Tag Archives: geometry

Opening Day as Opening Night

I really like our late start; why the hell are so many school districts kicking off in early August? (They want higher test scores, Ed.)

Anyway, I’m teaching trigonometry for the first time. In every course, I assess my kids on algebra I, varying the difficulty of the approach based on the level of math. What to do with trig? My precalc assessment was too hard, my normal algebra assessment too easy—or was it? I didn’t want to discourage them on the first day, but I also didn’t want to give a test that gave them the wrong idea about the class’s difficulty level. After much internal debate, I created a simplified version of an early algebra 2/trig quiz. I dropped the quadratics (we only had 45 minutes). Then, just to be safe, I made backup copies of my algebra pre-assessment. If the kids squawked and gave too much of the “this is too hard” whine, I’d be ready.

And so in they came, 23 guys, many of them burly, a few of them black, none of them both, and 11 girls. Fully half the students I’d taught before, two of them I was teaching for the third time. (one poor junior has only had one high school math teacher.) Perhaps their familiarity with me helped, but for whatever reason they charged right in and demonstrated understanding of linear equations, systems, a shaky understanding of inequalities, and willingness to think through a simple word problem. Good enough. Great class—rambunctious, enthusiastic, way too talkative, but mostly getting the job done.

I’m still not much of a planner, which is why I gave no thought to my trig sequencing until I saw how they did with the assessments. If they’d tanked, I would have done a simple geometry activity to give me time to regroup, start after the weekend with some algebra. But they didn’t tank, so how did I want to start?

Special Rights. Definitely. I would use special rights to lead to right triangle trig. All clear. But how to get to special rights? Algebraic proof of the ratios. But why special rights? It seems random to start there. As long as I’m going to be random, and since trigonometry has something to do around the edges with right triangles, why not start with right triangles? At that moment, this image popped into my head:

Hey. One step back to geometric mean, and I’ve got a nice intro unit all set up.

So the next day, I started with this:

geomeanquestion

Note: I told them the questions were separate—that is, the square was equal only to the area in #1 and only to the perimeter in #2.

I wasn’t happy with the questions. They gave too much away. But every rewrite I tried was even more confusing, and in a couple cases I wasn’t sure it was an accurate question. Besides, on the second day of school, you want to release to something achievable. Better too straightforward than have the kids feel helpless this early.

And it went great. Top kids finished in under five minutes; I had them test out the process for cubes vs rectangular prisms. All the rest completed the work in 15 minutes or less, with some needing a bit of reassurance.

I had to prompt them to recognize that the perimeter to side relationship is the “average” algorithm (that is, the arithmetic mean). “If I add two numbers and divide by 2, what is the result?” I think I noodged for a few minutes before someone ventured a guess.

I followed with a brief description of geometric mean, reminded them of the various measures of central tendencies, pointed out that now they all knew why the SAT followed “average” with arithmetic mean. Finished up with practice problems.

I was stumped briefly when a student noticed that the arithmetic mean always seemed larger. Argghh, I’d mean to look that up. I told them I’d look up the answer and get back to them. Meanwhile, I wondered, could the two means ever be equal? I made the stronger kids do some algebra, and let the others just talk it through.

Great lesson, not so much from the content, but from the energy. Look, I was winging it. I do that when I have a good idea that isn’t fully fleshed out. I cut back goals, keep things very simple, and watch for opportunities. I always advise new teachers to avoid mapping things out—they are often wasting time, because things will go off the rails early in some cases. Keep it broad, tell the kids that you’ll adjust if needed, and go.

The rest of the opening “unit”: a brief review of similarity and then use of geometric means in right triangles, leading to my favorite of the Pythagorean proofs. Then onto special right triangles, deriving the ratios algebraically. This puts things nicely in position for introduction of right triangle trig and I can drop in a quiz. Well, I’ll probably put in a day of word problems first.

After school today I ran into a group of football players waiting for practice to start, many of them previous students and two of them currently in that trig class. After hearing what they were all up to, how their summers had panned out, what the team’s chances were, Ronnie, one of the two current students, said, “I’m glad I have you; I would hate to be dumped for low grades my senior year.”

“Ah, yes, that’s my claim to fame. I’m not a great teacher, but by golly, I give passing grades.”

Shoney, the other of the two, a big, burly, not black senior, was laying along a school bench calmly watching the conversation, and spoke for the first time.

“You know. Trig was….fun today. It really was.”

Ronnie nodded.

The point is not oh, gosh, Ed is a fabulous teacher who makes kids love math. That’s never my goal, and it’s not what Shoney meant.

Recently, Steve Sailer writes that “school teaching can be thought of as a very unglamorous form of show biz, which involves stand-up performers (teachers) trying to make powerful connections with their audiences (students)”. He’s right. Education and entertainment are both, ultimately, forms of information transmission.

His next paragraph is dead on, too:

We are not surprised that some entertainers are better than other entertainers, nor are we surprised that some entertainers connect best with certain audiences, nor that entertainers go in and out of fashion in terms of influencing audiences. Moreover, the performances are sensitive to all the supporting infrastructure that performers may or may not need, such as good scripts, good publicity, and general social attitudes about their kind of performance.

People tend to construe the “education as entertainment” paradigm as “show the kids movies all day” or “keep the kids laughing”, but just as all entertainment isn’t comedy and happy endings, so too is education more than just giving the kids what they want.

I’m a teacher. I create learning events. I convince my audience to suspend disbelief, to engage. Learning happens in that moment. Some of the knowledge sticks. Other times, only the memory of learning remains, and I’m starting to count that as a win.

And so the year begins.

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Core Meltdown Coming

I’ve stayed out of the Common Core nonsense. The objections involve much fuss about federal control, teacher training, curriculum mandates, and the constructivist nature of the standards. Yes, mostly. But so what?

Here’s the only important thing you need to know about Common Core standards: they’re ridiculously, impossibly difficult.

I will focus here on math, but I’m an English teacher too, and could write an equivalent screed for that topic.

I’m going to make assertions that, I believe, would be supported by any high school math teacher who works with students outside the top 30%, give or take.

Two to three years is required just to properly understand and apply proportional thinking–ratios and percentages. That’s leaving off the good chunk of the population that probably can’t ever truly understand it in non-concrete situations. Proportional thinking is a monster. That’s after two to three years spent genuinely understanding fraction operations. Then, maybe, they could get around to understanding the first semester of first year algebra–linear equations (slopes, more proportional thinking), isolating variables, systems, exponent laws, radicals—in a year or so.

In other words, we could use K-5 to give kids a good understanding in two things: fractions and integer operations. Put measurement and other nonsense into science (or skip it entirely, but then remember the one subject I don’t teach). Middle school should be devoted to proportional thinking, which will introduce them to variables and simple isolation procedures. Then expand what is currently first semester algebra over a year.

Remember, I’m talking about students outside the top 30% or so (who could actually benefit from more proportions and ratios work as well, but leave that for another post). We might quibble about the time frames and whether we could add a little bit more early algebra to the mix. But if a math teacher tells you this outline is nonsense, that if most kids were just taught properly, they could learn all this material in half the time, ask some questions about the demographic he works with.

Right now middle school math, which should ideally focus almost entirely on proportions, is burdened with introductions to exponents, a little geometry, some simple single variable equations. Algebra I has a whole second semester in which students who can’t tell a positive from negative slope are expected to master quadratics in all their glory and all sorts of word problems.

But Common Core standards add exponential functions to the algebra one course load and compensate by moving systems of equations and exponent laws to eighth grade while much of isolating variables is booted all the way down to sixth grade. Seventh grade alone bears the weight of proportions and ratios, and it’s one of several curricular objectives. So in the three years when, ideally, our teachers should be doing their level best to beat proportional thinking into students’ heads, Common Core expects our students to learn half of what used to be called algebra I, with a slight nod to proportional thinking (and more, as it turns out. But I’m getting ahead of myself).

But you don’t understand, say Common Core devotees. That’s exactly why we have these higher, more demanding standards! We’ve pushed back the timeline, to give kids more time to grasp these concepts. That’s why we’re moving introduction to fractions to third grade, and it’s why we are using the number line to teach fraction numeracy, and it’s why we are teaching kids that whole numbers are fractions, too! See, we’ve anticipated these problems. Don’t worry. It’s all going to be fine.

See, right there, you know that they aren’t listening. I just said that three to four YEARS is needed for all but the top kids to genuinely understand proportional thinking and first semester algebra, with nothing else on the agenda. It’s officially verboten to acknowledge ability in a public debate on education, so what Common Core advocates should have said, if they were genuinely interested in engaging in a debate is Oh, bullpuckey. You’re out of your mind. Four years to properly understand proportional thinking and first semester algebra? But just for some kids who aren’t “smart”? Racist.

And then we could have an argument that matters.

But Common Core advocates aren’t interested in having that debate. No one is. Anytime I point out the problem, I get “don’t be silly. Poor kids can learn.” I point out that I never mentioned income, that I’m talking about cognitive ability, and I get the twitter version of a blank stare somewhere over my shoulder. That’s the good reaction, the one that doesn’t involve calling me a racist—even though I never mentioned race, either.

Besides, CC advocates are in sell mode right now and don’t want to attack me as a soft bigot with low expectations. So bring up the difficulty factor and all they see is an opportunity to talk past the objection and reassure the larger audience: elementary kids are wasting their time on simple math and missing out on valuable instruction because their teachers are afraid of math. By increasing the difficulty of elementary school math, we will forcibly improve elementary school teacher knowledge, and so our kids will be able to learn the math they need by middle school to master the complex, real-world mathematical tasks we’re going to hand them in high school. Utterly absent from this argument is any acknowledgement that very few of the students are up to the challenge.

The timeline isn’t pushed back for algebra alone. Take a look at Geometry.

Geometry instruction has been under attack for quite some time, because teachers are de-emphasizing proofs and constructions. I’ve written about this extensively (see the above link, here, and here). Geometry teachers quickly learn that, with extensive, patient, instruction over two-thirds of their classes will still be completely incapable of managing a three step proof. Easy call: punt on proofs, which are hard to test with multiple choice questions. Skip or skate over constructions. Minimize logic, ignore most three dimensional figures (save surface area and volume formulas for rectangular prisms and maybe cylinders). Focus on the fundamentals: angle and polygon facts (used in combination with algebra), application of pythagorean theorem, special rights, right triangle trig, angle relationships, parallel lines, coordinate geometry. And algebra, because the train they’re on stops next at algebra II.

Lowering the course requirements is not only a rational act, but a sound curriculum decision: educate the kids in what they need to know in order to succeed pass survive have some chance of going through the motions in their next math class.

But according to everyone who has never worked with kids outside that 30%, these geometry teachers are lazy, poorly educated yutzes who don’t really understand geometry because they didn’t major in math or are in the bottom third of college graduates. Or, if they’re being charitable—and remember, Common Core folks are in sell mode, so charity it is—geometry teachers are just dealing with the results of low expectations and math illiterate elementary school teachers.

And so, the Common Core strategy: push half of geometry down to middle school.

Here’s what the Common Core declares: seventh graders will learn complementary and supplementary angles and area facts, and eighth graders will cover transversals, congruence, and similarity.

But wait. Didn’t Common Core standards already shove half of algebra down to middle school? Aren’t these students already learning about isolating variables, systems of equations, power laws, and proportions and ratios? Why yes, they are.

So by virtue of stuffing half of algebra and geometry content into middle school, high school geometry, as conceived by Common Core, is a stripped-down chassis of higher-order conceptual essentials: proofs, construction, modeling, measurement (3 dimensions only, of course), congruence and similarity, and right triangles.

Teachers won’t be able to teach to the lowest common denominator of the standards, not least because their students will now know the meaning of the lowest common denominator, thanks to Common Core’s early introduction of this important concept, but more importantly because the students will already know the basic facts of geometry, thanks to middle school. The geometry teachers will have no choice but to teach constructions, proofs, logic, and all the higher-order skills using those facts, the part of geometry that kids will need, intellectually, in order to be ready for college.

Don’t you see the beauty of this approach? ask the Common Core advocates. Right now, we try to cover all the geometry facts in a year. This way, we’re covering it in three years. Deeper understanding is the key!

High school math teachers treat Common Core much like people who ignored Obamacare until their policy got cancelled. We don’t much care about standards normally: math is math. When the teachers who work with the lower half of the ability spectrum really understand that the new, dramatically reduced algebra and geometry standards are based on the premise that kids will cover a good half of the math now supposedly covered in high school in middle school, that simply by the act of moving this material to middle school, the kids will understand this material deeply and thoroughly, allowing them, the high school teachers, to explore more important topics, they will go out and get drunk. I did that last year when I realized that my state actually was going to spend billions on these tests. I was so sure we’d blink at the money. But no, we’re all in.

Because remember, the low proficiency levels we currently have are not only based on less demanding standards, but they don’t include the kids who don’t get to second year algebra by their junior year. That is, of the juniors taking Algebra II or higher, on a much harder test, we can anticipate horribly low proficiency rates. But what about the kids who didn’t get that far?

In California (I’ll miss their reports), about 216,000 sophomores and juniors were taking either algebra I or geometry in 2012-2013. California doesn’t test its seniors, but to figure out how many seniors weren’t on track, we can approximate by checking 2011-12 scores, and see that about 128,000 juniors were taking either algebra I or geometry, which means they would not have been on track to take an Algebra II test as juniors. That is, in this era of low standards, the standards that Common Core will make even more rigorous, California alone has half a million students right now who wouldn’t have covered all the material by their junior year. So in addition to the many students who are at least on paper on track to take a test that’s going to be far too difficult for–at a conservative guess–half of them, we’ve got the many students who aren’t even able to get to that level of math. (Consider that each state will have to spend money testing juniors who aren’t taking algebra II, who we already know won’t be able to score proficient. Whoo and hoo.)

Is it Common Core supporter’s position that these students who aren’t in algebra II by junior year are by definition not ready for college or career? In addition to the other half million (416,000 or so) California students who are technically on track for Common Core but scored below basic or far below basic on their current tests? We don’t currently tell students who aren’t on track to take algebra II as juniors that they aren’t ready for college. I mean, they aren’t. No question. But we don’t tell them.

According to Arne Duncan, that’s a big problem that Common Core will fix:

We are no longer lying to kids about whether they are ready. Finally, we are telling them the truth, telling their parents the truth, and telling their future employers the truth. Finally, we are holding ourselves accountable to giving our children a true college and career-ready education.

If all we needed to do was tell them, we could do that now. No need for new standards and expensive tests. We could just say to any kid who can’t score 500 on the SAT math section or 23 on the ACT: Hey, sorry. You aren’t ready for college. Probably won’t ever be. Time to go get a job.

If we don’t have the gumption to do that now, what about Common Core will give us the necessary stones? Can I remind everyone again that these kids will be disproportionately black and Hispanic?

I can tell you one thing that Common Core math was designed to do—push us all towards integrated math. It’s very clear that the standards were developed for integrated math, and only the huge pushback forced Common Core standards to provide a traditional curriculum–which is in the appendix. The standards themselves are written in the integrated approach.

So one way to avoid having to acknowledge a group of kids who are by definition not ready for career and college would be to require schools to teach integrated math, as North Carolina has done. That way, we could mask it—just make sure all students are in something called Integrated Math 3 or 4 by junior year. If so, there’s a big problem with that strategy: American math teachers and parents both despise integrated math. I know of at least one school district (not mine) where math coaches spent an entire summer of professional development trying to convince the teachers to adopt an integrated curriculum. The teachers refused and the district reluctantly backed down. Few people have mentioned how similar the CC standards are to the integrated curriculum that Americans have consistently refused. But I do wonder if that was the appeal of an integrated curriculum in the Common Core push—it wouldn’t increase proficiency, but would make it less obvious to everyone how many students aren’t ready. (Of course, that would be lying. Hmm.)

At around this point, Common Core supporters would argue that of course it’s more than just not lying to the kids! It’s the standards themselves! They’re better! Than the lower ones! That more than half our kids are failing!

And we’ll only have to wait eight years to see the results!!!

Eight years?

Yeah, didn’t anyone mention this? That’s when the first year of third graders will become juniors, the first year in which Common Core magic will have run its full reign, and then we’ll see how great these higher standards really are! These problems—they just won’t be problems any more. These are problems caused by our lower standards.

Right.

Or: As we start to get nearer to that eight year mark, we’ll notice that the predictions of full bore Common Core proficiency isn’t signaling. With any luck, elementary school test scores will increase. But as we get nearer and nearer to high school, we’ll see the dreaded fadeout. Faced with results that declare a huge majority of our black and Hispanic students and a solid chunk of white and Asian students are unready for career and college, what will we do?

Naw. That’s eight years out! By that time, reformers will need a next New Thing to keep their donors excited, and politicians will have figured out the racial disproportionality of the whole college and career ready thing. We barely lasted ten years with No Child Left Behind, before we got waivers and the next New Thing. So what New New Thing will everyone be talking about five to six years out, what fingers will they be pointing, in which direction, to explain this failure? I don’t know. But it’s a good bet we’ll get another waiver.

Is it at all possible that the National Governors Association thought up the Common Core as a diversion, an escape route from the NCLB 100% proficiency trap? It’s not like Congress was ever going to get in gear.

But it’s an awfully expensive trap door, if so. Much cheaper to just devise some sort of Truth In Education Act that mandates accurate notification of college readiness, and avoid spending billions on tests and new materials.

Notice how none of this is a public conversation. At the public debate level, the only math-based Common Core opposition argues that the math standards are too easy.

At which point, I suddenly realize I need more beer.


Isometries and Coordinate Geometry

Michael Pershan’s post on teaching congruence reminded me that way back in the beginning of summer, I’d been meaning to write up some of my geometry work, which I think is pretty unusual. Still on the list is the lesson sequencing, but here is some thoughts and sample problems on integrating Isometries and coordinate geometry.

To summarize my earlier work, explicated in Teaching Congruence, or Are You Happy, Professor Wu?, I was unhappy with the circular reasoning that geometry books present in congruence sections. Triangle ABC is congruent with Triangle DEF because all their sides and angles are congruent, and congruence is when the shapes have congruent sides and angles. Professor Wu’s writing taught me the link between congruence, similarities, and isometries (aka, transformations, or translations, rotations, and reflections). I’d previously skipped isometries, since the kids don’t need them much and they’re easy to figure out, but this discovery led me to use isometries as an introduction to congruence and similarity.

But all book chapters on isometries are very thin, or they rely on non-coordinate shapes, which is largely a waste of time. Was there any way I could bring back some other concept while working with isometries, particularly with my top students?

Which leads me straight to coordinate geometry. The most immediate tie-in is helping students figure out rotation, the most difficult of the transformations. A 90 degree rotation around a point involves perpendicular lines (“..and class, what is the relationship of perpendicular slopes, again? Class? Waiting!”). Moreover, the kids learn that the slope of the line connecting a point and its reflection must be perpendicular to the line of reflection. Finally, dilations involve all sorts of work with parallel lines. All of these reinforcements are excellent for weaker students, and are yet another reason to introduce transformations, even if only as a prelude to congruence.

But I wanted a meaningful connection for my top students, who usually grasp the basics quickly. What could I give them that would integrate algebra, coordinate geometry, and a better understanding of transformations?

Over the summer, I taught an enrichment geometry class to seventh graders whose parents got mad because I wasn’t assigning enough homework. My boss backed me–thanks, boss!—and the kids did, too—thanks, kids!—and not for the usual reasons (these are not kids who celebrate a lack of homework). The kids all told the boss that they were surprised that they weren’t able to just follow the pattern and churn out 50 problems of increasing difficulty in the same vein. “I have to really think about the problem,” said more than one, in some astonishment.

So, for example:

Homework: Reflect Triangle LMN [L: (-1,4), M: (0,7), N: (-4, 10) over line y=x+2. Prove it.

So we discussed the steps before they left. I actually posed it as a couple of questions.

  1. If you sketched this and just estimated points the reflection, what would be the key information you’d need to pin down to go from “estimation” to “actual answer”?
  2. Can you think of any coordinate geometry algorithms that might help you find these points?

And working with me, they came up with this procedure for each point:

  1. Find the equation of the line perpendicular to the reflection line.
  2. Find the solution to the reflection line and the perpendicular line. This solution is also the midpoint between the original point and its reflection.
  3. Using the original point and the midpoint, find the reflection point.
  4. Prove the reflection is accurate by establishing that the sides of the original triangle and the reflection are congruent.

And here it is, mapped out in Desmos—but honestly, it was much easier to do on graph paper. I just wanted to increase my own Desmos capability.

reflectionoverxplus2

This is the cleaned up version. Maybe I should put the actual work product here. But I’m not very neat. Next time I’ll take pictures of some of the kids’ work; it’s gorgeous.

When we came in the next day, the kids excitedly told me they’d not only done the work, but “figured out how to do it without the work!” Sure, I said, and we then predicted what would happen with the reflection of y=x+3, y=-x + 4, and so on.

But what about reflecting it over the line y=-2x?

reflectionover2x

Gleesh. I didn’t have time during summer to investigate why the numbers are so ugly. The kids got tired after doing two points, and I told them to use calculators. But we did get it to work. We could see the fractions begin in the perpendicular line solutions, since we’re always adding .5x to 2x. But would it always be like that?

However, I’ve got one great activity for strong kids done–it reinforces knowledge of reflection, coordinate geometry, systems of equations, and some fairly messy algebra. Whoo and hoo.

Down side–for the first time in two years, I’m not teaching geometry this year!

All the more reason to document. Next up in this sequence is my teaching sequence. But if anyone has ideas about the translation that makes the second reflection have such unfriendly numbers, let me know.

Hey, under 1000!


Two Math Teachers Talk

Hand to god, I will finish my post about the reform math fuss I twittered in mid-week, but I am blocked and trying to chop back what I discuss and I want to talk about something fun.

So I will discuss Dale, a fellow math teacher who was a colleague at my last job. Dale is half my age and three days younger than my son. Yes. I have coworkers my son’s age. Shoot me now.

He and I are very different, in that he is an incredibly hot commodity as a math teacher, whose principal would offer him hookers if he’d agree to stay, and gets the AP classes because he’s a real mathematician who majored in math and everything. He turns down the hookers because he’s highly committed to his girlfriend, who is an actual working engineer who uses math every day. I am not a hot commodity, not offered hookers, and not a real mathematician. I also don’t have a girlfriend who is an actual working engineer using math every day, but there’s a lot of qualifiers in that last independent clause so don’t jump to too many conclusions.

He and I are similar in that we both were instantly comfortable with teaching and the broad requirements of working with tough low income kids who don’t want to be in school, and extremely realistic about cognitive ability. We also don’t judge our students for not liking math, or get all moral about kids these days. (Of course, he is a kid).

We are also similar in that we like beer and burgers (he has a lamentable fondness for hops, but no one’s perfect), and still meet once or twice a month at an appropriate locale to talk math. I tell him my new curricular ideas, which he is kind enough to admire although his approach is far more traditional, and ask him math questions, particularly when I was teaching precalc; he tells me that most of the department wants him to be head, despite his youth and relative inexperience. We also talk policy in general. It’s fun.

“I have some news for you,” I told him, “but you will laugh, so you should put down your beer.”

He obligingly takes a pull on his schooner of Lagunitas IPA and sets it down.

“A new study came out,” I said, “and apparently, many high school algebra and geometry courses have titles that don’t actually match the course delivered.”

Dale, who clearly thought I was going in a different direction, did a double take. “Wait. What?”

“The word used was ‘rigor’. Like, some Algebra I courses don’t actually cover algebra I. Same with geometry.”

He looks at me. Takes another pull. “Like, not all algebra teachers actually cover the work formula?”

“Like, not all algebra teachers cover integer operations and fractions for two months. Like not all algebra teachers spend two weeks explaining that 2-5 is not the same as 5-2.”

“Uh huh. Um. They did a study on this?”

“They did.”

“They could have just asked me.”

“They can’t do that. They think math teachers are morons. But there’s more.”

“Of course there is.”

“Apparently, the more blacks and Hispanics and/or low income students are in a class, the less likely the course’s rigor will match the course description.”

He sighs. “I need more beer. Ulysses!” (that’s actually the bartender’s name.) “I’m assuming that nowhere in this study did they even mention the possibility that the students didn’t know the material, that the course content depended on incoming student ability?”

“Well, not in that study. But you know what happens when we point that out.”

“Oh, yeah. ‘It’s all that crap they teach in elementary schools!’ Like that teacher in that meeting you all had the year before I got here. ‘Integer operations and fractions! Damn. Why didn’t I think of that?‘”

“Yes. Actually, the researchers blamed the textbooks, which was a pleasant change from the platitude–and-money-rich reformers who argue our standards are too low.”

“Did anyone ever tell them if it were that simple, whether textbook or teacher, then we could cover the missing material in a few weeks and it’d all be over? Wait, don’t tell me. Of course they told them. That’s the whole premise behind….”

Algebra Support!” we chorused.

“But then there’s that hapless AP calculus teacher stuck teaching algebra support. He spent, what, a month on subtraction?”

“And the happy news was that at the end of the semester, the freshmen went from getting 40% right on a sixth grade math test to 55%.”

“The bad news being at the end of the year, they forgot it all. Net improvement, what–2 points?”

“Hell, I spend the entire Algebra II course teaching mostly Algebra I, and while they learn a lot, at the end of the course they’re still shaky on graphing lines and binomial multiplication. And I don’t even bother trying to teach negative numbers, although I do try to show them why the inequality sign flips in inequalities.”

“But it’s our fault, right?”

“Of course. But that’s not the best part.”

“There’s a best part?”

“If you like black comedy.”

“The Bill Cosby sort, or the Richard Pryor catching himself on fire sort?”

“Someone doesn’t know his literary genres.”

“Hey, we can’t all be English majors. What’s the best part?”

“The best part is that Common Core is supposed to fix all this.”

“Common Core? How?”

“By telling us teachers what we’re supposed to teach.”

I’d forgotten to warn Dale, who was mid-gulp. “WHAT???”

I handed him a napkin. “You’ve got beer coming out your nose. Yes. Checker Finn and Mike Petrilli always use this example of the shifty, devious schools that, when faced with a 3-year math requirement, just spread two years of instruction over three!”

“Wow. That’s painful.”

“Well, they don’t much care for unions, either, so I guess they think that when faced with a mandate that’s essentially a jobs program for math teachers, we teachers use it as an opportunity to kick back. But that’s when they are feeling uncharitable. Sometimes, when they’re trying to puff teachers up, they worry that teachers will need professional development in order to know the new material.”

“How to teach it?”

“No. The new material.”

“They think we don’t know the new material?

“Remember, they think math teachers are morons. On the plus side, they think we’re the smartest of teachers. (Which we are, but that’s another subject.) There’s still other folks who complain because ed schools don’t teach teachers the material they’re supposed to be teaching.”

“But we know that material. That’s what credential tests are for. You can’t even get into a program without passing the credential test.”

“Do not get me started.”

“So when the test scores tank, they’ll say it’s because teachers don’t know the material?”

“Well, they’ve got the backup teachers don’t have the proper material to teach the standards, in case someone points out the logical flaws in the ‘teacher don’t know the material’ argument.”

“Sure. If it ain’t in the textbook, we don’t know it’s supposed to be taught!”

“Don’t depress me. Yes, either we don’t know what’s supposed to be taught or we don’t know how to teach it without textbooks telling us to.”

Dale starts to laugh in serious. “I’m sorry, Governor. I would have taught vectors in geometry, but since it wasn’t on the standards, I taught another week of the midpoint formula.”

“I’m sorry, parents, I would have dropped linear equations entirely from my algebra two class, but I didn’t know they were supposed to learn it in algebra one!”

“Damn. A whole three weeks spent teaching fraction operations in algebra when it’s fifth grade math. I could have spent that time showing them how to find a quadratic equation from points!”

“I didn’t know proofs were a geometry standard. Why didn’t someone tell me? Here I had so much free time I taught my kids multi-step equations because my only other option was showing an Adam Sandler movie!”

“Stop, you’re killing me.”

“No, there’s too many more. Who the hell went and added conics to the standards and why wasn’t I informed? Here I spent all this time teaching my algebra II kids that a system of equations is solved by finding the points of intersection? Apparently, my kids didn’t bother to tell me that they’d mastered that material in algebra I.”

“I can’t believe it! Four weeks killed teaching kids the difference between a positive and a negative slope! Little bastards could have told me they knew it but no, they just let me explain it again. No wonder they acted out–they were bored!”

My turn to snarf my beer.

“Jesus, Ed, I’ve wondered why we’re pulling this Common Core crap, but not in my deepest, most cynical moments did I think it because they thought we teachers just might not know what to teach the kids.”

“That’s not the most depressing, cynical thought. Really cynical is that everyone knows it won’t work but the feds need to push the can—the acknowledgement that achievement gaps are largely cognitive—down the road a few more years, and everyone else sees this as a way to scam government dollars.”

“New texbooks! New PD. A pretense that technology can help!”

“Exactly. I’d think maybe it was another effort to blame unions, but no.”

“Yeah, Republicans mostly oppose the standards.”

“Well, except the ‘far-seeing Republicans’ who just want what’s best for the country. Who also are in favor of ‘immigration reform’.”

“Jeb Bush.”

“Bingo. You’ll be happy to know that libertarians hate Common Core.”

“Rock on, my people!”

“Yeah, but they want also want open borders and privatized education.”

“Eh, nobody’s perfect.”

“But all that depressing cynicism is no fun, so let me just say that I would have taught sigma notation except I thought that letter was epsilon!”

“Hey, wait. You do get sigma and epsilon confused!”

“No, I don’t, or I wouldn’t call the pointy E stuff sigma notation, dammit. I just see either E shape out of context and think epsilon. Why the hell did Greeks have two Es, and why couldn’t they give them names that start with E? Besides, the only two greek letters I have to deal with are pi and theta, and really, in right triangle trig there’s no difference between theta and x.”

“Well, you’re going to have to stop making that mistake because thanks to Common Core, you’ll know that you’re supposed to teach sequences and series.”

“Damn. So I won’t be able to teach them binomial multiplication and factoring and let them kick back and mock me with their knowledge, which they have because they learned it all in algebra I.”

“Here’s to Common Core and math research. Without them, America wouldn’t be able to kid itself.”

We clinked glasses just as Maya, Dale’s girlfriend walked in, a woman who actually uses centroids, orthocenters, and piece-wise equations in her daily employment. The rest of the evening was spent discussing my search for more real-life models of quadratics that don’t involve knowing the quadratic formula first. She offered road construction and fruit ripening, which are very promising, but I still need something organic (haha), if possible, to derive the base equation. So far area and perimeter problems are my best bet, which gives me a good chance to review formulas, because until Common Core comes out I won’t know that they learned this in geometry. I wondered if velocity problems could be used to derive it. Dale warned me that it involved derivations. Maya was confused by my describing velocity problems as “-16 problems”, since gravity is either gravity is either 32 ft/sec/sec or 9.8 m/s/s. Dale interpreted. I’m like Jeez, there are people who know what gravity is off the top of their heads? This is why I don’t teach science. (edit: I KNEW I should have checked the numbers. I don’t do physics or real math, dammit. Fixed. )

But all that’s for another, happier, post.


Algebra 1 Growth in Geometry and Algebra II, Spring 2013

This is part of an ongoing series on my Algebra II and Geometry classes. By definition, students in these classes should have some level of competence in Algebra I. I’ve been tracking their progress on an algebra I pre-assessment test. The test assesses student ability to evaluate and substitute, use PEMDAS, solve simple equations, operate with negative integers, combine like terms. It tiptoes into first semester algebra—linear equations, simple systems, basic quadratic factoring—but the bulk of the 50 questions involve pre-algebra. While I used the test at my last school, I only thought of tracking student progress this year. My school is on a full-block schedule, which means we teach a year’s content in a semester, then repeat the whole cycle with another group of students. A usual teacher schedule is three daily 90-minute classes, with a fourth period prep. I taught one algebra II and one geometry class first semester (the third class prepared low ability students for a math graduation test), their results are here.

So in round two, I taught two Algebra 2 courses and one Geometry 10-12 (as well as a precalc class not part of this analysis). My first geometry class was freshmen only. In my last school, only freshmen who scored advanced or proficient on their 8th grade algebra test were put into geometry, while the rest take another year of algebra. In this school, all a kid has to do is pass algebra to be put into geometry, but we offer both honors and regular geometry. So my first semester class, Geometry 9, was filled with well-behaved kids with extremely poor algebra skills, as well as a quarter or so kids who had stronger skills but weren’t interested in taking honors.

I was originally expecting my Geometry 10-12 class to be extremely low ability and so wasn’t surprised to see they had a lower average incoming score. However, the class contained 6 kids who had taken Honors Geometry as freshmen—and failed. Why? They didn’t do their homework. “Plus, proofs. Hated proofs. Boring,” said one. These kids knew the entire geometry fact base, whether or not they grokked proofs, which they will never use again. I can’t figure out how to look up their state test scores yet, but I’m betting they got basic or higher in geometry last year. But because they were put into Honors, they have to take geometry twice. Couldn’t they have been given a C in regular geometry and moved on?

But I digress. Remember that I focus on number wrong, not number right, so a decrease is good.

Alg2GeomAlg1Progress

Again, I offer up as evidence that my students may or may not have learned geometry and second year algebra, but they know a whole lot more basic algebra than they did when they entered my class. Fortunately, my test scores weren’t obliterated this semester, so I have individual student progress to offer.

I wasn’t sure the best way to do this, so I did a scatter plot with data labels to easily show student before/after scores. The data labels aren’t reliably above or below the point, but you shouldn’t have to guess which label belongs to which point.

So in case you’re like me and have a horrible time reading these graphs, scores far over to the right on the x-axis are those who did poorly the first time. Scores low on the y-axis are those who did well the second time. So high right corner are the weak students at both beginning and end. The low left corner are the strong students who did well on both.

Geometry first. Thirty one students took both tests.

Spring2013GeomIndImprovement

Four students saw no improvement, another four actually got more wrong, although just 1 or 2 more. Another 3 students saw just one point improvement. But notice that through the middle range, almost all the students saw enormous improvement: twelve students, over a third, got from five to sixteen more correct answers, that is, improved from 10% to over 30%.

Now Algebra 2. Forty eight students took both tests; I had more testers at the end than the beginning; about ten students started a few days late.

Spring2013A2IndImprovement

Seven got exactly the same score both times, but only three declined (one of them a surprising 5 points—she was a good student. Must not have been feeling well). Eighteen (also a third) saw improvements of 5 to 16 points.

The average improvement was larger for the Algebra 2 classes than the Geometry classes, but not by much. Odd, considering that I’m actually teaching algebra, directly covering some of the topics in the test. In another sense, not so surprising, given that I am actually tasked to teach an entirely different topic in both cases. I ain’t teaching to this test. Still, I am puzzled that my algebra II students consistently show similar progress to my geometry students, even though they are soaked in the subject and my geometry students aren’t (although they are taught far more algebra than is usual for a geometry class).

I have two possible answers. Algebra 2 is insanely complex compared to geometry, particularly given I teach a very slimmed-down version of geometry. The kids have more to keep track of. This may lead to greater confusion and difficulty retaining what they’ve learned.

The other possibility is one I am reminded of by a beer-drinking buddy, a serious mathematician who is also teaches math: namely, that I’m a kickass geometry teacher. He bases this assertion on a few short observations of my classes and extensive discussions, fueled by many tankards of ale, of my methods and conceptual approaches (eg: Real-life coordinate Geometry, Geometry: Starting Off, Teaching Geometry,Teaching Congruence or Are You Happy, Professor Wu?, Kicking Off Triangles, Teaching Trig).

This possibility is a tad painful to contemplate. Fully half the classes I’ve taught in my four years of teaching—twelve out of twenty four—have been some form of Algebra, either actual Algebra I or Algebra I pretending to be Algebra II. I spend hours thinking about teaching algebra, about making it more understandable, and I believe I’ve had some success (see my various posts on modeling).

Six of those 24 classes have been geometry. Now, I spend time thinking about geometry, too, but not nearly as much, and here’s the terrible truth: when I come up with a new method to teach geometry, whether it be an explanation or a model, it works for a whole lot longer than my methods in algebra.

For example, I have used all the old standbys for identifying slope direction, as well as devising a few of my own, and the kids are STILL doing the mental equivalent of tossing a coin to determine if it’s positive or negative. But when I teach my kids how to find the opposite and adjacent legs of an angle (see “teaching Trig” above), the kids are still remembering it months later.

It is to weep.

I comfort myself with a few thoughts. First, it’s kind of cool being a kickass geometry teacher, if that is my fate. It’s a fun class that I can sculpt to my own design, unlike algebra, which has a billion moving parts everyone needs again.

Second, my algebra II kids say without exception that they understand more algebra than they ever did in the past, that they are willing to try when before they just gave up. Even the top kids who should be in a different class tell me they’ve learned more concepts than before, when they tended to just plug and play. My algebra 2 kids are often taking math placement tests as they go off to college, and I track their results. Few of them are ending up in more than one class out of the hunt, which would be my goal for them, and the best are placing out of remediation altogether. So I am doing something right.

And suddenly, I am reminded of my year teaching all algebra, all the time, and the results. My results look mediocre, yet the school has a stunningly successful year based on algebra growth in Hispanic and ELL students—and I taught the most algebra students and the most of those particular categories.

Maybe what I get is what growth looks like for the bottom 75% of the ability/incentive curve.

Eh. I’ll keep mulling that one. And, as always, spend countless hours trying to think up conceptual and procedural explanations that sticks.

I almost titled this post “Why Merit Pay and Value Added Assessment Won’t Work, Part IA” because if you are paying attention, that conclusion is obvious. But after starting a rant, I decided to leave it for another post.

Also glaringly on display to anyone not ignorant, willfully obtuse, or deliberately lying: Common Core standards are irrelevant. I’d be cynically neutral on them because hell, I’m not going to change what I do, except the tests will cost a fortune, so go forth ye Tea Partiers, ye anti-test progressives, and kill them standards daid.


Spring 2013: These students aren’t really prepared, either.

I’m teaching Geometry and Algebra II again, so I gave the same assessment and got these results, with the beginning scores from the previous semester:

AlgAssessspr13

I’m teaching two algebra II classes, but their numbers were pretty close to identical—one class had the larger range and a lower mode—so I combined them.

The geometry averages are significantly lower than the fall freshmen only class, which isn’t surprising. Kids who move onto geometry from 8th grade algebra are more likely to be stronger math students, although (key plot point) in many schools, the difference between moving on and staying back in algebra come down to behavior, not math ability. At my last school, kids who didn’t score Proficient or Advanced had to take Algebra in 9th grade. I’d have included Basic kids in the “move-on” list as well. But sophomores who not only can’t factor or graph a line, but struggle with simple substition ought not to be in second year algebra. They should repeat algebra I freshman year, go onto geometry, and then take algebra II in junior year—at which point, they’d still be very weak in algebra, of course, but some would have benefited from that second year of first year.

Wait, what was my point? Oh, yeah–this geometry class class is 10-12, so the students took one or more years of high school algebra. Some of them will have just goofed around and flunked algebra despite perfectly adequate to good skills, but a good number will also be genuinely weak at math.

On the other hand, a number of them really enjoyed my first activity: visualizing intersecting planes, graphing 3-D points. I got far more samples from this class. I’ll put those in another post, also the precalc assessment.

I don’t know if my readers (I have an audience! whoo!) understand my intent in publishing these assessment results. In no way am I complaining about my students.

My point in a huge nutshell: how can math teachers be assessed on “value-added” when the testing instrument will not measure what the students needed to learn? Last semester, my students made tremendous gains in first year algebra knowledge. They also learned geometry and second year algebra, but over half my students in both classes will test Below Basic or Far Below Basic–just as they did the year before. My evaluation will faithfully record that my students made no progress—that they tested BB or FBB the year before, and test the same (or worse) now. I will get no credit for the huge gains they made in pre-algebra and algebra competency, because educational policy doesn’t recognize the existence of kids taking second year algebra despite being barely functional in pre-algebra.

The reformers’ response:

1) These kids just had bad teachers who didn’t teach them anything, and in the Brave New World of Reform, these bad teachers won’t be able to ruin students’ lives;

2) These bad teachers just shuffled students who hadn’t learned onto the next class, and in the Brave New World of Reform, kids who can’t do the work won’t pass the class.

My response:

1) Well, truthfully, I think this response is moronic. But more politely, this answer requires willful belief in a delusional myth.

2) Fail 50-60% of kids who are forced to take math classes against their will? Seriously? This answer requires a willful refusal to think things through. Most high schools require a student to take and pass three years of math for graduation. Fail a kid just once, and the margin for error disappears. Fail twice and the kid can’t graduate. And in many states, the sequence must start with algebra—pre-algebra at best. So we are supposed to teach all students, regardless of ability, three years of increasingly abstract math and fail them if they don’t achieve basic proficiency. If, god save us, the country was ever stupid enough to go down this reformer path, the resulting bloodbath would end the policy in a year. We’re not talking the occasional malcontent, but over half of a graduating class in some schools—overwhelmingly, this policy impacts black and Hispanic students. But it’s okay. We’re just doing it for their own good, right? Await the disparate impact lawsuits—or, more likely, federal investigation and oversight.

Reformers faithfully hold out this hope: bad teachers are creating lazy students who could do the work but just don’t want to. Oh, yeah, and if we catch them in elementary school, they’ll be fine in high school.

It is to weep.

Hey, under 1000 words!


Algebra 1 Growth in Geometry and Algebra II

Last September, I wrote about my classes and the pre-algebra/Algebra 1 assessment results.

My school covers a year of instruction in a semester, so we just finished the first “year” of courses. I start with new students and four preps on Monday. Last week, I gave them the same assessment to see if they’d improved.

Unfortunately, the hard drive on my school computer got wiped in a re-imaging. This shouldn’t have been a problem, because I shouldn’t have had any data on the hard drive, except I never got put on the network. Happily, I use Dropbox for all my curriculum development, so an entire year’s worth of intellectual property wasn’t obliterated. I only lost the original assessment results, which I had accidentally stored on the school hard drive. I should have entered the scores in the school grading system (with a 0 weight, since they don’t count towards the grade) but only did that for geometry, the only class I can directly compare results with.

My algebra II class, though, was incredibly stable. I only lost three students, one of whom got a perfect score—which the only new addition to the class also got, so balance maintained. The other two students who left got around 10-15 wrong, so were squarely in the average at the time. I feel pretty comfortable that the original scores didn’t change substantially. My geometry class did have some major additions and removals, but since I had their scores I could recalculate.

Mean

Median

Mode

Range
Original

just above 10

9.5

7

22
Recalculated

just below 10 (9.8)

8

7

22

I didn’t have the Math Support scores, and enough students didn’t take the second test that comparisons would be pointless.

One confession: Two Algebra II students, the weakest two in the class, who did no work, scored 23 and 24 wrong, which was 11 more than the next lowest score. Their scores added an entire point to the average wrong, increased the range by 14 points, and you know, I just said bye and stopped them from distorting the results the other 32 kids. (I don’t remember exactly, but the original A2 tests had five or six 20+ wrong scores.)

So here’s the original September graph and the new graph of January:

AlgtestAlgAssessyrend

The geometry class was bimodal: 0 and 10. Excel refused to acknowledge this and I wasn’t sure how to force it. The 10s, as a group, were pretty consistent—only one of them improved by more than a point. The perfect scores ranged from 8 wrong to 2 wrong on the first test.

geoalgclassgrowth

In short, they learned a lot of first year algebra, and that’s because I spent quite a bit of time teaching them first year algebra. In Algebra II, I did it with data modeling, which was a much more sophisticated approach than what they’d had before, but it was still first year algebra. In geometry, I minimize certain standards (proofs, circles, solid shapes) in favor of applied geometry problems with lots of algebra.

And for all that improvement, a still distressing number of students answered x2 + 12 when asked what the product of (x+3) and (x+4) was, including two students who got an A in the class. I beat this into their heads, and STILL some of them forget that.

Some folks are going to draw exactly the wrong impression. “See?” these misguided souls will say, nodding wisely. “Our kids just aren’t being taught properly in early grades. Better standards, better teachers, this problems’s fixed! Until then, this poor teacher has to make up the slack.” In short, these poor fools still believe in the myth that they’ve never been taught.

When in fact, they were taught. Including by me—and I don’t mean the “hey, by the way, don’t forget the middle term in binomial multiplication”, but “you are clubbing orphan seals and making baby Jesus cry when you forget the middle term” while banging myself on the head with a whiteboard. And some of them just forgot anyway.

I don’t know how my kids will do on their state tests, but it’s safe to say that the geometry and second year algebra I exposed them to was considerably less than it would have been had their assessment scores at the beginning of class been the ones they got at the end of class. And because no one wants to acknowledge the huge deficit half or more of each class has in advanced high school math, high schools won’t be able to teach the kids the skills they need in the classes they need—namely, prealgebra for a year, “first year” algebra for two years, and then maybe some geometry and second year algebra. If they do okay on the earlier stuff.

Instead, high schools are forced to pretend that transcripts reflect reality, that all kids in geometry classes are capable of passing a pre-algebra test, much less an algebra one test. Meanwhile, reformers won’t know that I improved my kids’ basic algebra skills whilst still teaching them a lot of geometry/algebra II, because the tests they’ll insist on judging me with will assume a) that the kids had that earlier material mastered or b) that I could just catch them up quickly because after all, the only problem was the kids’ earlier teachers had never taught them.


Teaching Congruence, or Are You Happy, Professor Wu?

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

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

More:

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

Abuse victims grow up to be abusers, you know?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

RotationPos1RotationPos2RotationPos3

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

RotationBeforeAfter

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

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

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

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

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

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

WuSimilarity

Day One: Review of Proportionality, then onto dilation

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

DilationSimilarTransversals

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

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

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

Day 2: Review of Dilation, then onto Similarity.

Done.

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

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

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

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


Geometry: Starting Off

The first day or two of geometry is always point line plane. We never really use it again. Geometry has mostly been subordinated to algebra in high school, as I’ve written before, and my geometry class is best thought of as algebra applications with geometry. Or is it the other way round? Purists see geometry as the medium for introducing proofs, logic, and construction. To which I say pish tosh. Most of them are never going to see those subjects again. “But if they don’t learn rigorous logic in geometry, they won’t be able to learn advanced math!” Yeah, that’s moronic nonsense. What is “solve for x”, if not a proof?

But I love history, so I always start by telling them to put their pencils down and just listen as I explain the significance of Euclid’s Elements and the wonder of a book written 2300 years ago. Three hundred years ago is older than our country. Euclid wrote Elements 300 years before the birth of Christ, so Christ’s contemporaries (the educated ones) thought of Euclid much the way we think of Alexander Hamilton or George Washington. Take seven additional chunks of people looking back 300 years and here we are. A book written that long ago was “in print” over 1000 years before “print” existed, and since then, is second only to the Bible in published editions—not just in the English language, which had to wait another 100 years after the Latin version was published, but in all languages.

As to writing another book on geometry [to replace Euclid] the middle ages would have as soon thought of composing another New Testament.–Augustus de Morgan

Why? Because he* nailed it. For over 2000 years, his model met the world’s requirements, and when the world finally found limits to his model, it wasn’t because he was wrong.

Euclid was nagged by his “fifth postulate”, which is easier to sketch than describe:

That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

If you’re not a mathematician—and I am not—you’re like, um, duh? What else is going to happen? The lines will meet up. But Euclid and other early mathematicians knew the fifth postulate wasn’t the same as the other four, and that’s almost certainly why he established the first 28 theorems without reference to it. For the next couple millennia, mathematicians tried to prove the fifth postulate using the other four, and failed. This collected history of effort around that single postulate ultimately led to the realization that there other, non-Euclidean geometries, many of which (if I understand this properly) begin with the negation of the fifth postulate. This discovery rocked the world, robbed it of a truth previously assumed absolute, and ultimately contributed a bit to Einstein’s theory of relativity.

And 2300 years ago, Euclid needed the fifth postulate to complete his model, furrowed his brow and said, “Yeah, hmm. Something’s not right about that one.”

I tell my students that I’m not a mathematician, and that they don’t need to be, either, in order to realize what a stunning achievement Elements is, and to realize the significance of math in our world that thousands of years ago, mathematicians knew enough to be bothered by a postulate that seemed obvious but was yet somehow different from the others needed for the model. That they, my students, are studying an incredibly old math, one that holds up for our ordinary requirements to this day, but also created the foundation for deeper, more complex models. That if they don’t like math, don’t like geometry, to at least appreciate it from a historical standpoint.

I am probably fooling myself a little bit, but the kids always seem interested. Which is all I’m looking for. Just to show I’m not making this up, here are my board notes:

(Yes, my board work sucks. It’s something I build as I go through a lecture most of the time, a document in progress. I’ve started taking pictures of my boardwork to get a better sense of what I said, what I emphasized, and what I could do to improve boardwork next time.)

Then I go onto undefined terms—not just the terms for geometry, but the meaning of undefined terms. Here, again, Euclid nails the building blocks for his model. (Geometry books give point, line, and plane as the three undefined terms, but I also spend time on “congruence” and “between”.)

Then I show how the building blocks of the undefined terms allow us to define everything else in the Eucliden model. I usually use ray, segment, and angle just to give the the idea.

This year, I decided I wanted to do more with 3-dimensional graphing (xyz) and introduced it as part of this lecture. First, the students learned to represent three dimensional planes without a coordinate system, and see for themselves what happened when two planes intersected. The kids had fun with that; here’s one of the best:

Then we went into more formal xyz graphing. I’m including more 3-d graphing this year to help prepare students for 3 variable systems next year, and also to give the students more variety in visualizing images. Click on the board work below to see that I draw in the rectangular prism, which helps students grasp the difference between 2-axis graphing, in which any two points are a diagonal in a rectangle, and 3-axis graphing, in which any two points are the vertices of a rectangular prism. I heard a lot of “ahas” as I went through this. Not sure what the next 3-d graphing activity will be, but I think I’ve started with a good foundation.

So that was the first day, really. Then I went into the meat of unit one: angle types, angle pairs, perimeter and area formulas, and as always, using these relationships to set up equations and solve with the ever loved algebra.

Here’s the first test. I think I caught all the glitches after I captured this. But I’m sure I missed something; I have a pathological tolerance for typos.

*I’m assuming it was just Euclid. More fun that way.


My math classes: are they prepared? Um. No. So what?

After over a month talking about policy, it’s fun to write about math classes for a change.

I’m teaching geometry, Algebra II, and Math For Kids Who Haven’t Passed The State Graduation Test Yet.

I’ve given this algebra readiness test to all my students for the past three years. I got it from a senior teacher at my last job, and it’s an excellent assessment of a students’ basic numeracy and first semester algebra skills. Can they substitute? work with negatives? multiply binomials? factor a quadratic? I don’t much care about second semester algebra (graphing parabolas, quadratic formula); my geometry students won’t need it, and my algebra II students will be reviewing the material again.

I know what some of you are thinking. “Why the heck are you giving your geometry and algebra II students a test in pre-algebra and first semester algebra? They already know that material, don’t they?”

This is me laughing at you naive folks. Ha ha!

Or, I could show you a graph of the results.

So the algebra II kids have taken a full year more of math than the geometry kids, and both groups have passed algebra. But the algebra II class is usually taken by kids who made it this far by their toenails, got low scores in both algebra and geometry. Geometry 9 is ninth graders who passed algebra the first time but chose not to take the honors course—and who aren’t in A2/Trig.

My Algebra II class is substantially stronger on average than last year’s class, which averaged around 20 wrong. I’ve got about 8 kids that got 0-2 wrong after finishing the test in ten minutes and should be taking A2/Trig, or even Honors. The Geometry 9 class is slightly stronger on average than my class from last year, which averaged around 12-13 wrong. My geometry classes last year were the strongest I’ve ever taught, but had a much weaker bottom than this class does. The strongest students in the Math Support class got 13-15 wrong, which is impressive.

For the uninitiated, there are two big pieces of info in this graph:

In all three classes but particularly the A2 and Geo, the range of scores on what should be an easy test is huge. The weakest students in both geometry and A2 got barely half right. I’m used to this—handling wide ranges in ability is probably my greatest strength as a teacher, although administrators don’t value it much. However, really think about that range and what it represents in terms of the ability gap within one classroom. And remember–this is a school that provides honors courses, so it tracks much more than my last two schools. The gap this year is considerably less than the scores from last year (which I can’t find, so you’ll have to take my word for it).

The other news is, of course, that the average score for the geometry class should be 5-6 wrong, and the algebra II students should, in a world where we worry more about what kids learn than what their transcript says, knock the test out of the park.

About half the kids in both geometry and algebra II classes should not be taking formal college prep courses, but rather an interesting math applications course, in which they continue to apply what they’ve already learned, rather than pile on new stuff.

Oh, well. That’s what we get for pretending ability doesn’t matter.

I don’t want to sound cynical or discouraged. I’m pumped. They’re a great group of kids and are stronger on average than my last year’s kids, with lots of high achievers. But what’s “normal” for me is clearly not anything that reformers understand or anticipate when they talk about high expectations or proficiency for all.