Back to Not Teaching English

I taught ELL all last year, which isn’t really teaching English, which we don’t really know how to do, as it is drenching the students with as much language exposure as you can and hoping they’ll pick it up with their peers.

From January through June I had 18 kids, six of whom had better language skills than than the bottom 10-15% of my US history class of mostly Americans, two others who had no desire to learn English, and three or four who I couldn’t give much attention because their English was too weak and the middle six, the kids who genuinely benefited from my class, were too quick on the uptake by this point.

I was also just genuinely bothered by the reality of ELL. With so many American kids at risk, not getting specialized attention, why are we giving the equivalent of one full-time English teacher to non-citizens who’d just arrived? For free? I’ve always been able to shrug off the policy implications of my job, though.   Ideological concerns disappear once the bell rings.  By far the most nagging concern was my feeling I wasn’t helping the kids.

Like the legal requirement for ELL language support, which trumps all other concerns.  A month into last year, I’d realized that Charlotte wasn’t so much ELL as special ed. Her English was as good as it was going to get. She couldn’t read, couldn’t process complex thoughts, and couldn’t write. Her spoken language was pretty fluent. The other two teachers agreed, so we asked that she be put into the special day class–a request that had been made the previous year (2015-16) and had gone nowhere. We pushed for months, and finally towards the end of the year, Charlotte was given two blocks of special ed. She will get a certificate this June , and rumor has it she will be marrying a 32 year old. I would be unsurprised if a payment to her father was involved. After all, the man has two wives over here to support.

Then there’s the placement itself, which is absurdly slow and demanding. I spent the last three months going through all the procedures to place out Anj, Tran, Juan, and Mary and put them in regular classes, and Marshall and Kit down to one class of ELL instead of three.The students actively cooperated, eager to get out of ELL hell–can you imagine being stuck in three long English classes a day, with very little control over your choices in the last class? They went to their counselors, were denied choices, came back to me, I’d call and clarify and the counselors would reluctantly agree.   I rarely take on this sort of activist role, but I’d been assured that teacher recommendation would trump procedural requirements, so I kept plugging away.

It was all for nothing. All the students were forced into two English classes a day the following year, only because funding for the third class dried up for ELL 2s. Nothing I’d done.

See, this is why I don’t do activism, not being a fan of aggravation and disappointment.  I apologized to the six kids, but they thanked me. “I’m happy we tried,” said Ang, and the others nodded.

The work did eventually pay off, although it took a year. All but one of my students from last year have been promoted at least one grade. Marshall and Kit are down to one class.  Ang and Juan are in regular English and Bob, one of the other two teachers from last year, is now agitating to get those two in his AP English course, which may give you a sense of how idiotic it was for them to be in ELL to begin with. Most of these advancements would not have happened without my efforts and recommendations.

ELL wasn’t terrible, mind you. I loved the kids. I enjoyed spending 90 minutes a day on language.  I loved picking random discussion topics (food was a regular, also movies. Not beer, alas.)

The fourth day of school (fall 2016), my first with them, I wrote the words to the Pledge of Allegiance on the whiteboard. Our school recites the pledge daily; you’ll see kids stand at attention facing the direction of the school flag, whether they can see it or not.  (I’ve been really annoyed at the Pledge fuss this year because kids never groused about standing until now.)

ellpledge

I didn’t have that many white boards, so you can see where I’d use the board for other quick notes over the year. But the pledge words stayed up the whole year; we ceremoniously erased them on the last day.

As I mentioned in the essay above, I started to feel a nagging sense of dissatisfaction with the curriculum (or lack thereof). The texts I’d purchased for sustained silent reading while teaching freshman humanities came in handy, in addition to some of the discontinued texts I found laying about the room. What wasn’t useful at all was the assigned texts, which were far too difficult.

I’d also make up activities like this one, called “Welcome to English”

ELLactivity

This is the completed activity. I’d just write these up with blanks on the spur of the moment, helping them discover and categorize the  absurd range of spellings and pronunciations. Here’s a students’ response to a different day’s sentences:

ELLStudentWork

You’d think there’d be a curriculum with sentences like these already designed with different sound groups. Alas.

One utterly delightful day, I gave them the lyrics to “Song Sung Blue” to read and discuss. We went through the notion of “blue” being a sense of sad, of down.  We constructed an understanding of what they thought was a poem, how the act of singing when you’re sad will make you feel better. We talked about weeping versus crying, why the willow weeps (with pictures), what “subject to” meant, and after an hour, by god, most of the kids truly thought they understood the poem. Then they learned it was a song, and their delight is one of my great memories of that year. On our last day, as we cleaned up the classroom, we played and sang it out again. I saw Marshall the other day. “Song Sung Blue! He’s sick!”

Delightful though it was, the aforementioned concerns had me in no hurry to return to ELL. But spring 2018 seemed a near lock for an empty 90 minutes.  As I mentioned in Twitter, I’ve taught three years of four classes, no prep. Three times, I’ve had a prep period scheduled and then an administrator appears in my classroom a couple days before school or the new semester begins, and asks me if I mind taking an extra class, and the 33% salary boost that comes with the additional load.

The new semester  was drawing nearer and no rescue on the horizon. I’d originally been scheduled for a history class, when a senior teacher announced last fall she’d be retiring mid-year, rather than at the end of the year. That meant hiring a new history teacher meaning (sob) no extra blocks. So I was resigned to taking a pay cut, and consoling myself with plans to investigate robotics. Bart and I want to start a program, maybe.

Sure enough, though, in walks an AVP the Friday before the semester ends. Prep period to disappear in a big bundle of cash. Well. It’s teaching, so “big bundle” is relative.

I’ve got six kids: two from Guatemala, one from El Salvador, one from Mexico, one from China, and one Brazilian who grew up in Germany. They were shocked to learn I spoke no Spanish, but otherwise they approve of me considerably over the predecessor.  Elian is my only repeater and,  as I suspected, without other more fluent English speakers to translate into Spanish, he’s improving rapidly. Gia, who went to a local middle school last year, is the strongest.

This year I’m teaching the conversation section, which is nice. I don’t have to pretend to teach reading, but can still focus heavily on reading, writing, and discussion. I’ve started them off on this book, which has nothing to do with ELL but has all sorts of neat reading activities. Today, we went through the recipe for smoothies and pizza.

So back to not really teaching English. Keep your fingers crossed the group doesn’t grow on me, and I have a chance to develop them all at the same pace.

Pre-calc, trigonometry, and algebra 2 fill out my schedule.

 

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“Good Teaching” and the Failure of Education Reform

 

 Student achievement is soundly measured; teacher effectiveness is not. The system is spending time and effort rating teachers using criteria that do not have a basis in research showing how teaching practices improve student learning.”–Mark Dynarski, Brookings Institute

Goodbye Mr. Chips. Up the Down Staircase. My Posse Don’t Do Homework. To Sir With Love. Dead Poet’s Society. Mr. Holland’s Opus. The 4th season of The Wire.

The “great teacher” movie has become a bit of a cliche. But decades of film and movies work on our emotions for good reason. That reason is not “Wow, this teacher’s practice is soundly based in practice that research shows improves student learning!”

“You cannot ignore facts. That is why any state that makes it unlawful to link student progress to teacher evaluations will have to change its ways.”–President Barack Obama, announcing Race to the Top

 

Reform movies usually fail. Won’t Back Down, a piece of blatant choice advocacy, bombed at the box office. Waiting for Superman was a big hit in elite circles but for a film designed as propaganda, it notably failed to move people to action, or even win considerable praise from the unconverted.

In general, performance-obsessed folks are the villains in mainstream movies and TV.

In Pump Up The Volume, the villain was a principal who found reason to expel teens whose lack of motivation and personal problems would affect her school’s test scores. This was before charters, when such practices became encouraged.

In Searching for Bobby Fischer (the movie, as opposed to the book), the parents reject the competition-obsessed teacher who wanted the boy to spend all his waking hours on chess, giving equal time to a homeless street guy who advocates a more open, aggressive, impulsive approach to chess. The parents preferred a son with a happy, rounded life to a neurotic who wouldn’t know a normal life. (Their son is, today, a happy well-rounded brilliant man who never became Bobby Fischer. In every sense of that meaning.)

In the famous season 4 of The Wire, AVP Donnelly tries hard to “juke the stats” by gaming the test, “spoonfeeding” the “Leave No Child Behind stuff”. Prez rejects this approach: “I came here to teach, right?”

I can think of only one movie in which a teacher was judged by his test scores and declared a hero:  Jaime Escalante in Stand and Deliver.

But most people throwing about Escalante’s name and achievements don’t really understand that  it took  fourteen years of sustained effort, handpicked teachers, legally impossible demands of his students, and a supportive principal to get 73 kids to pass the AB Calculus exam, with another 12 passing the BC, with around 140-200 in his program, out of a student population of 3500 . Once Escalante lost his supportive principal, he  was voted out as department chair because he was an arrogant jerk to other teachers, and handled defeat by  leaving the school.

Escalante’s story, channeled through Jay Mathews, thrilled policy wonks and politicians, and the public was impressed by the desire and determination of underprivileged kids to do what it takes to get an opportunity they otherwise wouldn’t have. But those same wonks and politicians wouldn’t have tolerated Escalante’s tracking, and 2% would have been an unacceptably low participation rate. He rejected a lot of kids. Mine is a contrarian view, but I’ve never though Escalante cared about kids who couldn’t or wouldn’t do the work he demanded.

“Teachers should be evaluated based on their ability to fulfill their core responsibility as professionals-—delivering instruction that helps students learn and succeed.”–The Widget Effect ((publication of the National Council for Teacher Quality)

In the book We Need To Talk About Kevin, the teacher Dana Rocco makes two brief appearances. The first is in a parent-teacher conference with Kevin’s mother:

danarocco

We don’t know how Dana Rocco’s students’ performed on tests, or even how she taught. But purely on the strength of this passage, we know she is passionate about her subject and her students, who she works to reach in ways straightforward and otherwise. And in the second passage, we learn that she kept trying to reach Kevin right up to the moment he split her head open with a bolt from crossbow while she was trying to carry another of his victims away from danger.

In Oklahoma, a hurricane blew down a school, and they pulled a car off a teacher who had three kids underneath her. Teachers were pulling rubble away from classrooms before the rescue workers even got there. Were they delivering on their core responsibility as professionals?

The Sandy Hook teachers died taking bullets for their students.

Were they fulfilling their core responsibilities as professionals? Would NCTQ celebrate the teachers who abandoned their students to the deranged young gunman, who left their students to be buried in rubble? Could they argue that their efforts were better spent raising test scores for another ten years than giving their lives to save twenty students?

“Most notably, [the Every Student Succeeds Act} does not require states to set up teacher-evaluation systems based in significant part on students’ test scores—a key requirement of the U.S. Department of Education’s state-waiver system in connection with ESSA’s predecessor, the No Child Left Behind Act.–Stephen Sawchuk, “ESSA Loosens Reins on Teacher Evaluations”

ESSA is widely acknowledged to have ended the era of education reform, started in the 90s, hitting its peak in the Bush Obama years. Eulogies abound, many including prescriptions for the future by the same people who pushed the past policies that failed so completely, so spectacularly. In future years, the Bush-Obama choice/accountability reforms will ever more be accompanied by the words “roundly repudiated”. The world we live in going forward is as much a rejection of Michael Petrilli, John King, and Michelle Rhee as the “Nation At Risk” era was to the wasteful excesses of the 70s. The only real question left is why they still have billionaires paying their salaries.

They failed for many reasons. But chief among their failures was their conviction that public education is measured by student outcomes. This conviction is easily communicated, and allowed reformers to move politicians and policy in directions completely at odds with the public will. Reformers never captured the  hearts and minds of the public.  They failed to understand that student academic outcomes aren’t what the public thinks of when they think of good teaching.

The repudiation of education reform policies and preferences in favor of emotion-based, subjective expectations is one of the most comforting developments of the past twenty years. Go USA.


Algebra 2, the Gateway Course

I’ve been teaching a ton of algebra 2 the past three years.  I squawk periodically, and the admins give me variety for a semester or so, but then the classes come back. Back in 2016, I taught 5 classes, all of them full, over the two semester block courses, or about 160 kids. Last year, I had just one course of 30 kids. This year, I’ve already taught three and one coming up.  I also get a steady flow of trigonometry classes–not as many, but three or four every year.  I’ve requested more pre-calculus every year;  they’ll give me one every so often, like a bone to a cranky dog.

In my early years here, I taught far more pre-calculus. From spring 2013 to spring 2014, I taught five pre-calc courses. From fall 2015 to now, I’ve had three.

Why? Because Chuck got his way. Chuck came to our school determined to upgrade the math department. He wanted to make it possible to get a committed kid from algebra 1 freshman year to AP or regular calculus senior year. As I pointed out at the time, this goal is incompatible with helping more kids attain advanced math. You can increase standards or increase inclusion, but not both.

Chuck knows this, and so every semester, particularly the midterm when we finish a “year” and do the turnover to new courses, he starts noodging us for the lists. Kids are often scheduled in two consecutive math courses, so Chuck wants to make sure that the kids who get Ds or Fs in the first course are removed and rescheduled into a repeat. Every year he sends out an email to the algebra 2 teachers, nagging them to give him a list of kids who are failing so he can get them rescheduled. Every year, I ignore him, because I find this activity unseemly and cruel.

I take this task on far more personally and by age. Seniors are given a C if they work hard but can’t pass the tests. Juniors get a choice: retake the course if I think they have the ability to learn more, or take our stats course  (which is designed for very weak kids, lots of project courses). All sophomores get this conversation: you don’t quite grok this material, and you should take it again. Ideally, with me, but either way, take it again. I’ll give a passing grade so you’ll get the credits. But you’re going to  fail if you move forward, and retaking trig is a waste of time, while you will learn more if you retake algebra 2.

But this year, Chuck turned into a wily bastard and instead of asking me for the list, got it from the counselors. He then emailed a list to me and  Benny , the other two teachers covering non-honors Algebra 2:

Hi, can you tell me which of these students won’t pass, so I can email the counselors?  Here’s all the algebra 2 non-honors students who either have a D or F right now, or who got an NOF at the last notification:

Benny (teaching one class of 30):
list of 12 students

Chuck (teaching one class of 30):
13 students

Ed (teaching three classes of 35, or 105 students):
15 students

(I don’t know why Chuck put his own students on the list, maybe to remind me that he was living by his own rules)

So a student in Benny or Chuck’s class had a 1 in 3 chance of failing algebra 2 with a D or F. In mine, their odds were 1 in 7. I was teaching three times as many kids but kept back half as many as they did combined.

Benny, Chuck, Steve, and Wing, the upper math teachers, complain constantly about the seniors they get stuck with, kids forced into a math class by the administrators, even though they hate math and don’t need the credits. The students sit in class every day and refuse to work. Their parents either support this choice or shrug in defeat. The kids have an F by the first quarter. They get bored and disruptive.  The kids waste an entire semester (our year) in their classes, sitting there doing nothing.

I find this akin to malpractice, and say so–well, I don’t say “That’s malpractice.” But I point out how odd it is that I never have this problem, despite being assigned many seniors with similar objections. Most end up like Wesley, learning more math than he ever dreamed.

I was reminded of this recently when going through my desk, cleaning out stuff for the new semester, and coming across Estefania’s note. I give an assessment test on the first day,  and discovered Estefania ignoring the test, writing on a slip of paper. I took the paper away from her, told her to give that test her best effort. I  was going to toss the note but then noticed it was a form of some sort, and opened it:

Estefanialet

Estefania came up after class. “I tried on the test, but I didn’t know a lot of it. Can I have my note back?”

I handed it to her. “I don’t think you should turn it in. I think you should take the class.”

“I’ll fail.”

“No. You won’t. I promise.”

“Math teachers always tell me that, like I’ll finally get math and be good at it. But I’m not any good and I’ve already failed twice.”

“You don’t understand. Come to class. Try. I will give you a passing grade. I don’t care if you fail every single test. I guarantee you will get a passing grade. And odds are really good you’ll also learn some math.” I held out my hand for the note. She hesitated, and then handed it back. And stayed. She did pretty well, too, well enough that she smiled whenever I reminded her about that note.

When I found the note in my drawer, I looked up some of her work on the finals.

Exponential functions:

Estefaniaexp2Estefaniaexp1

I forgot to take a picture, but she did quite well on the log questions, understanding that log base 2 of 16 is 4.

Here she is on quadratics, her best subject (she got an A, flat out, on her parabola graphing quiz.):

EstefaniaquadEstefaniaproj

She received a 60 on the first part of the final, putting her in the bottom third (most of my fails were between 42 and 60). I haven’t graded the second part,  although she clearly knew the quadratics. Girl learned some math, y’know?

Chuck and my four colleagues sometimes suspect that I dumb down my course. In fact, thanks to the epic teacher federalism agreement, my course is considerably harder and more cognitively complex than it was three years ago.

A month ago, Chuck trumpeted the results of his project. Six students entered at Algebra 1 or lower in their freshman year, and succeed in taking  AP Calculus their senior year. (One of them was Manuel.) Eight students entered at the same level took regular calculus. So fourteen students were not identified as honors students, took no honors classes, yet had made it to calculus by their senior year.

Of those fourteen students, I’d taught ten of them twice in their progression through algebra two, trigonometry, and pre-calcululus. Two others I taught once. Of the fourteen, only two had never been in my classroom.

The  road to Chuck’s dream runs directly through Ed.

Now you know why I get all those algebra 2 students. Because our administrators want to sign up for Chuck’s dream, but they don’t want a bloodbath. No one says so directly. They don’t have to. My schedule says it all.

In prior years, I was teaching more precalculus for a similar reason, as far too many students who’d made it that far were wasting their last year of high school math. But when Chuck unrolled his initiative, my principal realized that algebra 2 was going to be the new choke point. Well, not so much realized it as heard it straight from Chuck’s mouth, as in “More kids will fail algebra 2 because it’s going to be a much harder course if we’re going to  achieve this goal.” Rather than tell Chuck no–because it is indeed a worthy goal–our principal threads the needle between achievement and equity by adopting Chuck’s goals but assigning me the lion’s share of students in a critical gateway–or gatekeeping–course.

If I want to teach more pre-calculus, I need more colleagues with my methods and priorities teaching upper-level math. I spent three years mentoring Bart to share the teaching load, an objective I made clear to both Bart and the principal. Bart liked that idea. The principal did, too. But Bart wanted to teach physics, too, and we have a new science initiative, and now Bart teaches freshman physics. I am still pissed about that, but hell, we drink beer together so I can’t kill him.

In the meantime, our department chair is retiring. So I need to request input into the hiring decision for his replacement.

Yet I pause just for a moment to celebrate the Estefanias in my world, and remind everyone again that as teachers, we owe our first loyalty to the students, not the subjects.

As Joe said in All That Jazz when Victoria wanted to quit:

Victoria: I’m terrible. I know I’m terrible. I look at the mirror and I’m ashamed. Maybe I should quit. I just can’t seem to do anything right.

Joe Gideon: Listen. I can’t make you a great dancer. I don’t even know if I can make you a good dancer. But, if you keep trying and don’t quit, I know I can make you a better dancer. I’d like very much to do that. Stay?

Or take this question, which I first asked four years ago:EdRomanticism

If you teach at-risk, low-skilled kids and don’t struggle with this question, you aren’t really teaching them.

*******************************************************************************

My standard disclaimer: all my colleagues are good teachers who want the best for the kids. I disagree with their philosophy. They disagree with mine. No criticism intended, other than, you know, they still kill the bulls to worship Mithras while I’m Zoroastrian. (Also, all names are pseudonyms.)

The great Ben Orlin recently mused on this, giving birth to my take. Robert Pondiscio argues that education reform’s “underperformance” lies in their assumption that policy, not practice, is the key to drive “enduring improvement”. I don’t know that reformers will get anywhere until they realize that the facts on the ground say we’re teaching kids at capacity and that “enduring improvement” is likely a chimera. (Note: I edited this paragraph slightly to correct my characterization of Pondiscio’s argument.)

Previously, I’ve described my outrage at college policies that abandon remediation , conferring college-readiness on people who can’t manage middle school math. Anyone want to know how I square that needle with what I’m writing here? It’s an interesting question. I’ll get to it later.


“Get Out” a scathing satire? Get Out.

(Note: this is outside my “all things education” brief,  but squarely in my “most folks in the media simply don’t understand what diversity means anymore” zone, so it’s a wash.)

I enjoyed this movie far more than I  expected based on the rapturous reviews, which promised a scathing satire on race relations in America cloaked in a horror film an  exquisite comedy of manners,   an alarming portrayal of white racism.  As is so often the case, the movie’s creator doesn’t come near to achieving his stated goals. Jordan Peele isn’t the compleat observer of American mores that he–and many others–think he is, so his movie fails to uncover the “insidious qualities” of white liberals.   This is just a nice, tight horror film with some clever touches and a few non-fatal flaws.

The revealed plot (do not read if you have not seen):

Dean and Missy, a neurosurgeon and his hypnotherapist wife, have developed a means of personality transfer, probably at the instigation of Dean’s father. Missy can submerge an individual’s personality to “the sunken place”, suppressing his or her ability to control his body or mind. The helpless host is then surgically implanted with an invader’s brain matter, giving the invader’s personality full control.

Dean and Missy’s two children, Jeremy and Rose, procure the hosts.  Rose serves as honey pot, taking several months to woo a new lover and then “bring him home to the family”, which includes Walter and Georgina, the family’s “servants”, and always takes place the same weekend as the family’s yearly party, conveniently enough. Except Walter and Georgina are actual hosts for Dean’s parents, who invaded the bodies when their own wore out, and the party guests are actually bidders for the new host.

Jeremy brutalizes and kidnaps his victims off the street. There’s no fake party, no need for Walter and Georgina to pretend to mow the lawn and make the beds.  Rose’s method seems unnecessarily time-consuming; I don’t see why they don’t just send Jeremy out twice as often.

The plot is revealed through the eyes of Rose’s new unsuspecting prospect, Chris, a young photographer with mommy issues. He avoids his fate because his cellphone uses a flash, his best friend is a TSA agent, and because once aroused, he is capable of ferocious self-defense.

Notice how easy it is to describe this story without ever mentioning race. The tale hangs nicely without knowing that Jeremy only kidnaps, Rose only entraps, black people. It doesn’t suddenly make more sense, closing some puzzling plot loophole.

In fact, the interpretation as offered up by Jordan Peele and his following in elite circles makes the movie absurd.

Given a surgical procedure that implants their consciousness into another body, guaranteeing virtual immortality, rich white people would say “Great! Now find me a cute/buff white body that no one will miss.” (They’d also demand a plastic surgeon get rid of the scars.)

Rich white people do not want to be black. Nor do they want to be Hispanic, southern or eastern Asian, of course, but Peele’s horizons don’t extend that far.

Happily, the movie itself makes no such claims. The movie portrays members of this weird, creepy organization who want to be black. The (largely pointless) video forcefed to Chris makes  no mention of race. We only learn that the Armitages limit their procedures to black folks through Chris’s discovery of Rose’s photo album, coupled with Jeremy’s takedown of Dre.  Jim (the only authentic rich white guy to be found in the film) confirms that only black people are hosts, and he makes it clear that the “organization” has some sort of fetish on the topic.

That these particular white folks aren’t normal is supported by the party scenes themselves. Look, I worked almost exclusively for rich white people as a tutor for four years, including  for folks who have been at one time or another on the Forbes 400.  Rich white liberals from the boomer generation on down just aren’t that gauche. The Armitage guests are creepy,  touching hair, feeling biceps, asking about his sexual prowess.  Their cars are all wrong, too.  But my experience isn’t necessary here. Only idiots with critical faculties completely removed would see these cultists as typical rich white folks.

And here’s the thing: the movie thinks so, too. What else is the point of Jim Hudson, played by the always note-perfect Stephen Root? Jim isn’t a cultist. He’s the real thing: a rich white bastard  in all his authentic, heartless glory. He says so expressly in the video, but we don’t need to be told. At the “party”, Jim is the only one who treats Chris like a human. He’s a rich white bastard, but he’s no racist.  More importantly, he’s not a cultist.

I kept wondering throughout why so many critics–and Peele–invoked the Stepford Wives until I realized that they were referring to the cheerful black servants Walter and Georgina. Just as the men of Stepford turned all their womenfolk cheerful, sex-ready, and compliant by making them all robots, so too did Dean and Missy turn black people into servile peasants, eager to please their masters.

But Walter and Georgina aren’t servants. They’re just pretending to be servants for Chris. Walter and Georgina are Grandma and Grandpa, pretending to be servants to fool Chris. They are fully empowered players in this horrific game, welcoming the bidders to the new auction, messing with Chris’s phone, doing everything they can to kill Chris when he escapes. All we’re seeing is the facade. Homage to Stepford, certainly, but Walter and Georgina aren’t even remotely parallel.

Of course, the entire “servants” fakeout is a giveaway of itself. Rich white people don’t employ blacks as servants. That’s what they have Hispanics for, and why so many white elites resist any sort of immigration restriction.  Maybe people were so eager to see racism that they missed the obvious, but  I was instantly skeptical. Liberal white guilt about black servants reigns supreme; no Obama liberal would have them. By the time Walter was chopping wood–I mean, really. Chopping wood? For what, exactly? –I’d called the plot twist. Walter was a white guy in a black man’s body. Betty Gabriel, singlehandedly responsible for every jump-scare in the film, impeccably represents as a little old white woman who can’t quite get comfortable around “colored people”.

So I already had the plot figured out 30 minutes in, which left me plenty of time to wonder not only where the Hispanic maid was, but where the Hispanics were, period. The Northeast, where I’m told this movie takes place, has more Hispanics than blacks. New York, New Jersey, Connecticut, even New Hampshire have a higher percentage of Hispanics than blacks. Yet there isn’t a Hispanic to be found onscreen, not even in the police station. Maybe one walked by the TSA guy and I missed it.

At least there’s a Japanese guy bidding for the right to invade Chris, but that seems to be just one more homage to Rosemary’s Baby. Besides, Asians bidding for the right to be black? Look, I can believe in a cult of weird rich white people with a black fetish. (Cf Rachel Dolezal and Shawn King.)  But Asians, particularly  “fancy” Asians , are racist to levels that your average neo-Nazi can’t even conceive of.

But that’s all beside the point. The movie was fun and the performances note perfect.  None of Peele’s ideological agenda made it on the screen. Race just adds a delightful, even gorgeous, frisson of subtext. The cops laughing at the worried friend are all black, although how this casting does anything but make a joke of Peele’s grand designs is left for better minds than mine. Best of all is  Peele’s use of the “black boyfriend”, with Chris constantly worried about making the wrong impression, overreacting to seeming insanity–maybe this is how white folks do things. Then the finale–oh, the finale. I don’t enjoy watching violence, but Chris’s escape is ferocious righteousness that simply wouldn’t have played as well with any other race.

As for flaws, Everything Wrong with Get Out in 15 Minutes or Less picked up most of the flaws I found with the actual movie, as opposed to my complaints about the absurd interpretations. I’d add that Alison Williams would have been considerably more terrifying if she’d maintained her loose, “all American” persona after the reveal, rather than becoming a freakazoid terminator. How scary would that have been, coming after you with a shotgun?

Another nit:   Richard Herd, playing Roman Armitage, was born in 1932, just the right age to be Bradley Whitford’s dad, but just four years old when Jesse beat the Germans in 1936. A Jesse Owen contemporary would have been 60 years old when son Dean was born, and unlikely to be alive when the grandkids were born, much less old enough to lure unsuspecting African Americans into sexual relationships to bring them home for invasion.

Why not Harrison Dillard? He won in 1948, tied the existing world record just like Jesse Owens did. And he’s alive. Using Dillard as a plot point would have been more realistic, less trite, and maybe even brought the spotlight to a neglected black athlete.

So Jordan Peele may have had lofty goals for his little horror film, but thankfully they aren’t to be found in the actual movie, at least not for most white viewers, at least not for those who live in more racial diversity than the average reporter or movie critic. But Steve Sailer  was uncharacteristically harsh; he seems to have seen the movie Peele wanted to make. (I totally don’t see the Alvy Singer parallels).  Mine is–as always–a minority view.


What Counts as Teaching?

I met Dale, his girlfriend Maya, and their new puppy for lunch on Friday.  Dale and I worked together at my previous Title I school in his first year (my last). Within three years, Dale was department chair,  teaching AP Calc BC and AP Stats,  honors pre-calculus, designing his own courses, settling colleague spats. But Dale began his career teaching Algebra I, and even after taking a leadership position, he never hesitated to take on a low-level class to give his colleagues more variety. Given that Dale’s school, like mine, assigns teachers to categories of classes, he could easily have stayed in the stratosphere, but he easily handles tough kids.

Dale had planned to stay at that school forever. I remember one lunch when Maya and I pointed out he might be better moving to a different school, but he was adamant. However,  Maya (who makes a lot more money) has become exhausted and stressed by her commute. They decided to move closer to her job, which is in a much wealthier part of the world. Dale still isn’t thirty, and you don’t get hired into department head positions, particularly not in the wealthy suburban districts near his new place. Dale knew that, knew that  prioritizing location would limit his options, that he’d be unlikely to get the ideal course load he now had. My own school would have taken him in a heartbeat, given him the AP class load he preferred, but he wanted to be close to home.

We knew all this. We’d talked about it last year, when he was looking. When Dale accepted a job, we talked about his schedule: a bunch of algebra I classes and computer science. All freshmen. More money, but not enough more to get excited about. Dale’s a cheery guy and showed no resentment, at most mildly philosophical about the change.

End preamble. Nod to Dale’s awesome relationship priorities.

So we’re talking about Dale’s new job at lunch. He is using the district curriculum that is mapped out by day. He spends no time at all developing his own lesson plans, but since the required curriculum has regular quizzes and tests, he spends far more time grading than he used to. He loves it, though. The students  (mostly white) are capable and interested and far advanced of anything he’d seen at his last school. They do homework. They ask in depth questions, demand deeper understanding.

As he told me all this, Dale was grinning. I knew he was remembering something I told him early on in our acquaintance, when he was a first year and I was a third: “Look, I’m not sure what you call it when people deliver centrally planned lessons to prepared kids, but it sure as hell isn’t teaching. Because if it is, what the hell is it we’re doing?”

“So,” I said now. “Is it teaching?”

“Sure feels like it. But yeah, it’s different. Their questions are amazing. We’re really discussing in-depth math. The kids enjoy it. They aren’t obsessed about grades. I’d rather plan my own lessons, although I can see how I’d get used to just downloading the worksheets for the day. But the kids, they want to know about math. And I get to tell them all about math. What is that, if not teaching?”

Every so often, I can get my head around this notion. When I think of reformers going on and on about the importance of curriculum, how arrogant teachers are to think they can create their own, I realize these are people whose notion of teaching is something like Dale’s classroom. When I read a paper stating that to improve teaching and advance student learning requires weaving together the curriculum that students engage with every day with the professional learning of teachers“, I realize they, too, think that my world is just a slightly more chaotic, lower-level version of Dale’s universe.

But see, it goes beyond curriculum. That’s just my pet peeve. Think about what a PE teacher does every day. A first grade teacher. A high school special ed teacher managing twelve aides and eight severely disabled “students” who have to wear mitts to stop from hurting themselves, who will be attending “school” all year round until they’re twenty one because Congress gets crazy some time. Think about any teacher in a poorly managed inner city school, where chaos reigns. Or what about an ESL teacher who only speaks English, but has eighteen students from Congo, India, Salvador, China, Afghanistan, and three other countries?

What counts as “teaching”?

It’s a few hundred thousand blind men and a camel.

I’ve written various renditions of this point throughout the years. And I know teachers aren’t unique. Lawyers, engineers (if there is such a thing), doctors–all the professions have similar width and breadth.

But.

A surgeon can fix the ailment of a patient who sleeps through the operation, and a lawyer can successfully defend a client who stays mute throughout the trial; but success for a teacher depends heavily on the active cooperation of the student…Unless [the] intended learning takes place, the teacher is understood to have failed.–David Labaree, The Trouble With Ed Schools

Labaree goes on to observe that the aspects of health that require patient cooperation–obesity, smoking, addiction–have extremely low success rates. Doctors have offloaded these responsibilities onto therapists, who aren’t expected to have a fabulous success rates.

But when the country began actively forbidding both students from quitting and teachers from limiting their student population, teachers weren’t allowed to offload their responsibilities onto some lower career ladder occupation, or even lower expectations for success. In fact, governments became ever more quantitative in its demands for educational outcomes.

Don’t bleed too hard for teachers. We mostly ignore the outcome expectations. And it’s pretty good pay for most of us, as well as a great working schedule. But for any number of reasons, the public debate and the absurd expectations is a huge part of the job in my region of Teacher World, and not even a tiny blip on the horizon of Dale’s.

That is, if what we do is called teaching.


Wherefore and Whither The Teacher Shortage?–Supply

I will never leave my current position. I won’t ever risk leaving the tenure cocoon until I’m ready to leave teaching. I expect to stay seven more years in this job and then move to another state, where I will try to find get a teaching job but will weep no tears if I don’t.

Why? I’m too old to risk being on the market. Barry Garelick couldn’t find a full-time job, and I believe he still works just part-time with kids at a middle school. I doubt I agree with one in five words the man writes but that’s not why he didn’t find work.  I guess he needs an American version of the program Lucy Kellaway has put together  for fifty somethings in the UK.  But in the US, if you’re highly educated and past 40 but think being a teacher sounds rewarding, network like mad, get an internship job first, and don’t spend a fortune on tuition, is all I can say.

But didn’t I just say there’s a teacher shortage?  Well, sure. But age discrimination is everywhere. And there are other caveats.

Ed schools seem to be producing too many elementary school teachers.   One of the first big pieces I ever wrote referenced a study showing that just 77,000 of the 186,000 teacher class of 2010 took a job teaching, and it’s likely that most of the ones who didn’t were elementary school teachers. The best piece on this teaching shortage, skeptical and thorough (and, alas, 4 years old) is still Stephen Sawchuk’s Colleges Overproducing Elementary Teachers, Data Finds.

Notice Sawchuk points out that the data shows this. Note he also points out that “states flush with elementary teachers can face shortages, particularly in urban and rural areas.” And there are many irregularities. For example, Maryland produced 1000 elementary school teachers and hired 1,100, 723 of them new teachers. But over half of the new teachers came from out of state. Wait, what? As Sawchuk notes, teacher supply is complicated.

Another reality: teaching jobs are vastly different depending on the adjectives describing the schools and the students. Some principals can be picky and selective, rejecting and selecting based on their personal preferences. Other principals are the employer equivalent of  drunken out-of-towners looking for love in the bar at closing time: taking what they can get.

These analogical drunks run schools with the wrong adjectives. The schools or their districts are extremely rural, extremely urban, extremely poor, extremely expensive or, of course, two or more combined.  While charter schools definitely contribute to the increase in teaching populations, I don’t think heavy charter presence leads inevitably to a shortage for that state or district, but that’s a data analysis beyond my scope.

So here’s something I always wonder: when those desperate principals are looking, where are the English teachers, the history teachers, the elementary school teachers who change careers or work as substitutes when they can’t find jobs? Are they applying at the schools with the unattractive adjectives?

Reporters don’t often make this clear. In my case, I wanted to work in high poverty districts, although I never got desperate enough to apply to charters or inner city schools. If I couldn’t nail down employment among the suburban poor, I vowed to  move to North Dakota. My kid’s grown and in another state, so I could pretend it was an adventure. Fortunately, I was able to stick within those parameters and find tenure without relocation. Eventually, I probably would have sought out inner city schools, if I had to.

But that was me, with $50K in loans that I could only write off with a teaching job, loans I had no intention of paying off my own self. Suppose you’re a young credentialed college graduate with only a moderate ambition,  no major loans and a parent to help pay them off. Would you be willing to take that 2 am job in the extreme zones? Or were you just interested in teaching with reasonable kids, reasonable pay, reasonable cost of living and reasonable locale offered up by an attractive employer well before the midnight hour?

Then there are the college educated folks working as adjunct professors or barristas or bus drivers that I wrote about last time. Why aren’t they considering teaching?

Well, there’s this other thing to remember: Teaching is brutally hard for some people.

Many otherwise ordinary people can succeed and thrive in teaching. But the job can also crush a healthy ego as easily as an egg.  Who is going to have an easy time, who is going to struggle and thrive, who’s going to slink away with a mental scar that they often flinch away from? Typically, although not always,  counted among the failed are  many Dewey- and dewy-eyed idealists who envision themselves enrapturing a group of wide-eyed, attentive diamonds in the rough who had never once encountered a teacher who really cared. Then they face reality, often with no mentoring and no support. The results are ugly. Some recover. Some don’t.

I wrote about a breakdown I witnessed with a long term sub, one who’d taught in India and had originally planned to teach here in America.  Every teacher has these anecdotes, I think. If you want to read a book about the breakdown of an inept teacher whose psyche was severely unglued by teaching, I suppose you can suffer through Ed Boland’s Battle for Room 314, a truly revolting book about a terrible person whose choices are inexplicable, not least because he clearly despises his students. Upside: the self-absorbed, condescending little jerk  who thought he’d become a Great Savior, is permanently defeated by his shortcomings and other inabilities and he’ll be cringing from his failures for the rest of his life. Downside: the horrible little man got a book deal out of it, as well as all sorts of positive attention.

Sorry. I didn’t write a whole piece on that book because, well, you can see. Where was I?

College-educated people who are unhappy with their economic lot in life choose under-employment or insecure employment are already dealing with a sense of failure. Maybe they know they aren’t cut out for a job that can wreak psychological devastation, even one with  tremendous job security, enviable union benefits, and fantastic vacation time. Teachers  who abandon the field when they can’t find the job they want could be making the same calculation: why risk  soul-wringing failure when they already feel unwanted?

And so, the disconnect. Teaching seems to be easy, seems to offer a  ready supply of jobs, but  the jobs in classrooms with capable kids and involved parents  are much harder to come by, and hey, maybe  it’s not always so easy.

Sing me no happy talk about charters and choice. Megan McArdle’s mea culpa got some pushback, but the public has made it fairly clear it’s not on board with education reform.  Eventually, everyone will realize that lousy teachers don’t cause low test scores. And as I observed in my last piece, even assuming charters are making it more desirable to teach in low income areas, they’re consuming more teachers and burning through them at a faster rate.

TFA found another way to convince people to enter the field: make it a resume boost, but that story stopped selling, eventually.

There’s one sure way to convince more people to take on teaching pay a whole lot more. Pay so much the salary makes it worth while for people to move to Alaska, teach in Detroit, take a crappy one-bedroom apartment in Salinas for a teaching job in Palo Alto, risk losing tenure, or live in the remote frozen lands of North Dakota.

And none of these measly 20% increases, or $4K signing bonuses. It’s going to take six figures to get reliable sourcing for the schools with unattractive adjectives.

But that makes no sense. (Ha, had you fooled, didn’t I?)

Leave aside the states that need to revisit pay because they are constantly losing teachers to their higher-paying neighbors. Understand that the expensive  districts are paying through the nose already (average teacher salary in the Bay Area is nearly 6 figures as it is).  Understand that 3.5 million people seem to find the existing pay just fine.

Remember, too, that these are government jobs, with government pensions, and our state government pension commitments give responsible people nightmares.  As it is, pensions are a looming threat. Shortages are already leading to higher salaries in many states, and that only makes the commitments even more threatening.

Then there’s the fact that increasing the teacher supply invariably means lowering teacher standards even further–which, of course, I’ve never argued against. But lowering standards while dramatically increasing pay is just adding insult to injury for the taxpayers. And the reformer path to improved teacher quality–threatening teacher pay and benefits, offering merit pay,  reducing job security–only makes the supply problems more severe, as the charter experience shows. That is, of course, what education reformers have never really grasped, and it’s why their efforts have mostly failed.
 

I prefer reducing demand, preferably without too much of a baby bust. Faithful readers can probably remember when I’ve discussed that before, but everyone else will have to wait for the next post.

PS–I don’t want to make this whole series sound deeper than it is. I don’t have any hard research under wraps. I just wanted to discuss the various lines of thought that others have offered, and respond.


Wherefore and Whither The Teacher Shortage?

The Teacher Shortage Myth is typical of the scoffing view that says “How can there be a teacher shortage when the teacher population is increasing at a faster clip than the student population?

We have a teacher shortage.  Debate the breadth and depth as you will. But if teachers were thick on the ground, would principals be so reluctant to give bad reviews to marginal teachers? Would  Race to the Top have failed so satisfactorily if parents and districts alike weren’t worried that their teacher pool would be threatened? Why are California and Nevada and a number of other states bringing in teachers from the Philippines? 

So for the purposes of this article, accept the fact that we’ve got a teacher shortage.

I don’t see why anti-teacher folks make so much hay out of the discrepancy between student and teacher growth. I can think of several reasons why schools would need more teachers despite a stable or even declining student population.  From least to most determinative of obvious explanations:

  1. Since 2002 or so, middle school teachers have had to be credentialed at the more demanding high school standard. Existing teachers were grandfathered in, but as they retire, the newer hires are being drawn from the high school pool. At the same time, education policy in practice now requires all students to take four years of math, although likewise in practice this often means students take four years to pass the required two or three years officially mandated. In any case, there’s been a largely unreported increase in the need for academic subject teachers.
  2. A good chunk of that student growth is in immigrants and non-English speakers. We public schools are required to take all kids of school age the minute they set foot on our soil, regardless of their previous education or English ability. An explosion in immigration keeps the need for ESL teachers growing, and those classes are very small. Our school keeps the equivalent of one full-time hire for about 16 kids.
  3. Similarly, a good chunk of the student growth involved more special education students, another population that has a legal ability to demand small classes.  Special ed students cost, on average, twice that of a regular student (the last we checked, which was 16 years ago). In high school, most of them are eligible for what’s basically a study hall, meaning we pay teachers to do a lot of case management, parent meetings, and run classes of 8-10 kids doing very little most of the time.
  4. The big one, of course, is charter schools. According to Edweek, the public school teacher population grew from 3,385,200 to about 3,800,000 in 4 years, with 218,500 of that group teaching at charters. But what the Edweek reporter doesn’t mention is that those same four years earlier, from the same report, the charter teacher population was just 115,600, or an 89% increase. Public school proper teachers grew to 3,581,500 from 3,269,600, or just 9.5%–a believable increase given the first three points. Of the 400,000 teachers added in 4 years, 25% of them work at charters, teaching just 6% of the population.

    I do not understand why people don’t understand the obvious impact of using public dollars to fund what are basically small private schools–with capped enrollment, no less. We have X students. Educating X students at Y schools requires Z teachers. Educating X students at Y + some number of schools will require Z+ some bigger number of teachers. Full stop.

 

Leaving demand behind, let’s take a look at supply.

Do not expect, dear reader, a jeremiad about the woeful lack of respect the public has for teachers. The public gives us plenty of respect. If you want a fairly well-thought out, pro-union, pro-teacher case for the teacher shortage, here’s Peter Greene from a couple years ago. I don’t know that I agree with all his reasoning, but he’s thorough and logical and despite the anti-reform rhetoric, reasonably compelling. (Remember that I, too, think reform is useless, but our reasons differ, as do our proposed fixes.)

But I think about this differently. Never mind pay, respect, anti-teaching rhetoric. Take a look at these anecdotes from the last few years, all of which describe very current reality:

Millennials are Screwed:

millennialscott

Why So Many Higher Ed Professors Make So Little

adjunctteachercomparison

The Problem With Being Just a Teacher

adjunctteacherslam

I am an adjunct professor who teaches five classes. I make less than a pet-sitter.

adjunctwoes

College graduates working as bus drivers, barristas, or waiters. PhDs desperately trying to cobble a living out of college teaching contingent jobs.

I am reliably informed that teaching has a low barrier to entry. We don’t learn anything useful in classes, the argument goes, and there’s no knowledge base to memorize. GPAs are ludicrously low. The credential tests are a bit of a facer, but certainly not for the well-educated folks of the stories above.  There’s job security so tenacious it’s given birth to a dozen laws intended to dislodge it–laws that have, for the most part, failed, in a world that otherwise sees the tie between employer and employee disintegrate. Benefits are disappearing. Increasing numbers of jobs are contingent.

But a millennial with a business degree chose to be a bus driver when he couldn’t get a bank job. Adjuncts are so easily found begging for work that colleges can treat them like disposable diapers.

I’m not offering criticism or suggestion, just describing a choice that shouldn’t make sense, given the public policy discourse about teaching. These aren’t entrepreneurs choosing a career that will reward ambition and excellence, sneering at the “widget” teacher. These are people who, for whatever reason, aren’t succeeding in their post-college career. Whether  teachers are underpaid, disrespected, or undeserving mediocrities ensconced in a sinecure,  non-trivial chunks of people with college degrees are rejecting these secure teaching jobs with more pay and security for widget jobs with less pay, less respect, and no future.

Why?

So there you go. We have a teacher shortage. I offered some reasons why the growth in demand for teachers might outstrip the growth in students. I’ve pointed out at least two sources of college educated strugglers that apparently aren’t turning to teaching despite its offering of better pay and more security.

I have no evidence for any of this but then, when did that stop anyone else?

Part 2 coming soon, I think. If not, just consider this mind food and mull it over your own self.

Note: I borrowed the feature image, the one that comes through on Twitter, from here

.

 


The Structure of Parabolas

A year ago, I first envisioned and then taught the parabola as the sum of a parabola and a line.The standard form parabola, ax2 + bx + c, is the result of a line with slope b and y-intercept c added to a parabola with a vertex at the origin, with vertical stretch a.   This insight came after my realization that a parabola is the product of two lines (although I wrote this up later than the first).

I didn’t teach algebra 2 last semester, so I’ve only now been able to try my new approach. I taught functions as described in the second link. So the students know the vertex form of the parabola. Normally, I would then move to the product of two lines, binomial multiplication, and then teach the standard form, moving back to factoring.

But I’ve been mulling this for a few months, and decided to try teaching standard form second. So first, as part of parent functions, cover vertex form. Then linear equations. As part of linear equations, I teach them how to add and subtract functions.  As an exercise, I show them that they can add and subtract parabolas and lines, too.

So after the linear equations unit, I gave them a handout:parabolastructure

I don’t do much introduction here, except to tell them that the lighter graphs are a simple parabola and a line. The darker graph is the sum of the parabola and the line. What they are to do is explore the impact of the line’s slope, the b, on the vertex of the parabola, both the x and y values. We’d do that by evaluating the rate of change (the “slope” between two points of a non-linear equation) and looking for relationships.

Now, I don’t hold much truck with kids making their own discoveries. I want them to discover a clear pattern. But this activity also gives the kids practice at finding slopes, equations of lines, and vertex forms of a parabola. That’s why I felt free to toss this activity together. Even if it didn’t work to introduce standard form, it’d be a good review.

But it did work.  Five or six students finished quickly,  found the patterns I wanted, and I sent them off to the next activity. But most finished  the seven parabolas in about 40 minutes or so and we answered the questions together.

Questions:

  1. Using your data, what is the relationship between the slope of the line added (b) and the slope (rate of change) from the y-intercept to the vertex?
    Answer: the slope (b) is twice the slope from the y-intercept to the vertex.
    b2= rate of change
  2. What is the relationship between the slope of the line added (b) and the x-value of the vertex?
    Answer: the x value of the vertex is the slope of the line divided by negative 2.
    b⁄-2= x value of the vertex
  3. What is the relationship between the y-intercept of the line and the y-intercept of the parabola?
    Answer: they are the same.

Note: I made it very clear that we were dealing only with a=1, no stretch.

The activity was very useful–even some strong kids screwed up slope calculations because they counted graph hash marks rather than looking at the numbers. Some of the graphs went by 2s.

So then, they got a second handout: parabolastructure2

Here, they will find the slope (rate of change!) from the y-intercept to the vertex and double it. That’s the slope of the line added to the parabola (b!). The y-intercept of the line is the same as the parabola.

The first example, on the left, has a -2 rate of change from vertex to y-intercept. Since a=1, that means b=-4. The y-intercept is 8. The equation in standard form is therefore
x2 -4x + 8. In vertex form, it’s (x-2)2 +4.

Tomorrow, we’ll finish up this handout and go onto the next step: no graph, just a standard form equation. So given y=x2 -8x + 1, you know that the rate of change is -4, and the x-value of the vertex is 4. Draw a vertical line at x=4, then sketch a line with a slope of -4 beginning at (0,1).

This may seem forced, but students really have no idea how b influences the position of the vertex. I’m hoping this will start them off understanding the format of the standard form. If not, well, there’s the whole value of practicing slope and vertex form I can fall back on. But so far, it’s working really well.

By late tomorrow or Monday, we’ll be formalizing these rules and determining how an increase or decrease in a changes these relationships. So I hope to have them easily graphing parabolas in standard form by Monday. Yes, I’ll show them they can just plug x to find y.

Then we’ll talk about factored form, and go to binomial multiplication.

I’ll try to report back.


The Evolution of Equals

High school math teachers spend a lot of time explaining to kids that all the things we told you before ain’t necessarily so. Turns out, for example, you can subtract a big number from a smaller one.  Fractions might be “improper” if the numerator is larger than the denominator, but they’re completely rational so long as both are integers. You can take a square root of a negative number.  And so on.

Other times, though, we have to deal with ambiguities that mathematicians yell at us about later. Which really isn’t fair. For example, consider the definition of variable and then tell me how to explain y=mx+b. Or function notation–if f(x) = 3x + 7,  and f(3) = 16, then what is f(a)? Answer: f(a) = 3a+7. What’s g(x)? Answer: A whole different function. So then you introduce “indeterminate”–just barely–and it takes a whole blog post to explain function notation.

Some math teachers don’t bother to explain this in class, much less in blogs. Books rarely deal with these confusing distinctions. But me, I soldier on. Solder? Which?

Did you ever think to wonder who invented the equal sign? I’m here to wonder for you:

Robert Recorde, a Welsh mathematician, created the equal sign while writing the wonderfully named Whetstone of Witte. He needed a shortcut.

“However, for easy manipulation of equations, I will present a few examples in order that the extraction of roots may be more readily done. And to avoid the tedious repetition of these words “is equal to”, I will substitute, as I often do when working, a pair of parallels or twin lines of the same length, thus: = , because no two things can be more equal.”

First of his examples was:  or 14x+15=71.

Over time, we shortened his shortcut.

Every so often, you read of a mathematician hyperventilating that our elementary school children are being fed a false concept of “equals”. Worksheets like this one, the complaint goes, are warping the children’s minds:

I’m not terribly fussed. Yes, this worksheet from EngageNY is better. Yes, ideally, worksheets shouldn’t inadvertently give kids the idea that an equals signs means “do this operation and provide a number”. But it’s not a huge deal. Overteaching the issue in elementary school would be a bad idea.

Hung Hsi Wu, a Berkeley math professor who has spent a decade or more worrying about elementary school teachers and their math abilities, first got me thinking about the equals sign: wuquotenu2

I don’t think this is a fit topic for elementary school teachers, much less students. Simply advising them to use multiple formats is sufficient. But reading and thinking about the equals sign has given me a way to….evolve, if you will…my students’ conception of the equals sign.  And my own.

Reminder: I’m not a mathematician. I barely faked my way through one college math course over thirty years ago. But I’ve found that a few explanations have gone a long way to giving my students a clearer idea of the varied tasks the equals sign has. Later on, real mathematicians can polish it up.

Define Current Understanding

First, I help them mentally define the concept of “equals” as they currently understand it. At some point early on in my algebra 2 class, I ask them what “equals” means, and let them have at it for a while. I’ll get offerings like “are the same” and “have the same value”, sometimes “congruent”.

After they chew on the offerings and realize their somewhat circular nature, I write:

8=5+2+1

8=7

and ask them if these equations are both valid uses of the equal signs.

This always catches their interest. Is it against the law to write a false equation using an equals sign? Is it like dividing by 0?

Ah, I usually respond, so one of these is false? Indeed. Doesn’t that mean that equations with an equals sign aren’t always true? So what makes the second one false?

I push and prod until I get someone to say mention counting or distance or something physical.

At this point, I give them the definition that they can already mentally see:

Two values are equal if they occupy the same point on a number line.

So if I write 8=4*2, it’s a true equation if  8 and 4*2 are found at the same point on the number line. If they aren’t, then it’s a false equation, at least in the context of standard arithmetic.

So if the students think “equals” means “do something”, this helps them out of that mold.

Equals Sign in Algebraic Equations

Then I’ll write something like this:

4x-7=2(2x+5)

Then we solve it down to:

0=17

By algebra 2, most students are familiar with this format. “No solution!”

I ask how they know there’s no solution, and wait for them all to get past “because someone told me”. Eventually, someone will point out that zero doesn’t in fact, equal 17.

So, I point out, we start with an equation that looks reasonable, but after applying the properties of equality, otherwise known as “doing the same thing to both sides”, we learn that the algebra leads to a false equation. In fact, I point out, we can even see it earlier in the process when we get to this point:

4x = 4x+17

This can’t possibly be true, even if it’s  not as instantly obvious as 0=17.

So I give them the new, expanded definition. Algebraic equations aren’t statements of fact. They are questions.

4x-7=2(2x+5) is not a statement of fact, but rather a question.

What value(s) of x will make this equation true?

And the answer could be:

  • x= specific value(s)
  • no value of x makes this true
  • all values of x makes this true.

We can also define our question in such a way that we constrain the set of numbers from which we find an answer. That’s why, I tell them, they’ll be learning to say “no real solutions” when solving parabolas, rather than “no solution”. All parabolas have solutions, but not all have real solutions.

This sets me up very nicely for a dive back into linear systems, quadratics with complex solutions, and so on. The students are now primed to understand that an equation is not a statement of fact, that solutions aren’t a given, and that they can translate different outcomes into a verbal description of the solution.

Equals Sign in Identity Proofs

An identity equation is one that is true for all values of x. In trigonometry, students are asked to prove many trigonometric identities,, and often find the constraints confusing. You can’t prove identities using the properties of equality. So in these classes,  I go through the previous material and then focus in on this next evolution.

Prove: tan2(x) + 1 = sec2(x)

(Or, if you’re not familiar with trig, an easier proof is:

Prove: (x-y)2 = x2-2xy+y2

Here, again, the “equals” sign and the statement represent a question, not a statement of fact. But the question is different. In algebraic equations, we hypothesize that the expressions are equal and proceed to identify a specific value of x unless we determine there isn’t one. In that pursuit,  we can use the properties of equality–again, known as “doing the same thing to both sides”.

But  in this case, the question is: are these expressions equal for all values of x?

Different question.

We can’t assume equality when working a proof. That means we can’t “do the same thing to both sides” to establish equality. Which means they can’t add, subtract, square, or do other arithmetic operations. They can combine fractions, expand binomials, use equivalent expressions, multiply by 1 in various forms. The goal is to transform one side and prove that  both sides of the equation occupy the same point on a number line regardless of the value of x.

So students have a framework. These proofs aren’t systems. They can’t assume equality. They can only (as we say) “change one side”, not “do the same thing to both sides”.

I’ve been doing this for a couple years explicitly, and I do see it broadening my students’ conceptual understanding. First off, there’s the simple fact that I hold the room. I can tell when kids are interested. Done properly, you’re pushing at a symbol they took for granted and never bothered to think about. And they’ll be willing to think about it.

Then, I have seen some algebra 2 students say to each other, “remember, this is just a question. The answer may not be a number,” which is more than enough complexity for your average 16 year old.

Just the other day, in my trig class, a student said “oh, yeah, this is the equals sign where you just do things to one side.” I’ll grant you this isn’t necessarily academic language, but the awareness is more than enough for this teacher.

 


Teaching with Indirection

GeoTrigRep1Technology is a great illustrator and indispensable for presentation. But as a student tool? Eh, not so much. Certainly not laptops.   I found laptops very useful in my history class, but primarily as a delivery and retrieval mechanism, or for their own presentations.  I haven’t found that a compelling reason to submit to the logistics of handing out and collecting laptops. But then, I’m a Luddite on this.  Recently, some colleagues were jazzed with several thousand dollars of cool science tools which I oohed and ahhed over politely. But….? Basically data collection. Fast data collection, which the students can analyze.  I guess. I don’t really do science.

A couple months ago, I used laptops and Desmos to teach transformations, and after twoGeoTrigRep2 blocks that went….well, I suppose, I used whiteboards to do the same lesson in the last block. Far superior. I wouldn’t have even considered the hassle, but last year the school decided all algebra 2 teachers warranted a laptop cart and I want to occasionally acknowledge a gift intended to be useful. I would never–I mean no excuses never–book a laptop cart from the library to teach a lesson. But if it’s sitting around my classroom, I’m bound to try and find a way to use it. Still, even if I had a lesson that would be guaranteed superior to the same lesson on paper, I’d be tough to convince. Taking them out and putting them away takes up close to 15 minutes of classtime. Wasted. If all of my GeoTrigRep3students had their laptops with them at every minute, waiting to be used….maybe. I’ve certainly found uses for phones on an occasional basis. But it’s not a huge gap I’m longing to fill.

Teaching is performance art. Sometimes the art lies in holding students’ attention directly, taking them point by point through a new topic. Other times, it lies in making them do the work. In both cases, the art lies in the method of revealing, of making them come along for the ride of understanding–even if it’s just in that moment.

It’s hard to do that if you put technology in the students’ hands. First, they’re too easily distracted. Second, it’s too easy to do without understanding.  A colleague of mine simply worships Dan Meyer, and loves all the Desmos activities.  They are neat. Without question or caveat. But I have limited time, and I’d rather have my students doing math directly, by hand even, than have them work on laptops or phones. Some Desmos activities do, absolutely, require the kids to work or show their math directly. Others are an interesting form of guess andGeoTrigRep4 check, designed (hopefully) to help kids understand patterns. The first, I like, but am unconvinced that the time and distraction suck are an improvement over handwritten work. The second, no. Not generally interested unless I have time for games, and I don’t.

This piece is only partially about technology, though. I wanted to talk about designing experiences, and for me, technology doesn’t give me the freedom to do that. Not with my kids, ability levels, and existing technology, anyway.

But how can I claim that technology is a distraction if I’m busy performing for the students?

Well, recall I said it was great for illustration and presentation. I love my smartboard, although I move pretty effortlessly between smartboards and whiteboard walls.

GeoTrigRep5I have learned it’s very simple to screw up a lesson by speeding it up, but far more difficult to do slowing it down. I like introducing a topic, sometimes in a roundabout way, and having the students do the work alongside. Consider the example displayed here. These aren’t power points of my lecture. I start with a blank screen. I give the instructions, give the kids time to follow along, then use their input to make my own diagram. That way I can circle around, see that everyone’s on track, understanding the math, seeing connections.

I spend a great deal of time looking for ways to build instruction step by step, so that the vast majority of my students have no reason to refuse the effort.GeoTrigRep6 Draw a square. How hard is that? Besides, most of them enjoy drawing and sketching, and this beats posters.

Ideally, I don’t want them to see where we’re going. But then, remember I’m teaching advanced high school math. At various times, I want students to understand that math discoveries don’t always go where they were expected. The best way to do that, in my experience, is give them a situation and point out obvious things that connect in not so obvious ways.

Thus, a trigonometry class is a great place to start an activity that begins as a weird way of breaking up a square into similar triangles. The sketches in the first steps are just a way to get them started, suspend their disbelief.  The real application of knowledge begins at this step, as they identify the equivalent ratios for the different triangles. A geometry-level skill, one from two years ago, and one we try to beat into their heads. Proportionality, setting up cross products,GeoTrigRep7 is also something students have been taught consistently.  A trig class is going to have a pretty high percentage of functional students who remember a lot of what they’ve been taught a lot.

Which is important, because this sort of activity has to be paced properly. You have to have a number of pauses while students work independently. The pauses can’t be too short–you have to have time to wander around and explain–but not explain everything to everyone, which would take too long and kill the mood. Can’t be too fast, either, or why bother?

Ideally, students should be mildly mystified, but willing to play along. As I wrote several years ago, start slow, build student trust in your wild notions. If you keep them successful and interested, they’ll follow along working “blind”, applying GeoTrigRep8their existing knowledge without complaint. Don’t deliver and they won’t follow. Which is why it’s important to start slow.

So in this particular activity, the students drew a square, some triangles, and found ratios without knowing when, or if, this was going to relate to trigonometry. Now, finally, they are using class-related knowledge, although SOHCAHTOA is technically covered in geometry and only reviewed in the early months of the year. But at least it does have something to do with Trig.

I’ve only done this once, but I was surprised and fascinated to note that some students were annoyed that I reminded them about the 1 unit substitution after they’d built the proportion statements.  I liked the structured approach of two distinct moves. They didn’t. “Why you make us do this twice?” griped Jamal, who is better at math than you might expect from his pants, GeoTrigRep9defying gravity far south of his pelvis, much less his perpetually red-eyed stupor and speech patterns. (“He’s a c**n,” he informed me about a friend a month ago. I stared at him. “It’s okay. I’m half c**n, so I can say  that.  Like, my family, we all light-skinned but we c**ns.” I stared at him. “OK, I ain’t no c**n in your class.” I mentioned the discussion to an admin later, suggesting perhaps Jamal needed to be told that c**n isn’t n****r , and is an insult in any vernacular. “C**n?” she said, puzzled.  “Like….raccoon?” It took me a few minutes to realize that she was a Hispanic, so it was indeed possible she had no idea what the word meant. I should have gone to our African American admin.)

It’s not obvious to all students that the ratio labeling each triangle side is the length of that side. That is, if the base is one, then the length of the secant line will be the exact value of the secant ratio, and so on. Breaking the diagram into three distinct triangles helps, but I do recommend spending some time on this point.

So, for example, say if the angle is 30 degrees, what length would the side labeled sine be? What about cotangent? They already know about sine and cosine lengths, since GeoTrigRep10I introduce this after we’ve covered the basics of the unit circle. But it helps to prod them into realizing that the cosecant length would be 2 units, and so on.

My students are familiar with my term “mother ship”. I use it in a number of contexts, but none so commonly as the Pythagorean Theorem. I ask them if they’ve seen Independence Day,  or one of the other zillions of alien invasion flicks in which the little independent saucers  all go back to the big behemoth. Because aliens will centralize, else how could humans emerge victorious? Just as all these little buzzing pods lead back to the big one, so too do so many ideas lead back to Pythagorean. Even its gaps. The Pythagorean Theorem doesn’t do angles, I point out. That’s why we started using trigonometry to solve for sides of right triangles. Originally, trigonometry was developed thousands of years ago to explain planetary GeoTrigRep11motion, and was defined entirely in terms of spheres and chords. Not until Copernicus, a few hundred years ago, did we start to define trigonometry primarily in terms of right triangles.

Until this activity, I’d always taught the Pythagorean identities algebraically. I start, as many do, by reminding or introducing them to the equation for a circle, then talk about a radius of one, and so on. Then I derive the secant/tangent and cosecant/cotangent versions, which is pretty simple.

But I really like the geometric representation. The three triangles are spatial, physical artifacts of what is otherwise a very abstract concept. Ultimately, of course, these identities are used for very abstract purposes, but whenever possible, links to the concrete are welcome.GeoTrigRep12

Besides, isn’t it cool that the three triangles reflect what the algebra shows? I suppose the fact that the triangles are all similar plays into it, but I’m not enough of a mathie to grasp that intuitively. The students, of course, don’t yet know the algebra. The Pythagorean identities are the one new fact set this lesson delivers.

Remember, I don’t use these images you see here in the lesson; rather, they represent a combination of what I say and draw during the lesson, pausing as the students work things out themselves.  Could I do this with technology? Sure. Could they? In my opinion, no. But it’s debatable, certainly. BUT–I also couldn’t do this with a book.
GeoTrigRep13Is it just me, or do students take an absurdly long time, over many lessons and with many reminders,  to memorize the unit circle? I mean, my god, there are five values for each ratio. They go in order–big to small, small to big. How hard could it be? But after a couple years of students looking at me blankly at the end of the term when asked what the sine of pi over 6 is, I’ve learned to beat it into their heads. Some teachers never use the unit circle to teach ratios. I do not understand this. Steve teaches it all with co-functions and trig tables; I have taught any number of his students who know vaguely what it is, but have no conceptual understanding of it. They know the values, their operational ability is no different, but where’s the fun? The unit circle is an amazing entity.

I am a big fan of Desmos. At algebra 2 and higher, I ask my students to download the Desmos app. My students learn how to graph, how to create functions, how to explore functions. I want them to know Demos as a tool when it makes sense. Really.

So eventually–although I haven’t done it yet–I’m going to show my students this puny effort to automate the concepts we explored manually in this lesson.  Hey, I can use the laptops! It will be a great example of inverse calls.

But not right away. Look, my classes do a lot of repetition.  Plenty of worked problems. It’s not all discovery or exploration–in fact, relatively little time is spent on these. My students need to know how, building capacity. Why is the glue. GeoTrigRep14The better a student is at the basics of math, the more important it is to smack them around with why, occasionally.

But I’m a performer.  English teachers talk about grabbing up front with the hook. But in math, ending big, revealing the path they’ve been wandering, is my goal. So when I draw in the circle, put in the coordinates, and hear “Holy sh**!” and various stunned gasps, following by a smattering of applause, I know my planning paid off.

“The f***? Damn. This been the unit circle all along. Shee-it.” That would be Jamal.