Wednesday, February 21, 2018

Lesson 11-3: Equations for Circles (Day 113)

Lesson 11-3 of the U of Chicago text is called "Equations for Circles." In the modern Third Edition of the text, equations for circles appear in Lesson 11-6.

This is what I wrote two years ago about today's lesson:

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. [2018 Update: At this point two years ago I got into a long discussion about the PARCC and SAT. But due to my subbing from yesterday, the test on my mind right now is the IB exam. Two years ago, I looked at the PARCC and SAT from a traditionalist perspective -- and this year, I'm in the mood to write about the IB from a traditionalist point of view as well. So even though we just had a traditionalists post last week, today's post gets the "traditionalists" label as well.]

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.

2018 Update: OK, let's cut this old discussion off and start talking about the IB. Wow, so back then SteveH used to post on other websites besides Barry Garelick's blog. Of course, I'm not sure whether Garelick even had a blog back then.

Here is a post from last October on Barry Garelick's blog. SteveH comments there as usual, and here he mentions the IB exam. (There's no need to discuss what Garelick writes in this post, since this entire post is about SteveH's comments.)

The fundamental issue they never address is that integrated (or whatever) math tried and lost in high school. IB (which leads to taking the AP test) and the AP Calculus tracks have won. That’s what colleges want to see for any STEM program and many non-STEM degree programs. This track starts at the 7th grade math track split to a proper algebra class in 8th grade. Vocational schools and community colleges place students using the Accuplacer test, not CCSS.

There are several issues here. First of all, SteveH states that IB leads to taking the AP test. OK, my subbing from yesterday verifies his claim here. The students appear to be taking Honors Algebra II as sophomores, IB Math Studies as juniors, and then AP Calculus AB as seniors.

SteveH also tells us that the "integrated math" track lost, while the "IB and AP Calc tracks" won. In a way, yesterday's subbing supports him as well. The IB sophomores are taking Honors Algebra II, not Integrated Math (which clearly doesn't exist in this district).

But hold on a minute here. IB, again, stands for International Baccalaureate. As its name implies, students around the world take the IB exam. And students outside the United States (and Vietnam) take Integrated Math, not Algebra I, Geometry, or Algebra II. Thus, contrary to SteveH's remark, plenty of students take Integrated Math to prepare them for IB courses.

I decided to perform a Google search for the math program at British IB schools. Most schools just simply call the class "math" (or "maths," I guess, since that's the international term). I did find one Scottish school that explicitly describes the math classes:

We notice that there are two levels of classes in Grades 8-10 -- C (core) and E (extended). I assume that these paths lead to the two levels of IB exam, SL (standard level) and HL (higher level). Of course, HL is recommended for prospective STEM majors.

There is one class listed here that SteveH would definitely like:

Maths 8E (Extended - Algebra 1)

This course is designed to provide a foundation for IB and IGCSE Math courses and to teach students to be successful at solving mathematical problems.  Emphasis is placed on problem-solving in the context of real-life situations as well as integrating technology into everyday life and using it as a problem-solving tool.  Some of the topics covered include: properties of the real numbers; graphing and solving linear equations and inequalities; the concept of a function; quadratic and exponential expressions and radicals; and connections with geometry.  Students passing Math 8 Extended earn a High School Math credit for this course.
We see that this class mentions Algebra I, but not Geometry. Thus it's not an Integrated Math class, but more like our Algebra I classes that SteveH wants eighth graders to take. And even though the Maths 8C course is more like our Common Core 8, it's Maths 8E that leads to HL and STEM. And so this is what SteveH wants to see -- eighth grade Algebra at the start of a rigorous track.

But even with this Maths 8E course corresponding to Algebra I, we notice that the classes for both freshmen and sophomores include Algebra and Geometry. Thus even Maths 9E and 10E are truly integrated courses. This counters SteveH's claim that the "integrated track" and "IB track" are two separate tracks, with the former "losing" and the latter "winning":

Maths 9E* (Algebra & Geometry)

* Extended
This course is the first of two in a sequence designed for students that enjoy the challenge of a more rigorous math course, whose foundational math skill are in place, and whose interests may lie more in business administration, chemistry, physics, biology, engineering and mathematics.  Students must be willing to accept the challenge of a more rigorous math course and the rapid pace that comes along with it.  Students will acquire and further develop mathematical skills and learn to apply them to other subjects and to real world problems.  Students should have, and will be expected to use a graphing calculator on a regular basis.
The first portion of the IGCSE International Mathematics 0607 Extended syllabus will be studied.  Topics of coverage will include, but not necessarily limited to, number, algebra, functions, geometry, two-dimensional transformations, mensuration and coordinate geometry.

Then again, I can't help but think back to the local STEM magnet high school. A few years ago, students were required to take eighth grade Algebra I in order to be admitted to the school. But the high school itself used to offer IMP, an Integrated Math program from Grades 9-11. Then seniors would take Calculus. At this school, IMP no longer exists -- ironically, this school used the Common Core transition to return to traditional Geometry and Algebra II, at the same time that other schools were introducing Integrated Math. (The eighth grade Algebra I requirement still exists.)

Nonetheless, both the old magnet requirements and the new Scottish IB classes suggest a compromise between the traditional and integrated pathways -- teach Algebra I in eighth grade and then Integrated Math in Grades 9-10. Hopefully SteveH would accept this compromise, since it at least encourages K-6 math to prepare students for eighth grade Algebra I.

I don't link to the above school website since the Integrated Math program no longer exists there (and it was never an IB school anyway). I did search on Google for a school right here in Southern California that offers both Integrated Math and IB. The school I found is Corona Centennial High:

We see that this school teaches three sections of "Integrated Math I Enhanced/IB." Hence it's another counterexample to SteveH's claim that Integrated Math doesn't lead to IB. According to the following link from this district, eighth graders can take Integrated Math I, so a path to Calculus is possible:

This link also lists several college acceptances, including the Ivy League. Otherwise, SteveH could counter that Integrated Math students are taking IB, but aren't doing well enough to pass the exams or get into elite universities. Indeed, in that same October comment, he writes:

Many of these pedagogues now talk about “math zombies” who get good grades, but still really don’t understand what they are doing, which is very normal because there are different levels of understanding. OK, so where are their other ideal students? What exact curriculum gets them there? Show how those students do on SAT II Math and AP Calculus tests. I might be their biggest fan. No. Their dream students don’t exist, so now they’re left promoting the benefits of engagement and conceptual understanding which are agnostic to pedagogy and curriculum. However, to promote their individual products (not curricula), they have to trash “rote” traditional math using rote, and wrong, justifications. It’s quite ironic.

Unfortunately that last link doesn't list actual SAT II and AP Calc test scores (or IB test scores, for that matter), the final piece of evidence SteveH needs to prove that Integrated Math works.

Back when I was a student teacher, my school had an IB program -- and indeed, I taught an Algebra I class for IB freshmen. Presumably, some IB freshmen also took Geometry, but I only taught the Algebra I class. But if I'm not mistaken, the IB program at this school no longer exists -- it was replaced by an AP-based magnet.

Here are the worksheets for today:

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