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.

 

 

 

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Melanie Wilkes, Feminist

Recently, Richard Brookhiser (an essential follow for history buffs) tweeted:

I had a number of picks, none of which was Gone with the Wind, which I consider excellent moviemaking, but risible history. But naturally, someone mentioned it and Brookhiser opened it to the crowd, appropriately mentioning its “lost cause” ideology, while praising it as moviemaking.

I mentioned that Mellie was one of the greatest feminist characters of all time, and someone caviled. Mellie? Scarlett is the one feminists love.

True, Scarlett is the character more typically celebrated by feminists, at least before GWTW became off limits to praise. But when I first read Gone With the Wind in my teens, I was appalled. Scarlett is a loathsome, selfish, vain, cowardly little monster, a characterization the movie does little to soften.

So for a good decade or more, I shrugged off Gone with the Wind.  Eventually I saw it on the big screen and tempered my dislike; the story is beautifully told, the acting from top to bottom is tremendous, and as spectacle it’s impressive.

I’m not sure when I first realized that Melanie Wilkes, played by the great Olivia De Havilland, was the tremendous feminist model that others saw in Scarlett. I do know that from the first time I watched it to now, I preferred Melanie.  Sometime in the 90s, though, I realized that she, not the tempestuous Scarlett, is the exemplar of a powerful female character. De Havilland’s Melanie is, in my view, one of the five great feminist movie roles of all time (the others: Bette Davis in Now, Voyager, Faye Dunaway in Network, Meryl Streep in Out of Africa, and Sigourney Weaver in Aliens.)

You can easily watch GWTW and see only Melanie’s story. She is a happy woman who was given the gift of a man she adored above all others. From the movie’s first moments to the last of her life, her face lights up in his presence. She wants only to give Ashley a home and a family, and to be a mother and wife. But Ashley must  fight for The Cause, so she must  support him and the other brave men fighting for their country and the right to own slaves. She nurses and works endlessly to the extent her body will allow. She gives every bit of her strength to help in the war so that her Ashley can come home. She is warm and accepting based on others’ character and motives, unheeding of social standing.  She deliberately chooses to get pregnant in the middle of a war despite the risk to her health and financial security, because Ashley’s legacy must be carried on. (The movie is quite frank about Ashley and Melanie’s intention on Christmas Eve ; note Scarlett’s reaction as the two of them go up to bed.) She gives up her wedding ring, a cherished symbol of her marriage, to help support the Confederate Army (although Rhett Butler gets it back). And despite Ashley’s helpless infatuation with Scarlett, the Wilkes’ marriage is a strong one. Notice that Scarlett, for all her protestations of devotion, doesn’t waste a second glance on the ex-soldier she thinks is coming to beg. It’s Melanie who instantly recognizes the distant, shabby figure as her beloved husband, in a homecoming ranked second or third to  Sounder’s “running scene” and the two from Best Years of Our Lives.

She’s “just” a wife and mother. What else was possible in the 1860s south?

But Melanie’s onscreen action rarely involves cooking or childcare. She works with Rhett Butler to stage manage a show that gives the wounded Ashley an alibi, protecting him from Ward Bond’s Yankee captain. None of it is planned. She’s using pure wits while following Rhett’s cues. Watch the fine, upright, honest Mellie lie serenely to keep her husband from a Yankee prison, after Ashley, along with Scarlett’s husband Frank Kennedy, went to clear out the shanty town where Scarlett was attacked–“what a great many of our Southern gentleman have been called upon to do for our protection.”

And when those Southern gentlemen aren’t around? Well, notice that same night, when her husband isn’t home, she has a gun nearby to protect her guests. Or, most notably , she grabs her father’s sword to protect her baby from a raiding Yankee soldier who’d fought to free their slaves. Scarlett got there first, of course. But Mellie was ready to fight for her own, despite being “weak as a kitten” from blood loss during child birth.

“Scarlett, you killed him! I’m glad you killed him.”  And then Mellie, still  faint and dizzy, helps Scarlett pilfer his pockets and hid the body, being quickheaded enough to lie about the noise. (She also takes off her nightgown to absorb the soldier’s blood, giving her the movie’s only nude scene.)

After the war, Melanie more won’t hear a word bad about Scarlett,  not from  Belle Watling, who helped save her husband, not from her other sister-in-law India, not from anyone. The only cross word she ever has for her husband came when he tried to escape Scarlett’s attentions and move his family away from Atlanta. Scarlett wouldn’t hear of it and starts to cry. Melanie is outraged at her husband’s ungrateful, ungentlemanly behavior.

Melanie’s staunch, unquestioning devotion to Scarlett leads many to dismiss her as saccharine, but she is manifestly not a a goody goody. When her sister-in-law and husband are caught in a compromising position, and Rhett forces his wife to go to Ashley’s birthday party, Melanie shoves all that social disapproval right back in the town’s face, insisting they recognize Scarlett’s existence.

For some number of years, I considered Melanie’s only weakness to be her bizarre refusal to see Scarlett as evil, conniving, and weak.  Then one day I suddenly noticed that Melanie never once called Scarlett nice, or warm, or loyal, or any of the qualities that she herself had. Instead, she talks about Scarlett’s bravery. Scarlett saved them in Atlanta. Scarlett protected Melanie and her son. Scarlett found a way to help the families survive. Scarlett rebuilt their family’s fortune by marrying her sister’s fiancee.

I realized that all of these things were, well, true. Scarlett could have left Melanie in Atlanta.  She could have been the model of helpless futility through Mellie’s childbirth, running screaming from the house at the sight of bodily fluids just as she did from the hospital during a soldier’s amputation. But she boils water and hangs tough, encouraging the nobler Mellie to scream as loud as she wanted. She could have ignored Mellie’s desire to preserve her father’s sword but wraps it up to give her some peace of mind.  She saved Melanie and Beau, and got them all out of Atlanta, including the annoying Prissie. She could have run away from her family, leaving them all there to starve.  Instead she rebuilt her family’s fortune, which her sister never could have done with Frank Kennedy. She went to Ashley’s birthday party, even expecting Mellie and the town’s society to wither her with rejection. She did almost nothing without a hefty dose of whining. She did many loathsome things, hiring prisoners and hitting slaves. But everyone who hated her benefited from her courage and furious willfulness, and only Melanie understood that.

Through Melanie’s admiration, I’ve come to a reluctant and qualified admiration of Scarlett herself. Scarlett was awful, yes, but her actions do show her to be worthy of Melanie’s trust and support. And yes, precisely because she continually does the right thing even when she longs not to, Scarlett is a great feminist character, for good and bad. (She’s still not in my top five, though.)

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Gone with the Wind is going through hard times right now.  If in a year or three it’s banished from TCM and movie revival houses and the AFI top 100 (much less the top 10 position it now holds), I won’t be surprised. It’s not a perfect film. But Ingrid Bergman has very little to do in Casablanca, and Diane Keaton even less in The Godfather , both perfect films about men.  It’s another dozen movies down the AFI list until we find All About Eve and Double Indemnity, movies where women as heroes and villains drive the plot.

If I were going to show GWTW in a history class, I would use it as a means of exploring early movie attitudes on race. When I teach US history I focus more attention on the early 1900s than 1965 and beyond. The debate between Booker T. Washington and WEB Dubois is still compelling and relevant to our lives today. Booker and W.E.B. and black society had to build political power during the Jim Crow era, without TV, before the Great Migration had transformed cities and created voting blocs.  We explore the degree to which blacks could simultaneously use their political clout yet be virtually banned from voting in many states.

My students were surprised to learn that in the Jim Crow era, African Americans were able to protest Birth of a Nation in Boston and other cities, leading several states to ban the movie. The NAACP grew its membership considerably during this era, and while the racist Woodrow Wilson may or may not have called the movie “history written in lightning”, the resulting tensions led him to retract his endorsement.  Warren G. Harding was in many ways a terrible president, but his Birmingham civil rights speech, followed by Calvin Coolidge’s open and engaged support also enabled further extension of civil rights, although the Depression and Herbert Hoover led blacks to switch to the Democrats. But by 1938, blacks had sufficient political and ecnomic clout that David O. Selznick sought script approval from the NAACP, and (reluctantly) dropped the use of the n-word. (Selznick thought that it would be okay if blacks used the word.)

I’d be much more tempted to show GWTW in an English class, though. I want my students to feel the kind of passion that leads people to debate and care and argue about fictional characters, and I think this film inspires that kind of passion.

Yes, GWTW promotes the  false “lost cause” view of the Civil War. Yes, the characters hate the Yankees who  died to free the slaves. Yes, Prissie is a terrible caricature. But Hattie McDaniel, who beat out De Havilland for Best Supporting Actress, creates a spectacular character in Mammy, a performance that black actors weren’t really allowed to match until Sidney Poitier’s work in the 50s and 60s. Watch this amazing, single shot up the stairs by Mammy and Mellie. We never see the rage between Rhett and Scarlett as they mourn their daughter’s death. We don’t have to. It’s all in McDaniel’s voice and de Havilland’s muted reactions. How can we show our students the amazing talent of African Americans shining through despite their restricted opportunities if we demand all the movies meet our current norms?

And now, some words about De Havilland who, unlike every other white starlet in Hollywood, wanted to play Melanie, not Scarlett. A much bigger star than Vivian Leigh ever was, de Havilland uses her charisma and presence in service of Melanie’s character, which she  well understood, to create a warm and compelling character out of a role that many other actresses made vapid and sickly sweet.

At 101, she’s still with us. She’s suing the producers and director of  Feud;  her case has survived an attempt to dismiss and begins in November. She changed the history of Hollywood  by suing, so who knows how this will turn out?

She is amazing as Melanie and charming as Maid Marian, but her finest performance is as Catherine Sloper in The Heiress, for which she received her second Oscar. Both movies originate from American fiction to which both stayed amazingly faithful, but while GWTW never aspired to be literature, Washington Square by Henry James is an American classic. The movie is quotable, beautiful, and one of the most psychologically painful movies you will ever run into.  Don’t take my word for it. Ask Martin Scorcese.

 

 


Coaching Teachers

In 2011’s Personal Best, Atul Gawande recounts his desire to “up his game”, by hiring a retired surgeon who had once trained him, Robert Osteen, to act as a coach.  I often reread the article just for the best passage in an already great piece: when  Osteen gives Gawande feedback for the first time.

Prior to his own coaching experience, Gawande explores the difference between “coaching” and “teaching” in the teaching career itself. He sits in on a lesson and coaching session with  an 8th grade math teacher. One of the coaches was a history teacher, the other a math teacher who’d given up teaching to work at the district. While Gawande implies coaching is unusual, many school districts have coaching staffs, usually made up of history teachers and middle school math teachers, just like this one.

Everything that crackles and glows when Gawande describes Osteen’s observations falls with a thud in the teaching section. The lesson on simplifying radicals sounded fairly traditional, but seemed dull in the telling. The coaching feedback was similar to what I’ve experienced–banal platitudes. Socratic questioning. “What do you think you could do to make it better?” (Translated: I personally have no idea.) Not the same assertive advice Osteen gave Gawande, but carefully scripted prompts. Critzer seemed to like the “feedback”, such as it was, but I found the whole exchange extremely antiseptic. In no way were the two coaches “operating” (heh) on the same level as Osteen’s expert.

In 2011 I was a newbie. Now I’m edging towards a full decade of teaching and have now mentored  three teachers through induction and one student teacher. I’m better prepared to think about coaching, both as provider and recipient, and the stark differences in those two passages keep coming back to me.

My ed school supervisor , a full-metal discovery proponent, gave me one of the great learning experiences of my entire life. She never tried to convert me or push particular lesson approaches.  I can still remember the excitement I felt as she pushed me to think of new methods to achieve my goals, while I realized that regardless of teaching philosophy, teaching objectives remain resolutely the same: are the kids engaged? Are they learning, or parroting back what they think I want to hear? Am I using time effectively?  Osteen’s feedback reminded me of those conversations, and as I moved into a mentor role, she became my model.

A couple weeks ago, a district curriculum meeting ended early and I went back to school just in time for fourth block to observe my newest induction mentee.  This was an unscheduled observation, but she welcomed me into her pre-algebra class for a lesson on simplifying fractions prior to multiplication. Through the lesson, the students worked on this worksheet. The concepts involved are not dissimilar from the ones in Jennie Critzer’s lesson.

Here’s my feedback, delivered immediately after the bell rang.

“Okay, I’m going to split my feedback into three categories. First up are issues involving safety and management that you should take action on immediately. Everything subsequent is my opinion and advice  based on my teaching preferences as well as what I saw of your teaching style. I will try to separate objective from method. If you agree with the objective but not the method, then we’ll brainstorm other ideas. If you disagree with the objective, fine! Argue back. OK?” She agreed.

“For immediate action, make students put their skateboards under that back table, or in a corner completely away from foot traffic. The administration will support you in this in the unlikely event a student refuses to obey you, I’d also suggest making all the students put their backpacks completely under the desk. It’s like ski week around here, you nearly tripped twice. Now for the suggestions…”

“Wait. That’s the only mandatory change? My classroom management is good?”

“Yes. Kids were attentive and on task. But I want you to move about the room more, as you’ll see, and the way your kids strew their stuff around the floor, you’ll kill yourself.”

“I was worried about management because the students often seem…slow to respond.”

“We can talk more about your concerns before our formal observation so I can watch that closely. I’d like more enthusiasm, more interest, but that’s a subjective thing we’ll get into next. They listen to you and follow your requests. They’re trying to learn. You’ve got buy-in. You’re waiting for quiet. All good.”

“Phew. I’m relieved.”

“Now, some opinions. I’d like you to work more on your delivery and pacing.  You are anchored to the front of the class during your explanation time. Move about! Walk around the room. Own it. It’s your space.”

“I am never sure how to do that.”

“Practice. When you have a few sentences nailed down, just walk to the back by the door,  stand there for a minute or so, then move to another point, all while talking. Then go back up front. Do that until it feels comfortable. Then ask a question while away from the front. Then practice introducing a new topic while away, and so on.”

“I didn’t think of practicing. I thought it would come naturally.”

“I’m as big a  movie star teacher as they get, and what I just described is how I escaped the front-left cellblock.”

“OK.”

“Next up: you’re killing the flow of the lesson.  Here’s what you did today: give a brief description of method, work an example, assign two problems, go around the room looking at student work, come back up, work the problems. Then assign two more, go around the room looking at student work, come back up, work the problems. Lather, rinse, repeat. This precludes any concentrated work periods and it’s hurting your ability to help your top students. It’s also really boring.”

“Yes, many of my students have worked all the way through the handout. But I have to help the students who don’t get it right away and that takes time, right?”

“Sure.  So give a brief lecture with your own examples that illustrate two or three key concepts–NOT the ones on the worksheet. And while that lesson is going on, my advice is to insist that all students watch you. Right now, the strong students are completely ignoring your lesson to work the handout–and from what I can tell, occasionally getting things wrong.”

“Yes, they don’t know as much as they think they do in every case. But it’s good that they’re working, right? They’re interested?”

“Not if they aren’t paying attention to you. You are the diva. Attention must be paid.”

“But if they know it all…”

“Then they can finish it quickly after your lesson–as you say, they sometimes make mistakes you covered. So do an up front lesson of 15-20 minutes or less, depending on the topic. Then release them to work on the entire page or assignment. Let them work at their own pace. You walk around the room, giving them feedback. Don’t let the stronger kids move ahead in your packet. Have another handout ready that challenges them further You might have an answer sheet ready so kids can check their own work.”

She was taking notes. “How do I get these more challenging handouts?”

“Ask other teachers. Or I’ll show you how to build some. I know you’re using  someone else’s curriculum, but you can have additional challenges ready to keep your top kids humble. Math gets much harder. They need to be pushed.”

“So then I teach upfront and give them 30-45 minutes to do all the work, giving the kids who finish more work. Maybe a brief review at the end.”

“Bingo.”

“Got it. I’m going to try this.”

“Last thing on delivery: you’ve got a Promethean. Use it. It will free you from the document camera.”

“I don’t know how. I asked the tech guy for guidance and he said you were one of the most knowledgeable people on this brand.”

“Well, let’s do that next. Now, onto the much more difficult third topic: your curriculum. I could see you often backtracking from your own, authentic instruction method to return to the worksheet which forcefeeds one method: find the Greatest Common Factor or bust.  I could tell you didn’t like this approach, because you kept on saying ‘they want you to use GCF’, meaning the folks who developed the worksheet.”

“Yes, I kept forgetting to avoid my own method and  support the worksheet’s method.”

“Why?”

“Well, I have to use that worksheet.”

“Toots, you don’t have to use a thing. You’re the teacher. They can’t require you to teach it. I don’t dislike the curriculum, but that particular worksheet is flawed. As I walked round your room, I saw kids who just cancelled the first factor they saw, and then had an incomplete simplification. So 9/27 became 3/9 because the kid turned 9 into 3×3 and 27 into 9×3.”

“Yes, that’s what I saw, too. They didn’t realize it wasn’t fully simplified, because they weren’t realizing the need to find the GCF.”

“That’s because the method isn’t as important as the end result.  Who cares if they use that method? That’s what the one student said who challenged you, right? You were trying to push her to find the GCF, and she pushed back, saying ‘what difference does it make?’ and you were stuck because you agreed with her, but felt forced into this method.”

“God, that’s so right,” she groaned.

“But you weren’t giving them any plan B, any way to see if they’d achieved the goal. How much advanced math have you taught? Algebra 2, Trig, Precalc? None? You should observe some classes to see how essential factoring is. I talked to many of your students, and none have any real idea what the lesson’s purpose was. Why do we simplify at all? What was the difference between simplifying fractions and multiplying them?  What are factors? Why do we use factors?  I suggest returning to this tomorrow and confess that the student was correct, that in the case of simplifying fractions by eliminating common factors, there are many ways to get to the end result. Acknowledge you were trying to be a good sport and use the method in the handout, but it’s not the method you use.”

She wrote all this down. “And then I need to tell them how to know that they have fully simplified.”

“Exactly. Here’s what I saw as the two failures of the worksheet and your lesson: first, you didn’t tell them how they could test their results for completeness. Then, you didn’t tell them the reason for this activity. Namely, SIMPLIFY FIRST. When using numbers, it’s just an annoying few extra steps. But when you start working with binomials, failing to factor is disastrous for novices.”

“OK, but how can I circle back on this? Just tell them that I’m going to revisit this because of what I saw yesterday?”

“Yes! I recommend a simple explanation of  relatively prime. That’s the goal, right? The method doesn’t matter if that’s the end result.  And then, here’s a fun question that will startle your top kids. Given “two fourths”, why can we simplify by changing it to 2×1 over 2×2 and ‘canceling out’ the twos, but we can’t simplify by changing it to 1+1 over 1+3 and ‘cancel out’ the ones? Why don’t we tell them to simplify across fractiosn when adding? ”

“Wow. That’s a great question.”

“Yes. Then come up with a good, complicated fraction multiplication example and show them why all these things are true. Make them experience the truth by multiplying, say, 13/42 and 14/65. They might not retain all the information. But here’s what’s important, in my view: they’ll remember that the explanation made sense at the time. They’ll have faith. Furthermore, they’ll see you as an expert, not just someone who’s going through a packet that someone else built for her.”

“Ouch. But that’s how I feel.”

“Even when you’re going through someone else’s curriculum, you have to spend time thinking about the explanation you give, the examples you use. This isn’t a terrible curriculum, I like a lot of it. But fill in gaps as needed. Maybe try a graphic organizer to reinforce key issues.  Also, try mixing it up. Build your own activities that take them through the problems in a different way. Vary it up. You’ve got a good start. The kids trust you. You can push off in new directions.”

I then gave her a brief Promethean tutorial and told her I’d like to  see a lesson with some hands on activities or “cold starts” (activities or problems with no lecture first), if she’s interested in trying.

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

Mid-career teachers, like those in any other profession, are going to vary in their desire and interest in improving their game. Twitter and the blogosphere are filled with teachers who write about their practice.  Perusing social media is a much better form of  development than a district coach that isn’t experienced in working with the same population and subject. Conversations with motivated colleagues interested in exploring their practice, but hared to find the time or interested participants.

But  unlike other professions, we teachers are given ample, and often paid, opportunity to be coaches, and not the weak-tea district sorts. Induction and other new teacher programs give us a chance to push others to find their best.  I find these activities also lead me to review and improve my own practice.

If you’re tasked with helping beginning teachers, then really dig in. Challenge them. Encourage them to push back, but do more than ask a few questions. They’ll thank you later. Often, they’ll thank you right away.

 


Restriction of Range

I read Scott Alexander because he’s a pretty good weathervane for insight into the respectable crowd. For reasons I don’t understand, he periodically gets raves from writers way up the food chain, so he’s clearly writing about sensitive subjects without activating their panic buttons.  I once read this book on Highly Sensitive People, and the author was like “OK, this may be painful, so stop and take a breath before you move on. Sense how you’re feeling. Breathe again. Now turn the page.” I found this extremely irritating, and Scott reminds me of that author. Who, by the way and despite the offputting habits and an entirely unscientific theory, provided me with a successful frameworks and some useful tips. Yes,  I am a Highly Sensitive Person. Go ahead, laugh; it’s 20 years and I still think it’s funny.

Anyway. While this may seem like insider baseball, I’m writing this because the issue at hand illustrates an important point.

Recently, Scott wrote a soothing reassurance to the many people writing him “heartfelt letters complaining about their low IQs”.

See, the correct response to “heartfelt letters complaining about their low IQs” is a gagging noise or, perhaps more maturely, a discreet eye-roll. But that’s just me.

Scott quotes a Reddit commenter echoing a typical concern:

I never got a chance to have a discussion with the psychologist about the results, so I was left to interpret them with me, myself, and the big I known as the Internet – a dangerous activity, I know. This meant two years to date of armchair research, and subsequently, an incessant fear of the implications of my below-average IQ, which stands at a pitiful 94…I still struggle in certain areas of comprehension. I received a score of 1070 on the SAT, (540 Reading & 530 Math), and am barely scraping by in my college algebra class. Honestly, I would be ashamed if any of my coworkers knew I barely could do high school-level algebra.

Scott does something like five paragraphs on the measurement and meaning of IQ and how it’s great for groups but not terribly valuable for the individual. All that is just duck and weave, though, because basically, his response is “Well, your IQ test wasn’t accurate”.  But Scott’s worried that if he says that, it will undo all the hard work he’s put in convincing people that IQ has meaning.

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

So reading the post, the reddit thread, and the comments, I’ve concluded that my–well, somewhat undue–frustration has two sources. First, I  believe abrupt, brusque and occasionally rude responses are not immoral and frankly necessary. But more importantly, I’m dumbfounded that Scott would treat these queries as worthy of a treatise, so I’m wondering why.

I don’t usually quote Malcolm Gladwell unless it’s his ketchup piece, but this is instructive:

Of course, Gladwell was actually quoting someone with actual expertise, Arthur Jensen:

While individual IQs are irrelevant, the tiers are pretty useful. Those who interact regularly with all three tiers can place people pretty accurately in those tiers.  My various occupations have given me access to the entire range of  IQs, from the occasional low 80s to third standard deviation and possibly beyond. As a result, I don’t know a 98 from a 105, but I would never place either in the below 90 or above 115 group.

And from that vantage point, I can’t figure out why Scott is equivocating, because there is simply no way the Reddit poster, or indeed anyone who reads Scott’s blog, has an IQ much south of 115. The idea is ludicrous. Instantly risible.

Alexander is clearly aware of this. His characterization: “Help, I got a low IQ score, I’ve double-checked the standard deviation of all of my subscores and found some slight discrepancy but I’m not sure if that counts as Bayesian evidence that the global value is erroneous” oh so gently mocks his emailers–and mocks them in a manner that only higher IQs could understand.

But why would he spend so much time on the topic? Maybe it’s my (extremely low) opinion of the SSC groupies, but it’s pretty obvious that the emailers are looking for validation from their hero.

“I’ll tell Scott or random people on the internet that I’ve got a low IQ and they’ll go, pish tosh! and tell me how smart I am.” . Write an intellectual email, tossing in all the right buzzwords, worrying about their IQ, in order to get a reassuring  “Don’t be silly! You’re far too intelligent for a 90 IQ!” that they can brag about.

In short, I think Scott’s emailers are lying to get an ego boost.

Sure, it’s possible that IQ tests are routinely handing out scores of 90 to  people with 80th percentile SAT results. It’s just extremely unlikely.  Alternatively, these folks could be IQ-denialists lying to seed doubt and confusion about IQ tests. “We’ll be, like Russian agents and post fake news through Scott. No one will trust these foul instruments!”

I’ll take “Needy Validation” for $1000, Scott.

He may simply be too polite to say “I don’t believe you”. But no one else did, either, in all the megabillion comments he gets on each blog. Some of the reddit folks gently pointed this out, but their views didn’t catch on.

Hence I wonder about restriction of range. Are the people in the discussion, from Scott Alexander on down, so unfamiliar with the intellectual capabilities of a 94 IQ that he thinks it merely unlikely that the IQs are inaccurate, as opposed to a possibility that can be instantly dismissed?

Maybe that’s it. After all,  most of the educated world is setting their intellect standards like the second graph of this grip strength study illustrating the essay title:

 

restrangepic

As the author says, note the change in the x axis.

In perhaps his most famous piece, Scott characterizes the other, the people outside his inadvertently constructed social bubble as “dark matter”. These people exist. They are legion. But somehow he never runs into them, never has any contact.

It’s a neat little metaphor, but really all he’s describing are social bubbles that restrict your range pf experience or understanding. Just as most progressives never run into a conservative, so too are most college graduates who aren’t teaching in high poverty districts rarely going to meet an average IQ,  much less sub-90 intellects.

Steve Sailer, with the ruthless accuracy and snarkiness that (wrongly) inspires disdain for his excellent observational skills,  once observed that Rachel Jeantel, who testified at George Zimmerman’s trial  was a high school student. Steve, who notices things, was pointing out that our expectations for high school students must include Jeantel, when in fact most people yapping about at risk black high school students have Will Smith in mind. Wrong. Smith is a bright guy.

Rachel was 19 when she testified, and graduated the next year from high school at 20. The media reports that “extensive tutoring” helped her graduate, but high schools will graduate anyone who tries hard enough. In my opinion, the support and the attention, not the tutoring, is what helped Jeantel graduate.  I can’t find much about her life since then, but no news in this case is pretty good. I’d guess Jeantel below the 90 tier, but she might be right above it. She’s pretty functional. She’s savvy about how to handle her moment in the sun. She took advantage of the support offered her.

Listen to some of Jeantel’s testimony. Go back up and read that Reddit post that Scott says is typical of the worried emails he gets from people who are saying that they have roughly the same IQ as the young woman in that video.

Perhaps then you’ll see why I think the emailers deserve derision, gentle or otherwise.

Derision not because a low IQ is to be mocked or dismissed.  Derision in part because I believe these people are seeking validation and ego boosts. But mostly, derision to reinforce  and educate people about these tiers. The more people understand the basic realities of a 90 IQ as opposed to one of 115, the more we’ll understand the challenges of educating and employing them. The more people who engage in these debates understand how cocooned they are, the less foolishly optimistic they’ll be in considering education policy debates.

Educators, the peasants of the cognitive elite, can offer some guidance. Many educators deliberately ignore cognitive reality; I’m not saying we all have the right answers, or that I do. But I would like all educated people who think they understand American education to look at the whole picture, rather than be allowed to ignore the “dark matter”.

I really don’t  know if Scott himself is refraining from mocking these IQ queries or if he really doesn’t understand that their fears are impossible.

Ending where I began: I read Scott Alexander because he’s a pretty good weathervane for insight into the respectable crowd that prides itself on its skeptical humanism.  Unfortunately, either interpretation of his behavior is consistent with that set.  I remain befuddled.

 

 

 


Killing My Own Snakes

When I was hired to teach at Southeastern in May, 1979, the Academic Dean at the time gave me only two pieces of advice: “Make your own way,” and “Kill your own snakes.”-Steven Fettke

One of the most valuable pieces of advice I received, from two different teachers in two different years (student teaching, first year), was that a new teacher had to know what “quiet” is.  If kids wouldn’t shut up, then kick them out until finally, the teacher experiences….silence. Without that baseline, a new teacher has no gauge to assess the ambient classroom noise.

I began teaching as a better than average classroom manager, and somewhat shrugged this wisdom off until I got the advice the second time after five particularly troublesome geometry students wouldn’t shut up during an entire lesson. So the next day, I warned them once and then tossed one then another off to the office. After two were gone, the other three realized I was serious and shut up, after growling a bit about unfairness. Turning back to the board, I suddenly heard…..silence. Utter, attentive, silence. And from that point on, I knew what silence was, and what to expect when I demanded it.

As a mentor, I always advise new teachers to err on the side of excess with disruptive students. If they have an entire class out of control, ask for help. If they have a few students misbehaving, toss them out after a warning. Screw fair. Get silence. Know what it sounds like.

New teachers are often fearful of  sending students out. They worry that administrators will judge them. They’re right to worry. Administrators often notice. At my last job, the volume of my referrals was  a constant source of tension.  In really poorly managed schools, the admins refuse to accept students and send them back. (Note: leave that school.)

This is where mentors come in. Mentors can, and should, give balance to new teachers. My induction mentor’s support and acknowledgement of my unimaginably disruptive students finally forced administrators to take action. If the teacher is weak, by all means help shore up the crumbles. But in the meantime, encourage the teacher to boot students who disrupt teaching time. I get impatient with people who bleat that removing kids from the class is depriving them of education. All students deserve an education. Students who are determined to prevent that can step outside.

In my experience, novice teachers stuck with unusually unruly students will improve their management skills if given the opportunity to remove the disruptors. As time goes on, these teachers will improve their handling of rambunctious students. Part of that improvement involves knowing what silence sounds like.

So new teachers should not try to kill all their snakes, particularly given the likelihood that they’ll have the toughest students.

I assume most teachers kill their own snakes after the first few years. But I’m often amazed at what senior teachers will tolerate. Sample statements, followed by my (usually unspoken) response.

“I’m teaching an Algebra 10-12 class, and the kids start packing up their stuff with fifteen minutes to the bell. Does that ever happen to you? What do you do to prevent that?”

I tell them to unpack their damn books and get back to work. Right now. And if they don’t start moving right away, oh my goodness, pop quiz.

“I’ve been having so much trouble with kids using cell phones constantly in class, not paying attention at all. What do you do?”

I take their damn cellphones away, giving myself extra points if I can swipe it from under their nose without signaling intent. Students who can’t keep off their phones lose them until the end of the day instead of the end of class. And they don’t dare complain, because I can always hand it over to the administrators, whose penalties are far more stringent.

“I have these two kids who constantly talk to each other, but when I try to separate them, they insist on sitting together. It’s so frustrating.”

Why the hell do you give them a choice? Tell them where to sit. In fact, tell everyone where to sit.

“I tell the kids not to bring food to the class, but what do you do when they’ve just bought lunch?”

You take the lunch away and tell them they can enjoy it cold later.

“I’ve tried taking away phones/telling them where to sit/taking their lunch but they refuse to give it over, and I don’t know what to do.”

You call and have them removed from the class.

“What? For something so minor?”

Listen well, little teachlings. Defiance of a teacher is not minor. It’s one of the few snakes that even experienced teachers should hand off to an administrator if they can’t convince the student to comply. Give the kid a chance to walk back. Offer alternatives. Draw a line, though, and if the line gets crossed, have the kid removed for the day.

And of course, logistics get in the way sometimes. More than once, I’ve picked up the phone to call for a supervisor to come take a defiant kid away–and no one answers the damn phone. So I have to call another number. Sometimes no one answers. All that drama and then….man, turning back around to face the class really sucks.

But well over half the time, simply picking up the phone has results, and the defiant one says something like “Well, you want me to give up my lunch AND my drink! No way!” and I say quickly, “No. Just the lunch. I insist on the lunch!” which leads to “Oh, I thought you wanted my drink, too. OK, have my lunch. BUT I KEEP MY DRINK!”

Other times,  the troublesome kid smirks. “Ha, ha, you can’t catch me, copper!” Shrug. Just shrug. And then later, call again, after the smirker has forgotten all about it, and have him pulled from the room, protesting. Don’t gloat. Just go on with the lesson like this is no big deal.

 

So you might be reading all this saying, wow, Ed’s a tyrant. Which is hysterical, because I’m one of the loosest teachers you’ll ever run into. Remember, I don’t assign homework. My kids sit in groups. I have a non-existent detention rate, the lowest in the school. I rarely give an F grade.  To my considerable pride, I’ve gotten the coolest of the Student Nominations three years running (best story teller, most unpredictable, most dramatic).  My classes are noisy and boisterous affairs. In many ways, my classroom environment is a progressive’s dream, the kind of place that Ed Boland dreamed of having before he realized he hated students.

I have five rules, handwritten seven years ago on still bright yellow poster paper. Students should avoid:

  1. arguing with the ref (me)
  2. eating, drinking, or grooming
  3. setting objects airborne
  4. travelling without consent
  5. incessant yammering

But bottom line, do what I tell you.  My lines are very clearly marked, albeit occasionally negotiable. Just pay close attention to when I say “when”. As  I tell my kids every year at syllabus time: in order for “all this”–school, teaching, classroom environment–to work, I have to be in charge. Students have to obey my direct orders.

I realize that many teachers feel that schools already exert a great deal of control over student lives. They feel that rules about eating, phones, and seating are an unfair imposition. These same teachers often feel that “consequences” must be “deserved”, that their restrictions on those who have made bad choices, are somehow more reasonable.

Shrug. I’m not saying there’s only one way. Other teachers can make their own choices. Me, I avoid morality plays. I don’t talk about what students deserve or earn, simply about what helps me teach and others learn.  I handle even cheating as a pragmatic issue, not a value judgment.

From students’ perspective, their least  favorite of my management techniques is  my yelling, specifically  calling out or putting a student on blast.  They prefer teachers who rebuke quietly and in private. But they also agree that when you aren’t being the one called out, it’s fun to watch me rant.

As I invariably mention when going through the syllabus, the only action a student can take to earn a permanent black mark is deliberate cruelty to another student. I will punish that and I’m much better at being mean.

Note that I prohibit being mean to other students.  Nowhere in my rules is it verboten to be mean to me, the teacher.

At least once a year, I (usually inadvertently) get a student furious, and the exchange goes something like this:

Student: “F*** YOU!!!!”

Me, unfussed and occasionally confused: “Sit down.”

Student: “NO!!! You F******* *****! F*** YOU!! F*** OFF”

Me: “Sit down.”

Student, walking to the door: “NO WAY. EAT SH**. I’m OUT! YOU #*@#W%@#W%!”

Me: “DO NOT WALK OUT THAT DOOR!”

Student: “WHY NOT?”

Me: “BECAUSE UP TO NOW, YOU HAVEN’T DONE ANYTHING WRONG!”

This usually stops the student for a minute or so, giving me a chance to calm things down. In every case, after a brief talk with a fascinated class watching on, the student sits back down and everyone gets back to work. Show’s over.

Which is not to say I let students take nasty potshots at me. Like I said, I’m much better at being mean than your average adolescent. But I don’t demand respectful behavior, and don’t get upset at rudeness.  This will not come as a shock to people who know me online.

Look. Teaching is very much an expression of personality.  Mine is a teacher-centered classroom. But nowhere is it written that teacher-centered classrooms must be ruthlessly controlled environments of churchlike stillness.  My classroom is, like me, loud and often disorderly, friendly, sarcastic. It sometimes changes on a dime. But its purpose is always there, driving things along, moving everyone forward.

New teachers: does your classroom environment reflect your personality, your values? Experienced teachers: are you setting rules that matter? Are you sure?

 


Modeling Rational Expressions

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

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

Starting Activity

ModRatExp1

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

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

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

“Twenty dollars.”

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

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

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

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

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

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

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

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

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

Silence.

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

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

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

“Right. Is there a horizontal shift?”

“No.”

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

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

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

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

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

ModRatExpGraph6

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

ModRatExp2

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

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

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

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

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

ModRatExpEquation1

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

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

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

ModRatExpGraph2

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

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

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

 

Suggestions? The ideas come fast.

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

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

We try $23/hoodie:

ModRatExpGraph3

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

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

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

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

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

Puzzled looks.

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

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

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

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

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

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

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

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

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

ModRatExpGraph5

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

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

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

ModRatExpEquation2

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

ModRatExpEquation3

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

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

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

ModRatExpGraph4

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

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

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

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


Three VIPs for New Teachers

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

The Tech Guy

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

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

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

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

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

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

The Principal’s Secretary

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

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

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

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

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

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

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

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

“THAT’S NOT MY JOB!”

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

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

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

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

The Attendance Clerk

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

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

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

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

“What? No, he’s not.”

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

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

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

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

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

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

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

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

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

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

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

What about…..?

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

Leaving administrators. Shouldn’t new teachers cultivate administrators?

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

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

Of course, that might just be me.

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

Here’s to a new year.

 

 

 

 


Great Moments in Teaching: Or, Browbeating Psychoanalysis

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

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

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

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

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

“I know.”

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

“I know.”

“No, you don’t.”

“I get it.”

“No, you don’t.”

“Yes. I see now.”

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

“OK.”

“YOU DON’T UNDERSTAND HOW TO DO THIS PROBLEM.”

“I know.”

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

“I know.”

“STOP SAYING THAT.”

“I kno…OK.”

“What’s OK?”

“I understand what you’re saying.”

“No, you don’t.”

“OK.”

“No, it’s not OK!”

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

“Listen to me.”

“OK.”

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

“I know…no. OK. I get it.”

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

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

Warren stood. Affect way off, smiling nervously.

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

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

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

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

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

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

“But I asked for help.”

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

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

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

Warren answered readily, “Amplitude. Period.”

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

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

“Which is also the…..”

“Vertical shift.”

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

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

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

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

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

“OK.”

“Really?”

“Um. No. I don’t know how to start.”

I waited. The class waited.

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

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

Pause.

“I could…graph it, maybe?”

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

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

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

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

“Thanks for yelling at me.”

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

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

“Don’t be a zombie.”

“Okay–wait. What’s a zombie?”

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

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

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

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

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

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

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

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

 


GPA and the Ironies of Integration

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

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

 

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

Pause. I’ll wait.

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

Good, you’re caught up.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

I do believe they call this a collective action problem.

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

See why I loathe grades?

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

 

 


Teaching Transformations

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

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

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

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

That looks like this:

transbasepoints

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

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

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

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

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

-5 -3 2

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

transplustwo

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

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

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

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

“Exactly. So let’s try that.”

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

transtableminusn

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

x f(x)
1 1
2 2
3 3

etc.

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

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

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

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

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

“Output!”

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

“f of x!”

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

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

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

transfxminusthree

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

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

tranformationsshifts

 

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

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

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

transcombined

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

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

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

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

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

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

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

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