter posits one response to an acknowledge racial IQ gap:

…are we going to have an arrangement where we allow that black people have these lower IQs and therefore give extra help to black people, with the presupposition that all the average black people aren’t as bright? I think the black people would take it as an insult.

Fifty years and counting on affirmative action (which McWhorter famously opposes), and McWhorter thinks black people would take extra help as an insult? Seriously?

Even more surprisingly, Glenn Loury doesn’t point out this obvious hole.  Or maybe not so suprisingly, I’m a huge Loury fan, but as I’ve mentioned, he’s got some blind spots on education.

The most notable one I can recall is in this  excellent 2010 discussion Loury hosted with Amy Wax, author of Race, Wrongs, and Remedies:

Wax’s central point both in the discussion and in her book is that black academic underperformance is due to the wholesale collapse of the black culture and black family.

Loury pushed back hard (emphasis mine):

“Well, gee, when I look at education, I think, true, what happens at the home is really important….but I do think that the fundamental problem with urban education is its political economic organization. It’s the unions. It’s the work rules. It’s the efficacy of teachers in the classroom, and so on. It may also be to some degree the resources.

And that’s something that government, with great difficulty, with some consternation, can change….that’s something that ONLY government can change, either by making resources available to parents so they can have outside options and not be reliant on a school system that’s failing, and/or by reorganizing the functioning of those systems that are failing so that the kind of things that reformers want to do, that are known to work, are permitted to be put into place.

But in either case, those are political public large government undertakings that need to be done, and to blame the failure of urban education on the culture of African Americans is, well, maybe just a little offensive….Why wouldn’t we want to think of that as a public problem in public terms rather than [blame] the people who are really the victims of failed public policy?”

So Loury holds on to some of the conservative tenets from his youth.  Like all conservatives, he wants to blame schools: crappy, incompetent teachers, unions, and by golly maybe more money would help, too.

Wax responds that blaming schools makes no sense, because black kids in suburban schools are doing poorly. They do poorly in the same excellent schools that whites do well in, and she argues again that it’s culture.

Somewhat surprisingly, Loury agrees. Suburban black kids are doing terribly, too, but not for the same reason:

“I would not identify underperforming middle class African American performance in good suburban schools, which is a real thing, with the absolute wasteland in terms of the cognitive development of the students who have no alternative but to attend the failed inner city public school system. Both are problems, but the latter is just a huge problem and I say it’s mainly a problem of failed institutions not of inadequate culture.

So Loury has constructed  an absurd dichotomy: Middle class blacks and urban blacks have weak academic performance and he tacitly agrees with Wax that the problem is cultural. But urban black academic performance is about ignorant teachers and union work rules.

Wax comes back with an answer that any teacher would thank her for (which is why I’m quoting it in full)

“I would really argue, are the teachers really not trying to teach them basic stuff, or is it that the students, for whatever reason, are just so ill-equipped when they come in, so indifferent?….We know, from Roland Fryer…that these students are coming in significantly behind their counterparts, so they’re already behind the eight ball…then to turn around and say it’s the school’s fault that they can’t learn arithmetic, or they can’t learn to read. I really question that. I just am not sure that that is the locus of the problem or even the main problem.

Of course, then she ruins all that good sense by saying that the locus of the problem is  “the abject failure of the family….just collapse.”

Little evidence for either Wax or Loury’s position exists. Roland Fryer’s research on Harlem Children’s Zone ( the 2010 version) gets a mention by Loury as evidence that charters can do a better job. That’s odd, because Fryer’s research has widely been cited as evidence that HCZ does only an adequate job as a charter school, far less impressive academically than KIPP or the others–and of course, their “improvement” has a whole bunch of caveats. But Wax appropriately observes that the results, back in 2010 were new and fadeout often occurs.  As the medium term results from HCZ show,  lottery losers indeed  “caught up” in many ways.

Loury often swats his discussion partners for “asserting from faith”, but his demonization of  “failing schools”is exactly that.   Without any evidence, and considerable evidence against his position, Loury argued two separate culprits for black underachievement, depending on the SES category. Moreover, he uses a big ol’ group of conservative shibboleths to justify his position. That is not the Loury I know from other conversations.

For entirely understandable reasons, both Loury and McWhorter see any discussion of the IQ gap as a personal affront. They both interpret “racial IQ gap” as “blacks are inferior” and I’m sure that there are people who push the topic who see it that way. But one can angrily reject average group IQ as a sign of inferior or superior status while still acknowledging the hard facts of cognitive ability–Fredrik deBoer does it all the time. Whenever they discuss race and IQ, Glenn and John jokingly  mention how smart they are–which they are! But they don’t ever acknowledge that their tremendous intellects aren’t a rebuttal to the discussion at hand.

Men are, on average, taller than women. Michelle Obama is taller than Robert Reich. Both statements are true. Why more people can’t apply this to IQ is a mystery.

I don’t know all the corners of IQ science history, but I’ll stipulate that many unpleasant people discuss IQ gaps with a disgusting glee. I find it incredibly troubling how many people use “genetically inferior” as an equivalent term for “blacks have lower IQs on average”.  But I spend too much time with students of all abilities, and all races, to consider race as the logical grouping for IQ.  I’m more interested in IQ, or more generally the cognitive ability discussion, as a starting point to correctly frame what is now cast as a public education failure.

Our schools “fail” to educate many students. We tend to focus only on the black and Hispanic students–and not as individuals, just as data points that would push the average up nearer to that of whites.  We use white average academic achievement as the standard for success.  We began comparing racial groups back when it was primarily blacks and whites, in that optimistic era after Jim Crow ended, confident that the data would show blacks catching up to whites. If blacks had just caught up, if we just had the same amount of students “failing” to be educated, we’d have moved on.

But blacks never caught up. Since at the national level, we’d begun with the presumption that the gap was caused by racist oppression, we continued with that assumption as long as possible. Over time, other culprits arose. It’s the parents. It’s the culture. It’s the schools. People who offered cognitive explanations were ostracized or at least subject to a barrage of criticism. It’s always odd to hear Loury talk about the Bell Curve era as a traumatic time, when his peers were coolly discussing racial inferiority, while almost everyone else recalls it as a time when Murray was nearly banished from the public square.

As time went on, despite our failure to close “the gap” or, more accurately, despite our failure to provide a needed education for all students, demands went up. High school transcripts got more impressive, more loaded with “college prep” courses and, in many high schools, more akin to fraudulent documents, all designed to push all kids into college in equal proportions. Colleges have obligingly obliterated requirements. Schools have increasingly come in for blame from the political and policy folk, but all attempts to penalize schools for their failure have, well, failed. The public likes their schools and the public, frankly, is more willing to consider cognitive ability relevant than the political and policy folk are.

I’m always reminding myself that most  people see it in, literally, black and white terms for very good reason. But I only discuss IQ in terms of race because society insists on grouping academic achievement by race. Ultimately, I see IQ discussion as an effort to correctly categorize the “failure” of some students, regardless of race. I see it as a way of evaluating student achievement, to see how to best educate them to the extent of their ability and interest. I am well aware that these questions are fraught with reasonable tension.

But I worry, very much, that we won’t take needed steps both in education and immigration policy (as well as a host of other areas, no doubt) if we don’t stop insistently viewing cognitive issues through the prism of race. That is, as I first wrote  here, we need to consider the possibility that the achievement “gap” is just an artifact of IQ distribution. 

I would be pleased to learn this is not the case, as I wrote then. But if in fact IQ distribution explains the variations in academic achievement we see, then we need to face up to that. This facing up does not mean “well, your people are good in sports and music”. The facing up means asking ourselves regardless of race, how do we create meaningful jobs and educational opportunities for everyone?


I hope they change their minds. Because we are putting millions of kids in schools each year, making them feel like failures. Yes, some are black. Others are white, Hispanic, Asian. And we’ve spent no time–none–trying to figure out the best ways to educate them.  We’ve only looked for causes, for the right groups to blame.

Of course,  maybe we could trade: no more talk of the achievement gap in exchange for no more talk of race and IQ. That’s not the best approach, though, because in today’s employment environment, we need to educate everyone, not write off “failures”.

But I guess Glenn and John–along with a lot of other people–are still trying to wish reality away.

Note: The piece  Philip Dick, Preschool, and Schrodinger’s Cat is still canonical Ed on IQ.

 1: since I wrote this, McWhorter published the piece in the National Review. I made some additional responses directly on Twitter.

Teacher Federalism

A year or so ago, our school’s upper level math teachers met to define curriculum requirements for algebra two.

I’d been dreading this day for several weeks, since we agreed on the date.  I teach far fewer Algebra 2 topics than the other teachers. Prioritizing depth over breadth has not made me terribly popular with the upper math teachers–who of course would dispute my characterization of their teaching. There were three of them, plus two math department leaders who’d take their side. I’d be all alone playing opposition.

Only two possible outcomes for this meeting. I could, well, lie. Sign off on an agreed curriculum without any intention of adhering to my commitment. Or I could refuse to lie and just and fight the very idea of standardization The good news, I thought, was that the outcome would be my choice.

Then the choice was taken away from me.

Steve came into my room beforehand. Steve is the member of the upper math group I’m most friendly with, which means we are, well, warily amicable. Very different characters, are we. If you’re familiar with Myers-Briggs, Steve is all J and I’m as P as P can be.  But  over the years we realized that while our approaches and philosophies are polar opposites, we are both idiosyncratic and original in our curriculum, more alike than we’d imagined. He was interested by my approach to quadratics and his approach to transformations is on my list of innovations to try.

So Steve tiptoed into my room ahead of time and told me he wanted the meeting to be productive. I went from 0 to 95 in a nanosecond, ready to snap his head off, refusing to be held responsible for our departmental tensions, but he called for peace. He said it again. He wanted this meeting to be productive.

I looked, as they say, askance. He asked me if I would be willing to settle for good, not perfect. I said absolutely. He asked me to trust him. I shrugged, and promised to follow his lead.

For reasons I won’t go into, no one expected Steve to run the meeting. But in the first five minutes, Steve spoke up. He said he wanted the meeting to be productive. He didn’t want the perfect to be the enemy of the good.

We all wanted what was best for our students, he said. We all thought we knew what was best for our students. But we had very different methods of working. If we tried to agree on a curriculum, we’d fail. Eventually, someone in power, probably at the district, would notice, and then that someone might make the decision for us.

So rather than try to force us all to commit to teaching the same thing, why not agree on the topics we all agreed were essential, “need to know”?  Could we put together a list of these topics that we’d all commit to teach? If it’s not on the list, it’s not a required element of the curriculum. If it was on the list, all teachers would cover the topic. We’d build some simple, easily generated common assessments for these essential topics. As we covered these topics–and timing was under our control–we’d give the students the assessment and collect the data. We could review the data, discuss results, do all the professional collaboration the suits wanted.

If we agreed to this list, we would all know what’s expected. All of us had to agree before a topic went on the “need to know” list. No teacher could complain if an optional topic wasn’t covered.

I remember clearly putting on my glasses (which I normally don’t wear) so that I could see Steve’s face. Was he serious? He saw my face, and nodded.

Well. OK, then.

Steve’s terms gave me veto power over the “need to know” list.

Wing and Benny were dubious. What if they wanted to teach more?

As requested, I backed Steve’s play.  “We could make it a sort of teacher federalism. The “Need to know” list is like the central government.  But outside these agreed-upon tenets, each individual teacher state gets complete autonomy. We can teach topics that aren’t on the list.”

“Exactly,” Steve added. “The only thing is, we can’t expect other teachers to cover things that aren’t on the list.”

In other words, Steve was clearly signaling, no more bitching about what Ed doesn’t cover.

We agreed to try building the list, see if the results were acceptable. In under an hour, we all realized that this approach would work. We had 60-80% undisputed agreement. At the same time, Wing and Benny had realized the implications of the unanimous agreement requirement. A dozen or more items (under topics) the other three teachers initially labeled eessential) were dropped from the “need to know” list at my steadfast refusal to include them.  Steve backed me, as promised.

While all three raised their eyebrows at some of the topics downgraded to the “nice to have” list, they all listened carefully to my arguments. It wasn’t just “Ed no like.” As the day went on, I was able to articulate my standard–first to myself, then to them:

  1. we all agreed that students had to come out of Algebra 2 with an indisputably strong understanding of lines.
  2. We routinely have pre-calc students who need to review linear equations. In fact, I told them, this realization was what led me to dial back algebra 2 coverage.
  3. Non-honors students were at least a year away from taking precalc, which was where they would next need the debated skills. If some of our students weren’t remembering lines after three years of intense study, how would they easily remember the finer points of rational expressions or circle equations, introduced in a couple weeks?
  4. This called for limiting new topics to a handful. One or two in depth, a few more introduced.
  5. Our ability to introduce new topics in Algebra 2 was gated by the weak linear knowledge our students began with. If we could convince geometry teachers to dramatically boost linear equations coverage, then we could reduce the time spent on linear equations in algebra 2.

Once I was able to define this criteria, the others realized they agreed with every point. Geometry priorities were a essential discusison point, but outside the scope of this meeting and a much longer term goal. That left all debate about point 4–how much new stuff? How much depth?

This reasoning convinced them I wasn’t a lightweight, and they all knew that my low failure rate was extremely popular with the administrators. So they bought in to my criteria, and were able to debate point 4 issues amicably, without loaded sarcasm.

I knew I needed to give on topics. At the same time I was shooting down topics, I was frantically running through the curriculum mentally, coming up with topics that made sense to add to my own curriculum, making  concessions accordingly.

The other teachers looked at the bright side: I’d be the only one changing my curriculum. Every addition I agreed to had to be carefully incorporated into my already crowded Algebra 2 schedule. I did have some suggested additions (a more thorough job on functions, say), but none of mine made the cut. The other teachers’ courses were entirely unaffected by our “need to know” list.

At the end of the day, we were all somewhat astonished. We had a list. We all agreed that the list was tight, that nothing on the “like to know” or “nice to have” list was unreasonably downgraded. I want to keep this reasonably non-specific, because the issues apply to any subject, but for the curious: rational expressions were the most debated topic, and the area where I made the most concessions.  They covered addition and subtraction, multiplication and division, graphing. We settled on introduction, graphing of parent reciprocal function and transformations, multiplication and division. Factoring was another area of dispute: binomial, of course, but I pushed back on factoring by groups and sum/difference of cubes. We agreed that exponential functions, logarithms and inverses must be covered in some depth, enough so the strongest kids will have a memory.

“What about grades?” Benny asked. “I don’t want to grade kids just on the need to know list.”

“But that’s not fair,” I objected. “Would you flunk kids who learned everything on the need to know list?”

“Absolutely,” Wing nodded.

I was about to argue, when Steve said “Look, we will never agree on grading.”

“Crap. You’re right.” I dropped the subject.

In a justly ordered world, songs would be sung about “That Day”, as we usually call it. Simply agreeing to a federalist approach represented an achievement of moon walk proportions. Then we actually built a list and lived by it, continually referring to it without the desire to revisit the epic treaty. Stupendous.

I  didn’t write about the agreement then because I worried the agreement would be ignored, or that other senior math folk would demand we revisit. Instead, our construction of the  “Need to Know” list shifted the power base in the math department in interesting ways.   Our point man on these discussions did indeed express displeasure with the Need to Know list. It’s too limited. He wants more material on it. He expected us to comply.

Wing, Benny, and Steve could have easily blamed me for the limits. “Oh, that’s Ed’s doing. We all want more on the list.” Instead, upper math folk presented an instantly united front and pushed back on incursion.  No. This works for us. We don’t want to break the agreement. We like the new productivity of our meetings. Team cohesion is better. Wing and Ben still think I’m a weak tea excuse for a math teacher, but they understand what we’ve achieved. With this unity, we are less vulnerable.

In short, we’ve formed our own power base.  As I’m sure you can guess, Steve is the defacto leader of our group, but he gained that status not by fiat, but by figuring out an approach to handle me that the others could live with. No small achievement, that.

Will it last? Who knows? Does anything? It’s nice to watch it work for the moment. I’ll take that as a win.

We’ve used that agreement to build out other “need to know” lists for pre-calc and trigonometry. They aren’t as certain yet, but Algebra 2 was the big one.  Worth the work it took to update my curriculum.

Our teacher version of federalism has allowed us to forge ahead on professional practices, lapping the lower level crew several times. In fact, on several department initiatives, the upper math department has made more progress than any other subject group, something that was duly noted when hot shot visitors dropped in on our department meeting. The other groups are trying to reach One Perfect Curriculum.

I’m not good at describing group dynamics unless it’s in conversational narrative. But I wanted to describe the agreement for a couple reasons.

First, some subject departments  operate in happy lockstep. But many, even most, high school math departments across the country would recognize the tensions I describe here. .  I recommend teacher federalism as an approach. Yes, our agreement may be as short-lived as some “universal curriculum” agreements. But the agreement and the topics list are much easier to agree to, and considerably more flexible. I’ve seen and heard of countless initiatives to create a uniform curriculum that foundered after months of work that was utterly wasted. Our group has had a year of unity. Even if it falls apart next year, that year of unity was purchased with a day’s work. That’s a great trade.

But in a broader reform sense,   consider that none of the four teachers in this story use books to teach algebra 2. Not only don’t they agree on curriculum, but they don’t use the same book. Some, like me, build from scratch. Others use several books as needed.  Our epic agreement doesn’t fundamentally change anyone’s teaching or grading. We simply agreed to operate as a team with a given set of baselines.  Noitce the words “Common Core” as the federal government (or state, your pick) defines it never made an appearance. It was simply not a factor in our consideration.

Does this give some small hint how utterly out of touch education policy is? How absurd it is to talk about “researching teacher practice”, much less changing it? I hope so.

The Invisible Trump Voters

According to Google, only  Steve Sailer and  Alexander Navaryan have pointed out that Bret Stephens’ call for  mass deportation of Americans was actually a diatribe against blacks and Hispanics.

But just imagine trying to point that out in a public venue:

“You’re denying you were calling for blacks and Hispanics to be deported? Why would anyone believe you were referring to white people? They don’t have the highest illegitimacy rates, the highest incarceration rates, the worst test scores….”

As Steve pointed out a few years ago, noticing things is a problem. In this particular case, noticing Bret Stephens’ callous provincialism would cause far too many problems. Anyone who dared point out the obvious, if unintended, target of the slur would be risking media outrage–all the more so because the media wouldn’t want anyone wondering why they hadn’t noticed the attack on African American and Latino honor. That’s probably why Navaryan hastened to add that most of the outrage seemed to be from media outlets popular with “right-leaning whites”.

Damon Knight intro to a 1967 Robert Heinlein collection that’s often proved illustrative:

People are still people: they read Time magazine, smoke Luckies, fight with their wives.

Knight, one of the great science fiction editors, wrote this essay  two years before his wife Kate Wilhelm became one of the first female Nebula winners. Knight and Wilhelm led widely acclaimed writer’s workshops for years. (The great Kate is still writing and running workshops. Bow down.)

In short, Knight wasn’t particularly sexist. But   when he wrote “people”, he meant “men”.

Bret Stephens and most of the mainstream media aren’t particularly racist. But when Bret wrote about deporting “Americans” and  “people”, everyone read “whites”.

So this whole episode reminded me of the invisible Trump voter. Not the ones people usually mean, like the ones discussed in this  article on journalism’s efforts to find Trump voters.  Everyone talks about the downscale white voters, but they aren’t invisible anymore. Those white voters, many of them recently Democrats, finally turned on the party and put Trump over the top. I’m talking about the Trump voters still unseen.

Consider the Republican primary results by county:


That’s a lot of counties Trump won. New York and New Jersey went for Trump, as did Virginia and Massachussetts. He won California with 75% of the vote, after Kasich and Cruz had withdrawn but were still on the ballot.  (Trump also had a commanding lead in the polls, for what they’re worth, when the race was still in play.)

Trump did very well in high immigration states during the primaries. At a time when Never Trumpers were attempting a convention coup, Californians could have given them ammunition by supporting Kasich or Cruz with a protest vote.  Arnold Schwarzenegger put it about he was voting for Kasich. No dice. Trump won every county.

All these states that ultimately went commandingly blue, of course. But Trump voters are white voters. Hillary Clinton won thirteen of the states that had exit polls, but only won the white vote in four of them:

Clinton state, Trump won white voters Clinton state, Clinton won white voters
Virginia (59%)   Washington (51%)
Nevada (56%)  California (50%)
New Jersey (54%)  Oregon (49%)
Minnesota (53%)
 Maine (47%)
Illinois (52%)
New York (51%)
New Hampshire (48%)
Colorado (47%)
New Mexico (47%)

These are the Clinton states that didn’t have exit polls, with her percentage of the votes and the state’s percentage of white non-Hispanics (not the percentage of white votes, which isn’t available):

State % NHW
Hawaii 63% 26%
Maryland 60% 44%
Massachusetts 60% 74%
Vermont 57% practically everybody
Rhode Island 54% 76%
Connecticut 54% 71%
Delaware 53% 65%

Hard to see how Trump lost the white vote in Maryland, Connecticut, or Delaware. But if you give her all seven, she still only won the white vote in eleven states total. A more realistic guess is eight or nine. And for a liberal bastion, California’s white vote was surprisingly close. California has fewer working class whites than New York and New Jersey, but  California’s white voters supported Romney in 2012 and Bush in 2004. Perhaps a lot of Republican voters stayed home rather than vote for Trump.

Why so much support? Well,  in September 2016, a California poll showed whites were almost split on immigration–only 52% saying immigrants were a boon, 41% saying they were a drain on public services. I looked for similar polls for other blue states and couldn’t find any. But that’d certainly be an avenue to explore.

These are big states, and 30% of big states is a big ol’ number of voters. The LA Times observed that only Florida and Texas gave Trump more votes than California. Nearly a million people voted for Trump in Chicago and the “collar counties”, as many as the entire state of Okalama,  The San Francisco Bay Area counties and Los Angeles County contributed roughly 600,000 each, slightly less than Kansas gave Trump or the combined Trump votes of Montana and Idaho. New York City counties kicked in close to half a million, slightly more than the combined vote of the Dakotas.

Consider, too, that these voters knew full well that their vote wouldn’t matter and they went out and voted for Trump anyway. If every Trump voter in California, New York, New Jersey, and Illinois had simply stayed home, he’d still be President, most of the local races would have unchanged results, and Hillary’s popular vote margin would be four or five million more.

These voters pay too much rent to be hillbillies.  They live in some of the most expensive real estate in America, so they’re not likely to be poor or unsuccessful. Vox’s condescending tripe about the home-town losers voting for Trump because they’re racist, sexist losers afraid of change doesn’t  explain the millions of voters in high immigration areas who voted for Trump. Emily Ekins typology of Trump voters doesn’t seem to cover these voters, either. Why would Staunch Conservatives who could afford the high rents of blue states continue to live in places so at odds with their values? Free Marketers wouldn’t have voted so enthusiastically for Trump in the first place.  I suspect Ekins has defined American Preservationists too narrowly.

How can anyone argue that Trump’s support in Deep Blue land is racist? Huge chunks of white Trump voters in blue states work, live, send their kids to school with a range of diversity in culture, race, and economics that elites like Bret Stephens can’t even begin to comprehend. They often live cheek and jowl with people who speak no English at allwho speak no English at all, and have to handle endless cultural issues that arise from having Russian, Chinese, Syrian, or/and Congolese neighbors, usually uninterested in assimilating and often with no visible means of support.  They see schools struggling with policies designed for a much simpler bi- or tri-racial country, policies designed with the expectation that most students would be Americans. They see immigrants qualifying for tremendous educational expenditures, guaranteed by law, supported by a court that shrugged off the cost   of guaranteeing all immigrants access to public schools. They see the maternity tourism that will allow yet anothe generation of Chinese  natives gaining access to public universities while not speaking any English.

They see immigrants voting by race, supporting Democrats despite a generally tepid lack interest in most progressive causes,  simply to assure themselves the ability to bring in relatives (or sell access through marriage or birth certificate fraud). They’re used to white progressives imposing near total rule on the government using the immigrant citizens voting strength to enact policies that the immigrants themselves will ignore or be unaffected by, but the white citizens, in particular, will pay for.  These Trump voters watch immigrant enclaves form and slowly gather enough voters to vote in politicians by race and religion.  They might worry that white progressive rule will give way to a future of a parliamentary style political system in which various immigrant political forces who don’t consider themselves American, but only citizens, combine to vote not for progressive or conservative values, but some form of values genuinely alien to Americans.

They think it’s hilarious, but not in a good way, when reporters earnestly reassure their readers that immigrants don’t qualify for benefits, or that non-citizens aren’t voting. They see tremendous fraud and illegal behavior go unpunished. They know of huge cash only malls run by immigrants, and know the authorities will never investigate. Fortunately, the authorities do find and prosecute all sorts of immigrant fraud rings, but that only makes them wonder why we bring in so many immigrants to begin with.

I suspect that between thirty and fifty percent of white people living in high immigration regions voted for Trump. But if they see the worst of intensive immigration, they also haven’t chosen to leave it. They don’t say “people” and mean “whites”, like Bret Stephens.

Ironically, Bret Stephens is furious at the downscale “losers” who voted for Trump–the voters who don’t usually vote Republican, or even vote at all. He’s too ignorant, too blind to realize he’d also have to deport millions of invisible Trump voters, the voters he might grudgingly concede are successful, who pay more in taxes than they cost the government, who start successful businesses, who have children they can support. The voters who have been voting Republican a long time without any real enthusiasm, who have always been less than enthused about the values he arrogantly assumes are universally held by Republicans. The white voters whose existence he doesn’t understand enough to write about.

These invisible Trump voters have a lot to risk by going public. But reporters should seek them out. How many of the Trump voters in Deep Blue Land, the ones making it in the high-rent, high-immigration, highly educated regions, how many made him their first choice? And why?

So c’mon into Blue Land, Salena, Chris. Talk to some of the invisible Trump voters that haven’t really been considered yet. Let them add to the story.



The Trump Effect: Reboot or Yesterdays Enterprise?

The first Star Trek “reboot”  took the bold act of altering the past in a famous fictional timeline. The new movies have the freedom to reinvent, while we watch the movies, fully aware what “really” happened. This got taken to extremes for “Into the Darkness”, when the last half hour echoed word for word the greatest Star Trek movie ever made with a character swap, but it’s still pretty clever.

Ever since Trump won in November, I’ve felt like we’re all living through an alternate timeline. Like Tom Hanks’ “Doug” said in that sublime Black Jeopardy skit, “Come on, they already decided who wins even before it happens”. Everyone of any importance knew Hillary would win.  Jobs were accepted. Plans were made.

But while I see it as a reboot, an opportunity to rewrite the future, all the people with any voice or influence think of the election as Yesterday’s Enterprise. Just as the Enterprise C slipped through the temporal rift and forestalled the truce between the Klingons and the Federation, so too did a whole bunch of voters escape the notice of the Deep State. Which is a good thing, because otherwise the Deep State have taken action before the election . Trump would have been doped up and stuck in bed with a dead transgender Muslim and a live boy peeing on him. Not that this would have cost him the election, but at least they’d have grounds for an arrest.

Instead, most of the elite institutions were stunned by the actual voters making a choice that defied all their warnings, their  manifest horror at Trump’s candidacy, never mind his primary triumph. They haven’t stopped trying to convince us of our mistake.

A couple weeks ago, I was really upset at the many corners of the media openly and excitedly debating whether it’d be better to use impeachment or the 25th Amendment to rid themselves of this meddlesome Trump–where even the opponents to the idea concurred that Trump was a witless boob, inept and obviously unfit, that impeachment was reasonable or that the fish rots from the head.

When I realized that the feeling was….familiar. Flashback to a year earlier, back in March and April, when anti-Trump elite GOPs were debating the best way to rig the convention,  gleefully mocking Trump and his voters as Cruz stole his delegates,  happily contemplating a Kasich-Cruz alliance.  Deep in the stunning beauty of central Idaho, I was struggling to enjoy spring break because I knew, beyond any doubt, that the media and institutional powers of the conservative movement would do anything within their power to deny the voters’ choice.

At some point, I realized the idiocy of letting this nonsense get to me and went hiking. Well, walking around a mountain and going up a few hundred feet. It felt like hiking.

But  Trump triumphed.  We got to mock Nate Silver’s open dismissal of Paul Manafort’s prediction of locking up the nomination as “delusional” when in fact the job got done earlier than expected. We had the fun of watching the delegates boo Ted Cruz. We all enjoy reminding Jonah Goldberg that he followed Bill Mitchell on Election Day “for kicks”,confidently expecting to retweet Bill’s pained realization of Trump’s obliteration.

Despite all those earlier outrageous, determined efforts,  here we are on what, the fifth catastrophe that the media predicted will wipe out Trump’s presidency? Shrug. They’ll find something else. Why get angry? It didn’t work last time. So I let go of the anger, and enjoyed the drama queen Comey telling his tale.

I don’t understand those who are disappointed in Trump’s achievements. Bush 43 had near total control of Congress and got No Child Left Behind. After 2002 he had full control of Congress and passed Medicare Part D. From 2005-2007, he did everything possible, including race-shaming, to pass “comprehensive” immigration reform. A few days after 9/11, he arranged a photo op with Muslims to make sure no one had Bad Thoughts.

Meanwhile, Trump is appointing judges, deporting illegal aliens, and building the wall.He’s letting the military take it to ISIS and Syria.  He’s rolling back environmental policies and stepped out of the Paris Accords. He’s ringing employment to the industrial regions that supported him–maybe not as much as they need, but more than they had. I don’t like Betsy much, but at least she’s doing some interesting evasions on IDEA and special ed.

How much virtual ink has been spilled on the deportations, on Paris, on the environmental policies? How many politicians before Trump wouldn’t risk media disapproval? He’s shown what can be done. That’s an invaluable service.

Much of the rest is noise.  Turns out  many important people aren’t really concerned about what a president does, so long as he only has one scoop of ice cream at dinner while he carefully discusses his hopes for Michael Flynn’s future. Whatever charge the media flings, there’s a countercharge about a prior presidency.  If I am too cynical about Washington, if there is a measurable difference between Trump and his predecessors in terms of the venal opportunism found in his government officials,  you’ll forgive me if I’m unconvinced by the assurance of those “experts” who called for impeaching Bill Clinton, invading Iraq or Afghanistan, and/or electing the incompetent naif Barack Obama on the country.

Is Trump suited to be President? Beats me. Should he be hiring more people? Maybe. Is he upsetting European leaders? I certainly hope so. I don’t see him as a bully or a dictator. I’ve never been convinced by those who do.

Do I want more? Sure.  Like most of his immigration restrictionist supporters, I’m unhappy that he’s still approving DACA waivers and extensions.  I hope his daughter and son-in-law go back to New York. Would I like less tweeting, more thoughtfulness? Yes.  Do I wish his cabinet didn’t look like the Goldman Sachs retirement weekend? Absolutely. Less emphasis on tax breaks and other GOP wishlist items? Indeed.  But as far as hard asks go, just one: cease and desist any talk of firing Jeff Sessions.

Still,  if Trump were note-perfect, he’d still be facing a huge, hostile force. Of  all the institutional wisdom that Trump showed up as canard, the media’s power took the biggest hit. Trump showed conclusively that the media is only speaking to half the country (usually the left). No other conclusion is possible. The media has no influence over the people; it’s just preaching to its believers. Worse, the people now know that the media didn’t change a single mind.  Profits are up, because their half of the country is enraged and active. But they’ll never again be able to pretend their reporting speaks to the entire country, or that they influence public opinion. They keep trying–the sob stories about the deportees, the stenography of various government leakers, the outright fake news (tells us again how Trump was under investigation, guys!). But the whole of the public remains curiously unmoved, despite the hype.

The media wants to change the world back to way it was.  What’s happening now is all wrong, they’re not supposed to be here, they have to  fix it.  If they can just keep the pressure on and play for time, someone who “wasn’t supposed to be here” will drag the wounded Enterprise C back a hundred years to be destroyed.  The timeline can be restored.

So it’s  ungrateful and even a bit stupid to demand Trump alter every personality trait that got him this far.  Trump has the perfect characteristics for moving America in spite of  media outrage.  He’s sublimely unconcerned about how things are done, comfortable with violating norms. Crass. Obnoxious. Unflinching. Self-absorbed. They might not be comfortable qualities in a roommate, but they’ll do nicely to protect him during the onslaught.

Because it’s going to get time to get everyone accepting the reboot. Note that political pundits still fixate on approval numbers. You know, the kind that comes from polls. Like the polls that predicted Hillary would win.  Paul Ryan and other respectable Republicans are still trying to figure out how they can win media approval, win support from moderates, and improve their polling numbers.

They should take a page from Mitch McConnell’s book. Back when Ryan was playing Hamlet, McConnell quietly told his senators to do whatever they needed to do, and held on like grim death to that empty Supreme Court seat. These days, McConnell refuses to be gobsmacked by the intemperate Trump. Sure, he’d like less drama. But in the meantime, he’s getting it done.

I wish everyone in GOPVille would do the same.  What I want, of course, are more people  following Trump’s example. The first one to violate expectations had to be a billionaire who didn’t need donors with a willful desire to offend people. But with time, others will be able to build on his first steps. Others might be equally willing to brave disapproval but, dare I say, more temperamentally suited to government. Many of Trump’s policies have already become  accepted–if not respectable, at least not reviled.  Over time, more will.  That’s my hope–that others build on his success, the knowledge that his policies have tremendous support.  Embrace the alternate timeline.

That’s the best way of ensuring the changes will hold, that calls to end Trump’s presidency fade.  Sure, the pendulum will swing back. I’m just hoping for more changes that permanently alter the landscape. Don’t let the media win and enforce the pretense that the alternate timeline didn’t ever happen. Let this be a genuine reboot where Christopher Pike gets a better death, rather than a temporary odd happenstance that had no effect once Enterprise C went back.

Of course, this advice could be coming from a Klingon who’d rather achieve  total victory over the Federation than a treaty in which both sides move forward in peace.   You takes your chances.


In case you’re new and missed my other political pieces (I usually do education):

Note from a Trump Supporter: It’s the Immigration, Stupid!

Citizens, Not Americans

This Great Election

Celebrating Trump in a Deep Blue Land

(destiny quote from R. Stevens,


The Product of Two Lines

I can’t remember when I realized that quadratics with real zeros were the product of two lines. It may have been
this introductory assessment that started me thinking hey, that’s cool, the line goes through the zero. And hey, even cooler, the other one will, too.

And for the first time, I began to understand that “factor” is possible to explain visually as well as algebraically.

Take, for example, f(x)=(x+3) and g(x)=(x-5). Graph the lines and mark the x-and y-intercepts:


Can’t you see the outlines of the parabola? This is a great visual cue for many students.

By this time, I’ve introduced function addition. From there, I just point out that if we can add the outputs of linear functions, we can multiply them.

We can just multiply the y-intercepts together first. One’s positive and one’s negative, so the y-intercept will be [wait for the response. This activity is designed specifically to get low ability kids thinking about what they can see, right in front of their eyes. So make the strugglers see it. Wait until they see it.]

Then onto the x-intercepts, where the output of one of the lines is zero. And zero multiplied by anything is zero.

Again, I always stop around here and make them see it. All lines have an x-intercept. If you’re multiplying two lines together, each line has an x-intercept. So the product of two different lines will have two different x-intercepts–unless one line is a multiple of the other (eg. x+3 and 2x+6). Each of those x-intercepts will multiply with the other output and result in a zero.

So take a minute before we go on, I always say, and think about what that means. Two different lines will have two different x-intercepts, which mean that their product will always have two points at which the product is zero.

This doesn’t mean that all parabolas have two zeros, I usually say at this point, because some if not all the kids see where this lesson is going. But the product of two different lines will always have two different zeros.

Then we look at the two lines and think about general areas and multiplication properties. On the left, both the lines are in negative territory, and a negative times a negative is a positive. Then, the line x+3 “hits” the x-axis and zero at -3, and from that zer on, the output values are positive. So from x=-3 to the zero for x-5, one of the lines has a positive output and one has a negative. I usually move an image from Desmos to my smartboard to mark all this up:


The purpose, again, is to get kids to understand that a quadratic shape isn’t just some random thing. Thinking of it as  a product of two lines allows them to realize the action is predictable, following rules of math they already know.

Then we go back to Desmos and plot points that are products of the two lines.


Bam! There’s the turnaround point, I say. What’s that called, in a parabola? and wait for “vertex”.

When I first introduced this idea, we’d do one or two product examples on the board and then they’d complete this worksheet:


The kids  plot the lines, mark the zeros and y-intercept based on the linear values, then find the outputs of the two individual lines and plot points, looking for the “turnaround”.

After a day or so of that, I’d talk about a parabola, which is sometimes, but not always, the product of two lines. Introduce the key points, etc. I think this would be perfect for algebra one. You could then move on to the parabolas that are the product of one line (a square) or the parabolas that don’t cross the x-intercept at all. Hey, how’s that work ?What kinds of lines are those? and so on.

That’s the basic approach as I developed it two or three years ago. Today, I would use it as just as describe above, but in algebra one, not algebra two. As written,I can’t use it anymore for my algebra two class, and therein lies a tale that validates what I first wrote three years ago, that by “dumbing things down”, I can slowly increase the breadth and depth of the curriculum while still keeping it accessible for all students.

These days, my class starts with a functions unit, covering function definition, notation, transformations, and basic parent functions (line, parabola, radical, reciprocal, absolute value).

So now, the “product of two lines” is no longer a new shape, but a familiar one. At this point, all the kids are at least somewhat familiar with f(x)=a(x-h)2+k, so even if they’ve forgotten the factored form of the quadratic, they recognize the parabola. And even better, they know how to describe it!

So when the shape emerges, the students can describe the parabola in vertex form. Up to now, a parabola has been the parent function f(x)=xtransformed by vertical and horizontal shifts and stretches. They know, then, that the product of f(x)=x+3 and g(x)=x-5 can also be described as h(x)=(x-1)2-16.

Since they already know that a parabola’s points are mirrored around a line of symmetry, most of them quickly connect this knowledge and realize that the line of symmetry will always be smack dab in between the two lines, and that they just need to find the line visually, plug it into the two lines, and that’s the vertex. (something like this).

For most of the kids, therefore, the explanatory worksheet above isn’t necessary. They’re ready to start graphing parabolas in factored form. Some students struggle with the connection, though, and I have this as a backup.

This opens up the whole topic into a series of questions so natural that even the most determined don’t give a damn student will be willing to temporarily engage in mulling them over.

For example, it’s an easy thing to transform a parabola to have no x-intercepts. But clearly, such a parabola can’t be the product of two lines. Hmm. Hold that thought.

Or I return to the idea of a factor or factoring, the process of converting from a sum to a product. If two lines are multiplied together, then each line is a factor of the quadratic. Does that mean that a quadratic with no zeros has no factors? Or is there some other way of looking at it? This will all be useful memories and connections when we move onto factoring, quadratic formula, and complex numbers.

Later, I can ask interested students to sketch (not graph) y=x(x-7)(x+4) and now they see it as a case of multiplying three lines together, where it’s going to be negative, positive, what the y-intercept will be, and so on.


At some point, I mention that we’re working exclusively with lines that have a slope of positive one, and that changing the slope will complicate (but not alter) the math. Although I’m not a big fan of horizontal stretch outside trigonometry, so I always tell the kids to factor out x’s coefficient.

But recently, I’ve realized that the applications go far beyond polynomials, which is why I’m modifying my functions unit yet again. Consider these equations:


and realize that they can all be conceived as as “committing a function on a line”. In each case, graphing the line and then performing the function on each output value will result in the correct graph–and, more importantly, provide a link to key values of the resulting graph simply by considering the line.

Then there’s the real reason I developed this concept: it really helps kids get the zeros right. Any math teacher has been driven bonkers by the flipping zeros problem.

That is, a kid looks at y=(x+3)(x-5) and says the zeros are at 3 and -5. I understand this perfectly. In one sense, it’s entirely logical. But logical or not, it’s wrong. I have gone through approximately the EIGHT HUNDRED BILLION ways of explaining factors vs. zeros, and a depressing chunk of kids still screw it up.

But understanding the factors as lines gives the students a visual check. They will, naturally, forget to use it. But when I come across them getting it backwards, I can say “graph the lines” instead of “OH FOR GOD’S SAKE HOW MANY TIMES DO I HAVE TO TELL YOU!” which makes me feel better but understandably fills them with apprehension.