At some schools, the Big March doesn't start until next week, because students get the entire week of President's Day off from school. This is true in New York City, and last year I linked to a Northern California teacher whose school observes a week of February break. Actually, this year I found a Southern California district that has a mid-winter break as well. But that district clearly isn't one of the two districts where I'm employed as a sub.
In fact, sometimes I wonder whether things would have gone better for me two years ago if my old charter school had a February week off. Many of the problems that I described earlier on the blog actually occurred on the first four days after President's Day.
After that first week, I wrote that I'd stop blogging for the rest of the Big March. If I were teaching full-time this year, I might have taken off the entire Big March again this year, just to make sure that I work hard to survive it rather than spend so much time posting on a blog. Since this year I'm just a sub, I'll continue to blog throughout the Big March.
Indeed, today I subbed at a high school. Since this was not a math class, there will be no "Day in the Life" today. Instead, it's a special ed class. This is my second visit to this class -- my first was back in December. I write more about that class in my December 10th post.
Like many special ed teachers, this regular teacher has one period of co-teaching (a sophomore World History class). His own classes are one section each of Government and Econ (the two one-semester senior social science class) and two sections of Integrated Science.
(Recall that in this district, "Integrated Science" is for special ed students only -- even though I myself took Integrated Sci as a young high school student. There are students of all grade levels in these classes, but the majority are juniors.)
All four of the regular teacher's classes have the same assignment on Mondays -- a current event, which they look up on Chromebooks. This is after the students watch "CNN 10," a daily ten-minute news video intended for school students. (Since there was no school yesterday, the students must do their weekly Monday assignment today.)
There will be no "Day in the Life" since most of the classes have a special aide. But seventh period, the last class of the day, has no aide. It's an Integrated Science class with mostly juniors.
(At least I've finally broken out of that "English" funk -- and there's no reason to mention the big assignment that took place in those classes. But I'm still subbing in many special ed classes -- plus as I start my second year in the district, I'm now making second visits to several classes.)
Back on December 10th, I wrote about how I ran out of current event forms in seventh period -- and that resulted in chaos. This time, I count the forms during fifth period to be sure that they would last all through seventh period.
As class begins, I write on the board that today, the students are expected to do the current event. I do so because of December 10th -- some students might remember me and how much they got to play around that day, so now I dispel any notion of that day repeating. I'll be checking throughout the period to make sure that the students are working on the current event, and I'll write down the names of any students who refuse to work.
And on the board, I mention that this is the start of the Big March. (In my new district, the Big March is indeed the longest stretch of school days without a holiday.) I tell them that they should just deal with it -- just wishing won't make the Big March disappear.
But there are still a few problems in this class -- beginning with tardies. (Obviously, nothing I write on the board can eliminate tardies.) Nearly as many students arrive after the bell rings (seven) than before (eight). We expect tardies in the morning, but not at the last class of the day.
Also, it's interesting how so many students claim that they're "tired." In my own experience as both a student and teacher, I was most alert on Mondays and most tired on Fridays. I felt refreshed coming back from a holiday and most exhausted leading up to the holiday. So it's interesting that students complain of being tired on the first day of the Big March. (Perhaps my experience as a student was atypical because I ran track -- I was alert on Mondays after resting with no workouts on Sundays, and then I was tired on Fridays after the races on Thursdays.)
In the end, I catch four students who fail to do the assignment -- or even start it, due to either talking with each other or playing games on phones. The best class of the day turns out to be fourth period, followed by fifth period.
There are still a few things I could have done better today. (Yes, I think I was better today than on December 10th, but I must always look up to the next level.) I try to catch the attention of some students who are talking or playing games, but fail to do so. Since there is no seating chart, I don't know the students' names. In this situation, I should have addressed the students with something like "young man" or "young lady" to get their attention. (I also could have gained their attention by yelling, but I already know that it's wrong to do so.)
Let's get to today's lesson. I've written above that one of the most difficult units always seems to begin right around the start of the Big March. Many students have trouble with graphing throughout Chapter 11, and furthermore, today we learn the equation of a circle, which just a few years ago was part of Algebra II! (Meanwhile, English classes tend to read Shakespeare during the Big March, for example.)
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 last year 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. [2019 Update: At this point two years ago I got into a long discussion about traditionalists. This year -- I'm going to have another traditionalists' post, because some of them posted over the long holiday weekend.]
Actually, I wasn't sure whether I wanted to have another traditionalists' post this week. Yes, our favorite traditionalist Barry Garelick blogged over the weekend, but our favorite commenter SteveH hasn't responded yet. I fear that he might comment tomorrow, and then we're suddenly having multiple traditionalists' posts in a row again.
But then again, there are other traditionalists in the comment thread. And besides, sometimes SteveH adds nothing to the discussion but just repeats his usual canards (how he'd gotten to AP Calculus without any help, yet needed tutors for his son to make up for poor K-8 instruction from using U of Chicago elementary texts, and so on).
And besides, our first traditionalist today isn't a Garelick commenter, but the tweeter CCSSIMath.
CCSSIMath (February 15th tweet):
This doesn't answer the question at all, but perhaps HS students should not be doing "transformations". Here are translations, rotations, and reflections from consecutive pages in a Japanese #math book. 7th grade. Cue the panda bear.
Of course, we know that some traditionalists would agree that high school students shouldn't be doing transformations at all during their careers (most notably Ze'ev Wurman). But here, CCSSIMath is making the usual complaint about the Common Core teaching math later than in other countries such as Japan.
There are actually two gaps to discuss here. Notice that transformations first appear in Common Core 8, so there's only one year between Japan and the Core. The second gap is about why, since these transformations appear in Common Core 8, do they appear again in high school Geometry.
Let's attack the second gap first. For one thing, students don't remember everything that they learned in previous years, and so many texts review old material. This is most notably true for Algebra II, which reviews much material from Algebra I. It would be disingenuous for CCSSIMath to present an Algebra II lesson that's clearly Algebra I review next to a junior high text from Japan and "cue the panda bear" as we see the huge gap between junior high in Japan and Algebra II in the Core.
CCSSIMath's tweet is in response to another Twitter user who discusses the following standard:
CCSS.MATH.CONTENT.HSG.CO.A.5
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
And here's my solution -- tracing paper is clearly for introducing transformations. Since said intro occurs in eighth grade, this should be dropped down to Common Core 8. High school Geometry should be for reviewing the transformations and then using them in proofs.
But we still have the other gap between Japan 7 and Common Core 8. Well, we know that the traditionalists like to see Algebra I in eighth grade. The eighth graders who take Algebra I must have seen all the Math 8 standards in an Accelerated Math 7 course. Thus such students really would have seen transformations in seventh grade.
Yet they'll have to review transformations in eighth grade anyway before the SBAC. I've written a possible solution in previous posts -- eighth graders (or seventh, if there are any) in Algebra I should not be required to take the SBAC. (After all, ninth and tenth graders in Algebra I don't take it.)
OK, let's get to Garelick:
https://traditionalmath.wordpress.com/2019/02/17/misunderstandings-about-understanding/
I had the great fortune to attend the researchED conference in Vancouver on Feb. 10, 2019. I was also honored to give a presentation there. Here is a summary, followed by a link if you are interested in the presentation. It is a PowerPoint which when viewed in “Notes” format contains the script that accompanies each slide.
The presentation is about the dichotomy between fluency and understanding. Traditionalists like Garelick tell us that the former shouldn't be sacrificed for the sake of the latter.
We'll now read the comments. First up is Wayne Bishop, a traditionalist whose commented before:
Wayne Bishop:
Absolutely correct. That said, how receptive was your audience? Were there a bunch of math ed experts who never learned (and never will learn) any mathematics beyond the algebra level much less a good proof-based Euclidean geometry course or anything but the most monkey-see/monkey-do calculus?
First of all, I must ask, what exactly is "monkey-see/monkey-do calculus"? In particular, is the AP Calc class that high school students take a "monkey-see..." class? Perhaps Calc AB is "monkey-see," but Calc BC is deeper. Since Bishop is clearly distinguishing between math ed and true mathematics majors, maybe the classes beyond "monkey" are the ones that only math majors take (Multivariable Calc and so on).
Ironically, "monkey-see/monkey-do" sounds like mindlessly repeating procedures as opposed to a deeper understanding -- which is exactly what Garelick's presentation is supporting.
Since we don't know what exactly Bishop means with the Calculus part of his comment, let's focus on the part that we should be familiar with -- Geometry:
Wayne Bishop:
Absolutely correct. That said, how receptive was your audience? Were there a bunch of math ed experts who never learned (and never will learn) any mathematics beyond the algebra level much less a good proof-based Euclidean geometry course [...]
[emphasis mine]
I've devoted several posts over the years to what constitutes a proof-based course. In theory, even math ed majors must have taken three years of math, including Geometry, to be admitted to college in the first place. Thus Bishop here is telling us that not all of these courses are proof-based.
I've written about David Joyce and his lament that a certain Geometry text isn't proof-based. The modern gold standard for proof-based texts is Dr. M's online text. In the past, I linked to Dr. M, since he described how to write proofs in Common Core Geometry. But since then, his state (Indiana) dropped the Core, and thus the latest edition of his text no longer mentions transformations. (This is why I don't actively link to or write about Dr. M anymore, but sometimes I still cut-and-paste old posts mentioning him.)
Ze'ev Wurman tells us that proof-based Geometry and transformations are incompatible. Even though Bishop doesn't directly state this, his comment seems to be distinguishing between proof-based traditional Geometry on one side and transformation-based Common Core Geometry on the other.
But it is possible to have rigorous proofs in Common Core Geometry -- indeed, the Reflection Postulate of the U of Chicago text allows us to prove SAS, ASA, and SSS as theorems instead of assuming them as postulates (as most "proof-based" texts do). And Hung-Hsi Wu of Berkeley takes this a step further -- he uses 180-degree rotations to prove the Alternate Interior Angles Test as a theorem rather than assuming one parallel test as a postulate (as most "proof-based" texts do.)
I once devoted several posts to proving the properties of parallel lines using translations instead of 180-degree rotations. But I don't like what I came up with, and so Wu's proofs are still the best way to develop Common Core Geometry. In New York State, the official curriculum (Eureka) is based on Wu's recommendations.
I decided to revisit the Eureka website to see how Wu's Geometry is being taught:
https://www.engageny.org/resource/high-school-geometry
Wu's proof of Alternate Interior Angles is in Lesson 18, and his proof of SAS is in Lesson 22.
But unfortunately, there are very few proofs in Module 1, Topics A and B. Topic A is constructions, and this reminds us of Chapter 1 of the text that Joyce criticizes. Actually the first lesson is on the equilateral triangle construction, which is OK since Euclid himself starts with it! But subsequent lessons are constructions that need proofs to justify. (The U of Chicago text, meanwhile, doesn't discuss constructions much at all.)
And Topic B is on unknown angles. These lessons assume many angle theorems, including those related to parallel lines and transversals (which, as I said, aren't proved until Lesson 18). The reasons for stating these theorems before proving them is that students have previously learned these in eighth grade or earlier.
OK, I'll give that one to the traditionalists, since I myself just stated that we shouldn't rely on what students "remembered" (or forgot) from previous years. Instead, we should review and prove what students learned in previous years before using them. Anyone reading Topics A and B would thus conclude that Eureka is not proof-based, regardless of how rigorous Topic C is. A solution would be to fix Topics A and B while keeping the excellent Topic C and beyond.
Let's conclude this post with a quick look at the other commenter, traditionalist Tara Houle:
Tara Houle:
There was a lovely young trainee teacher, who grew up in China. She gave a very meaningful, and vivid illustration about how math was never difficult for her in school, but her mother insisted that the basic facts were already instilled BEFORE she started school, so she could then focus on the more abstract aspect of mathematics in the classroom. She also said there was one aspect of a problem that she struggled with. So over the course of a month, she spent 1000s of hours of practicing this one aspect of the problem, in order to really understand what it was about. And didn’t think twice about dedicating that much time to the problem, because that is what mathematics, like many abstract subjects, require.
Notice that this supplemental instruction in "basic facts" (addition?) is needed for a girl who grows up in China, far beyond the influence of Common Core. Also, notice that a 31-day month contains only 744 hours, so she couldn't have practiced for thousands of hours in one month.
But the key part of her post is the last sentence:
And didn’t think twice about dedicating that much time to the problem, because that is what mathematics, like many abstract subjects, require.
Yet many of the students actually sitting in our math classes not only think twice about dedicating that much time to studying math -- they refuse to do it. Traditionalism might work if teachers could be guaranteed a class full of students as dedicated as this Chinese girl.
But as I see in class on this day -- the first day of the Big March -- some students will refuse to spend time on a current event assignment even when I'm standing next to them. How much less time, then, will such students spend studying math?
Here are the worksheets for today:
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