Wednesday, April 6, 2016

Lesson 11-3: Equations for Circles (Day 134)

This is what I wrote last year about today's lesson. I had to edit out references to the PARCC PBA and EOY exams, as there is now only a single PARCC EOY exam:

Now that spring break is over, we've now reached the "Long April/May" -- since for the school whose calendar this blog is following, this stretch, from now until Memorial Day, is actually the longest stretch without a break, as opposed to the "Long March" most other years.

Today's lesson continues a short mini-unit on circles and spheres, to reflect the fact that these lessons on circles appear on the PARCC End-of-Year (EOY). There are three lessons that need to be covered here.

The first circle lesson is on Lesson 11-3 of the U of Chicago text, on Equations of Circles. I mentioned that I wanted to skip this because I considered equations of circles to be more like Algebra II than Geometry. Yet equations of circles appear on the PARCC EOY exam.

Furthermore, I see that there are some circle equations on the PARCC exam that actually require the student to complete the square! For example, in Example 1 of the U of Chicago text, we have the equation x^2 + (y + 4)^2 = 49 for a circle centered at (0, -4) of radius 7. But this equation could also be written as x^2 + y^2 + 8y = 33. We have to complete the square before we can identify the center and radius of this circle.

In theory, the students already learned how to complete the square to solve quadratic equations the previous year, in Algebra I. But among the three algebraic methods of solving quadratic equations -- factoring, completing the square, and using the quadratic formula -- I believe that completing the square is the one that students are least likely to remember. In fact, back when I was student teaching, my Algebra I class had fallen behind and we ended up skipping completing the square -- covering only factoring and the quadratic formula to solve equations. And yet PARCC expects the students to complete the square on the Geometry test!

I also wonder whether it's desirable, in Algebra I, to teach factoring and completing the square, but possibly save the Quadratic Formula for Algebra II. This way, the students would have at least seen completing the square in Algebra I before applying it to today's Geometry lesson. Notice that the Quadratic Formula doesn't appear on the SAT, since the SAT assumes that students took Algebra I as freshmen and are halfway through Algebra II when they take the test as juniors. So any topic that appears during the second semester of Algebra II -- including the Quadratic Formula -- doesn't appear on the test. [2016 Update: All of this refers to the last year's version of the SAT. I'll have more to say about the SAT later in this post.]

But what about the PARCC test for Algebra I -- does the Quadratic Formula appear there? I took a quick look at the EOY test for Algebra I, and at least one question that asks a student to convert a quadratic equation from standard into vertex form, which is often done using completing the square (but this could also be done by using x = -b / 2a, plugging it into the original equation to find y, and then letting these values be h and k in the vertex formula). I also saw a few problems that appeared to be inappropriate for an Algebra I test and looked more suitable for a higher-level class.

So this goes right back to the Common Core debate. What level of math should students be expected to master at each level? There is a poster who goes by the username SteveH, who posts at the traditionalist website Kitchen Table Math. Here's a detailed discussion of this issue by SteveH:

They could have found schools that produce good numbers of Calc AB and BC students with scores of 3 or higher and detailed their high school math curricula in terms of specific textbooks and syllabi. They could show the number of students who got 3’s or higher on the AP Calc or AP Stat tests who did NOT take algebra in 8th grade. Then they could ask the parents of the successful students what specific support they had to provide at home or with tutors to even get their kids to algebra in 8th grade. This is the hidden tracking and mapping that educational pedagogues specifically overlook with weasel word mappings. They just point to successful students and claim them for their own. My son must be his old school’s poster boy for Everyday Math.

Like many other traditionalists, SteveH wants to make sure that students are able to reach AP Calculus in senior year. One problem with these current PARCC tests, by including some of these harder problems on the Algebra I test, is that schools then say that the Common Core Algebra I test is too difficult for eighth graders, so they wait until ninth grade to let them take Algebra I. Then the students can never reach AP Calculus.

So we see SteveH's proposal here -- he writes that there should be a survey of students, who not only took AP Calculus but passed with with a 3 or higher, that asks them what they math they took prior to Calculus to attain that goal. Then one should have written the math standards to reflect the levels of math given by students in the survey. SteveH's mention of "specific support they had to provide at home of with tutors" refers to students whose elementary schools offer progressive math curricula, such as the U of Chicago's elementary texts, so that parents would have to supplement this with traditionalist (instructivist) math lessons at home. The idea, of course, is that the elementary standards should be rewritten to support more strongly a traditionalist pedagogy.

SteveH's idea, on one hand, is appealing. One criticism of the conversion to Common Core is that parents feel that their students are being treated like guinea pigs. Of course, whether we have Common Core or another set of standards, some class of students has to be the first to use the standards, and the parents of the first class will feel that their students are "guinea pigs" for being the first to use such untested standards -- so there could be no innovation without guinea pigs. But suppose we were to replace Common Core with a SteveH Core based on the survey mentioned in the paragraph that SteveH wrote. Since the SteveH Core Standards would be based on what actual students said they took in the survey, they wouldn't be untested standards -- so the first class of students who learned them would not be guinea pigs!

On the other hand, here are a few things I have to say about the SteveH proposal:

-- SteveH mentioned AP Statistics in his post. Is it possible for students to take Algebra I in ninth grade and still make it to AP Stat? Of course, that's what the survey would find out.
-- Would Integrated Math still exist under the SteveH Core? I bet it's possible for a homeschooled student to make it to AP Calculus, yet learned under the Singapore or Saxon math curricula, which favor the integrated pathway.
-- Why does SteveH find it so important for students to reach AP Calculus, anyway? He writes:

The low expectations start in Kindergarten and that creates adults who will never have that opportunity. By seventh grade it’s all over for most students.

That is, math standards that don't lead to Calculus end up closing doors for students, since it's unlikely for a student to get into a competitive college and attain a STEM major, and thus a STEM career, without having had Calculus senior year.

But a counterargument could be that forcing students to take Algebra I in eighth grade, Algebra II in tenth grade, and so on, actually closes doors for students. For example, a student who plans on having a non-STEM job that requires no math higher than arithmetic may wish to participate in sports or other extracurricular activities, but can't because the low Algebra II grade in sophomore year is pushing the GPA below 2.0. Or the student may want an after school job, but the parents won't let their child get one after they see the "D" or "F" in math on the report card.

I have no problem with wanting to get students to Calculus, but I wonder whether it's possible to keep the doors leading to STEM open without closing any non-STEM door.

And suddenly this has turned into yet another unexpected post about traditionalists. This often happens on the blog -- I want to reuse material from last year to set up this year's curriculum, only to find that I wrote about traditionalists. I suppose that this week's material lends itself to traditionalist criticism -- first yesterday's lesson on Cavalieri's Principle and then today's where we're pushing Algebra II material down into Geometry.

First of all, SteveH does still occasionally post at the Kitchen Table Math website. His most recent post there was in a long heated debate spanning from November to January:

The original post was about the phrase "sit and get," a phrase that progressives sometimes use to denigrate traditionalist pedagogy, along with the more common "drill and kill," and "guide on the side" as a phrase progressives use to describe themselves. On the other hand, traditionalists use the phrases "drill and practice" and "sage on the stage" to refer to their own pedagogy.

The progressive Michael Goldenberg writes:

Further, how would teachers who never heard of or considered any other sort of classroom develop slogans about such classrooms? Only those of us who experienced the traditional approach and saw that it was wanting would have motivation for summarizing in a few words what it was, with the implication of what was wrong with it implicit in those words. I guess you can call those "slogans," but I think that's being overly polite. 

On the other hand, no one awake in the 1990s who was interested in mathematics education is likely to have missed all the demeaning and dismissive slogans that groups like Mathematically Correct and NYC-HOLD [both traditionalist -- dw] came up with in regards to "reform" math education. Indeed, the old 2+2=4 website of Mathematically Correct had an entire page of such epithets, as I believe you know perfectly well.

And then SteveH takes up the traditionalist side of the debate:

But they were entirely correct considering curricula like MathLand, which was so bad that it disappeared with nobody claiming responsibility for it only to be replaced by curricula like Everyday Math [U of Chicago elementary texts -- dw] that tells teachers to "trust the spiral" and allows kids to get to fifth grade without knowing the times table. And now we have CCSS implementations like PARCC which officially declare that K-6 is a NO-STEM zone and their top level ("distinguished") only means that students are likely to pass a college algebra course - all the while talking about "problem solving" and "understanding" as if they've figured out some royal road to math. All of this just means that only affluent or educated parents have a chance to prepare their kids for a STEM career. Is that "polite?" My "math brain" son had to have a lot of help at home from his parents to survive math in K-6 math and now his schools claim that he is an exemplar of Everyday math. Nobody asked his parents. They are not interested in the truth.

The problem is that parents and students have no choice in the matter. It's not like future teachers in ed school all "discover" this sort of educational philosophy. No. They are directly taught the pedagogy by rote. It now defines their turf, which isn't about mastery of anything close to STEM-level content and skills in K-6 math.

All of this would be a non-issue if people had choice. Is it "polite" to force all students to accept one approach to math?" It wasn't very polite many of the comments made to my wife and I about education. They were specifically designed to get us to go away. They were personally demeaning.

Your attempt to position modern reform math as unappreciated and repressed by others is a complete failure. This has never been just a war of slogans. There have been very many specific arguments against the math curricula in schools, but they have been ignored. Schools can do whatever they want. Is that "polite?"

After this the debate ends up degenerating into name-calling, politics, and racial comments, so I won't quote them any further. Other than that, it covers some of the same material that I mentioned back in my April Fool's Day post. In particular, SteveH agrees with the traditionalists who claim that tracking already exists -- except that the tracking occurs outside the schools, via parents and outside tutors.

I'm not exactly sure what SteveH would want to see in a K-6 curriculum to make it pro-STEM. I've driven past elementary schools that declare themselves to be STEM schools, but I'm not quite familiar with their STEM curriculum.

As you keep in mind that I agree with traditionalists like SteveH when it comes to strengthening the elementary math curriculum (but disagree with him regarding Calculus), I must point out that "choice" mentioned in these debates is a red herring. SteveH only favors "choice" because he wants parents and students to choose a traditionalist curriculum. If most schools had a strong traditionalist curriculum, it would be Goldberg arguing to allow parents and students to "choose" the progressive curriculum and SteveH defending the status quo.

Meanwhile, yesterday I mentioned a New York Times article about the SAT and ACT. I was trying to avoid long traditionalist debate posts so soon after the April Fool's Day post, but now I've already reopened a can of worms by mentioning last year's SAT in this post. So I might as well go back to yesterday's article:

Recall that I like the idea of having states adopt the ACT or SAT as a Common Core test. It reduces the amount of testing that juniors have to take. Instead of taking both a Common Core test and a college entrance exam, they only have to take one test.

But this article undermines this idea -- many colleges no longer require the ACT or SAT. So now the ACT and SAT are as meaningless for these students as the PARCC and SBAC are.

Some of the comments to this post so far are interesting. Under "NYT picks," dcl writes:

I predict that the new SAT will show an even stronger correlation with socioeconomics, especially math, which is now much more subject matter-focused. Kids in poor schools that have weaker math programs will be at a distinct disadvantage. Also, with the stress on reading (CC stresses reading, no data-driven reason why), students who are strong at math but not reading (coders, immigrants, etc.) will do more poorly.

I wonder whether by "weaker math programs," dcl is referring to the same weak programs as SteveH, such as the U of Chicago elementary texts. Also, the commenter's lament about the Core's emphasis on reading has also been discussed by other traditionalists.

Another NYT-picked commenter, Lou, writes:

No once a year test is a competent measure of school success...If states want meaningful growth measures to provide information about student progress in schools, they must administer tests multiple times per year in order to establish annual student baselines and then growth during the school year. Anything else is a very expensive sham that taxpayers are forced to support.

But there are several problems here. Some people might point out that if the current tests are "a very expensive sham," testing them more often, as Lou recommends here, is even more expensive.

Of course, the idea is that if the tests provide useful information to parents, students and teachers, then they will want the testing to occur more often. But they must be convinced that the tests actually provide useful information.

This is why, in previous posts, I recommend that the tests be given less often. Some traditionalists have suggested that the tests be given once in grades 3-5 and once again in middle school. I recently saw someone recommend fourth and eighth grades as the two specific years for testing. Sometimes I wonder whether a different grade would be better for testing. Sixth grade would be a better year to test math under the SteveH plan -- we can tell whether the students have mastered K-6 math, as SteveH prefers, and thus are ready for genuine Pre-Algebra courses in seventh grade.

Once the tests can be trusted to produce useful information to parents, students, and teachers, only then can the tests be given more often, to once a year or even several times per year. Certainly, before we can even consider giving the test more often:

-- The turn-around time should be reduced. If we are giving computer-based tests, then they should report the scores instantly (at least for math). Any question that can't be graded by the computer instantly has no business being on a math test.
-- The amount of time devoted to testing should be reduced. The math test, at least in middle school, should be reduced to 30 minutes, so that testing doesn't affect the bell schedule. But some people will include "test review" or "test prep" as part of testing time, so the exams should be designed in a way that so much "test prep" time is not necessary.

Okay, here are the worksheets:

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