Differentiating by Assessment

Bless me Father for I have sinned. It’s been four and a half years since my last curriculum article.

My entire teaching career has been spent navigating a student ability gap that got noticeably wider sometime before the pandemic. The strongest kids are getting better, the weaker ones spent middle school not learning a thing because they’d pass anyway. (Thank you, Common Core.)

I am constitutionally incapable of giving an F to kids who show  up and work. Call it a failing if you must. So yes, the gap grew, but I adjusted and got back to moving the unengaged kids forward, building the confidence of the kids willing to play along, and pushing the top kids as far possible, given the ability ranges.

But tests were increasingly a problem. My long-standing approach was to allow all students extra time to finish tests or ask questions. But this was premised on a much smaller gap and by this time the gap is a  canyon. Add to that the fact that cheating is just an enormous issue post-pandemic, so giving kids access to the tests after school and at lunch increases the chance of them getting a camera on the questions, posting it on Discord, using photomath, memorizing the question to ask a classmate, whatever. Once upon a time, kids were just grateful to have the chance for extra time, but lo, those were the days of yore. So I want more kids finishing all or most of the test within a single class, and if they need extra time I want it wrapped up quickly.

To understand the nature of my dilemma: If I build a quiz that my weakest kids (10-20% of the class) can get a high F, D, or C after the entire class period and an hour of additional supported work, the top kids (30-40% of the class) will  finish it in 10 minutes. I’ve given quizzes that the top kids are handing in before I’ve finished handing them out.

That’s just straightforward quizzes. How do you build a cumulative unit test that challenges your best kids that your weakest kids won’t flunk by giving up  20 points? Or takes them eight hours to work through and then flunk with 40 points? How do you make something manageable for your strugglers that isn’t a five minute exercise for your strongest students? 

The idea hit me in November when I was building the unit two test. I’d pulled up last year’s version. It was a good test, but the strongest kids had finished it in 40 minutes, and the weakest students from that class were far better than my 30 weakest students (out of 150) from this year. So I’d be giving a test that the best kids would blitz through, the middle kids would struggle with, and the weakest kids would give up and start randomly bubbling.

As mentioned, I’ve got just the one prep. So instead of this two or three weak students, 10-12 strong students, it’s 40 weak students, 60 strong ones, and 40 midrange.

Never mind the aggravation, it’s just crap, pedagogically speaking. My weaker students need math they can do, build up their confidence. My strong kids need…doubt. Challenge. Something more than the straightforward questions that strugglers need.

Suddenly a familiar thought snuck in: I could create two tests. Not two different versions of the same test. Two tests of entirely different questions on the same topics, one much less challenging than the other.

Familiar because I’ve wanted to do this …well, since before the pandemic, at least five years ago, when I first accepted that the gap wasn’t getting any better. But every other time I dismissed the notion. I usually teach from five to seven different classes a year, so that would be 10 to 14 tests, some of which would only be taken by five or six students. Not worth it.

But hey. I’ve got just the one prep.

Maybe I could make a little lemonade.

Requirements:

1. The tests had to have the same number of questions, the same answer choices, tests the same topics. I build my own scan sheets and I wanted to be able to make each test a different version on the same scantron.
2. Questions had to be on the same topic. I organize my tests by section, with each section having from 4 to 6 questions. The situations presented in each test had to be on the same general topic, and the questions asked had to explore the same knowledge base.
3. My weak students couldn’t get As or Bs on their tests. I couldn’t say “an A on this test is really a C”. The easy test had to be hard enough for my weak students to get a C or B- at best. I wouldn’t have even considered taking this on without confidence in my test development skills, but admitting failure had to be on the table
4. I had to be accurately categorizing students. The “easy test” scores had to be overwhelmingly C or lower with only a few outliers doing well, and no As. The “hard test” could have more variance, as I already suspected some degree of cheating at the high end of my class, but well north of half the students needed to come out of the first test with an 85% of higher. If I met this objective, then I’d decide what to do with the kids who scored too high or too low. But if I had too many students flunking the hard test or acing the easy test, then my whole theory of action was flawed and I had to give up on the idea.

So I made two tests–well, technically, two tests with three versions. Two versions of a challenging test, one I thought would really push the median B student in my class.  One version of a test that I thought would be manageable but tough for the strugglers.

About a hundred students got the hard tests; forty something got the easy test.

Results of the first scan:

• Half of the “hard test” students got an A. Another 10% got a high B, which I counted as an A (this was a tough test). Another 10% got between 75 and 85%, indicating they knew what they were doing and just had a few fixes. The rest tanked badly, scores of 30-60%. More false positives than I expected, but enlightening. These students clearly knew more than my strugglers but there was a clear ability line separating them from the top students. I had a bigger middle than I knew.
• All but two of the weak kids got above 50% on the easy test (two didn’t finish), but only three got 80%, and about 10 got above 70.

I was accurately categorizing students. Well enough at least to continue. And I have, since then, on both tests and quizzes.

The “middle” students needed a mama bear test, and I built a new one that did the job. From that point on I always created three tests, which allows me more flexibility in moving students around. I only build two quizzes (again, I’m talking about difficulty levels; I still build multiple versions of each quiz to cut down on cheating). About 60 kids take the hard test, 40 taking the middle and 40 taking the easy version.

I’m extremely pleased with the results. First, my top kids love the harder tests. A number of students who just thought they were top kids got a dose of humility and started paying more attention. I’ve made math harder for them in, I think, a positive way. They can’t complain that I’m being unreasonable when a fourth of the students are acing the harder tests.

But the real impact has been on my strugglers, who range in motivation from absolutely none to never stops plugging away. They work harder on the tests and quizzes instead of just giving up. They work more in class, finally seeing a link between their effort and achievement. While they all still have difficulty on unit tests and integrating their knowledge, their quiz scores have seen considerable boosts. I’ve actually been able to make the “easy” quizzes more difficult and still see high passage rates.

Regular readers will note a recurring theme of mine: My weaker students are learning more not because I raised standards, but because I lowered them.

Allow me to quote the degenerate wise man, Joe Gideon, once more: Listen. I can’t make you a great dancer. I don’t even know if I can make you a good dancer. But, if you keep trying and don’t quit, I know I can make you a better dancer.

Rigor and high expectations are, forgive me, not the way to make kids better students.

Anyway. It’s working. My weak kids are doing better and my top kids are getting stretched. Score a single lonely point for giving me just one prep.

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Some notes:

Back in 2015, I wrote a lot about my “multiple answer math tests“, as I called them–inaccurately. I still use them, updated them to a scan sheet. Given how essential and permanent these tests are in my work,it’s odd I haven’t written about the format since. To understand this post, you may want to read about the test structure, which is somewhat unusual, I think. I’ve changed the format somewhat since then, with less T/F, but the organization of multiple broad topics with many questions per topic is the same.

While students get the same points for questions, I weight some free response questions with more points on the harder tests. Not always, but sometimes the questions are in entirely different orbits of difficulty. Still, I keep an eye out for students who are getting all the questions right on the easier tests, as it’s a sign they need to move up.

Students have moved up and down. Three or four strong students were coasting off of the easier tests and like their phones way too much. Some figured this out and got serious, others didn’t. Some students have moved permanently from the low test to the mid-range test.

Ever since I found my test on Discord last year I’ve quit returning tests, which makes all this much easier. The top students have realized there are different tests beyond just versions, but after a few questions early on they quit mentioning it. They all seem to like it.

I almost gave up on this post twice because I couldn’t figure out how to display images of the different difficulty levels on the page. Here are some examples:

• Segment Addition with Algebra
• Area and Perimeter
• Angle Pairs and Algebra
• Triangle Centers and Coordinate Geometry

All Geometry, All the Time

Here’s the thing: I only have one prep this year. The whole year.

I have a lot of credentials. I can teach any subject except science.

(OK, language and PE as well, but they’re not real subjects, just things we make kids do because no one wants to acknowledge that school has no impact on physical activity decisions later in life and colleges aren’t even making students know math and grammar anymore, much less a foreign language.)

Put me in a classroom teaching damn near anything and I won’t complain, provided  you don’t put me in a classroom teaching the same damn thing. This is a topic I have opined on at great length ere now (erenow?) so if you want the whys and wherefores, check the links. For now, know that the tremendous depression I’ve been working with this school year has been exacerbated by teaching the same thing over and over and over and over and over again.

If you really want to turn the screws on my agony, make the one class geometry. We can discuss the merits of any math branch in the abstract, but in the world of high school there isn’t a single topic in geometry that’s more important than linear equations and factoring are in algebra 1, algebra 2, and precalc. You could argue, and I often do, that an entire year of math is wasted on geometry when we could just roll Pythagorean theorem into algebra one, go straight into algebra 2, and then devote a whole year to trigonometry (which by definition brings in similar triangles and special rights, the other two essential geometry topics) to give more advanced students more relevant algebraic proofs and a year of applied algebra.

So the year begins with all geometry all the time. Except my prep. Oh, yes, the year started with a prep period.*  It’s really a banner year.  Huge pay cut and that’s not even the worst, which is the gap in the day with nothing to do but walk to Starbucks to kill time and get some cardio.

The grading. Grading 30 tests is fine. Grading 60 tests is manageable. Grading 150 tests all the same is….well, up to now my definition of hell is a noisy, crowded bar with a ska band playing at deafening volume with flashing lights in otherwise total darkness where I can’t move, can’t hear, and can’t talk. I don’t even know what ska is, but it’s for damn sure I wouldn’t like it. Right now, though, noisy ska bar hell has been upgraded to purgatory.

I am nonetheless grateful for this job this year, and every complaint I make is privately cocooned in gratitude. But that doesn’t make it less difficult. I’ll survive. Probably. That said, I was just at a department meeting listening to the others discuss next year’s schedule and someone said well, Ed can cover Lucy’s classes next year when she retires. It’s only the one. And I said one what? And some other person said Lucy only teaches one course. And I said HAHAHAHAHAHAHAHAHAHAHAHAHA no. I will shoot myself. Or retire before Lucy does.

I have scheduled a meeting with the principal to request written confirmation of no fewer than three preps next year.  If for no other reason than it has taken me FIVE MONTHS to finally link every student correctly to their name. Apparently, I mentally organize my students first by topic. Like, this is how the kids in my algebra class sit, this is how the kids in my trig class sit, and then I can start to put the names together. But all the classes are geometry and the names all run together.

You might think I’m being punished for poor performance but no, it’s just part of what happened.

Things are improving. The principal came to my room in December and asked me if I’m available to work my prep and I said yes, please, even if it’s geometry but a merciful boss wouldn’t make it geometry. And it wasn’t. It’s a really sweet algebra support class of strugglers who actually want to pass algebra, ask questions, work hard, and are learning. I learned their names in just a few days. Extra courses don’t come your way unless you’re a really good teacher, so the minute that extra course was available, I didn’t have that horrible hole in my day. Evidence I still got it, performance-wise, and none of this hell is punishment.

I try to look for the positives.

Example: I keep coming across insights and patterns that a teacher with four preps doesn’t stumble on because 35 kids aren’t enough for the information to sink in, whereas teaching 150 students day after day after day after day or grading 150 tests hour after hour after hour after hour is the mental equivalent of a sledgehammer banging the information into your head. When two or three kids of the thirty five make the same mistake you notice but think it’s random. When 15 of the 150 show the same error in thinking, the sledgehammer registers as an action item.

Example: There’s a difference between kids who genuinely have cognitive issues with math (as opposed to general cognitive issues) and kids who’ve been cheating for four years to get a passing grade and have no plan B once they can’t lift someone else’s work or turn in late homework or get extra credit in a class that has neither.

Example: A non-trivial chunk of A students just use multiple choice answers (even when I don’t include the correct result) to backsolve their way into a right answer. Most common case: they can’t solve a system, but just backsolve. Boyo, they can just bugger that for a lark. I’ll squish that dead.

The insights are valuable. But it’s a high price.

One thing I haven’t been doing is leaving work earlier. The last two years, I haven’t left before 6 most nights and usually was working until 7 or later. Given my ridiculously easy schedule, I should be leaving right after school most days, but I’m not, even though I often have nothing to do or just some copies to make.  I should leave. But I sit at my desk, checking twitter, thinking through curriculum, listening to podcasts, rather than just go home. I set that as a New Year’s Resolution: at lunch, ask myself if I know what I’m doing the next day. No? OK, make the worksheet or plan the lesson and be out by 5. Yes? Pack up and leave with the kids. Managed it two days in a row so far. Let’s see how that goes.

Another change, a big one, is something I’ve wanted to do for years: entirely different tests based on ability. This was originally my objective in this article, but I decided to postpone that because in putting all this pain down, I came to a new realization.

My schedule this year is as lousy as I’ve ever had. I am deeply unhappy with it. And yet.

Before this happened, I would not have thought I could tolerate an entire year teaching only one course several times a day. Now it’s clear I’ll not only make the year but also–reluctantly–have to acknowledge some positives that my preferred schedule of endless variety….well, may not provide.

Better yet, I’m realizing that certain elements of teaching are immune to prep concerns. No matter the degree to which my psyche is screaming please god not this same lesson again, I still love my job. I still love my geometry students, even the ones whose names I only mastered three days ago. They all still laugh at my jokes and quite a few of them thank me for emphasizing algebra instead of proofs.  Building curriculum is still a beloved challenge. Explaining is still my go to strength.

It’s still teaching. A job formed in no small part by personality. Some actors probably avoid the theater–do the same thing night after night? No, thanks. Others thrive on the familiarity. There’s a reason some doctors are anesthesiologists and others are work the emergency ward. One of the best teachers I knew taught algebra all day, every day, and literally had a lesson plan for day 117 because she had a lesson plan for every day that never varied.

Me, I need variety in my work day something ferocious. And I don’t have it right now.

But I’m not just slogging through day by day but by golly finding the essence of the job I love and keeping it aflame  despite the wildly infelicitous prep allocation. And just over halfway through the year, it’s clear I’ll survive and even learn some things.

Yay, me.

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*If you’re confused by the difference between one prep and a prep period, you aren’t a teacher and skipped by one of my links. Tsk, task. Read the first couple paragraphs of  Handling Teacher Preps.

Ten Most Read, Ten You Should Read

Eight years ago, on the second anniversary of my blog, I asked, “Am I a hedgehog or a fox?”  Hilarious, that I could ever be so deluded. I understand why my brain thinks itself a hedgehog, but it will just have to cope with reality.

I am a fox. Even at my lowly level of the word, this is a list only a fox could produce.

Ten Most Read Articles:

1. More than Gotcha: Kamala’s Busing Blunder— June 28, 2019
   The only item past its sell date. Most of my work maintains its relevance. But this article, outdated though it is, has a good number of my strengths on display. First, unlike the entire media class, I know how to search for and use relevant history. No one listening should have thought anything other than “that’s bullshit” when she claimed to have been on the frontlines of segregation in Berkeley, CA. But no journalist bothered to do the research. Next, I understood as no one else seemed to that she was essentially coming out in favor of busing.  At a time when most of the media (and all of Twitter) was wowed, I  pointed out she’d almost certainly have to walk that comment back. The other strength: sometimes I really hate people while many other folks are like, man, why is Ed hating on her and then later they go oh, I get it.
   ignore me, lazy way to space
2. Asian Immigrants and What No One Mentions Aloud–October 8, 2013
   I’ve kind of cornered the market in Asian immigrant criticism–not of the people, but of the culture, which I think is very damaging to American education. I wouldn’t make such a big deal out of it if everyone else weren’t determined not to notice. This was the first time I wrote about it.
   ignore me, lazy way to space
   
3. Functions vs. Equations: f(x) is y and more — May 24, 2015
   A math curriculum piece in third place? Blew me away. But as I mentioned, curriculum searches are specific and get through Google’s recency bias, so they’re the one article category that still gets fed via search engines. I keep meaning to revisit this article because it had a very bimodal reaction. Mathy readers who didn’t teach were aggravated and confused by the article and told me I didn’t understand the math. On the other hand, a number of professors on Twitter understood my point  instantly and were very appreciative (and some later commented as well). I think the mathy folks thought I was confusing a system with a function, whereas the professors understood I was using an example of multiple equations that wasn’t a system to show students a difference they hadn’t seen before.
   ignore me, lazy way to space
4. Homework and grades–February 6, 2012
   I have relatively few strong views about what teachers should do. Homework is the exception. Homework is insane. Grades are fraud.
   ignore me, lazy way to space
5. Algebra and the Pointlessness of The Whole Damn Thing— August 19, 2012
   My first really huge piece, and one I’m still quite fond of. It’s getting harder to find data easily; more states are hiding racial and economic distinctions. But if you look at current data, you’ll see the same pattern: poor whites do about as well as non-poor blacks and Hispanics. Been like that for decades.
   ignore me, lazy way to space
6. Philip Dick, Preschool and Schrödinger’s Cat — April 5, 2013
   Canonical Ed on IQ.
   ignore me, lazy way to space
7. Binomial Multiplication and Factoring Trinomials with The Rectangle— September 14, 2012
   Another curriculum piece. I took a long time to make sure the figures and explanations were thorough. I hope other teachers get good use from it. Still the best way to teach factoring, even if your kids don’t use it.
   ignore me, lazy way to space
8. The myth of “they weren’t ever taught….”— July 1, 2012
   This is one of my favorite pieces. It’s all true, still. Every word. And new teachers have to come to grips with it every year.
   ignore me, lazy way to space
9. The SAT is Corrupt. No One Wants to Know.–December 31, 2014
   I am adamantly opposed to grades-based college admissions. But the College Board is corrupt. The international SAT is corrupt. And they’ve changed it in ways to make it far less useful, all in the hopes of ending the score gap, which was never going to happen.
   ignore me, lazy way to space
10. The Gap in the GRE–January 28, 2012
    Another of my favorite pieces that asks a very good question: why are genuine high achievers in verbal tests so less frequent than in math tests? Note that in the intervening years, the College Board and the ETS have eliminated all the verbal difficulty in the SAT and the GRE.

So there’s my ten most popular.

Then I just looked over all my articles and looked for favorites that also captured my zeitgeist (can people have zeitgeist?). I was particularly looking for self-contained articles–a lot of time I go down one rabbit hole and then get to the main point. (Yes, I’m thinking of those for my rewrite plans.) I also wanted a good sample.

1. Teacher Quality Pseudofacts, Part II–January 15, 2012
   This is a top 20 all-time post and was a steady performer for years. I almost didn’t include it; today it seems kind of old hat. But in fairness, that’s like saying 1933’s 42nd Street is cliché because it uses all the old tropes about movie musicals. It didn’t use them. It invented them. When I wrote this article, it was common wisdom that teachers were low-skilled, low-quality, and not very bright. Only the terminally uninformed, the amateurs and the hacks,  have made that claim in four or five years.  I like to think Pseudofacts has had something to do with that change,  because of the very easily found data I brought to light.
   ignore me, lazy way to space
2. The false god of elementary school test scores–July 30, 2012
   Another one I almost didn’t include because it definitely has the rabbit hole problem about Rocket Ship at the beginning. However, like Pseudofacts, it’s an early example of my actually looking at readily available information and pointing out the obvious. Plus, great title.
   ignore me, lazy way to space
3. The Fallacy at the Heart of All Reform–September 7, 2012
   I wrote a history of modern education reform throughout much of 2020-21. This was a history of earlier policy. But the definition of fallacy I include here holds for the entire era.
   ignore me, lazy way to space
4. The Day of Three Miracles— April 28, 2015
   I don’t often talk about colleagues, mainly because for years my relationship with them was….fraught. Not bad, just…there. But this is not only a colleague story, it captures a conundrum that few people in education policy seem to understand. Access or rigor. Not both.
   ignore me, lazy way to space
5. Citizens, Not Americans— June 16, 2016
   I love this piece. By the way, Dwayne is married, has a kid, and is in the military. Abdul went to a top tier school and majored in pharmacy, and when he told me I want “Gack!” and he said “yeah, I know. Stupid move.” and now he’s getting an MA in nurse practitioner, or whatever it’s called. Haven’t heard from Chuy. Wing and Benny still teach. One of them is now department chair, and I had a lot to do with it.
   ignore me, lazy way to space
6. “Get Out” a scathing satire? Get Out.–January 22, 2018
   I love movies, and I know as much about American diversity as anyone in the country, and I think this is a terrific review that isn’t at all what you’ll expect.
   ignore me, lazy way to space
7. Algebra 2, the Gateway Course–January 28, 2018
   Another story about colleagues, students, and really stupid education policy.
   ignore me, lazy way to space
8. Making Rob Long Uncomfortable–December 24, 2018
   Silly title, but you can listen to the podcast and see what I mean. It’s well-written, and captures a certain mindset among the centrist conservative punditocracy. As I wrote: “You could practically hear Rob’s toenails shrieking against the tiles as he braked to a stop.  This was not the conversation he’d signed up for. He was there to lightly mock feminists and social justice nuts, not crack witty, on-the-nose jokes with Heather about the racial skills deficit.”
   ignore me, lazy way to space
   
9. The Students of My Christmas Present— December 25, 2018
   I don’t often get sentimental. And I’ve put up Christmas trees most years since.
   ignore me, lazy way to space
10. Idiosyncratic Explanations for Teacher Shortages–May 31, 2019
    Here I raise an issue that seems quite obvious, but isn’t. We have thousands, if not millions, of unemployed PhDs who will never get a tenured job and work as poorly paid adjuncts. Why don’t they become teachers? After all, everyone says we need smarter teachers, right? There’s a cognitive dissonance revealed in the fact that everyone understands that a poorly paid PhD is acting rationally in refusing to take a better-paid, more secure job with great benefits.
    ignore me, lazy way to space
    I thought I was done, but 2017 spoke up, really pissed off. Why nothing? I tried reassurance. It was nothing personal. I wrote some good shit that year. Besides, 2020 and 2021 aren’t represented either. But it would not be assuaged and as my mother isn’t doing well, and this is a not only an ode to American schools but also a lovely story about my mom, an extra…
    ignore me, lazy way to space
11. What the Public Means by “Public Education”–March 19, 2017
    When education reformers wonder why everything went wrong, they should think about the thoughts expressed here.

Thanks for reading.

What I Learned: Year 2

Year 2 was all algebra, all the time. Few things are as draining—or as revealing of the utter emptiness of our educational policy—as teaching algebra I in high school.

The epiphany moment: Realizing that kids could distribute and combine terms but lack the skills to work the steps in combination. I noted this in first year, and just the hint of its return in the fall led to the development of the Multi-Step Equation Drill Down and the mantra “Distribute-Combine-Isolate”.

Technically, this process is called a “reteach”, but another thing I learned that year is that done well, a reteach is much more than reviewing the same material. Once I realized, less than a month after covering equations, that most of the kids could solve 3(x+5)=18 but not 3x+2(x-7) + 6 = 6x – 2, I didn’t drop everything. I finished up with linear equations and considered the best way to move on this. The problem wasn’t the individual steps, but the confusion that resulted when using the processes in combination. So why reteach? Instead, I just explained to the kids where their confusion started, by using a problem set similar to the one above.

When done properly, the kids are totally on board. The minute I put the two problems up and demonstrated common mistakes, I had their attention because they were bothered by it, too. Which is a bit weird, because trust me, I can list a million things that kids don’t get about algebra. And yet, the kids know the difference between something they completely don’t understand and something they vaguely feel they should understand, but don’t. The confusion they felt during a complicated multistep equation fell firmly into the latter area.

I learned that even unmotivated students have this, dare I say, inchoate yearning to having confusing areas clarified. Spotting these areas and drilling down gives the students faith in their teacher. They can see the teacher is paying attention to mistakes, distinguishing between utter confusion and “man, I almost got this but sojmething’s wrong”.

And the performance pop from these clarifications is huge. The majority of mid-level ability kids never lost the DCI clarity. I attribute this to the time spent in the darkness, followed by the blazing light of knowledge. Or maybe I just pick my moments well.

Never fear, most of them still struggled with “is this slope negative or positive?” Alas, that confusion has nothing to do with procedural confusion.

Other things I learned:

• Data Collection and Analysis: as a long-time analyst, I knew how to evaluate the data, but this was the first year I had the opportunity to gather and compare the data.
• I always differentiate instruction, but in Year 2, I ran 4 different lesson plans at all times. This led me to realize that teachers can pick their discomfort. For me, watching kids be either lost or bored leads to far greater stress than the work involved in running four classes simultaneously.
• Low ability students: In algebra I, you often get kids who literally do no work, or who scribble frantically but clearly have no clue. At the end of the first semester, I put them on a contract. I would pass them if they demonstrated competency in five high-leverage areas: linear equations, quadratics, multistep equations, and simple systems.

  I didn’t need to use it in Year 3, but I strongly recommend this approach. Motivation among my low ability kids increased dramatically. Even the kids who ultimately failed due to attendance issues saw a big bump in their ability to solve multi-step equations and factor quadratics.

Some earlier posts that expand on these issues: Teaching Algebra I and Teaching Algebra, or Banging Your Head with a Whiteboard

Teaching Humanities, History of Elizabethan Theater, (III)

Days 1 and 2, and 3 and 4.

Day 4 part I: Recreating History
Students use their notes and documents from Day 3 to sketch the details of an Elizabethan era theater. Fishbowl discussion: what did they sketch for seats? What about curtains? What did “backstage” look like? How do historians “fill in the blanks” when they don’t have primary evidence to point them in the right direction?

After the discussion, students took a virtual tour of the Globe theater. How would someone go about establishing the source material and accuracy of this tour?

No deliverable here. I just wanted the kids to grasp the choices involved in recreating history, whether it be for a book, a movie, or simply an image. I thought something familiar and specific would give them a better idea of how many thousands of decisions are involved in filling in those blanks. And while I can offer no tangible proof that this worked, I can say that every student had that “aha” moment, when they realized how much they didn’t know, and how every decision in a recreation can further affect our general understanding of history. The discussions were active and everyone was engaged; we had some great exchanges. One of my delightful ditzes suddenly realized she couldn’t assume that there would be bathroom stalls.

“But….the plays could be hours. What would they do? Go back outside? What if they were way up front?”

“Maybe they had pots,” offered a classmate.

I broke in with a brief history of the chamberpots.

Another student’s eyes widened. “Hey, maybe that’s where the phrase comes from–they didn’t have a”

“pot to pee in!” the class choruses.

“What about the actors, though? They had to have at least one bathroom. They didn’t even need girls’ bathrooms, right?”

One of my top historians pointed out, “But look, they didn’t even have running water back then. Did they even have toilets?”

“When did they get toilets?”

I gave them the story of Thomas Crapper and ended the segment during the ensuing hilarity.

Underneath all the fun, they really did get an inkling of the challenges involved in understanding the past. And, of course, some potty jokes.

Day 4 part II: Sonnets
Lecture on the sonnet, including its history and the two major styles (Petrarchan and Shakespearean). Students listen to five sonnets written from Shakespearean to modern times, and write responses to each. They identify the link between one of the sonnets and a modern song.

I am not a cut-and-dried planner. I’d always known that this unit would have a sonnets lesson. I’d vaguely thought of them reading the poems, which seemed unsatisfactory but I figured something would occur to me. The “something” waited until 30 minutes before class time, when I suddenly realized how much the sound of the sonnets would add to the experience, and so spent a frenetic half hour hunting down them all (mostly on youtube).

After all that, though, it went beautifully.

Day 5 and 6: Shakespeare in Love

They got all the jokes. They didn’t giggle at the sex scenes. They were engrossed by the story. They enjoyed the movie, understood the movie, and were completely aware that enjoyment and understanding came from their new content knowledge. I could tell.

So if I have a disappointment, it’s only that I would have preferred they’d be blown away by the movie. I am a film propagandist who shows movies that students would never think of watching and are nonetheless enthralled. I’m very good at this, so when I could tell that they just enjoyed the movie, it felt like a letdown.

In retrospect, though, Shakespeare in Love isn’t really a propaganda film; I’d never deliberately try to sell it to early teens. In this case, it was curriculum. And from that perspective, it worked beautifully.

Note: My kids had all been approved for R-Rated films for health class. I doubt most teachers could get away with showing SiL to freshmen otherwise.

Day 7: Content Knowledge and Art
Students write an essay on this prompt: “To what extent did content knowledge help you appreciate Shakespeare in Love?”

In fact, I had seen the answer in their faces as they watched the movie, but I wanted them to think about it.

Teaching Humanities, History of Elizabethan Theater (II)

See Days 1 and 2, if you’re interested.

Day 3 Part I: Elizabethan Theater Who’s Who
Students, grouped roughly by ability and content knowledge, were given different readings about key figures in the era. After reading, taking notes, and discussing, they created posters about their subject(s). The lesson ended with a “gallery walk” in which the students take notes about the key figures who weren’t part of their reading.

Yes, posters. Given my druthers, I’d have given them all four readings and 25 minutes to peruse, followed by a class discussion. But there you go. Well over half of all students I’ve worked with love making posters, and I always commiserate with the ones who don’t.

This lesson uses a form of “jigsawing”; as I’ve mentioned before, while most trendy math teaching techniques are hooey, I’ve become fond of more than a few used in history and English. Jigsawing is a terrific way to provide content by ability group. In this case, it allowed me to make sure that the students focused in on the content area most appropriate to their abilities.

So my weakest kids got the Shakespeare reading, because I wanted them to get a solid grasp on who he was, what he’d written, and some important quotes. If that was all they got from the exercise, that was a good get. Next group of kids up, I figured would be able to get the key ideas about Shakespeare from the poster, so I had them focus on the other Elizabethan playwrights. That way, they’d get some solid new information on Marlowe, Kidd, and Webster, whilst still picking up the key facts about Shakespeare. Again, if that was all they got, terrific.

Next group up, I knew, would be interested in learning about the new playwrights and would pick up the content from the poster–so they got the actors. The strongest group got Henslowe and Tilney, and the responsibility of figuring out what it was they did.

I know I’ve said this before, but I really wish I’d had an android back then, since the posters were stunningly good—not just in terms of artistic value, but in terms of how the students incorporated the readings into their posters.

In the gallery walk, each group took turns explaining their subject to the others, so they could take notes. I quizzed everyone it later, but I can’t find that document. They all knew enough to laugh at John Webster as the young boy with a violence fetish, and several were sad knowing that Marlowe must die, so the content definitely filtered in.

Day 3, Part II: Theater and its Impact on the Economy
Students reviewed pages of Philip Henslowe’s diary. What evidence did this primary document provide to support the claim that Henslowe was a producer who worked with some of the key playwrights of the day?

Using diary entries, write an essay supporting or contradicting this assertion: “The rise of the theatrical industry probably had a positive impact on London’s economy.”

I’m not sure if the pages in the attachment the pages I actually used. I spent several hours looking for three or four pages that would give them a wealth of evidence for both parts.

This was definitely an activity I designed primarily for the stronger students, and their essays showed they appreciated the challenge. However, all the students were interested. Much chortling when they discovered how much time Henslowe spent with his lawyers and on “copywrighte”.

Day 4: History In Motion
Reading and lecture on the constantly changing nature of “history”. Not only does it keep on building up, but we keep discovering more about our past. Students learn of the Swan drawing, by Johanes de Witt, which wasn’t discovered until 300 years later. Then, 30 years ago, a routine building excavation led to the discovery of Philip Henslowe’s Rose Theatre–and then the Globe was quickly located as well. But is there a cost to these discoveries? Students discuss the impact of living in a historical site.

This is the third really cool “primary” document (well, a copy of one) that I found for this unit, and I spent much of my own time researching it because it’s exactly the sort of tidbit I find fascinating.

What we know of the London theatres of Shakespeare’s age is, to a disproportionately large extent, due to the records such as diary entries left by tourists….Two of these tourists are Johannes de Witt and Aernout van Buchell, friends from the city of Utrecht, the Netherlands….To them we owe the best piece of visual evidence of what an Elizabethan theatre looked like on the inside: the sketch of the Swan theatre on Bankside, along with a brief Latin text describing the London theatre scene….First of all, it is unclear precisely what each of the two friends contributed. We do know that de Witt travelled to London, probably at some time between 1596 and 1598, and visited some theatres there, including the Swan; at some later point, he apparently gave (or sent) his sketch and written ‘Observations’ to his friend van Buchell, who copied the drawing as well as the text–or part of it. It is van Buchell’s notebook that now survives in the Utrecht University Library……There is no evidence that de Witt’s information had an immediate impact on his own culture. Sure enough, van Buchell made a copy of (some of) the information, but he never published it. Not that he was merely a collector of information for his own amusement; the documents he gathered, the facts and impressions he noted down, were made available to fellow scholars on demand for future generations. But for practical purposes, van Buchell’s main function as a go-between lies in transmitting de Witt’s sketch and observations to posterity. For some 300 years, the notebook gathered dust in libraries, until it was discovered by the German scholar Karl Theodore Gaedertz in 1888.

(Source: Renaissance go-betweens:
cultural exchange in early modern Europe (page 79-81)

So. For centuries, we had no actual image of an Elizabethan theater. Then, for a few generations, the de Witt drawing was all we had, until The Rose was discovered, and once they had that location pinpointed, finding the Globe was pretty easy (hell, my kids found them using a 400 year old map). The past just won’t have the decency to sit still.

And yet, what of the developers? Lucky them! Their business plans had to be postponed; they had to incur additional expense to protect the Rose “until a future date”. Their building had to be suspended over the excavation. and what, exactly, will that do to the retail value of their property? Good, if the rich movie stars propose to buy it out. Bad, if the efforts never get far enough to the purchase level but stay at the annoyance level.

I told the kids about a hospital in my area moving from its old space to an empty lot that held the last orchard in an area that once was devoted to farming. I mean, couldn’t they have moved it anywhere? This orchard had somehow lasted that long, couldn’t we preserve it as the last little piece of heritage in the area?

On the other hand, the hospital can’t move there, it moves to another town, and bye bye jobs and property taxes. Discuss the degree to which we should interfere with business development in order to protect our past.

The essays were great. Heartless guttersnipes all. “Hey, it’s only trees!” “They lasted for 400 years without the theaters. What’s the big deal?”

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Teaching Humanities, History of Elizabethan Theater (I)

Writing up the Twelfth Night unit reminded me how much fun I’d had and how thoroughly the students had learned the material. I need no reminding with my Elizabethan history unit, which is still the single finest two-week period I’ve ever had as a teacher.

Learning Objective: Students can discuss the milestones of the transition from mysteries to plays, and provide the rationale for the development of physical theaters. They can identify the tensions involved when archaelogical remains don’t have the courtesy to become inactive, but just keep inconsiderately piling on history unto the present. They can identify the key Elizabethan playwrights and theater personnel. They know how many lines are in a sonnet.

Secondary Objective: They would understand every in-joke in Shakespeare in Love, which they’d watch at the end of the unit.

Day 1: Lecture on the development history of Western European theater, which disappeared entirely after the fall of the Roman Empire for over five hundred years, until it returned again in the form of “mysteries”, the performance of Biblical scenes, and slowly developed into the morality plays of the late Middle Ages. During the Renaissance era, the morality plays developed slowly into plays written more purely for entertainment, and thus were born both the acting profession and the theater (which allowed the actors to collect money from their audience). The lecture covered the development of the first theater (started by James Burbage, as we all know from SiL) through to the great razing of the theaters in the mid-1600s.

I can’t find my lecture notes, but the gist can be derived from the handout:
It must be said: I give a great lecture, and am particularly good, I think, at making the kids see developments as interesting when they’d never before given them a thought. Movies are a huge part of their lives, and before movies there were plays. But how did plays start? I often test attention by popping in questions during the talk, and the kids were all right there with me.

“So you had these acting troupes traveling from town to town, acting out these morality plays to big audiences, whereas before, the churches themselves put on the mysteries. How would that change things, Meg?”

“Well, the church is religious, and the actors aren’t.”

“Okay, and how would that matter? Renee, your hand is up.”

“The church probably had priests who acted!”

“Okay, and what does that mean? Think about the difference between a priest and and actor. Isaac?”

“The actors would need money, right? The Church had money already.”

Ian said, “Yes, they’d have to get money. How do you get money if it’s just a big crowd?”

“You pass the hat,” said Dom, “like they do in the movies.”

“You can’t pass the hat if you’ve got hundreds of people in the audience,” said Kayla.

“And what if no one wants to pay? How would they make money?” asked Sheena.

“Wow. You all are taking my lecture away from me. You’re right about the differences, the money, and passing the hat works well as a voluntary payment for small groups, but if actors were going to make a business of it, it’d be really convenient to have, oh, I don’t know, maybe a building? To let people in after they’ve paid? Otherwise known as….”

“Theaters!” they chorused.

The handouts, as well as their work over the next week, confirmed that this stuck with them.

Day 2: Using Primary Documents
Students had to map out the theaters of London during the Elizabethan era, using the “Agas Map” (Civitas Londinium). They had a handout with descriptions of theater locations, and they had to use the big map as well as the map on their handout to place all the theaters. When they finished, they checked their work against the placement based on current historical knowledge.

I printed out the Agas Map of London in 32 full page sections, which I then glued together. And trust me, crafts ain’t my thang, usually. But it was worth it, if only for the great visual. It’s a gorgeous map.

These were the clues:

Here’s the map they used to check their work:

Doing it again, I’d have magnifying glasses to read the Agas map more clearly. Still, it was a successful day. The students, working in pairs, found all the theaters and enjoyed the scavenger hunt aspect.

To be continued….

Test Pattern

The polynomial quiz results were fantastic. Over half the class got a perfect score; no one outright failed–that is, all of the students did at least one of the four problems correctly.

Which brings up something that I’ve been bothered by for a while: around a dozen of my algebra II students are nailing the quizzes and failing the tests, or close to it.

All of the students showing this pattern are hard workers. Some have shown strong math skills; a few of them struggle but patiently work things out. I know they are discouraged by their low test scores, and I’ve been encouraging, but puzzled.

Certainly, I expect some fall-off between quizzes and tests. My quizzes are directly on point with no surprises. I give problems exactly like the ones we’ve been working in class. My tests slant off sideways and crisscross (my geometry students constantly whine about this). Some of the strugglers might get flummoxed when the problems don’t appear in exactly the same form, sure. But others of these students are well beyond that. They work the toughest problems of the day with minimal guidance; when they have questions, they are logical and structured.

I don’t know what’s going on, but since all the students in question did excellent work on the last quiz, I decided it was time to step up to the problem. Rather than try to set up time with them individually, I just made an announcement in class when I returned the quiz.

“If you are a student who is looking down at an A on this quiz, but got a C or worse on the last test, I want you to know that I’ve noticed this pattern–great on quizzes, near-disaster on tests. So here’s the deal: I will be adjusting your grade on the next progress report. It’s now clear to me that something about the test is causing you problems, not the math itself.”

“BUT. You must schedule time to come in after our next test and work quietly, either at lunch or after school, so I can watch you and see what’s going on when you take tests. I’ve invited everyone to do that after every test, but for you guys, it’s mandatory if you want a grade adjustment. You can’t keep going through life tanking important tests when it’s clear you know more.”

In all three classes, I scanned the room as I said this, and every one of those dozen students looked up in hope and relief.

Which makes me a bit sad. It’s not like they haven’t had this option all along. Why don’t they take advantage of it? Why wait until I mandate it?

But it also reminds me that no matter how many times I think I’ve made it clear that my door is open, help is here, even on tests—I haven’t said it enough.

Discovery Doesn’t Work

I had trouble in ed school because (well, at least in my view of it) I openly disdained the primary tenets of progressive education. I am pro-tracking, anti-constructivist, and pro-testing, all of which put me at odds with progressives. Here is the irony: I mention often that I am a squishy teacher (squishy=touchy feely). I am not just squishy for a math teacher, I’m the squishiest damn math teacher from my cohort at the elite, relatively progressive ed school that made my life very difficult. My supervisor, who knew me first as a student in a curriculum class, was genuinely shocked to learn that I didn’t talk at my kids in lecture form for 45 minutes or more, given my oft-expressed disagreement with discovery. Even my lectures are more classroom back and forth than me yammering for minutes on end. (In fact, my teaching style did much to save me at ed school, but that’s a different story.)

Here is what I mean by squishy: My kids sit in groups, not rows. When I set them to practicing, which is usually 20-35 minutes of class, they are allowed to work independently, in pairs, or as a group of four. I often use manipulatives to demonstrate important math facts. My explanations are, god help me, “accessible”. I don’t just identify the opposite, adjacent, and hypotenuse and then lay out the ratios. No, I’ve been mentioning opposite, adjacent and hypotenuse for weeks, whenever I talked about special rights. I introduce trig by drawing a line with a rise of 4, a run of 3, and demonstrate how every right triangle made in which one leg is 3 and the other 4 (that is, have a “slope” of .75) must have the same angle forming it. I spend a great deal of time trying to think of a way to help kids file away knowledge under images, concepts, pictures, anything that will help them access the right method for the problem or subject at hand. (For more info, see How I Teach and The Virtues of Last Minute Planning.)

However, I am not in any sense a constructivist as progressive educators use it. I use discovery as illustration, not learning method. I don’t let kids puzzle over a situation and see if they can “construct” meaning. I explain, give specific instructions, and by god, my classroom is teacher centered. I am the sage on stage, baby. And that’s why I got in trouble in ed school, despite my highly accessible, extremely concept-oriented teaching style; I routinely argued against constructivist philosophy, and emphasized the importance of telling kids what to do.

Anyway. I was incredibly excited to read an article that openly states the obvious: Putting Students on the Path to Learning: The Case for Guided Instruction. This article is just so dead on right. To pick one of many great excerpts–click to enlarge, but why can’t I copy text from pdf files any more?:

Yes. Low ability kids like discovery; it is less work for them, yet they feel they are doing something important—but in fact, they aren’t learning very much. High ability kids tend to be “for chrissake, give me the algorithm”, when they would be better off puzzling through the math for themselves.

The article talks about the importance of worked-out examples. I read the article this morning and had a worked out example on the board the same day—step by step factoring of a quadratic. Here’s the weird thing: the kids who need the help with factoring had to be prompted to use the example, but the kids who got factoring were clamoring for worked examples in the area they had trouble with.

This would be a great thing for notebooks. But how do you get the kids who need help to keep the notebooks?

Great article, that changed my teaching immediately. How often does that happen?

Teaching Polynomials

When I met with my new supervisor in ed school (the second one), I told her that I didn’t feel like I was introducing topics well. She was extremely supportive, and it became one of our favorite discussion points. How do you introduce a math topic? And in my two plus years of teaching, I think I’ve become good at it–particularly in first year algebra, which I’ve taught more than any other. When I taught CPM Geometry, I hated everything about it except the way it introduced tangents (as a slope). I often spend several days mulling a good intro, and have been known to toss in a few days of review just to get my story right.

The story is usually a problem the kids can understand—and understand that they don’t have the tools to solve it. Or sometimes it’s a parallel. Either way, I try to give them an image, a reference, a bucket. Maybe it will help them trigger memories, because retention is a huge issue in teaching math.

It’s weird, the quick descriptors that teachers use. When I say I’m “teaching polynomials” in Algebra II, any math teacher knows I’m teaching everything but quadratics in their binomial/trinomial form, since quadratics is its own unit. Teaching polynomials means the kids are learning polynomial multiplication, polynomial division, synthetic division, maybe some binomial expansion, certainly some brute force factoring In general, the polynomials unit in Algebra II doesn’t have any obvious purpose other than to prepare the kids for pre-calculus. It’s just “let’s learn how to manipulate polynomials to no immediate purpose”. And that makes the intro tough.

Over half my students will not go on to pre-calc next year. Some will be taking Algebra II again, either with or without Trig. Others will be going into remedial college classes. So even leaving aside the intro, how do I help the kids make sense of this? I don’t care if they can expound on function notation or binomial expansion, but I do want to be sure they know the difference between multiplying two trinomials vs. two binomials, and when to factor. And for god’s sakes, I want them to know that they can’t “cancel out” the x² term when presented with a rational expression.

I started the unit by explaining the preparatory nature of some of this—that they won’t really see how it’s used until pre-calc, that they just need to recognize these equations and know what to do. Multiplication, they’ve been doing for a while. Factoring, too. But then there’s division proper, which most of them won’t use again. I thought about not covering it, until I realized that I could use the lessons as a way to get them to think about division and factoring.

And so, the introduction:

I don’t present this all at once. I start with the first fraction, then ask what could we do. Someone will reliably say “reduce it”, and so we’ll reduce it. I then introduce the term “relatively prime”.

So then, I say, we do the same thing with variables and fractions, and we go through this step by step:

This isn’t the actual whiteboard example; I just wrote it up and took a picture. But it’s the idea.

And it worked. It gave the kids a great point of reference and most of them were able to divide a simple polynomial on a quiz a few days later.

I’m trying to build on that now. Thus far, I’ve taught them two forms of division and factoring. Ideally, they should be able to identify when to factor, when to divide and when, please, synthetic substitution is a good idea. So I went through the pros and cons of each:

1. Factoring: the default. Pros: It’s fun to cancel out the common terms. In a test situation, you can pretty much assume that the terms will factor. Cons: Only works with first and second degree polynomials. After that, it’s brute force. Limited: if you can’t factor, you can’t. Nothing to tweak.
2. Division: you don’t have to use it much, unless asked. Pros: It’s the easiest to relate to–works just like number division (most know this already. Most). It’s extremely flexible, works in every situation. Cons: you don’t have to use it much.
3. Synthetic division: you never really have to use it. Pros: Incredibly useful for evaluating terms, which is what we’ve been using it for. Quickest method to find factors in higher order polynomials. Cons: It’s the most difficult to learn. Unless you use it often, it’s easy to forget. You have to know what it means in order to find it meaningful.

So they all copied it down. Did they get it? I gave them a quiz—a pop quiz, no less.

And with one exception, they did pretty well.

Question 2. It got to them. First, they saw the division sign. So they divide, right? No! Don’t they remember? “Division is…..” I prompt. “Oh, yeah, you flip it!” They flipped it. But then they multiplied, which made sense because they were being tested on that too, right?

Argggghhhh.

Still, it was a good quiz. Once I reminded them, they worked it correctly. Few misconceptions. I’ll need another week to beat in the triggers to tell them what to do when. But it’s working.

I think.