Algebra Student Distribution–An Example

I thought I’d provide some data that I never see in the discussions about student achievement or teacher evaluations that seems to me to be highly relevant.

Last year, I taught algebra I. I taught more algebra I than any other teacher: 4 sections. Three teachers had three sections; one teacher had one (and won’t be part of this data collection). One of my classes was intervention, a double period, and one of the other teachers who had three segments had that same double.

It became apparent to me and another teacher, listening to another colleague, that this third teacher had dramatically different students than we did. Or at least we hoped she did, because she mentioned casually once that she’d had an extra day and so introduced point-slope to the kids, and they got it fine. The other teacher and I whispered about that conversation later—would your kids learn point slope in a day? No, we agreed, our kids weren’t even clear about slope intercept, and wouldn’t be for a while. So we decided, tentatively, that the other teacher had much more advanced students than we did. Of course, we were both worried that she was just that much better a teacher than we were, which meant we were utterly disastrous. Because if she had the same population of kids we did, and was able to teach them point slope in a day, then we were very bad teachers indeed.

We weren’t resentful, we weren’t mad at the teacher at all—she’s an excellent teacher who always has time to answer questions. We were just confused—was there that much difference in our populations, or were we that horrible that we couldn’t get our kids to learn?

So it was a matter of considerable import to me to determine whether my kids were really that much weaker, or if I was just that rotten a teacher. We had some common assessments that year that reassured me—I was slightly below them, on average, but not unusually considering I was teaching intervention kids, and when I took those kids out of the results, I was roughly the same. And we were constantly hearing that our algebra scores were way above previous years, so given I taught about a fourth of all the students, I couldn’t be doing that bad.

Still, I wondered. And so, when the state test results came out—where we’d improved on last year’s pass rate by over 10 percentage points—I delved into the assessments database and pulled out all the data for last year and looked at the allocation by test scores.

A couple points to remember as you look at these numbers:

• They do not include all students. About 100 or more of the nearly 500 algebra students don’t have test scores from the previous year. Some are excellent, some are weak. Each teacher in the graph below is missing approximately 20-25 students.
• Over 75% of our algebra students took algebra the year before—that is, they were repeaters. Until this year, freshmen students were put in an algebra class if they hadn’t passed or taken the district test. It didn’t matter if they’d scored advanced or proficient on the state test. They could opt out, but they had to sign a document. Consequently, we had some very strong students repeating algebra. This year, the school discontinued that policy.
• Some small percentage of our students took pre-algebra. With some exceptions, these students were extremely weak.

Because of the pre-algebra issue, I broke down the scores twice. First up, the population of students who have test scores from last year, broken down by state score. This chart makes no distinction between students who just finished pre-algebra and those who are repeating algebra for the first, second, or third time—because the state reports don’t care, either.

I also put the average incoming scores of the students, both with and without pre-algebra students (that is, students who took pre-algebra, not algebra, the previous year).

Brief comments:

• If you paid attention above, you know that I’m Teacher 3, since that’s the teacher with the most students. Teacher 1 is who I nervously conferred with about the teacher who seemed to have entirely different students—who is Teacher 4. Teacher 2 is the teacher who, like me, has an intervention class.
• While Teacher 1 and I have an equally high total of FBB and BB students, my students’ average score is considerably lower than all the teachers. T2, who had the other intervention class, has a low average despite having fewer total students in the FBB and BB category.
• Teacher 1 and I were, apparently, correct. Teacher 4 had very different students than we did. Her average students’ score was a full 23 points higher than mine and 12-14 points higher than Teachers 1 and 2.
• One of the reasons I had so many weak students is that I told any student with an Advanced or Proficient state test score that, in my opinion, they should go on to Geometry. I followed the 14 kids that took this advice, and they all did well. The classes are then “rebalanced”, after all the adjustments happened. I picked up a number of wonderful kids, but only two of them were particularly good at Algebra (and neither of them had test scores from last year).
• These were the student populations we had at the beginning of the year. This graph is after the rebalancing, so almost no students switched teachers after this point. The only changes after this point were arrivals and departures.

Now, here’s the same picture except I pulled out the students who took pre-algebra. This will show graphically what percentage of each teacher’s population had taken pre-algebra last year.

Brief comments:

• I had the most pre-algebra students, but the percentage differences aren’t terribly large.
• Notice that the bulk of my pre-algebra students are in the BB category (see how the block shrinks from the first to second graph). I also had three pre-algebras with Basic scores, two of whom left and the third slept through class every day. T1 pick up a good chunk of FBB students in pre-algebra (although she had the only “Proficient” pre-algebra student with a test score).
• This again emphasizes that T1 and I were not wrong to wonder, because we not only had the weakest students, but a good chunk of our weakest students had been weak at pre-algebra, and hadn’t even had algebra before.

For those familiar with low ability algebra student populations, you might be wondering why I didn’t break it down by age. Answer: too much work and didn’t tell me all that much meaningful. About 80 of the kids are sophomores, juniors or seniors. These kids don’t fall easily into one category, but three. Some of them have good abilities and just goofed off too much. In some cases they passed algebra in eighth grade but missed taking the district test (see above) and then goofed around, didn’t do homework, and flunked their ninth grade year (Don’t get me started.) These kids were usually pretty good—they knew they’d screwed up and were determined to move on. Then there were the kids who passively sat and did nothing because they didn’t understand what was going on, and either didn’t understand or didn’t care that just doing some work would get them moved on. These kids have very low skills, but if you can reach them it’s satisfying. And then, of course, there are the kids who are simply waiting out the time until they go to alternative high school, and are behavior nightmares. Teachers 1, 3, and 4 all had around 26 non-freshman. Teacher 2 had about 12.

I also have the results from our state tests, comparing before and after for the students that have two years of test scores. In my opinion, all the teachers did well, but I won’t post those numbers without the teachers’ consent. I will say that, based largely on our algebra performance, our school had an excellent year.

But these graphs, in and of themselves, say quite a bit and attack many assumptions about what should and should not be reasonable expectations for student outcomes. For example, we are expected to move a large percentage of our students to Advanced or Proficient, something that I will now assert is only possible if a good chunk of your students enter with Proficient or Advanced abilities in the first place. I’d also add that the graphs seem to have something relevant to say in terms of student equity because seriously, what strong student would want to be in the class with 65% students that struggle with basic math facts? Oh, and yes—they just might have some relevance to the value-added conversation, since how is it fair to be compared on needle movement when some of your students have taken algebra three times and still failed, or just finished pre-algebra without basic abilities?

My test numbers as published in the reports looked….eh. Not disastrous, but nothing that says “Wow, this teacher sure knows how to raise test scores!” But I feel fine about my numbers. First, there’s the certainty that comes from the logic of “I taught the most algebra students, we had a great year in algebra, therefore whatever else is true, I couldn’t have done that badly.” But also, because I took them and broke them down by incoming category and compared them to the other teachers on the same basis, I have confirmable data that my students were far weaker, andI know that my students improved reasonably compared to the other teachers. To the extent they didn’t improve as much, I have to wonder how much is my teaching, and how much of it is the fact that I had to actively help far more students up the hill?

I would like to see more schools give teachers data like this but alas, it would require caring about incoming student ability. Administrators and reformers get offended if you bring the subject up—are you suggesting all students can’t achieve? Progressives hate tests and hold they’re irrelevant, so they aren’t going to argue for comparative ability loads.

I’m pro-testing, and have no problems with being evaluated by test scores. In fact, I’d probably be better off. But I’m going to be compared against another algebra teacher who had 65% below basic or lower students, right? Along with 30% students ready to learn algebra, so I had to constantly balance competing needs? I’m going to be compared against that teacher, right? Right?

We can’t have a meaningful conversation about student achievement, much less grading teachers on student achievement, until we know whether or not ability distributions like those in the graphs above have an impact on outcomes. And right now, we not only don’t know, but most people don’t care.

Teaching Humanities, Twelfth Night

Twelfth Night lesson plan and discussion.

This was a 100 minute block period daily, which was pretty cool. Rather than do history one day and English the next, we switched off between history and English. Block is pretty brutal for math, but it’s great for history and English. I have most of the assignment images at the bottom of this post.

As I mentioned in an earlier post, I went off the reservation for the last month of the school year. The original lesson plan called for an entire month of Twelfth Night, spent not reading and understanding it, but exploring issues of identity and gender and how the play helped the students see these issues in their own lives. Yeah, not my thing. I started out with good intentions, but by Day 3, I knew I’d done as much as I could. So I cut the time on Twelfth Night in half and then did a two week unit on Elizabethan theater. When I mention the “original lesson plan”, I’m referring to the plan I followed before jumping off.

Primary learning objective: Shakespeare becomes much more understandable when it’s seen and heard. Yes, the words can be simplified, but a great deal of the beauty and power is lost. Students who struggle with understanding Shakespeare can help themselves gain a better understanding of the material by hunting out a movie, talking it over with friends, and yes, reading synopses or translations. But they should also realize that the whole reason we still read Shakespeare has much to do with the beauty and strength of the words, and to make every effort to increase their understanding of those words.

Secondary learning objectives: Students will learn one of Elizabethan acting traditions (e.g., why was Viola dressing like a boy?). They will also form a greater awareness of the challenges and opportunities that emerge as the written word is translated to the stage.

Notes: Remember that a third of the class had reading skills at about the sixth grade level. Actually understanding Twelfth Night text was well beyond their abilities. The top kids, who read at college level (four of them) would find the assignments I was going to give intellectually interesting, even if we weren’t actually analyzing the literature–which they all would read. The mid level kids (half the class) probably would have benefitted from more close reading of the literature. However, as any English teacher can tell you, kids don’t usually do the reading at home, which means you have to schedule a lot of school time for it—and that would leave my low ability kids lost and bored and goofing around, all for the possibility that my mid-level ability kids might gain something from reading more Shakespeare. It was a tradeoff. Eh. Which is not to say the kids weren’t assigned the reading, and that my top level and probably half of my mid-level kids did. It’s just that I didn’t push the reading, as you’ll see.

Day 1-2: Watched the movie, a very good rendition. The kids had to create a “social network” of the characters.

This came from the original plan, it did help the kids focus on characters as well as plot. I then modified the assignment a bit to be less squishy, and off they went. I am not a teacher who goes to the art well much, and whenever I do I am reminded how much the kids love it. They were incredibly creative. I didn’t have an android back then, or I would have grabbed pictures of their work.

Day 3: Gallery walk through the networks and a reading of a Edith Nesbit’s short story of the play and then a fishbowl discussion of three questions. The questions were assigned by student ability (not obviously so). Students are graded in fishbowl on their participation and the strength of their discussion. While the discussion topics varied in complexity, students focused on the tradeoffs made in telling the same basic story in text, live action, or film.

It was the Nesbit story, in fact, that gave me my jumping off point and my new lesson objective. The only students who actually discussed the story were the lower ability students; the top students had to start wrestling with staging and technique.

It’s funny: I have little truck with math “techniques” (pair and share, blah blah blah). I am a big fan, however, of fishbowls and jigsaws in English and history. I suspect it’s because fishbowls and jigsaws work, whereas math techniques just make everyone feel better about trying something. No evidence to support this theory, though.

I thought the questions I came up with for this fishbowl were pretty strong; the kids liked them, too.

Day 4: Read the first act of Twelfth Night aloud. Discussed how much more difficult it would have been to understand if they hadn’t seen the movie first–or was there anyone who found it easier to read than watch? (there wasn’t) Reference back to the free write, short story, and staging.

Assignment: read Act II, scenes 1-3. In class, we looked at them to be sure everyone had a reference point from the movie. I told them yes, I knew of No Fear Shakespeare, and NOT TO USE IT RIGHT NOW. It was okay if they didn’t get everything; we’d discuss it tomorrow, and we would be talking about how to use NFS.

I did a longer stint with SSR today, because reading aloud can be deadly. The first day, the kids just took turns reading it aloud with no acting it out. I am reasonably sure that most of the kids did not use NFS to translate, given that I’d told them I wanted them to puzzle about it without that help and that I wasn’t criticizing NFS.

Day 5: Assigned SSR for 30 minutes–read the rest of Act II, looking for parts you remember from movie. Then we acted out most of Act III–not just reading aloud, but with emphasis and minimal staging. If a student muffed a delivery, I’d make him or her do it again. (Note: I had them skip through a lot of the boring parts.)

Weekend Assignment: Read Acts IV and V of Twelfth Night. Yes, No Fear Shakespeare was allowed, but as much as possible they are to look at both to gain a better sense of what those words mean. Extra credit: spot one scene that was definitely not in the movie. (four kids found some scenes that hadn’t been in the movie.)

As we read through Sebastian and Antonio’s scenes, one student asks “So, am I the only picking up on the whole gay thing? What were they doing on that ship, anyway?” “I think Antonio only helped Sebastian because he was hot for him” said another and the class quickly devolved into three minutes of ribald speculation until I reluctantly restored order.

Day 6: Freewrite: I gave them one scene in two columns, one of the original text, the other “translated” by NFS. It is my opinion, said the freewrite assignment, that NFS loses a great deal in translation. Do you agree or disagree? Be specific and discuss the use of descriptive language, particularly metaphors and similes.

They agreed in all cases (safe choice) but every student, from the strongest to weakest readers, gave thoughtful responses about what was lost in translation, along with some decent specific examples. I was pleased. By the way, I don’t have any documents from the past four days, because it was all done by my winging it through Days 4-6 while I planned out the next assignments, which start now.

Rest of Day 6 and Day 7: Gibberish assignment. No, not the game or the language, but literally words that couldn’t be understood. Students formed groups and chose from a list of scenes that I’d chosen. They had to stage the scene—props and costumes allowed—but they were only allowed to use nonsense language of their choice. They had to focus on making the staging as clear as possible to someone who wasn’t familiar with the play. They had close to two hours of class time to work on their staging, as well as during advisory and of course, after school.

I had no idea how this would work. I just didn’t want them spending time creating scenes that wouldn’t be very good to start with, and this promised to be funny

Day 8: Performance of the Gibberish Staging

This is undoubtedly the funniest 100 minutes of class time I have ever designed.

• Two boys enacted Olivia’s proposal to Viola saying “glockle blockel stoppel” over and over again, with Olivia looking mooningly into Viola’s eyes and Viola–who was much, much larger than Olivia—desperately trying to escape.
• Three girls acted out Malvolio’s yellow stocking scene in three languages: Tagalog (Malvolio), Japanese (Olivia) and Chinese (Maria).
• Four students acted out Sir Andrew’s letter writing scene with Sir Andrew (a girl) doing knee bends, fencing stabs, and muscle preening while Sir Toby reads the letter aloud, giving reassurances to Sir Andrew as to its ferocious law-abidingness, which exchanging snide looks with Maria–and then the duel itself with Viola played by one of the tallest boys in the class, wincing in terror–all done to the sounds of Dubdubdubbiddybub.

And those are just the three I remember specifically; they were all hysterical and in the main, very well staged. I dunno, maybe you had to be there. The students had a blast and wrote their own reviews.

HW: Final Essay–you have a friend who is complaining about how hard it is to read Shakespeare. Write a full-page letter (can be informal, but no text-speak) and give some advice with specific examples.

The results here were pretty much what you’d expect. Nothing spectacular, but they all got the idea.

Day 9 or Day 10—I can’t remember if I’m missing a day or if we had to do some assembly. In any event, the last day of the unit was the final. Students had to identify key plot points and quotes.

While I didn’t track specific results, all the students got a C or higher, and only 3 students got a C. I had spent the year emphasizing the importance of remembering content, not letting it just go in one ear and out the other, and it had really paid off. My favorite airhead got a B-, to her shock, and took a bow to the class.

Documents below.

It’s funny; I have many fond memories of the Elizabethan history unit, but not until I wrote this up did I realize that the Twelfth Night unit had also been quite successful. Learning objective definitely achieved by all students. I’m quite sure they increased their content knowledge and all of them, regardless of reading ability, have a decent memory of the main plot points of the play. All that and gender identity discussions, too!

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Teaching Humanities, Part I

I got lucky my first year out and was able to teach math and humanities.

Were I ever to get a full-time job teaching either English or history, I would feel guilty for abandoning math and taking the easy way out. That’s how much easier it is to teach either subject. Do not picture me as short-timer, stuck teaching math as some sort of dead-end, droning on praying for the day that I get to teach my true passion. No. I find teaching math, constantly struggling to find a way to make abstract concepts understandable to uninterested students with no inherent ability, to be one of the most fascinating tasks invented. I’m hooked. But teaching English or history is a hell of a lot easier, and my lord, is it fun.

I taught 9th grade humanities at an extremely progressive school. (You’re wondering why on earth they hired me. They were desperate and got rid of me as soon as they decently could.) I planned out the curriculum with two other teachers for most of the year. As a rule, I did tests, grammar, and history (I was the only certified history teacher of the three of us, had considerable experience with standardized tests and–also as a result of my test experience–a lot of background in teaching grammar). They did literature and most of the actual planning (weeks spent on each section and so on). It was all collaborative and lots of fun–I learned a lot about planning out a book, and they gave me great feedback on how to simplify a history lesson for freshmen, while I taught them how to keep the rigor in even if the vocabulary is simplified.

We did the history of India, history of the Philippines, a brief history of Russia from the freeing of the serfs to the Russian Revolution, and the Age of Enlightenment, Age of Discovery, and revolutions industrial and agricultural. The kids read Nectar in a Sieve, a book on post-colonial India, a choice of three books on the Philippines (can’t remember their names), Animal Farm, and Twelfth Night. They also did some sort of project on natural disasters, which interested me not at all except I learned a good deal about geography, and some sort of personal narrative.

There was a great deal of indoctrination in the course material from years past, but I convinced the other teachers to dump a lot of it (without ever mentioning my opinion of the indoctrination) and the rest of it I just cut from my lesson.

The kids’ reading abilities ranged from 6th grade to college level, as did their writing. We were supposed to do “sustained silent reading/writing” each day for 20 minutes (it was a block class of 100 minutes) and it was supposed to be based on student choice, but I realized that most of them were just sitting around doing nothing. So I instituted my own hand-made SRA program of three levels. I just went to the bookstore, picked out some enrichment materials at various levels (and yes, bought them with my own money), and made copies for the kids. The kids read nifty little passages on all sorts of subjects, answered questions, did crossword puzzles with new vocabulary, and had little tests at the end of each unit that I checked on.

The kids gained tremendously in content knowledge at all levels. My favorite example: when we were talking about Russian history and Trotsky, I mentioned in passing that Stalin hunted down Trotsky even after he fled to Mexico, where he lived for a while with two famous artists, Diego Rivera and Frieda Kahlo.

One of my weakest readers perked up. “Is that how you say her name?”

I laughed. “I think so, but I’m not sure. I’ll go look it up later. You know about Frida Kahlo?”

“She was in the reading packets. She was famous for painting herself, or something.”

“Self-portraits?”

“Yeah, that was it. She wasn’t happy with the Diego guy, right?”

“Oh, is that the one that had a car accident?” pipes up another weak reader.

Another boy pops in. “She crashed into something.”

A strong reader (who therefore had not read about Frida) was interested. “When was this?”

“Yeah, this was like….it was after 1900. I think it was a lot after 1900, but not like 1950 or anything.”

“I think her car crash was in the 1920s, but don’t quote me,” sez I. (It was.) “That was very useful information, and thanks for the interesting details. Back to Trotsky and the axe.”

Content knowledge, baby.

Anyway, the segment on Twelfth Night we were expected to do was designed by a student teacher, and it was all about identity and examining their own navel—exactly the kind of nonsense I don’t like to do. By now, I knew I wouldn’t be back next year and this was the last segment of the year, so I went off the reservation. I did two weeks on Twelfth Night and two weeks on Elizabethan theater, and every minute of it was joyous fun. For the kids sometimes, too.

This post is getting long, so I’ll put the lesson plans and a story about in subsequent posts.

Meanwhile, back in Geometry….

As much as possible, I want the students to know a few unifying ideas about triangles. That’s why I dumped medians, orthocenters, and a lot of the relatively obscure triangle facts—not because they aren’t useful, but because most of my students will never use them again and I want them to have plenty of time working with the geometry they’ll need forever. My top students get practice thinking through challenging problems but I don’t throw a lot of random facts into this mix.

This has a lot to do with my own preferences. I like to remember the big things and look up the little, and I don’t personally see much value in making students jump through hoops to remember obscure math facts. (History is a different matter.) I do want them to remember the important math facts (Pythagorean theorem, special right ratios, area formulas, and so on), so when I say, underscored five times, remember this, my students won’t treat it as one of a random flood. That doesn’t mean they’ll all remember it, alas. A distressing number of my students still look at me perplexedly when I ask them the formula for the area of a triangle. It is to weep. But that’s all the more reason to keep the memorizing to a minimum.

I taught special right triangles as a ratio, rather than a pattern. This worked very well for most students, and it also provided for continuity when we moved into similarity. Same process, but now the ratio isn’t fixed. (And this should make the move to trigonometry, which is also based on ratios, part of the same continuity).

I gave them a test after special rights and then a test after similarity. I tested much of the same material in both. (After break, I’m going to give them a test on other things from first semester, to see how their retention is.) Here are the tests and the correct percentages for each.

I just noticed that I forgot to fix the typo in question 14 before I made this image. The kids were given corrections before they started the test. (My tests often have minor typos, as I create them from scratch each time. My kids know this and are encouraged to check with me if they think they find something wrong. They are usually incorrect—that is, there was no typo—but I’m confident that no one is sitting silently perplexed for the wrong reasons.)

Overall comments:

• Questions 1 and 2: see what I mean about area? I was actually pleased with those numbers. They are still forgetting to take half.
• Questions 3 and 4: very good understanding of perimeter and algebra (I don’t think my diagonals are accurate on that pentagon).
• Question 5: A straightforward test of Triangle Inequality, something I’ve emphasized. It’s frequently on the SAT and ACT and besides, it’s a useful limit to remember. And they did!
• Question 6 and 7: The students had to know that triangles had 180 degrees AND that a “not acute” triangle must have one angle of 90 degrees or higher. Another 20% knew that triangles had 180 degrees, but didn’t catch the second criteria.
• Questions 8, 9, 10–straightforward special right questions.
• Questions 11 and 12 are a case of unintended consequence. I thought 11 would be the easy one, as we’d just reviewed congruent triangles and I’d reminded them (again) of the SSA (see the bad word? Yeah. Not a congruence shortcut). Most chose the SAS answer—the correct answers were all top students. While question 12 didn’t have great results, more students hroughout the ability spectrum answered it correctly.
• Questions 13-15—more special right questions. (see note above about the glitch on 14).

The last five were probably too difficult; I wanted to see how my top students would do. I ended up dropping the last five and grading students on the first 20.

Average curved score: 68% (51% uncurved)

This was a much more solid performance, I thought. I’d increased the difficulty, and the average was just a little higher (70% curved, 54% uncurved).

In both tests, the students had a terrible time dealing with radicals and the Pythagorean Theorem, which is what led to a review. But they showed increased mastery of special rights, and their understanding of similar triangles was better than I’d been hoping for.

These percentages are the actual answers given. I went through the D and F tests and reviewed questions 15, 17, and 20. In all three cases, I’d put a spin on the question. So if a D or F student saw that the angle was 60 degrees but didn’t realize I was asking about the unmarked part, did the proportion correctly in question 17 but forgot to convert to feet, or solved for x and then forgot to plug it in, I gave them credit for that work.

Any geometry teacher knows that I’m going veeeeeery slowly. I have two major topics to cover before state tests—trig and circles—and two minor ones, volume/solids and regular polygons. At the same time, I’m definitely not giving my students an easy time. I’m working them hard and I think they’re doing challenging math. In fact, that’s the most consistent beef my students have about me—they think my tests are “weird”. But they also appreciate my curving, and they also, I think, accept my assurances that this is what actual standardized tests look like, so they may as well man up.

Every so often….

I got into teaching to work with struggling kids. I’d enjoy working with an entire class of motivated and able kids, but it would also come as a shock. Most of my time I spend pushing kids up the hill and praying they don’t roll backwards.

But I do have exceptionally bright kids in my classes, too. My geometry classes in particular are a joy, since even the low ability, low incentive kids know they’ve finally escaped algebra and aren’t eager to repeat that experience by repeating geometry. So it’s a well-behaved group with decent motivation. On the other hand, 80% of my Algebra II classes scored Far Below Basic or Below Basic in Algebra two years earlier and a few of them are much harder to keep in line, even though the bulk of them are well-behaved and want to learn, not out of any love of math, but because they understand, thanks in part to my frequent rants, that they will be taking a placement test in a year or two and the only good thing about my class is that it’s free.

But I do have some students who chose an easier course because they are athletes, or because they struggled in geometry, or just because someone somewhere made an odd placement decision, and that are in fact quite strong in math—and of course, there are always a few students who finally “get it” and who actually start to grasp the material (this is extremely rare, and one major reason why value added is a problem in high school teacher evaluations).

Anyway, on Thursday, I had two interactions with bright students that stay with me because of their infrequency. I’m not complaining. It was just fun.

Incident #1

A solid geometry student had done horribly on her last test. She doesn’t often engage with me and remains a bit distant, but after I turned back the tests I checked in with her.

“I do not get special right triangles. I don’t understand how they work. I get similar triangles, but I can’t ever remember the ratios and I can’t see how it works.”

I drew a right triangle on the board with the two legs labelled “x” and the hypotenuse labelled “hyp”. “What kind of triangle is this?”

“Isosceles right.”

“Okay. Create an equation to solve for the hypotenuse using the Pythagorean Theorem. I’ll be back in a bit.”

I came back, and she was working through it, a classmate tossing in advice and argument.

“No, it’s square root of two! You took the square root of both sides!”

“Oh, that’s right.” and she had it solved, the hypotenuse equal to x * sqrt 2.

I pointed to the class notes which were on the board. The isosceles right triangle was labelled x, x, x sqrt 2.

“Oh! I get it.”

I then sketched out a 30-60-90, asked her what it was and she correctly said it was a half of an equilateral triangle. I told her to label the sides, making sure she put the x opposite the 30. Then I told her to solve for the second leg. When I came back, she’d finished that up and was in a great mood.

Had I taught the Pythagorean method before? Yes. Several times. Sometimes they just aren’t ready to get it until they’re ready to get it. But only a strong student can grasp the algebra of the Pythagorean proof and see how that knowledge can help her remember the ratios.

Incident 2

One of my strongest Algebra II students was struggling with synthetic division, which I introduced as a method of testing quadratic values (more on that later). She asked me to explain it to her again—not just the how, but the why and the what.

I began from the top and ran through it all. When she grasped that the division process revealed not only the quotient but that the remainder was the equivalent to evaluating the function at the divisor value, she said “Wow. You might almost think that was planned.”

I laughed in unexpected pleasure. I am not a believer in God nor a mathematician, nor am I a proponent of the “math is everywhere, math is beauty” propaganda that true math lovers preach. But the Remainder Theorem, like the Fundamental Theorems of both Algebra and Calculus, is indeed enough to make even me wonder if there’s some Grand Design. That a student of mine should reach that conclusion after a largely utilitarian but comprehensive explanation by yours truly was an unlooked for joy.

I was telling this to a colleague, and he reminded me of the famous quote: “God exists because Arithmetic is consistent. The Devil exists because we can’t prove it.”

Enrichment

I teach composition and book club on Saturdays at an SAT academy, which is codespeak for a near-entire Asian (ethnic Chinese, Korean, Indian, and Vietnamese) first generation immigrant parent clientele who want a place to send their kids from 7th grade on. My kids spend three hours every Saturday with me for a year, then I teach summer school, often to some of the same kids. Understand that most of them live in Asian enclaves in which they rarely run into white people, much less black or Hispanic. The public schools they go to are 80% Asian, then they go off to public or private universities that are 40-50% Asian, they (thus far) marry other Asians and will eventually form additional enclaves and renew. I always start off every new year by asking the kids to estimate the percentage of the American population that is Asian–the lowest guess ever has been 15%. Most of them guess 30%.

I love it. I have my doubts about the impact this same population is having on public schools and college admissions, but my affection for the kids themselves knows no bounds.

It’s a book club, and the primary emphasis is on vocabulary, reading comprehension, and writing in different forms. But I nonetheless include a great deal of instruction on “white people world*” and most of them soak it up eagerly. I am often the first person they’ve met who has told them that watching more TV is actually helpful, that good grades are nice but only if they are accompanied by actual knowledge and achievement which is not the same thing, and who understands but gently mocks their parents’ demands. I can only be satisfied by them thinking for themselves, and there are no grades—a topsy turvy world for these kids.

Each class quickly grasps that I will mention things that they’ve never heard of, and that they should know of, and that I think it’s a problem, or at least a deficit. And periodically, the deficit will be so significant that I immediately act to remedy it.

Which is what happened today, when we were going over the news of the week. They all knew that Whitney Houston had died. It took me a while to realize that none of them could identify a single song of hers.

“Seriously? I Will Always Love You? Never heard of it? Hmmf.”

“Was she really popular?”

“Oh, hugely so for about 15 years back in the 80s and 90s. She came from a talented family. You’ve probably never heard of her mom, and probably wouldn’t know Dionne Warwick, but Aretha Franklin was her godmother, and…..” I see the blank looks.

“Oh, come ON. You do too know Aretha.” They all shake their heads. “You have too. What’s annoying is that you’ve heard her and just didn’t know it was her, and you SHOULD. So I’m going to play her most famous song, you’re going to go ‘oh, yeah, I know that song!’ and from now on you are going to know who Aretha Franklin is.” I am thumbing through my Android as they assure me they have no idea who Aretha Franklin is. Their assurances last through the opening of “RESPECT” and then , as her voice comes on, sure enough….”

“Oh, is THAT Aretha Franklin! I know that song!” and they are all cracking up because they are doing exactly what I told them they’d do.

“Now. Never forget who Aretha is, okay?” They nod.

I then play two Whitney songs. Not only had they heard them before, but one of them had “I Wanna Dance With Somebody” on her Ipod.

Don’t worry, parents, we talked about art and Asher Lev, too.

*Yes, I know, there’s a certain irony in my calling it “white people world” when I’m explaining Aretha and Whitney.

Developing Curriculum

Ed schools, particularly elite schools, preach the need for teachers to create their own curriculum. We don’t get any lessons on how to take a few pages of a textbook and break it down into explanation and practice, or how to select a good range of problems from a textbook to assign for classwork. Many ed schools use the Wiggins and McTighe text Understanding by Design (and its follow up on differentiating instruction), which argues that adherence to textbooks lead to the sin of coverage. No, not only shouldn’t we follow the book, said my ed school (and many others) but we should throw the book out and work backwards from out learning goals.

I thought this an idiotic idea. Open the book, give some examples with a good explanation, and have them work some problems. I explain things well, and any decent textbook has a wide range of problems to assign.

I had good instructors, particularly in C&I, and I expressed this opinion frequently and openly. I was not shot down—ed schools are doctrinaire at the administrative level, but at the instruction level, I found all my professors to be open to challenge. In fact, when I look back, I’m struck by how often my instructors reiterated what Wiggins himself says time and again: Understanding by Design is a framework, not a philosophy.

But it was impossible for me to believe that because the examples in the readings, and the openly progressive politics of ed school always sold curriculum design as a way to indoctrinate. The UBD books use, as an example, a history teacher who created an elaborate project for students to design their own constitution, reflecting the needs and interests of everyone in the community, not like those racist Founding Fathers. Or the instructor might describe a math teacher with an equally elaborate projects for students to “discover” transformations of functions when none of them have the skills to solve the functions in the first place.

Or there was the time we had to listen to an absolute idiot of an English teacher at an inner city charter school yammer on ignorantly about the “culturally whitewashed curriculum” that gives urban kids Robert Frost, who (to paraphrase) didn’t know trouble, didn’t know suffering, and wrote peaceful rural poems this teacher’s unfortunate inner city students couldn’t relate to, instead of the “real” poetry of Gwendolyn Brooks. There I was, paying huge chunks of money to listen to a brain dead jackwit present Frost as an out of touch white guy who wrote pretty poems about happiness and peace. Surely there’s a teacher out there who has considered a lesson for inner city kids helping them to see Frost’s hidden bleakness by contrasting it with Brooks’ open despair, and surely that would be the teacher invited to lecture at an elite university instead of this buffoonish hack? But I digress.

Ironically, I learned that textbooks could be a problem when, my first year out, I worked with a famously progressive, constructivist text known as CPM. I’ve used CPM to teach geometry and both years of algebra, and all the books had a few moments of interesting brilliance, way way WAY too much text, not enough practice problems, insufficient respect for the real priorities of any subject and an ordering approach that drove me crazy. I hated it, just like the good little progressive teachers hate their cold, formula-laden traditional text, and so I learned to ignore the book and develop my own curriculum.

Three years out, I have a very different view of textbooks. Good ones are great tools for ideas and problem sets that I can dip into as needed. But all textbooks fall short in some ways, many of which aren’t their fault.

The big problem: I often teach kids who won’t use them.They certainly won’t take them home and back to school (most of them just leave the books at home). They won’t use them as a resource. In fact, many kids actively prefer a worksheet to a textbook, as they get a sense of completion from finishing a page of problems. A textbook never ends.

Another problem is, of course, the size. At more than one of my schools, the kids don’t all get lockers. So the books are either going to stay home or stay at school. Last year, (2011-2012), I asked my kids to keep the books in the classroom, so I could use them on and off as needed. The textbook supervisor at the school got very upset at this, for good reason, but it worked. I was able to pull in the books as needed, particularly for my strongest kids, and ignore them the rest of the time. (Update: this year, at a different school, all my kids have lockers. It’s very convenient–I just write BOOKS in big letters on the board, and the kids go back to their lockers to get the books on the days I need them.)

Another problem is that textbooks are designed for one audience. Progressive texts, like CPM, are primarily designed for low ability kids in a constructivist classroom. Not enough problems, very few challenging problems, too much text, usually strained efforts to connect math to “real-life”, way too much indoctrination, and an exhaustive preference for explanations over answers. Spare me.

So when I got to my current school, using Prentice for algebra and Holt for Geometry, I was happy to shake the CPM dust off and use textbooks daily. Alas, I realized that these books were the Papa Bear to CPM’s baby—a fire hose for all but a fraction of my kids. Most textbooks are designed for students who are actually ready for the material, with low ability kids an afterthought. These textbooks cover material far too quickly. In my current geometry book, three pages in one section covers both 30-60-90 and 45-45-90 triangles. Sure, because kids pick it up just like that. These books often include different worksheets on a CD for lower ability kids, but at that point, you’re not using the textbook anymore.

I also find books are too limiting. They rarely provide teachers with useful illustrating activities–sometimes the book will sketch out an interesting possibility, but leave the details to the teacher. For example, the Holt Algebra II text introduces complex numbers as if the topic is little more than a walk to the drugstore. I mean, the numbers are imaginary, for chrissakes, and the text just spells it out in a sentence and moves on. For as huge as math books have gotten, publishers still haven’t used any of that space to lay out an explanation that works for low ability kids. Of course, the kids wouldn’t use it anyway, since they’d avoid the textbook, but at least it would give me something to copy so I didn’t have to create my own or steal a good start off the Internet.

Then, the book ordering is often insane. The year I taught all Algebra I, the Holt book introduced rate problems and work formula problems in Chapter 2. I laughed. My kids are still shaky on subtraction, and I’m going to cover high-complexity word problems in the third week. Sure. Only my top students got these problems, and then only at the end of the year.

When I realized how advanced the book was, I checked with a senior teacher, and she snorted. “Oh, I don’t use the books.” At our school, in algebra and geometry, relatively few teachers use the books on a regular basis. They develop their own lessons, their own tests, they borrow worksheets, and cobble together a curriculum that, in their view, meets their students’ needs.

I know my experiences aren’t universal. I know many teachers teach from the book, and many teachers work collaboratively to produce a class taught over 13-15 sections by multiple teachers. I student taught at a school that planned course-alikes collaboratively, were faithful both to the (CPM) text and a common schedule for all classes. The next three schools I’ve taught at gave me a solid grounding in “teacher as island”; everyone does their own thing. Given my druthers, I’d rather the latter. While I do wish I worked in departments that did more course-alike planning, I’m becoming increasingly sympathetic to the teachers who resist lockstep synchronicity. (Update: Right now, I’m dealing with a math teacher who insists that everyone teach trinomial factoring in exactly the same way. Um. No. Unless y’all want to use my way.)

What spun off this post was a review of The Tyranny of Textbooks, a purported expose of the textbook selection committee, with proposals to change and improve the process.

But that makes me laugh. Improve the process? Tons of teachers don’t even use the books! Why waste billions on textbooks that go home to serve as doorstops? Pick a few approved texts. Buy a few sets of each. Let the teachers who want to use them get a class set, or (in the case of advanced classes) check them out to the students. In all but a few cases, schools could save money by using class sets—except, of course, many states legally require districts to give every kid a book. Taxdollars in action, baby.

Here’s the really funny thing: I’ve described what I do, and what many high school teachers do—develop our own curriculum to cover the standards. But it’s clear from even a cursory overview of the education debate, that a million teachers planning their own curriculum is not what eduformers foresee as the future of education in this country. It’s also clear, however, that most education reforms never make it to the classroom and don’t have a clue how teaching actually happens, particularly in high school. (Eduformers believe that once elementary school is fixed, all will be well. They’re wrong.)

So if you don’t like teachers overriding local and national priorities by developing their own curriculum, and using the books, too bad. First off, progressives own ed schools and they’re always going to be pushing teacher curriculum development.

But more to the point, the range of student abilities, and the expectation that low ability students are to be taught a college-prep curriculum, pretty much mandates curriculum development at the district, school, or even classroom level. You want lockstep classroom curriculum? Bring back tracking and develop different texts for different ability levels. Let’s all laugh at that idea.

And now, a mea culpa: given how much misery I caused my ed school, I feel it only fair to acknowledge that my disdain for the progressive agenda and my dislike of constructivism was drowning out my instructors’ message about Understanding By Design: Here’s a framework for building your own. Take what works and toss what doesn’t. While they might approve of a particular agenda, the framework is ideologically neutral.

My last three plus years of teaching have done much to increase my approval of progressives. Yes, their agenda is still overtly political and yes, they still ignore ability in much the same way that eduformers do. But progressives in ed schools know far more about teaching than eduformers will ever know, and buried underneath their squishy curricular nonsense is a core of useful knowledge that I tap into quite often.

(Note: I first wrote this while at my last school; the updates were made in mid-September at my third school.)

Sigh…..

My union dollars at work.

The lurker in the teacher quality debate

Just a few weeks after I complained that the debate on teacher quality ignores state certification tests, Ed Week steps up: Analysis Raises Questions About Rigor of Teacher Tests (Edweek has a paywall, but the synopsis I used includes a link to the original):

The average scores of graduating teacher-candidates on state-required licensing exams are uniformly higher, often significantly, than the passing scores states set for such exams, according to an Education Week analysis of preliminary data from a half-dozen states.

The pattern appears across subjects, grade levels, and test instruments supplied by a variety of vendors, the new data show, raising questions about the rigor and utility of current licensing tests.

I’m happy to see the attention, but what “questions about the rigor and utility”? The scores are high, so there’s a problem? Imagine for a moment that the average score was juuuuuust barely above the cut score. Would Edweek then congratulate the states for setting such an ambitious cut score that teachers barely qualify? I’m thinking not. Besides, this analysis is missing a key ingredient: without a benchmark, passing rates or cut scores tell us nothing about the test’s rigor or utility. The ETS provided that information in its teacher quality report a few years back, so it’s no secret. Without that information, I really don’t see the point of this analysis.

But if the people at Edweek are really wondering why the teacher certification cut scores aren’t higher given the high average score, I think I can provide illumination, thanks to the always useful ETS, an organization that really isn’t given enough credit for its useful information.

(Cite: Recent Trends in Mean Scores and Characteristics of Test-Takers on Praxis II Licensure Tests)

This particular data is for the Math Content Knowledge Praxis II test, but the report shows the same gap in all the Praxis II tests—African Americans who have passed the Praxis (both I and II) scored over one standard deviation below whites.

Average score and passing rates (again for math, but the link above has all the scores).

Praxis states don’t have substantial Hispanic populations, but California does, showing Hispanic pass rates that are, as always, in between blacks and whites. (this chart again is for math)

These pictures make it pretty clear that raising the cut scores dramatically wouldn’t affect passing rates for white teacher candidates all that much but would run a buzz saw through the prospective non-Asian minority teaching pool. And while no one appears willing to say so, I suspect that cut rates are set to allow some percentage of black and Hispanic teacher candidates.

Black and Hispanic teachers are severely underrepresented. Reporters and educational pundits go through a great show every so often of scratching their heads and wondering why—and then, the next day, interview eduformers demanding that we raise the bar on teacher qualifications without ever connecting the dots.

From another ETS report that combined extensive reporting on the teacher test score gap between blacks and whites with ed school student and faculty interviews, an observation I’ve never seen in a story on the missing minority teachers:

So two takeaways:

First, raising teacher quality, whether by requiring more education or higher qualifying test scores, would further reduce the ranks of black and Hispanic teachers and make the teaching pool much whiter and Asian that it already is.

Second, the evidence linking teacher credentials, whether it’s degrees or test scores, to student achievement is sketchy at best, non-existent in most cases.

So I ask again: How smart do teachers need to be? What proof is there that raising the teaching standards will lead to better educational outcomes?

Modeling with Quadratics

After my success (I hope) with linear equations, I started a unit doing the same thing with quadratics.

Days 1-3:

“A triangle’s height is three feet longer than its base. Create a table linking the height to the area. Graph.”

“A rectangle’s length is twice as long as its width. Create a table linking the width to its area.”

“A rectangle has sides of X+3 and X-2. Create a table linking X to the rectangle’s area.”

Just as with linear equations, the students really improved at generating values. They also, I think, quickly grasped that generating data for quadratics is considerably more complicated than linear equations. More than one pointed out to me that they couldn’t just “add three each time” as they could with linear equations.

I taught them how to break it down into parts and assign each part to a column. For example, the first triangle problem:

Base	Height	Base * Height	Half BH (Area)
1	4	4	2
2	5	10	5

The stronger students could see how this led to the equation; even the weaker students could see that each equation had steps, and they started to get suspicious if it got too easy. One struggling student called me over to tell me he must be doing something wrong because “look, it’s going up by the same amount”. He was linking length to width, rather than length to area.

Even if they don’t get much stronger in working with quadratics generally, this exercise clearly helped them gain competence at working through a problem. Next step: generating values quickly from an equation in standard form. Hello, synthetic substitution.