Category Archives: teaching

Building Narratives

I will get back to the ed school thoughts, I promise, but thought organization, she’s a bitch. In the meantime, I’m helping others organize their thoughts in my enrichment classes for Asian kids whose parents think free time is for Americans.

Our selected book for the week doesn’t offer much analytical fodder, so they wrote narratives—specifically, a first person account of an actual experience with only an occasional invented detail, no more than a handwritten page.

They came up with their ideas on Wednesday and workshopped them just before the end of class, coming back with a first draft on Thursday. They shared out in at least two pairings, while I reviewed them individually. It’s an hour plus of constant writing and talking. Below are authentic representations, with occasional invented details, of my feedback.

“Ellen, good story, well-executed, flows well, nice attention to detail. But everything needs tightening up. Take a look.”

No matter how badly I wanted to leave my sister to go to the concert by myself, I had to accept that she had a good point. I had been her age when I went to my first concert and I ….

Ellen blushed. “I’m an egomaniac!”

“Naw, I do it too. You just need to edit. Obviously, dump a lot of the glue. Be creative about pronouns. Show, don’t tell. What’s your sister’s name? ‘Jenny always wanted my opinion on new tunes, making it clear her big sister was the final authority. Refusing to take her along might save face with [can you name your friends, please?] but why deny Jenny the same opportunities I’d had? ‘ Something like that. It’s all there; just refocus the action.”

Next up was Ben, another strong writer with good story sense, and in about a minute I’d sent him over to Ellen with a similarly glue-id’ed paper so they could collaborate. On to David, whose closest friends had noticed, on their frequent outings to a local amusement park, that he never rode the roller coaster. Despite his assurances that he simply wasn’t interested, they figured out he was actually terrified and staged a supportive but forceful intervention. No longer afraid, he now likes sitting up front, the better to get the rush.

“David, where’s the essay you wrote the first day? Dig it up for me.”

While waiting, I scanned Jack’s story, which began:

The locked door blocked my last hope for escape. I pounded frantically, hopelessly, screaming for rescue, but my doom was sealed. My fate wasn’t just awaiting me. It was headed my way. One hundred and twenty pounds of hulking, angry sister.

“Jack, this is very funny. But keep the suspense going without identifying the villain. Go through all your efforts to escape, your despair, and then the one-two punch. First, the horrible fate is your sister. Second, with no other escape, you offer up her other common victim, Sister #2, the one who locked that door to leave you to your doom—and who has candy. So you two were always escaping the bully by handing her each other? Why was she always beating you up? She stopped, I hope?”

“Yeah, I was seven, she was fourteen. She’s in college now. I lied about the candy.”

“Very nice. Go change the pacing, and bring out the villain later. Don’t leave out the lie.”

David came back with his essay, and I found the line:

My primary goal in this class is to achieve an essential understanding: I am thoroughly, permanently screwed if I don’t stop playing video games and take school more seriously.

“Where’s that kid?”

“What do you mean?”

“Your quirky quality is clearly in that sentence and nowhere to be found in the narrative. With the voice, your story isn’t really about being afraid of roller coasters, but about you, a unique goofy guy, and your supportive friends who called you on your bs. Without that unique voice, this story isn’t much more than a confession: you’re a baby who’s afraid of roller coasters. Go find the voice. Irene?”

Irene had two typed pages. “I know! Ellen told me it has to be shorter. I’ve already cut eight sentences.”

“Try eight paragraphs.” I read the first couple of paragraphs, stop. “What do you mean, your mother wants you to get CM Level 9 this year in piano?”

“So it’s on my transcript. Otherwise, I’ve wasted the piano lessons.”

“Yeah, this is some annoying Asian thing that’s going to really tick me off, isn’t it?”

Irene, a passionate artist who sketches every spare minute, laughed. “Probably. I take lessons so that I can pass the level 9 test. So my transcript shows I have artistic ability. But mom wants me to pass it this year, before I start my SAT prep.”

I sighed. “Remember if you are writing an essay or story for a wider audience that Americans would disapprove of your mom’s priorities. Those details detract from the narrative. Why not just write about the unexpected outcome from all this?”

“My discovery of anime music?”

“Yeah. Cut everything else out.”

On to Jasmine, whose entire narrative read:

My family left on a 4-day trip to Reno, Nevada. On the way, the car’s air conditioner broke down. We were unbelievably hot. I could feel the heat beating down on my face as I stuck my head out the window, but it was the only way I could get some air. Finally, we arrived. The hotel was airconditioned. I lay back on the bed and felt my body slowly adjust to the cool. It was amazing. I had never felt my body adjust and cool down before. When I began the trip, I had a list of things I wanted to do, but now I was happy to just be in cool air.

“What’s the heart of this story?”

“Um. The heat?”

“Really? I see a huge boost in writing energy when you get to the hotel.”

“Well, that’s because the cool felt so good.”

“Exactly. A blissful feeling so great that you reassessed your goals for the trip. That’s the heart: negative, difficult experiences caused a change in your perspective. Now, what did you think I was going to say about this story?”

“It’s too short.”

“Not ‘too short’ in any absolute sense, but you have to share the suffering, make us feel that heat, feel the endless hours in the car. You are telling a tale of sensations. Give me the sensory info. I want you to sketch your memory of that day. Stick figures, whatever. Or make a list of all your sense memories from that time. How do you communicate that heat? Show your observations of everyone’s suffering. Think who, what, when–heart of summer?–how. Then do the same thing for the blissed-out time in the chill. Overdo it first—you can cut it down later.”

Then Nick, with a well-crafted first draft of his anxious excitement the day he presented his science project at a company-sponsored competition.

“Nick, the opening rocks. Then you lose focus a bit. I want details about the judging. You say he’s nice–what sort of questions did he ask? Was it as you’d imagined it in all your practice runs? And….what’s this?”

“The end? Is it wrong?”

“‘Best of all, this win will really look great on my college applications.’ Dear God. Please tell me you included that little gem to stop my heart.”

He looked puzzled. “That’s why I entered the contest.”

I banged myself in the head with his notebook. “So I’m feeling like I learned something about your love of science, when in fact you did all the work to get a win for your resume?”

“Well, science is okay. But…yeah, I picked the project because I thought it had the best chance of winning.”

“Not because you were interested. See how you’re looking a bit shamefaced? Because you know what I’m saying, right?”

He nodded.

“The thing is, I’m torn in these situations, just like I was with Irene. I’m glad you’re writing authentic emotions. But you are so wrapped up in your Indian cocoon that you have no idea how bad this looks to the Americans who aren’t a generation or less away from Asialand. To us, you come off as a slogger who is only interested in appearance, in faking it, not in pursuing excellence for its own sake. And of course because I’m American and you are, too, I want you to want to pursue excellence for something other than a resume bullet. And you don’t. Which is okay, I like you anyway. For now, though, this story gets much better if you dump this last line and allow your readers to delude themselves about your passionate love of science.”

Eddie up next with a story about three families in two RVs travelling to Banff, Canada to see the sights.

“So you were all related? These are your cousins?”

“No, we didn’t really know any of them.”

I was perplexed, but Ben chimed in. “We do that all the time. It’s an Asian thing.”

“You like big vacations?”

“No, cheap ones” chorused Serena, Ellen, Ben, Eddie, and Jerry.

“We either fly and stay in really cheap places, with all the kids camped out in one room,”…

“One time I was with twelve kids!” from Serena.

“But aren’t there occupancy limits?”

“We ignore them; most of us go out to dinner while one family checks in. Then we sneak in.”

“Indians, is this a thing for you?”

“God, no,” says Ace. “It sounds horrible.”

“My parents came here to escape lots of people, who wants to go on vacation with them?” from Jasmine.

“And you got thrown out of an RV park, Eddie?”

“We almost got thrown out of Canada.”

“All because you were teasing your cousin and he started swearing?”

“It was after midnight, and he was screaming, and Henry wasn’t my cousin, just this other kid.”

“And the adults didn’t stop him?”

“They were in another RV.”

“But they apologized?”

“When the ranger came, yeah. They apologized and promised to be quiet. Even tried to give the ranger money.”

“Bad idea.”

“She was furious, told us to go back to Montana. My parents don’t even know where that is. So she just told them to leave the RV park.”

“Huh. See, your family’s thoughtless cultural rudeness offers some great insights, but I’m not sure you learned anything from it.”

“Yeah, I did. I told them it was a bad idea to offer money. Plus I should have told Henry to shut up, or gotten adults involved.”

“Okay. So drop all the stuff about how pretty Banff was and how terrible the food, tell me about these weird cheap Chinese family vacations, and what you learned about Westerners—and where you slept after leaving the park. Ace, you’re next.”

Ace, the oldest kid in the class, shuffled up hesitantly. I read through the story once, read it again. Read it a third time.

“It sucks, right?”

“No. It’s great.” He looked at me in shock.

“It’s not perfect. It needs work. But four weeks ago, you could barely produce a coherent sentence, because you were just writing words to fill up paper. Now your sentences have subjects and verbs and purpose. Your thoughts are organized. You’ve just described a heartbreaking basketball defeat, made me feel your disappointment—and then you bring up the subsequent win to end on a high note. Beautifully structured. Now, go back and read this aloud, first to yourself and then to someone else, and listen to the words. Any correction you find yourself making as you read it aloud, stop and jot them down. Match the words to your memory, see if anything’s missing.”

Ace went back to his desk with new confidence and a purpose.

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

On ending the year

Year 2 I did finals on the last day of class, because the school required it and my room was in the center of campus. I was returning—probably. (I looked for new jobs; an offer came in too late to accept). Better part of something not to flout administration, so I did the final on the intended day.

Every other year, I’ve a series of finals or one big one in the days before, and show a movie during the two hour final period. I’d misplaced my copy of Rear Window the year before, so on Friday it was the featured film in all three classes.

I’ve mentioned before that I don’t really give a damn if my kids love math or just survive it, find “Hamlet” enthralling or torture, or are really interested in what the Founding Fathers thought of strict vs. loose Constitutional construction. But they will by god not turn up their noses at classic films.

And so three times, my film buff’s heart just went pitty-pat thump thump with satisfied joy as twenty to thirty kids shrieked in horror when Lars Thorwald came around the corner of the hall while Lisa was still in the apartment, or gasped and flinched when Burr realizes who’s been watching him.

One girl had seen it already, and she confided that the beginning was slow.

“That’s because you’re used to a different style of movies. But think of this as a novel you’re analyzing for lit class. Look for subtle changes in Stewart’s behavior, for the first time he openly reaches for Lisa instead of fending her off. Or look at the window stories and see how many of them are just reinforcing the different outcomes for women and relationships. And remember this: at the end of the first 30 minutes, Lisa asks if either of them can ever change. Consider that the rest of the movie as an answer to that question.”

She came up to me after it was over to say that she’d never before realized that movies were “just like books”.

“I could write an essay about Rear Window for the SAT!”

“You could indeed. Make a nice change from Martin Luther King.”

Anyway. A richly rewarding experience. Maybe even for my students.

On to the year-end check out.

While the last days of school are usually pretty easy, the very last day of duty is a hassle. Teachers have to get signed off on a bunch of things over the summer, turn in their keys, and leave. You can tell the teachers who count every second of the summer, who have been preparing their room for the end days for at least a week, who know the checkoff list by heart and have it all done before the last bell rings. They’re the ones waiting in line on Friday morning for an admin signoff so they can prove to the principal’s secretary that they’ve changed their voicemail password and turn in their keys.

Then there are teachers who make a day of it–eh, summer’s here, they won’t rush. These are teachers for whom the most significant task—room cleaning—is something they’d rather not think about. They come in late, sigh at the mess of their room, do some grading, get grades in, lackadaisically pack up a few boxes, go get coffee, come back and sigh at the mess of their room, shove a bunch of stuff into their car, go get lunch, toss a bunch of stuff they’d been saving in case they needed it, jam anything left over into their cars, and then look at the sign-off sheet to see what other tasks they need. By this time it’s usually late afternoon and everyone’s left, so they skate the things like turning in two copies of grades, turning in keys, changing voicemail and so on. They email the principal’s secretary and drop by a few weeks later to turn in their keys.

You’ll never guess which sort of teacher I am. Go ahead, guess.

Year 1 and Year 3, I was leaving the schools, so I’d taken all my belongings home earlier. The actual last day, I looked at the various things acquired at the school, remembered where I’d found them, realized no one would give a damn if they were gone, so shrugged and took them, too. Like the really cool geometry book I found stuffed into the corner of a box of books the previous teacher had left for the trash, or the white board found jammed into the back of a junk room that had “3/5/04″ on the meeting agenda written (but still removable) on it. Or the massive trunk of fantastic manipulatives, taken from a room stuffed with such trunks that the book clerk told me had been there for five years because “no one used them”. Indeed, at my school, I was the only one who used them, along with the set of 30 student-sized white boards that I talked up to all my colleagues, who all looked at me perplexedly. “They’d just use them to draw on, or scribble obscenities.” “Sure. but sometimes they do math.” No takers. They’ve been put to good use. Those years I was usually the last one out, or close to it, but would usually turn in my keys early and then just prop my door open.

Years 2 and 4, I didn’t leave schools but changed rooms, which required me to pack my car much as if I was leaving because lord knows what gets lost in a move. Last one out both years, had to drive back to the schools to turn in my keys later.

Year 5: I am staying for a third year. No room change. My summer job doesn’t start until Tuesday, so I have an actual brief break to enjoy. Vowing to commemorate the occasion with a behavior change, I stay late both Wednesday and Thursday, finishing all but a bit of my grading, and get much of my room boxed up. On Friday, little to do but finish up grading and pack the boxes, computers, printer, lots of books, office supplies into my storage closet—nothing to come home! Then I bought a lock at the Dollar Store. Last thing in the closet, just before the lock, my class rules sign, made in Year 2, a big piece of thin yellow paper. I don’t like throwing things away.

What was left? Grades done. What was the stuff I usually skated because I was late? Oh, print out copies of grades. Check. Change voicemail. Well, I never set it up to start with, so that should be easy. But the preset password didn’t work. Oh, that’s right, our voicemail system had crashed. Use the system default. Didn’t work.

I sat there, perplexed. Wait. Is it possible that I did set it up after the crash? I vaguely remember the principal’s secretary telling us that the system would be upset if passwords were left in default. Could it be that I’d complied? I never do voicemail. Never. But if I had done voicemail, my password would be….

“You have three messages.”

Only three? Pretty good. Does this mean I’d set up a voice recording?

“Hi, this is Ed. I’m happy to get in touch with you, but to ensure a record of all my interactions, I prefer that parents contact me via email. If this is a problem, please send a note in with your student and I’ll be happy to contact you. ”

The only three messages were system-wides from other teachers. It worked!

I must have set this up in an extra five minutes I had between classes. Not a single memory of it, though.

I was done! Everything signed off, grades done. Turned in the keys. People were still cleaning their rooms. It was 3:30. I wasn’t last!!!!

Then I realized I’d left my laptop and the few take-home things in the classroom. To which I no longer had the keys.

Thank god a custodian was walking by right then. He didn’t even laugh at me.

And so summer begins.

If you want to assist me in enjoying my summer:

Apple for Paypal, for Square Cash or Google Wallet are all welcome. Use the email educationrealist at the big G.

I vow to use all funds for nothing more than a better class of beer, more sushi, or airfare to see my adorable granddaughter.

I’m just happy you’re reading. But money’s nice, too. Thanks again to all who have contributed.

Learning from Mr. Singh

I first heard about Mr. Singh (not his real name) the first week at my school, working through a modeling problem with a student.

“Come on, you know the perimeter formula for a rectangle, don’t you?”

“No. I had Singh last year for geometry,” the kid says matter-of-factly. A nearby student rolls her eyes.

“Oh, I had him two years ago! He flunked me. He was making mistakes all the time, everyone told him, he said no, they were wrong.”

I was taken aback. I had never run into students who called teachers incompetent before. But then up to that point, I’d taught at much tougher schools, where the “bad teachers” were the ones who couldn’t control their classrooms full of kids who didn’t give a damn. (We have difficult kids here, but the ratio is something approaching a fair fight.) I was not in any way used to kids complaining that teachers didn’t know their subject.

I forget which student this was—it’s been almost eighteen months, and the whole pattern had yet to form. But I remember distinctly the kid wasn’t a math rock star. Just an ordinary student in algebra II, telling me he knew more than a math teacher, enough to realize the teacher was ignorant. I shrugged it off at the time, but I heard it routinely through the next semester. Occasionally, I’d get it from parents, “Well, my son had Mr. Singh two years ago and told me the man had no idea what he was doing.”

When I started teaching pre-calc, the occasional comments became a constant. I began with my usual response: state it’s unacceptable to criticize one teacher in front of another, whatever the reason. But at a certain point I flat out banned that anti-Singh jokes.

I don’t know Mr. Singh well; math teachers aren’t a chummy crew. He did not and does not strike me as incompetent in any way. Like at least half my colleagues, he privately thinks I’m a pushover, too willing to give kids passing grades. But when some members of the department pushed for a higher fail rate to ensure that we only had qualified kids in advanced math, he was on the side of the demurrers (I did more than demur, of course, because I’m an idiot). He is younger than I am, Asian, speaks English well, with only a slight inflection. I don’t know if the kids are openly disparaging him to other math teachers, and haven’t asked.

Last fall semester (for newcomers, we teach a year in a semester, then do the whole thing again, four classes at a time), I had a handful of very bright seniors who were refusing to go on to Calculus the next semester, because “Mr. Singh’s an idiot”. I got fed up and told the crew in no uncertain terms that they should all have taken honors pre-calc anyway, that I was tired of them not challenging themselves and using teachers as scapegoats, and they were to get their butts into Calculus. They gulped and obeyed, “but we’ll show you that he doesn’t have a clue; he’s just using the book as a guide!”

So the whole passel of them, along with a number of my precalc students from the previous spring, would occasionally drop by during lunch to tell me about how Mr. Singh was wrong, how everyone was telling him he was wrong, but he kept insisting he was right. What the hell is going on in these classes, that he’s arguing, I’d wonder, and tell the kids I didn’t believe them, that I found it incredibly hard to believe Mr. Singh was wrong and certainly wouldn’t take their word for it. If they were so sure, bring me a specific example.

A couple months ago, Jake came rushing into my room, triumphant. Very bright kid, Korean American (grandparents immigrated), and if you want to know how white folk world might change Asians over the generations, he’s a good place to start: refused to take honors pre-calc “because of Mr. Singh”, took Calculus at my orders only, and is going to a junior college (where he easily qualified to start in Calculus).

“I can prove it. I took a picture of the board!” This incident happened long before I’d thought of writing about it, and I can’t remember the specific problem. It was a piece-wise function, a complicated one, and he sketched out the graph he’d captured on his smartphone. “See? He’s saying it’s negative for x < -1, and it can’t be, because [math reason I don't remember].

I frowned at the board. “Hang on, let me think. I see what you’re saying, but I’m pretty sure there’s something wrong with that approach, like a mistake I’ve made before but can’t remember why.” Frowned some more. “Look, Mr. Singh knows way more math than I do. Why don’t you go ask him about this?”

“He doesn’t know what he’s talking about.”

“No, I don’t know what he’s talking about. Oh, wait. Duh.” and I turned back to my computer and brought up Desmos to graph the function. Desmos agreed entirely with Mr. Singh.

“Wow.” Jake is utterly gobsmacked. A world view shattered. “But how the hell does [technical math question I don't remember anymore]?”

“Here’s a thought, Jake: go ask Mr. Singh.”

“He hates me.”

“I can’t think why. You’re just this punky jerk who disrupts classes with arguments because the possibility that the teacher might just know more than you hasn’t crossed your peabrain.”

“Well, when you put it that way, I’d hate me, too.”

“So go up to him and say ‘Hey, Mr. Singh, I’ve been reviewing this function and I can’t figure out why the graph looks like this when x is less than negative one. Can you help me figure it out?’ He will like you for this. I promise.”

Jake had the conversation, reported back, explained to me why we both thought it should be something else (and the minute he mentioned the reason, which I still can’t remember, I went “yeah, that was it! I made that mistake before!”)

This happened periodically over the next two months, but Jake grew increasingly tentative, uncertain of his own certainty. Rather than rolling in confident he held evidence that would convince me of Singh’s stupidity, he was now doublechecking with me. Mr. Singh said this, but I think that, what do you think? Sometimes I knew the answer, in others I’d look it up, but I would always send him back to Mr. Singh for either more information or confirmation. Eventually, he started going to Mr. Singh first and then reporting the results to me.

His new data points had an impact. Now, when Jake and the others came in to say hi, they don’t have any tales of Mr. Singh’s errors but instead have all sorts of stories about how they pwned a classmate with their awesome math skills.

(Does this seem weird? Remember that at my school, Calculus is third tier from the top—AB and BC Calc are ahead of it. They’re all bright but not quite nerds. Many of them are my favorite sort of kid—more interested in learning than good grades. But they’re boys, so posture they will.)

Last Thursday Tom, a white junior who’d taken my precalc class as a sophomore, came by during our “advisory” (brief tutorial period after lunch).

“Do you know anything about L’Hopital’s Rule?”

“Vaguely. Something to do with limits. I have a Stewart Calculus text, and can inquire. Why?”

“Because he marked me wrong on a test. I got the right answer! But when I asked him about it, he said that I couldn’t use the Quotient Rule, that I had to use L’Hopital, and that it was a fluke I got the right answer.”

I looked up L’Hopital’s Rule, page 289. “If I understand this correctly, L’Hopital’s Rule is intended at least in part for cases where you can’t use the Quotient rule. If you have an indeterminate result, like dividing zero by zero or infinity by infinity, the Quotient rule won’t apply.”

Tom looked aghast. “It doesn’t?”

“Not according to this book and, I’m betting, not according to Mr. Singh.”

“It was a limit of sin(2x)/sin(3x).”

“Well, I know the limit of sine isn’t infinity, so I’m guessing it’s…”

“Zero. Oh, I can’t divide by zero. So he was right. It was just a fluke I got the answer.”

“Looks like it.”

“It’s so weird. There’s always like fifteen ways to do something in calculus, then sometimes, only one way.”

“Hah. But look. I don’t know much about this. I want you to go back to Mr. Singh. My guess is this test question was specifically designed to assess your understanding of the cases for L’Hopital’s Rule. But you need some clarity, and he’s the guy to explain.”



This story began nearly two years ago, and not until a few days ago, when I read this piece of utter Campbell Brown crap, did I think of writing about Mr. Singh, me, and his students. But at one point Brown quotes a student who said ““There were certain teachers that you knew, if you got stuck in their class, you wouldn’t learn a thing. That year would be a lost year” and I realized how often I had read that sentiment. Kids know who the bad teachers are. Parents know who the bad teachers are. They just know. Word gets around.

Well, no. They don’t. Students are, I think, the best judge of teacher quality in classroom management. They know when a teacher can’t control the kids. But they are usually incapable of evaluating teacher content knowledge. I hope this story shows that students can form fundamental received wisdoms that are simply false. From average to excellent, Mr. Singh’s students all thought they knew more than he did. And they didn’t. I’m pleased that I now have a knowledge base that allows me to do more than just tell the kids not to discuss Mr. Singh. I can laugh at them—“Yeah, I heard that before. Every time someone tells me Mr. Singh’s wrong, I ask for proof. Turns out the student doesn’t know what he’s talking about. You want to play?”

But my tale has a few more object lessons. First, teachers and parents, please note what I am proudest of. I sent the kids back to learn from Mr. Singh.

We want kids to form trusted networks. We want them to find resources when they feel lost or doubtful about education, so they don’t lose hope or quit because they feel isolated. And when they do come to their trusted resource, it’s incredibly tempting for that resource, whether teachers and parents, to regard the kids’ trust as an ego feed—see, I’m the one they really need, the safe place, the wise soul. This is particularly tempting for teachers, because it’s practically a job requirement that our personality type value trust and respect over pay. However, when a kid is using you as a resource not just to get more information or clarity, but as a substitute for the teaching process, you send him back. He or she has to learn how to use the educational process as it’s intended, to push the teacher for more information, to make sense of the unfamiliar. Ideally, students must learn not to just do what feels safe—complain to another teacher—but what feels terrifying, and ask for help. Sure, sometimes it won’t work. That’s a lesson, too. You’ll be there to help them figure it out, if needed.

Then please note what I have used everything short of neon signs to highlight: Mr. Singh knows far more math than I do (see the comments if you have issues with my description of L’Hopital’s Rule). The kids know this. I make it clear to them. Yet they still came to me for help.

And that, readers, is an important takeaway from this little essay, a truism people mouth without really thinking about what it means. Teaching involves trust. You can’t just have content knowledge and run a fair classroom. Your students have to trust your ability and your judgment. Your students’ parents have to believe that you have their interests at heart.

Reformers might do well to remember that, as they wonder what went wrong in Newark, in DC, in Chicago and Indiana. It’s not enough to tell everyone you want excellent schools. They have to believe you.

Yes, sometimes that trust will be misplaced. That is a huge reason why the charter market doesn’t work, in fact, because parents are taking schools they trust to keep their kids safe over the schools the charters want them to demand. No doubt, reformers in general think that misplaced trust is why teachers and their unions continually win the long game. But regardless, reformers aren’t trusted by the very populations they say they want to help. And alas, trust has nothing to do with test scores.

Finally, please note: in no way am I suggesting that I am a superior teacher to Mr. Singh. When I am tempted to that conclusion, I remind myself of the occasional students of mine who go running to other teachers (including, no doubt, Mr. Singh) to get a straightforward lecture or template. When I learn that students have done this, I always remind them that they can ask me, that if they need more structure, see me and I’ll give it to them. I wish those teachers would let me know when students come to them for help with my class. And then I remember that I haven’t said a word of this to Mr. Singh.


Postscript: The comments have been revealing of the way people are filling in gaps. First, my kids are doing well in Singh’s class. Most of them are getting As, the occasional B. They understand the math. Second, this is NOT a case of a teacher refusing to allow students to point out errors. Third, my students drop by for many reasons—it’s not like this is all a constant bitchfest about Singh. I’m just pulling out representative moments.

Multiple Answer Math Tests

As previously explicated in considerable detail, I’m deeply disgusted with the Common Core math standards—they are too hard, shovel way too much math into middle school. If I see one more reporter obediently, mindlessly repeat that [s]tudents will learn less content, but more in-depth, coherent and demanding content my head will explode.

Reporters, take heed: you can’t remove math standards. The next time some CC drone tells you that the standards are fewer, but deeper, ask for specifics. What specific math standard has been removed? Do students no longer have to know the quadratic formula? Will they not need to know conics? No, not colonics. That’s what you all should be forced to endure, for your sins. In all likelihood, the drone has no more idea than you reporters do about high school math, so go ask Jason Zimba, who reiterates several times in this interview that the standards are fewer, but go deeper. (He also confirms what I said about algebra, that much of it is moved to middle school). Ask him. Please. What’s left off?

Pause, and deep breath. Where was I?

Oh. Tests.

So the new CC tests are not multiple choice, a form that gets a bad rap. I give my kids in algebra one, geometry, and algebra two lots of multiple choice tests—not because I prefer them, and they aren’t easier (building tests is hard, and I make my own), but because my top students aren’t precise enough and they need the practice. They fall for too many traps because they’re used to teachers (like me) giving them partial or most of the credit if all they did is lose a negative sign. Remember, these are the top kids in the mid-level or lower math classes, not the top kids at the school. These are the kids who often can get an A in the easier class, and aren’t terribly motivated. My multiple choice tests attempt to smack them upside the head and take tests more seriously. It works, generally. I have to watch the lower ability kids to be sure they don’t cheat.

We’ve been in a fair amount of PD (pretty good PD, at that) on Common Core; last fall, we spent time as a department looking at the online tests. The instructors made much of the fact that the students couldn’t just “pick C”, although that gave us a chuckle. Kids who don’t care about their results will find the CC equivalent of picking C. Trust them. And of course, the technology is whizbang, and enables test questions that have more than one correct answer.

But I started thinking about preparing my students for Common Core assessments and suddenly realized I didn’t need technology to create tests questions that have more than one answer. And that struck me as both interesting and irritating, because if it worked I’d have to give the CC credit for my innovation.

On the first test, I didn’t do a full cutover, but converted or added new questions. Page 1 had 2 or 3 multiple answer questions and 3 was free-response, but on that first test, the second page was almost all multiple response:


I had been telling the kids about the test format change for a week or two beforehand, and on the day of the test I told them to circle the questions that were multiple answer.

It went so well that the second test was all multiple answer and free response. I was using a “short” 70-minute class for the test, so I experimented with the free-response. I drew in the lines, they had to identify the inequalities.



I like it so much I’m not going back. Note that the questions themselves aren’t always “common core” like, nor is the format anything like Common Core. But this format will familiarize the kids with multiple answer tests, as well as serve my own purposes.


  • Best of all, from my perspective, is that I am protected from my typos. I am notorious, particularly in algebra, for test typos. For example, there are FIVE equations on that inequality word problem, not four. See the five lines? Why did I put four? Because I’m an idiot. But in the multiple answer questions, a typo is just a wrong answer. Bliss, baby.
  • I can test multiple skills and concepts on one question. It saves a huge amount of space and allows the kids to consider multiple issues while all the information is in RAM, without having to go back to the hard drive.
  • I can approach a single issue from multiple conceptual angles, forcing them to think outside one approach.
  • It takes my goal of “making kids pay attention to detail” and doubles down.
  • Easier, even, than multiple choice tests to make multiple versions manually.
  • Cheating is difficult, even with one version.


Really, only one: I struggle with grading them. How much should I weight answers? Should I weight them equally, or give more points for the obvious answer (the basic understanding) and then give fewer points for the rest? What about omitting right answers or selecting wrong ones?

Here’s one of my stronger students with a pretty good performance:


You can see that I’m tracking “right, wrong, and omit”, like the SAT. I’m not planning on grading it that way, I just want to collect some data and see how it’s working.

There were 20 correct selections on nine questions. I haven’t quite finished grading them, but I’ve graded two of the three strongest students and one got 15, the other 14. That is about right for the second time through a test format. Since I began the test format two thirds of the way through the year, I haven’t begun to “norm” them to check scrupulously for every possible answer. Nor have I completely identified all the misunderstandings. For example, on question 5, almost all the students said that the “slope” of the two functions’ product would be 2—even the ones who correctly picked the vertex answer, which shows they knew it was a parabola. They’re probably confusing “slope” with “stretch”, when I was trying to ascertain if they understood the product would be a parabola. Back to the drawing board on that.

Added on March 7: I’ve figured out how to grade them! Each answer is an individual True/False question. That works really well. So if you have a six-option question, you can get 6/6, 5/6, 4/6 etc. Then you assign point totals for each option.

I’ll get better at these tests as I move forward, but here, at least, is one thing Common Core has done: given me the impetus and idea for a more flexible test format that allows me to more thoroughly assess students without extending the length of the test. Yes, it’s irritating. But I’ll endure and soldier on. If anyone’s interested, I’m happy to send on the word doc.

Note: Just noticed that the student said y>= -2/3x + 10, instead of y<=. It didn't cost her anything in points (free response I'm looking for the big picture, not little errors), but I went back and updated her test to show the error.

Most Popular Posts and Favorites

I had a huge month in April, over 25% larger than my last winner, November. My blog has a total of 121,000 page views (since January 1, 2012) and have 178 followers on Twitter. The last probably doesn’t seem terribly impressive, but I literally started with 0 followers. I told no friends or family of my blog, although three or four found me over the months. I had just 7000 pageviews in June 2012, when I created a Twitter account. (First follower: the hyperliteral Paul Bruno, of This Week in Education, who I argue with via twitter but quite enjoy as a writer.)

I have absolutely no idea what this means in relative audience size. What matters to me is that, in a loyal band of regular readers, interspersed between teachers, parents, and Dark Enlightenment folk, I count more than a few policy wonks and reporters—and even a publisher, apparently. I might not have a large crowd following my every tweet, but well over half of my followers do. I started this blog to inform and persuade. So far, so good.

I often check my top posts, reading the growing numbers in awe and wonder, because they, too, confirm that my blogging goals have been and continue to be met. The most popular posts cover pedagogy, policy, some unique data analysis or exposure, and my somewhat scathing opinions about the reform crowd. (I don’t much care for progressives, either, but plenty of people are around to debunk them.)

Since my audience has grown again, I thought I’d remind everyone of my most popular posts, in case someone wanted to check them out. Most of my essays represent at least five or six hours work (I worked on the Philip Dick essay for over a month, the algebra pointlessness one for two weeks), and I think any of the 1000+ view entries are worth a look for a general audience.

Title Views Written
Algebra and the Pointlessness of The Whole Damn Thing 4,733 Aug 12
Escaping Poverty 3,664 Nov 12
Teacher Quality Pseudofacts, Part II 3,417 Jan 12
The myth of “they weren’t ever taught….” 2,992 July 12
Homework and grades. 2,576 Feb 12
The Gap in the GRE 2,280 Jan 12
Why Chris Hayes Fails 2,240 June 12
Philip Dick, Preschool and Schrödinger’s Cat 2,102 April 13
The Parental “Diversity” Dilemma 1,907 Nov 2012
An Alternative College Admissions System 1,553 Dec 2012
Why Most of the Low Income “Strivers” are White 1,525 Mar 13
The Dark Enlightenment and Me 1,137 April 13

I left off my “About” page, but both it and “Who am I” right below were nowhere on the horizon last December, so more people are checking out my bio. Neat, if unnerving.

So then we have the 800-900 views, also worth a read for the general audience unless you really have no interest in math pedagogy or curriculum, in which case skip the obvious suspects. But I’m incredibly proud of those curriculum posts; googling modeling linear equations brings up my post in the top two or three as of this writing; likewise a search for binomial multiplication area model brings my post up right near the top.

Title Views Written
Who am I? 966 Jan 12
Plague of the Middlebrow Pundits, Revisited: Walter Russell Mead 918 Mar 13
Teaching Polynomials 917 Mar 12
Modeling Linear Equations 907 Jan 12
SAT Prep for the Ultra-Rich, And Everyone Else 871 Aug 12
What causes the achievement gap? The Voldemort View 820 Jan 12
More on Mumford 817 Nov 12
Binomial Multiplication and Factoring Trinomials with The Rectangle 790 Sept 12

And now the less viewed posts that represent my favorites of the rest. I really wish people would read more of these, particularly the Chris Christie post and the Fallacy at the Heart of All Reform. So pick a few to check out. You can also check my year in review for posts I’m fond of.


Title Views Written
Why Chris Christie picks on teachers 699 Aug 12
Radio silence on Clarence Mumford 660 July 12
Learning Math 605 Aug 12
American Indian Public Charters: What Word Are You Forgetting, People? 602 Apr 13
Acquiring Content Knowledge without Hirsch’s Help 555 Jan 13
Jo Boaler’s Railside Study: The Schools, Identified. (Kind of.) 548 Jan 13
Boaler’s Bias (or BS) 521 Oct 12
Picking Your Fights—Or Not 501 Apr 13
Those Who Can, Teach. Those Who Can’t, Wonk. 493 Dec 12
What’s the difference between the SAT and the ACT? 483 June 12
The Fallacy at the Heart of All Reform 454 Sept 12
The difference between tech hiring and teacher hiring 219 June 12

Pedagogy and Curriculum

Probably not too interesting unless you’re a teacher. But I have to say that Modeling Probability is pretty kick ass.

I realize these probably come off as vanity posts, but for me, they’re a great way to take stock. I have had a genuinely terrific year, between blogging and teaching, and it’s fun to write it all down.

Meanwhile….midterms again

I originally thought I could blog daily. Ha, ha. I managed close to that in January 2012, the first month of the blog, but it made me unhappy. Since then, I’ve usually managed an entry every five days or so. Sometimes I’d go 8-9 days after a really successful post, hiding away and watching reruns of Quantum Leap or The Mentalist, worried that I’d written my last good thing.

But this gap is more like the two-week plus hiatus I had in April of last year, when I spent most of a month working on a piece (under my real name). When I’d finally finished, I couldn’t even think of a thing to write about for a week or more. And so it was with my Philip K. Dick and preschool piece, which I’ve been thinking about since late December and working on for over a month. Kicked my ass, a bit. Happily, it was extremely successful—already #8 on my greatest hits list. My Twitter followers know that the piece had one fan that truly made it all worthwhile, and it kills me not to namedrop, but I won’t.

And then, midterms. My school covers a year in a semester, so we just finished the “semester” grades for the second “year”. We teach 4 block periods a day, so the kids can take 8 classes a year. Not great for test scores, since half the kids haven’t looked at the material in four months, and the other half haven’t entirely finished the courses. But it works in other ways.

One thing I noticed in my first “year” at this school: I wasn’t testing enough. In a normal schedule, I test or quiz kids every 8-10 school days, with the occasional longer gap. But in a block schedule covering a year in a semester, that’s the equivalent of testing kids every three-four weeks. Nothing terrible, but I decided I should up the pace slightly.

Except I’m teaching four classes and three preps—one of them for the first time. So I upped my quiz/test activity, thus increasing both my test creation and grading load, at the same time I was working a heavier teaching load, both in time and in preps. I work straight through from 8 to 3, in front of an audience of 35, with a brief 10 minute break at 9:30 and a half hour lunch at 12:35.

You know how teachers complain about grading, the tyranny of tests? I never have. I never even knew what they were talking about. I work fast, read fast, think fast, and I’m an insomniac. So the occasional two hour grading stint never struck me as particularly onerous. I’d generally schedule the grading session on a weekend, but if I had to grade things on a Tuesday or Wednesday every so often, I could manage easily. I pride myself on a rapid test turnaround, returning quizzes and tests within 3-4 days, a week at worst.

And suddenly, I was giving tests or quizzes to 140 students in three different subjects every week. Every time I turned around, I had a stack of tests to grade. At a modest estimation, I’ve spent 6 hours a week grading since February.

Then midterms. My geometry midterm is fifty multiple choice questions (multiple choice), my intermediate algebra 33. I have two sections of algebra, the test was four pages long. I graded it a page at a time on Sunday. Figure it took me 30 seconds a page to grade. 30*4*70 is 8400 seconds, or over two hours just to correct them. Then I had to calculate the curve and write the scores on each test, which takes another half hour or so, so three hours. And most teachers will tell you that’s a fast job.

Then my geometry test, which was five pages. I did two of them on Monday night, then just couldn’t take it any more. So Tuesday, I passed the tests out to the kids (geographically arranging them so no kid had his own test or a near neighbor’s), gave them red or blue pens, and we went through the whole test in 10 minutes. The kids loved the activity, were scrupulously exact, and asked why we didn’t grade tests this way all the time. I told them I usually look over the tests closely to see what kind of mistakes the kids make, what areas need revisiting, and so on. But I wasn’t going to be managing that little activity on this test.

Then Pre-Calc, the test that I spent five hours building the night before I gave it, and four hours grading on Wednesday night/Thursday morning (grades due 1:30 pm on Thursday).

The next day, Friday, I gave my intermediate algebra kids a quiz, so I have seventy quizzes to grade this weekend. I also have to build a pre-calc test (polynomials) and a geometry quiz (similar triangles).

I am not whining. Were I teaching a normal schedule this work would be entirely manageable. The longer day itself exhausted me for all of February, but by the first week of March I’d acclimated to the extra strain. I am getting well-compensated for the extra work, if you assume my normal salary is adequate, and I do. I get a 33% bump for the extra class.

But I finally know what teachers are talking about when they complain about the workload. It took an extra class and a concentrated effort on my part to assess my kids more regularly to get me to the point that many teachers reach on a normal schedule. Given that I’ve always had an inordinate ability to work long hours and do a lot of work without noticing (I worked full-time through all three of my degrees), I figure that I’m odd and the rest of teachers aren’t lazy.

Meanwhile, I’m beat.

Random notes on the classes:

Geometry: my most difficult class, academically speaking, since it’s a 10-12 class. Six kids are probably bored, because I can’t give them enough to do—the bottom half of the class is not unmanageable, but I can’t leave them unattended for too long without havoc ensuing. But they’re learning despite themselves, and the algebra progress is extraordinary. The kids right out of the top 6 but above the halfway mark are appropriately challenged, I think, and doing well. But I rarely feel good about the class.

On the other hand, last week, right at the time I was feeling really low, I gave the kids a worksheet (looked something like this, with a wider range of difficulty). I told them their task was twofold: 1) use what they knew about isosceles and equilateral triangles and 2) most importantly, build on the diagram, using what they knew, to solve for x. The second part, I said, was incredibly important. They couldn’t just look at the diagram and create an equation. They had to take the given information and work forward. The equation might be two or three steps away. I was not hopeful. And glory be, the kids surprised me by doing a bangup job—the weakest kids finished at least half the handout without begging for help every second. So I must have instructed them well! Need to remember what I said. Midterm performance: exceptional. They averaged 81% on the midterm (with a 15 point curve), and even the weaker kids did a great job on a long test.

Intermediate Algebra class: Having spent too long last “year” on linear equations, I used the same amount of time to cover linear equations, inequalities, and absolute values. We’re now wrapping up quadratics. Now that I’ve kicked linear equations and inequalities into submission, I’m struck by how damn complicated quadratics are. They don’t easily model. They have multiple forms and multiple methods of solving, all of which are complicated. But these kids, too, are doing well. Midterm: average 73% with almost no curve.

Precalc: longer post coming, eventually. Fun class. Midterm average was 69%. I have about 5 kids who simply do not understand the material, and I’m a tougher grader at the precalc level.

I was going to write this post first, but I’d seen one too many wails about American Indian Public Charter High, so I got back into it with a bang.

Modeling Linear Inequalities

I committed to making a big leap forward in inequalities this year. They’ve always been low priority in my curriculum, nothing more than a subset of equations, even though as a programmer, I can come up with fifteen real-life examples of working with them—much more than I can for linear equations. But once I kicked off linear equations with modeling, I could see some obvious introduction points that would help them fall into place. I committed the time to design some new lessons and build some new handouts. And I completely forgot to take pictures of the boardwork, dammit.


Back to tacos and burritos, but this time I took away the “spend all the money” constraint. Stan could either make the purchase or he couldn’t. I gave them some starting combinations to test, and then they came up with their own.

By this time, they are old hands at modeling and easily came up with more true and false pairs. I then graphed all the points on the board, using blue for “True” values, and red for “False”.


“See that space between the TRUE and FALSE values? You can see pretty clearly there’s a line separating them, like this:


“Take a second. What might that line be?” I’m pleased to see several hands shoot up, but I pick on Karl, slinking away from my glance up front. “Karl?”

“I don’t know.”

“What if it wasn’t an inequality?”

“Well, then it’d be 3x + 5y = 60….oh. It’s the same line?”

“Yeah,” chimed in three other students.

“Yes, exactly. We are now working with linear inequalities, not linear equations. Solutions aren’t defined by a single point, but by entire regions. The line is the same in both equations and inequalities, but in an inequality, the line acts as a border between the TRUE and FALSE values. Everything on one side of the line is TRUE, and everything on the other is FALSE.”

From there, I give them one of the new handouts:

I’m really pleased with this one, as a first pass. The students plot the TRUE and FALSE values in different colors. Then they determine which of the linear inequality borders will correctly separate the regions. In one exercise, they become more familiar with solution regions, while also improving their ability to visualize lines from an equation (especially, god help me, positive and negative slopes).

Next, I give them a notes handout I’ve modified over the years.


as well as board notes a few days later:


I made two HUGE changes this year from previous years, changes I suspect most math teachers will recognize.

First, I abandoned “above and below the line” test for the solution region. This seems so obvious to math teachers, but it’s really only meaningful to about half the students. Worse, the test only works with slope intercept form, and has no validity in standard form. The purple math link above tells the student to solve the equation for y—yeah, because solving inequalities is such a breeze, particularly with negatives. It’s a kluge. I didn’t even mention that test.

Instead, I had them use (0,0) for a test value and had them test the value for all forms. MUCH better. No “hey, remember the first way I taught you? Above and below the line? It doesn’t work in standard form, so here’s another way!” Testing values always works.

(Liam in the comments pointed out that the test value can’t be on the border, and I do cover this. I don’t put it in my notes, though. I used to, but the kids would get very confused at the caveat. So now I wait until they are confident of the (0,0) point test and then introduce an example in which the point is on the border equation. I’m discovering that it sometimes takes me a couple years through before I figure out the best way to create fully complete notes.)

Second, I had them write “true” or “false” by the test point, and then shade the same or opposite side of the line. This gave them a visual and kinesthetic step in the process: One, test the inequality with (0,0). Two, write “true” or “false”. Three, shade the correct side of the line.

I’m sure in later years I’ll further hone this, but this was the first time I felt good about my linear inequalities unit. From there, systems of inequalities was an obvious step:


I did a day of modeling systems, but no, I didn’t go onto linear programming.. Instead, I moved onto practicing graphing without the models, and they did great.

I used one of my favorite worksheets as a quiz. The average grade was a B+. Check out this sample—see the “true” markings? By golly, they listened.


I was so emboldened by my success I went onto absolute values, which I usually only cover for tests. They did all right with that, too.

I’m grading the second unit test, covering linear equations, inequalities, and absolute values. So far, it’s looking very good.

Modeling Linear Equations, Part 3

See Part I and Part II.

The success of my linear modeling unit has completely transformed the way I teach algebra.

From Part II, which I wrote at the beginning of the second semester at my last school:

In Modeling Linear Equations, I described the first weeks of my effort to give my Algebra II students a more (lord save me) organic understanding of linear equations. These students have been through algebra I twice (8th and 9th grade), and then I taught them linear equations for the better part of a month last semester. Yet before this month, none of them could quickly generate a table of values for a linear equation in any form (slope intercept, standard form, or a verbal model). They did know how to read a slope from a graph, for the most part, but weren’t able to find an equation from a table. They didn’t understand how a graph of a line was related to a verbal model—what would the slope be, a starting price or a monthly rate? What sort of situations would have a meaningful x-intercept?

This approach was instantly successful, as I relate. Last year, I taught the entire first semester content again in two months before moving on, and still got in about 60% of the Algebra II standards (pretty normal for a low ability class).

So when I began intermediate algebra in the fall, I decided to start right off with modeling. I just toss up some problems on the board–Well, actually, I start with a stick figure cartoon based on this lesson plan:


I put it on the board, and ask a student who did middling poorly on my assessment test, “So, what could Stan buy?”

Shrug. “I don’t know.”

“Oh, come on. You’re telling me you never had $45 bucks and a spending decision? Assume no sales tax.”

Tentatively. “He could just buy 9 burritos?”

“Yes, he could! See? Told you you could do it. How many tacos could he buy?”


At this point, another student figures it out, “So if he doesn’t buy any burritos, he could buy, like,…”

“Fifteen tacos. Why is it 15?”

“Because that’s how much you can buy for $45.”

“Anyone have another possibility? You? Guy in grey?”

Long pause, as guy in grey hopes desperately I’ll move on. I wait him out.

“I don’t know.”

“Really? Not at all? Oh, come on. Pretend it’s you. It’s your money. You bought 3 burritos. How many tacos can you get?”

This is the great part, really, because whoever I call on, and it’s always a kid who doesn’t want to be in the room, his brain starts working.

“He has $30 left, right? So he can buy ten tacos.”

“Hey, now, look at that. You did know. How’d you come up with ten?”

“It costs $15 to get three burritos, and he has $30 left.”

So I start a table, with Taco and Burrito headers, entering the first three values.

“And you know it’s $15 because….”

He’s worried it’s a trick question. “…it’s five dollars for each burrito?”

I force a couple other unwilling suckers to give me the last two integer entries

“Yeah. So see how you’re doing this in your head. You are automatically figuring the total cost of the burritos how?”

“Multiplying the burritos by five dollars.”

“And, girl over there, in pink, how do you know how much money to spend on tacos?”

“It’s $3 a taco, and you see how much left you have of the $45.”

“And again with the math in your head. You are multiplying the number of tacos by 3, and the number of burritos by….”


“Right. So we could write it out and have an actual equation.” And so I write out the equation, first with tacos and burritos, and then substituting x and y.

“This equation describes a line. We call it the standard form: Ax + By = C. Standard form is an extremely useful way to describe lines that model purchasing decisions.”

Then I graph the table and by golly, it’s dots in the shape of a line.


“Okay, who remembers anything about lines and slopes? Is this a positive or a negative slope?”

Silence. Of course. Which is better than someone shouting out “Positive!”

“So, guy over there. Yeah, you.”

“I wasn’t paying attention.”

“I know. Now you are. So tell me what happens to tacos when you buy more burritos.”

Silence. I wait it out.

“Um. I can’t buy as many tacos?”

“Nice. So what does that mean about tacos and burritos?”

At this point, I usually get some raised hands. “Blue jersey?”

“If you buy more tacos, you can’t buy as many burritos, either.”

“So as the number of tacos goes up, the number of burritos…”

“Goes down.”

“So. This dotted line is reflecting the fact that as tacos go up, burritos go down. I ask again: is this slope a positive slope or a negative slope?” and now I get a good spattering of “Negative” responses.

From there, I remind them of how to calculate a slope, which is always great because now, instead of it just being the 8 thousandth time they’ve been given the formula, they see that it has direct relevance to a spending decision they make daily. The slope is the reduction in burritos they can buy for every increased taco. I remind them how to find the equation of a slope from both the line and the table itself.

“So I just showed you guys the standard form of a line, but does anyone remember the equation form you learned back in algebra one?”

By now they’re warming up as they realize that they do remember information from algebra one and earlier, information that they thought had no relevance to their lives but, apparently, does. Someone usually comes up with the slope-intercept form. I put y=mx+b on the board and talk the students through identifying the parameters. Then, using the taco-burrito model, we plug in the slope and y-intercept and the kids see that the buying decision, one they are extremely familiar with, can be described in math equations that they now understand.

So then, I put a bunch of situations on the board and set them to work, for the rest of that day and the next.


I’ve now kicked off three intermediate algebra classes cold with this approach, and in every case the kids start modeling the problems with no hesitation.

Remember, all but maybe ten of the students in each class are kids who scored below basic or lower in Algebra I. Many of them have already failed intermediate algebra (aka Algebra II, no trig) once. And in day one, they are modeling linear equations and genuinely getting it. Even the ones who are unhappy (more on that in a minute) are getting it.

So from this point on, when a kid sees something like 5x + 7y = 35, they are thinking “something costs $5, something costs $7, and they have $35 to spend” which helps them make concrete sense of an abstract expression. Or y = 3×-7 means that Joe has seven fewer than 3 times as many graphic novels as Tio does (and, class, who has fewer graphic novels? Yes, Tio. Trust me, it’s much easier to make the smaller value x.)

Here’s an early student sample, from my current class, done just two days in. This is a boy who traditionally struggles with math—and this is homework, which he did on his own—definitely not his usual approach.


Notice that he’s still having trouble figuring out the equation, which is normal. But three of the four tables are correct (he struggles with perimeter, also common), and two of the four graphs are perfect—even though he hasn’t yet figured out how to use the graph to find the equation.

So he’s doing the part he’s learned in class with purpose and accuracy, clearly demonstrating ability to pull out solutions from a word model and then graph them. Time to improve his skills at building equations from graphs and tables.

After two days of this, I break the skills up into parts, reminding the weakest students how to find the slope from a graph, and then mixing and matching equations with models, like this:


So now, I’m emphasizing stuff they’ve learned before, but never been able to integrate because it’s been too abstract. The strongest kids in the class are moving through it all much faster, and are often into linear inequalities after a couple weeks.

Then I bring in one of my favorite handouts, built the first time I did this all a year ago: ModelingDatawithPoints. Back to word models, but instead of the model describing the math, the model gives them two points. Their task is to find the equation from the points. And glory be, the kids get it every time. I’m not sure who’s happier, them or me.

At some point in the first week, I give them a quiz, in which they have to turn two different models into tables, equations, and graphs (one from points), identify an equation from a line, identify an equation from a table, and graph two points to find the equation. The last question is, “How’s it going?”

This has been consistent through three classes (two this semester, one last). Most of the kids like it a lot and specifically tell me they are learning more. The top kids often say it’s very interesting to think of linear equations in this fashion. And about 10-20% of the students this first week are very, very nervous. They want specific methods and explicit instructions.

The day after the quiz, I address these concerns by pointing out that everyone in the room has been given these procedures countless times, and fewer than 30% of them remember how to apply them. The purpose of my method, I tell them, is to give them countless ways of thinking about linear equations, come up with their own preferred methods, and increase their ability to move from one form to another all at once, rather than focusing in on one method and moving to another, and so on. I also point out that almost all the students who said they didn’t like my method did pretty well on the quiz. The weakest kids almost always like the approach, even with initially weak results.

After a week or more of this, I move onto systems. First, solving them graphically—and I use this as a reason to explitly instruct them on sketching lines quickly, using one of three methods:


Then I move on to models, two at a time. Last semester, my kids struggled with this and I didn’t pick up on it until a month later. This last week, I was alert to the problems they were having creating two separate models within a problem, so I spent an extra day focusing on the methods. The kids approved, and I could see a much better understanding. We’ll see how it goes on the test.

Here’s the boardwork for a systems models.

So I start by having them generate solutions to each model and matching them up, as well as finding the equations. Then they graph the equations and see that the intersection, the graphing solution, is identical to the values that match up in the tables.

Which sets the stage for the two algebraic methods: substitution and combination (aka elimination, addition).


Last semester, I taught modeling to my math support class, and they really enjoyed it:


Some sample work–the one on the far right is done by a Hispanic sophomore who speaks no English.

Okay, back at 2000 words. Time to wrap it up. I’ll discuss where I’m taking it next in a second post.

Some tidbits: modeling quadratics is tough to do organically, because there are so few real-life models. The velocity problems are helpful, but since they’re the only type they are a bit too canned. I usually use area questions, but they aren’t nearly as realistic. Exponentials, on the other hand, are easy to model with real-life examples. I’m adding in absolute value modeling this semester for the first time, to see how it goes.

Anyway. This works a treat. If I were going to teach algebra I again (nooooooo!) I would start with this, rather than go through integer operations and fractions for the nineteenth time.

Modeling Probability

This is a lecture class, but I put all the instructions in the handout as well, mostly so I can remember the outlines of the activity.

In the lecture, I explain that video games run on probabilities, that balancing probabilities is an essential element for strategy video games. Any games that give the user a running series of choices has to balance outcomes and create choices with tradeoffs. Otherwise, the user learns that the “good” choices are and the game gets boring. (I have no idea if this is true, but it sounds reasonable.)

Obviously, if the students were really at a software company, these scenarios would be automated, but hey, it’s a math class.

Materials: Scintilla Handout (reproduced here), and 9 Oracle advice cards (3 gryphons, 3 dragons, 3 excaliburs)

Here’s the main handout.

Crossing the Scintilla

You’re an intern at a major software development company! You’ve been assigned to work with the team developing Sorcery & Shadows, a new game scheduled for fall release.

S&S is “retro”, harking back to the 70s and 80s fantasy games—less violence, more strategy. At various points, the player’s choice of avatar must consult the three Oracles for permission and guidance. The Oracles determine the character’s actions, but the response varies based on the avatar chosen:


The software team is working on the following scenario:

The player (Vlad, Dulcinea, or Chaos) is trying to cross the Scintilla River. The Oracles must be consulted. The Oracles will each advise one of the following options:

  1. The gryphon, which swims across.

  2. The dragon, which flies across.
  3. Excalibur, the sword, which magically transports the carrier.


Gryphon                                Dragon                                         Excalibur

The player will follow the Oracles’ advice if the right number of them agree. The Oracles are pleased when the player follows their advice and give the player five silver coins.

If the player can’t follow the Oracles’ advice, then the player must pay 1 silver coin to cross on the ferry.

The software team has already decided that the Oracles’ responses are randomly generated, and they need to determine the probability that each character gets across the river. They’ve asked you to work this out.

Before you start, have a brief group discussion and make your predictions.

    Is it equally probable that Vlad, Dulcinea, and Chao will be able to follow the Oracles’ advice?

  1. If not, who do you think is the most likely to be able to follow the advice?
  2. Least likely?

I wander around to listen in on the predictions. The two most common predictions are Vlad and Dulcinea, which is interesting.

I always do two trials as a class before I set them on their own, stressing that the actual advice—Gryphon, Dragon, Excalibur—is immaterial. They are tracking which character is able to accept the Oracle’s advice. Back to the handout—this is on the flip side.

Experimental Probability
Experimental Probability—Performing an event repeatedly and measuring the results as a ratio of a particular occurrence to the total number of trials: exprob.

Once a pattern has been established, we can rely on this data as an empirical probability.

You are going to perform 30 simulated trials of the event “Asking the Oracles for Advice”.

Each group has three sets of three cards: a gryphon, a dragon, and Excalibur. Three group members will “play Oracle”, by randomly and simultaneously throwing down one of the three cards. (IT MUST BE RANDOM!). The other group member tracks the outcome of each trial, using the table below.


When you have completed 30 trials, compute the experimental probability of each character’s likelihood of following the Oracles’ advice. Remember to keep careful track of how many trials you run, as that’s going to be your denominator.

Send someone from your group up to the front whiteboard and report your results. Report the totals, not the experimental probability percentages. We’re going to calculate the results for the class.


The kids love the trials. They are religiously random, and really get a kick as they see a clear pattern emerge in the results.

After all the results are on the board, I tote them up and calculate the experimental probability for the class.

Then I transition to theoretical probability and discuss the difference between experimental probability—what actually happens in a series of trials—and theoretical, what is expected to happen. I ask for examples of trials that would have no theoretical probability: medical trials, new treatments, new procedures. I point out that researchers run thousands of trials because they want to have a reliable experimental model in order to begin to build theoretical probability.

In other cases—coin tosses, lottery tickets, and asking the Oracles’ advice—the theoretical probability is easily modeled. And that’s what we do next. Back to the handout, page 3.

Who Crosses the River?–Theoretical Probability

In Part B, you ran trials and calculated the experimental probability of each character’s being able to follow the Oracles’ advice. Now you’re going to determine the theoretical probability for each character and compare them.

Theoretical probability can be calculated when the number of possible outcomes is fixed.

In this case, we can define the following:

Sample Space:
The set of all possible outcomes for a trial.
Particular outcome(s) of the sample space. An event may contain other events, each with its own sample space.
Target event:
The desired outcome to be tested.

For example, the sample space for one instance of “asking for an Oracle’s advice” is: gryphon, Excalibur, dragon.

But in this case, we are asking for all the Oracles’ advice. So how do we find all the combinations possible from three Oracle requests?

There are three useful tools to help you model theoretical probabilities for a multiple-event scenario.

  • A tree diagram can help you determine all the possible outcomes for a “compound” (multiple event) outcome, particularly complex events such as this one. Trees model a new “branch” for each component of an event.

  • An area model is simple and easy, but limited to two or three events.
  • A counting diagram is useful for ordering and calculating the number of outcomes.

See the flip side of this page for detailed descriptions of area models and probability trees.

Probability Tree

Working with your team, create a probability tree for all possible combinations of Oracular responses.

Using different colored pens for each character, trace the different outcomes: all three match (Vlad), two match (Dulcinea), no matches (Chaos).

Count up the possible crossings for each character, and the total possible outcomes.

How many different outcomes are possible? __________________

How many outcomes allow Vlad to cross? ________

Dulcinea? _______________

Chaos? _______________

Compare these numbers to your own experimental results, as well as the class totals. How do they compare?

I use a CPM handout (page 14), which isn’t all that great but gives visual examples of both models on one page. All I really want is the visual, which I haven’t gotten around to building for myself yet.

Building the probability tree diagram for “asking the Oracles” is a great activity that really brings home the difference between theoretical and experimental probability. The kids can see why Dulcinea gets the necessary agreement more often, and why Chaos wins more frequently than Vlad.

In earlier years I would have just had the students create the probability tree on paper, in their groups. But earlier in the school year, I came up with a fun way to work on bigger, multi-step problems. I have a lot of whiteboards. So in their groups, the kids go to a whiteboard section, and start working on the assigned problem(s). They have more room, I can see the work and be sure everyone’s got the correct answers. It shoves the math right under the nose of the weaker students, who might otherwise (ahem) sit quietly hoping I don’t notice they aren’t working or paying attention. And it mixes things up, which is always useful. I used this about three times during the year (our semester is a year); here’s a picture of them working on parabolas:


(Note: smudged the faces, took out the color, and asked the kids permission to use the photo.)

As I was driving to school the morning of this class, I suddenly realized that whiteboard work would be perfect for the probability trees. If I had them do it on paper, at least half of the kids would be looking on as someone else did all the work anyway, so I might as well be sure they were actually looking on, instead of tuning out. I usually use this method as review rather than for new concepts, but in this case it was the right call. Each of the groups were all involved in their own tree, and none of them were simply copying some other group’s work—some surreptitious checks to see if they were on the right track, sure. But that’s a bonus.

There were several gorgeous versions done in red, blue, green, and purple that I forgot to photograph before I erased them, but this one is nicely functional. Except the misspelling.

So we take a few minutes to compare the theoretical outcomes with the experimental, and since we have close to 200 trials (8 groups, 25 trials on average), they match up beautifully.

I tell them that science and research engage in experimental probability, but in math, we focus entirely on the theoretical by modeling the possible outcomes. I always start with the most flexible and visual, but the least useful, model. I then outline the other two models I want them to use, both of which I think have limitations, but are much more helpful: the area model and the “counting diagram”.

Counting Diagram

I always snicker at a formal name for a very simple concept. But it’s extremely useful as an organizer.

One blank line for every event. So asking the Oracles, there are three events.

______      ______      _______

How many outcomes are possible for each event? The events are independent of each other.

___3__        __3___        ___3___ = 27

Multiply across. That’s the total number of outcomes that can result in asking the Oracles.

Now, model each character. In these cases, the number of desired outcomes for subsequent events is conditional. (I point out that the Oracles’ response is still random and independent, but the desired response is conditional, and that’s what we’re counting).

Vlad is the easiest. The first event can be any of the three responses. But the second and third events must match the first, so there’s only one acceptable outcome for each.

___3__       __1___       ___1___ = 3

Dulcinea is more complicated, but the trees are very helpful in getting the kids to see that the first two events can have any outcome. The third event must match one of those first two.

___3__       __3___       ___2___ = 18

Finally, Chaos. Why is Chaos twice as likely as Vlad to get the correct Oracle response? The counting diagram helps students see that as each event occurs, he loses the possibility of that outcome—and yet, this gives him more outcomes than Vlad.

___3__        __2___         ___1___ = 6

The diagram can also calculate probabilities of multiple events, but it’s primarily useful for counting.

Right around now, I bring up lottery tickets, and we go through the hugeness of the numbers in a diagram. And here, I mention a key difference between theoretical and experimental probability, aka Why All Math Teachers Tell You Not To Buy Lottery Tickets.

Theoretical probability says it’s utterly pointless to buy lottery tickets. But every time the lottery runs, someone achieves the functional equivalent of getting struck by lightening while finding a four-leaf clover while getting abducted by aliens. Someone wins. Reality occurs. Gambling exists because of experimental probability. So my students won’t get the Lottery Tickets lecture from me. Go ahead and buy. Cross your fingers when your plane takes off, even though the car ride to the airport was the riskier trip. But if you blow your entire salary on poker, find a 12-step program.

Area model

My students are already

For this, I have a handout, because the area model makes the two most important probability operations beautifully clear: when to multiply probabilities, and when to add them.

Limitations—only two sample spaces, alas. You couldn’t model “asking the Oracles” with the rectangle.

So there’s the three basic models that we use for the rest of the unit. I often return to the “intern at the software shop” scenario, which gives me endless possibilities. Here’s a couple more.

This one is usually homework for the first day—what is the probability of getting various payouts? I don’t mean expected value—we go through that the next day, with the original Scintilla scenario and this one.

And here’s one that goes nicely with an area model, which can really help students visualize conditional probability.

Thanks to binomial expansion, probability and elementary combinatorics are sandwiched into second year algebra and it’s hard to go into the subject in depth. AP Stats is pretty joyless. Example 99,521,325 on the list of Why We Need to Offer a Broader Range of Math Classes.

Teaching Congruence, or Are You Happy, Professor Wu?

I first ran into the writings of Professor Hung-Hsi Wu in ed school, and never forgot him. How often do you see a math professor hyperventilating about elementary teachers as the abused children of math education:

For elementary teachers, there is at present a feeling that they have been so damaged by their K–12 experience…that we owe it to them to treat them with kid gloves…. Those that I have encountered are generally eager to learn and are willing to work hard. The kid-glove treatment would seem to be hardly necessary. …There is another school of thought arguing that for elementary teachers, one should teach them not only the mathematics of their classrooms, but at the same time also how children think about the mathematics. Again, I can only speak from my own experience. The teachers I observed usually had so much difficulty just coming to terms with the mathematics itself that any additional burden about children’s thinking would have crushed them.


There are ample reasons to believe that at present most teachers are operating at the outer edge of their mathematical knowledge. Now when one finds oneself in that situation, one is prone to being tense and inflexible, and is consequently not likely to create a friendly atmosphere for learning. There should be a study to look into how much of the so-called math phobia in this country can be traced to this fact (especially in elementary schools). The other simple reason is that no matter how elementary the topic, some students would bring up deep or at least non-elementary related questions.2 If the teacher fails to answer such questions too often, the students’ confidence in the teacher is eroded and, again, a non-productive learning atmosphere would result.

Abuse victims grow up to be abusers, you know?

He trains elementary school teachers in these special recovery workshops for abuse victims:

The main difficulty with the Geometry Institute, and the relative lack of success thereof, was the teachers’ unfamiliarity with anything geometric. With but mild exaggeration, some teachers literally trembled at the sight of ruler and compass or when they were handed a geometric solid. As mentioned in the preceding paragraph, we were prepared for teachers’ being ill-at-ease with geometric reasoning and lack of geometric intuition, but not for the degree to which both were true. School education in geometry is in deep trouble.

I know many who read Professor Wu and take his descriptions at face value, using him as evidence of teacher stupidity. Anyone who believes a math professor—a mathematician, that is—can be objective about elementary school teachers’ math knowledge has never met any mathematicians. Read the above link in which he describes the bare minimum of what fifth grade teachers need to know in order to prepare their students, and ask yourself if elementary school math teachers have ever known that much. How on earth did we all get to the moon and create the internet?

Still, his fulminations about geometry caught my attention over the summer, when I read this:

As to the subject of school geometry, the problem is that if universities do not teach it, or do not teach it well, then the only exposure to school geometry that geometry teachers ever have will be their own high school experience in geometry. The latter of course has been scandalously unsatisfactory for a long time, to the point where many school geometry courses cease to prove any theorems.

Well, hang on. As I’ve mentioned before, I don’t teach proofs and rarely prove theorems. I also typically dump transformations, most of construction, and solids. But I’m teaching non-honors geometry in a Title I school, where geometry is but a brief respite before the kids are dumped back into Algebra Hell. As a tutor, I’m very familiar with geometry as it is taught in the high-performance schools in the area, which are some of the highest performing schools in the country, and they are getting proofs a-plenty. Moreover, geometry hasn’t been “scandalously unsatisfactory” since we began forcing everyone into college prep math (an absurd notion in and of itself). What’s scandalously unsatisfactory is the idiocy of trying to teach proofs to low ability kids. But I digress. The point is, I dispute his notion that geometry is taught badly at all schools. A highly modified version of geometry is taught at some schools, for a very good reason, and given the kids involved there’s little evidence that it’s doing any damage, and regular “old school” geometry is routinely being taught to the top students.

But I was curious, nonetheless, as to what dire notions Prof Wu had about geometry as it is taught in schools, and so I googled. Teaching Geometry According to the Common Core Standards is the first article I read, but this interview with Rick Hess spells out the key point with the fewest words (would that I valued the same behavior in myself):

As another example, when state standards ask that the concept of congruence be taught in middle school, they do not realize that what students will end up getting is that congruence means same size and same shape. As a mathematical definition, the latter is completely unacceptable.

I sat up straight at this. Remember, I’m not a mathematician (which of course means that Prof Wu wouldn’t let me near fifth graders, much less high schoolers), and had never given much thought to congruence and similarity until I began teaching public school, as opposed to reviewing similarity for college admissions tests (congruence is not a tested subject). At that point, however, the book definitions seemed a tad circular. Check out the CPM section on similarity. It uses dilations, which (as you will see) is the right start, but at no point does the text explain the link between similarity and dilation. It’s all “cut out the figures, talk with your group, what’s the same, what’s different” crap. And when I taught geometry two years later using Holt, the official definition of congruence, straight from the book, was “Two polygons are congruent if and only if their corresponding sides and angles are congruent”. Really? Polygons are congruent if they are congruent?

I actually apologized to my class last year when we got to that point, because I hadn’t noticed this bizarre definition until the day before I taught congruence. I said yeah, a tad circular, and I hereby promise to investigate but for now, let’s go with it. (Sorry, Professor Wu, but most of them are never going to use congruence again.)

So here’s Wu saying that yes, this is a problem? How does he want it to be taught?

Holy Crap. Rigid motions? Isometries? (not isometrics. I make that mistake all the time) The sections I ignore? They have a purpose? I should have majored in math. No, not really.

This was a huge revelation, and incredibly easy to put into action. Most students got two “transparency triangles” and a white board. Some students used the graph paper with the transparency triangles. Three students (strong students) used the white boards with different colored pens and no manipulative. I wanted to see which methods worked best and if any problems came up with a particular method.

Day 1: Introduce translations and reflections, moving to increasingly complex reflection equations. Emphasize that reflections occur over a line; evaluate the change in coordinate points with pre-image and image, and then start calculating new coordinate values without using the manipulatives first.

Day 2: Rotations (by far the most difficult, in my mind). We focused on rotations of 90 degrees, and on reviewing the definition of perpendicular slopes, since that’s how the students found the new point—find the slope from the point of rotation (usually the origin) to the vertex to be rotated, convert to the perpendicular slope. Stressed that rotations were around a point, in contrast to reflections.

I foolishly didn’t take pictures in class that day—or if I did, I can’t find them. Here’s roughly what it would have looked like for a student using graph paper and the manipulatives, except they used colored pencils for the different slope connections. This is an example rotating a triangle 90 degrees clockwise.

First step (click on image) was to identify the slope from the point of rotation to each vertex. Then they identified the perpendicular slope for each of the rotation points—reinforcing perpendicular slope relationships being a big ol’ secondary point of the lesson—and sketched that line in as well. The students used different colors for each vertex, so they could easily see the before and after for each point, and recognize the 90 degree nature of the turn.

Then, with the points sketched, they did the actual rotation. Put the triangle on the original point, hold the manipulative at the point of rotation, and turn. Voila.


And then put the first manipulative in the original position to see what the rotation before and after looks like.


I’ve always had a difficult time teaching rotations, but the manipulative really helped.

I end the day pointing out that transformations preserve both degree and distance. They can see this because they are using identical manipulatives, but I have them calculate some side lengths and slopes to confirm.

Day 3: Congruence
And now, congruence. Instead of a circular definition, I have a clean syllogism:

If Polygon A is congruent to Polygon B, then A can be mapped onto B using a series of transformations. If the figures can be mapped into the same space, then their corresponding angles and sides are congruent, because the mapping preserves degree and distance. Therefore, congruent polygons’ corresponding angles and sides are also congruent.

From there, I go onto congruence shortcuts and proofs, blah blah blah. But it started much more cleanly. I taught transformations, reviewed perpendicular lines and other coordinate geometry formulas, and linked it all to congruence in a meaningful way.

A few weeks later, it was onto similar polygons. Again, instead of just saying “Similar polygons have congruent angles and proportional sides”, I can link it to dilation.


Day One: Review of Proportionality, then onto dilation

The kids did straight dilations as well as transformations and dilations in combination. I started with straight dilations, because I wanted the students to confirm the elements of similarity. The kids generally remember that parallel lines have the same slope, but I thought it would also be useful to see the transversal relationships with the parallel lines. We could prove, algebraically, that the lines of the dilated triangle were parallel to the original, and we could then extend those lines to prove that the corresponding angles on each triangle were congruent. Here’s an example (again, one I just sketched up) that shows how the kids determined the angles were congruent.


The kids colored the corresponding angles—there are three in each case (one of the green ones in my image is an error, you can see I xed it out, just too much hassle to draw again).

So again, the point was to algebraically and visually confirm the parallel relationship, and then follow the dual sets of parallel lines and transversals to confirm that the angles are congruent.

I had them do a combination transformation/dilation, confirming that order didn’t matter, and identifying which of the isometries had the parallel relationship.

Day 2: Review of Dilation, then onto Similarity.


Linking isometries to congruence and similarity was so much better, and whenever I tell math teachers about it they go oooh, ahhh and think about trying it themselves. And yet, I can’t point to why it’s so obviously superior. I can’t swear that my students learned congruence or similarity more thoroughly—in fact, I think they learned it as well but not any better than my students last year.

But it’s just more….cohesive, maybe? Not only am I finally linking in rigid transformations, which I never gave more than a quick review at test time (two of the three are intuitive, rotations are tough), to the rest of geometry, and creating an organic reason to review the relevance of perpendicular and parallel lines/transversals, but I am also linking both of these concepts to congruence and similarity, rather than just giving that annoying circular definition. While congruence doesn’t have much relevance past geometry, similarity runs through the next three years of math in a big way. So anything that makes the introduction more meaningful is probably a good thing. Moreover, transformations are easily grasped by even weak students, and their interest kept them going through the review of perpendicular and parallel lines.

None of this required complicated worksheets. I taught congruence using notes and Kuta worksheets; for similarity we used the book (Holt). I taught the transformations with boardwork–I really, really could have used a document camera, but that just came a couple weeks ago. Still, the kids got it well.

The really important thing, though, is that I have to feel mildly guilty about mocking Professor Wu. Next time I see him at Math Survivors Anonymous, I’ll grovel.


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