Sunday, June 24, 2018

Another Round...of Traditionalists' Debates

Table of Contents

1. Pappas Question of the Day
2. Fawn Nguyen: Another Round
3. Maureen Sikora's Response to Nguyen's Post
4. Barry Garelick: Singapore Math
5. SteveH's Response to Garelick's Post
6. Joanne Jacobs: Eighth Grade Math in SF
7. Traditionalists' Responses on the Jacobs Website
8. Case Study: My Old Sixth Grade Class
9. Progress on Translation Proof
10. Conclusion: Three Californias?

Pappas Question of the Day


Today on her Mathematics Calendar 2018, Theoni Pappas writes:

This wheel revolves (1/15)pi" per second. Convert this to degrees per second.

[Given information from diagram: the radius of the wheel is 0.5".]

Notice that if we simply divide the linear velocity by the radius, the result is in radians per second. I have seen this written using variables as follows:

v = r * omega
omega = v/r
omega = (1/15)pi / 0.5
omega = (1/15)pi / (1/2)
omega = (2/15)pi radians per second

All that remains is to convert radians/sec. to degrees/sec. as stated in the goal:

omega = (2/15)pi (180/pi)
omega = (2/15)180
omega = 24 degrees per second

Therefore the angular velocity is 24 degrees per second -- and of course, today's date is the 24th.

It's debatable whether I should even count this as a Geometry question -- after all, I only do the Pappas question of the day if it's Geometry. The question does involve the radius and circumference of a wheel as well as angle measures. But radians aren't studied until Precalculus (or maybe Algebra II if it has a trig component). Angular velocity is often covered along with radian measure. I suppose if we wanted this to be a true Geometry question, I could just find the circumference of the circle (pi inches) and then set up a proportion (pi inches is to 360 degrees as pi/15 inches is to how many degrees) without bringing up radians at all.

I could return to spherical trig in today's post. But instead, I prefer to alternate my summer posts between spherical trig and the math debates. Many math bloggers have posted recently, including both Fawn Nguyen and some of the traditionalists.

Fawn Nguyen: Another Round

Let's start with Fawn Nguyen's blog. Back on Thursday (the summer solstice, I might add), the Queen of the MTBoS made the following post:

http://fawnnguyen.com/another-round/

Exhausted and hungry, I walk to the restaurant a hundred feet from the hotel’s lobby. The hostess greets me and asks how my day has been. I tell her it’s been a long day, that I just came in from LA on a 5-hour-plus flight. She asks about my reason for being in Philadelphia. I tell her I’m here for a math conference, and she volunteers, “Oh… I’m not a big math person.”

And here's Nguyen's response -- at least in her mind:

Are you a small or minuscule math person then?!? I don’t care if you say that you’re not big on eating raw octopus or fried worms, but math??!!

Well, of course the server doesn't say "I'm not a big raw octopus person" -- after all, Nguyen doesn't say she's in Philly for an octopus conference.

Of course, it's easy to see what's going on here. Chances are that what the hostess would really like to say is "I hate math." In other words, "not a big" is just a way to avoid the h-word.

Indeed, Nguyen proceeds:

She doesn’t know that I hear the likes of that statement each and every time people learn I’m a math teacher.

And I believe the point Nguyen's making here is that, while "I'm an English teacher" might draw a response of "I'm not a big English person," and likewise "I'm a history teacher" leads to "I'm not a big history person," those responses don't come up quite as often as "I'm not a big math person."

Nguyen continues her post with a short story about a former Navy SEAL parent she once met. He tells her that he fears seeing her because he doesn't want to answer any of her math questions.

Let's think about this from the perspective of the restaurant hostess and the Navy SEAL. I don't know what their grades were in middle and high school, but two possibilities are most likely given their attitudes towards math:

  • Math was consistently their lowest grades throughout secondary school.
  • They received about the same grades in math as in their other classes, but more effort was required to maintain the math grades as compared to the other grades.
She continues:

I know I have students who may utter the same words leaving my class. While I believe I have made great strides in improving math learning and math teaching in my classroom, I haven’t done enough, there’s still a lot of work to do.


Yes, there's still a lot of work to do -- but what are we trying to accomplish? We're trying to make the response "I'm not a big math person" disappear (or at least not occur significantly more than "I'm not a big English person" or "I'm not a big history person"). We're trying to make it so that not as much effort is needed to maintain math grades as compared to other classes, so that adults no longer have unpleasant memories of math vis-a-vis other subjects.

Of course, a big reason that math grades take so much work to maintain are the problem sets (p-sets), since not only must they be completed as graded assignments on their own, but the only way to do well on the quizzes and tests is to do the p-sets. And the biggest advocates of p-sets in math classes are -- who else? -- the traditionalists.

Maureen Sikora's Response to Nguyen's Post

But before we look at the traditionalists, let's return to Fawn Nguyen's post. Her first commenter is from a math tutor, Maureen Sikora:

Maureen H Sikora:
This year I started tutoring a Junior in high school algebra 2. She has some learning issues, and difficulty with her memory. But I can tell you, from the way she had to learn algebra 2 this year, I can now understand the statement of that hostess. It was awful to watch this girl that I tutor, try so hard, and fail each test. No modifications made to help with her memory, no basic skills review to help her go over her weaknesses in simple pre-algebra concepts (like adding integers) that forced her to rely on a calculator. When she failed, there was no expectation that she go over the concepts that she didn’t understand, and the teacher just continued onto the next lesson as if everyone understood it. Her teacher often making comments that “if you fail the bench mark, you will probably fail the final.” What hope does that give any of those struggling?

And I can already imagine how the traditionalists would respond to Sikora. They'd immediately zero in on the following statement:

No modifications made to help with her memory, no basic skills review to help her go over her weaknesses in simple pre-algebra concepts (like adding integers) that forced her to rely on a calculator.

The traditionalists would wonder, why does this girl have trouble adding integers? What curriculum did her middle school use in sixth or seventh grade -- the years she should have mastered the addition of integers? She should have been given a "proper Glencoe text" -- or better yet, an "old Dolciani text" -- and been asked to do p-set after p-set in middle school. Then she would have been so successful that she'd have been placed in eighth grade Algebra I and onward to the AP Calculus track in high school.

Indeed, one fact the traditionalists would notice is that Sikora is a tutor. The traditionalists often write about the need for tutoring to supplement the non-traditional school curricula. Thus to them, Sikora was hired four years too late. Instead, she should have been hired when the girl was a seventh-grader, so that she could have learned how to add integers properly. They might even surmise that if this had happened, the student would have made it to "the land of reality" -- that is, the AP Calculus track -- and be so strong in math that she'd no longer need a tutor!

Maureen H Sikora:
Our schools need to do better to help people like her, and to help make math more meaningful and not just about trying to do well on a standardized test to get into college.

But of course, the traditionalists' argument is all about not only getting into college, but making it into elite STEM programs at those universities. Even if this girl has no intention of going to college, math class should be all about "keeping doors open." Anything other than the AP Calculus track is "closing doors" for the students.

Sikora laments that the girl's teacher won't spend any time reviewing Pre-Algebra skills. The traditionalists would reply that for this girl, it's too late -- "it's all over in seventh grade" because her middle school teachers didn't teach her traditional math. In fact, some traditionalists might even take it a step further -- the Algebra II is correct not to slow down the class for this girl because by slowing down, the AP Calc-bound sophomores in the class won't be prepared for AP Calc. The teacher should (rightfully, in their eyes) keep the class on the Calc-bound pace.

This whole comment is all about students who hate math. Sikora surmises that her student hates math because she's so far behind and not given any chance to catch up -- she's doomed to fail. So what do the traditionalists have that would make the girl hate math less? All they seem to have is that she should've been taught more traditionally in middle school -- with more p-sets.

But as far as I can tell, those middle school p-sets would have made her hate math more, not less. She might have already known as a seventh grader that she doesn't want to be a STEM major, yet the traditionalists would insist that she "keep doors open" by preparing for eighth grade Algebra I. To them, it's all about the AP Calculus track.

By the way, one commenter responded to Sikora:

Susan Townsend:
Soooo many ‘how not to teach math so students actually learn math not to mention LOVE math’ in your student’s story! Your story also tells me you DID provide some positive math memories for your student!

Barry Garelick: Singapore Math

I now let the traditionalists speak for themselves. Here is Barry Garelick's latest post:

https://traditionalmath.wordpress.com/2018/06/16/a-tale-told-by-an-idiot-dept/

Yet another article on Singapore Math, which in the end signifies nothing. In fact, this one could probably be used as a template for so-called education reporters when they write about math education.
It starts out well enough with a very concise history of the program that was developed in Singapore:
What is referred to as Singapore Math in other countries is, for Singapore, simply math. The program was developed under the supervision of the Singaporean Minister of Education and introduced as the Primary Mathematics Series in 1982. For close to 20 years, this program remained the only series used in Singaporean classrooms.

We already know that Singapore Math is popular with traditionalists, so it's no surprise that Garelick would endorse it. In this post, the traditionalist writes about the article which lists the pros and cons of Singapore Math. We see that he even disagrees with one of the listed pros of Singapore math:

But then we have the obligatory “not like traditional math” narrative:
  • Asks for students to build meaning to learn concepts and skills, as opposed to rote memorization of rules and formulas.
Yes, it does require students to learn concepts and skills and provides a contextual background for how the rules work. But so did many textbooks of the past which I’ve talked about extensively.  And it isn’t like teachers don’t teach the conceptual underpinnings.  Caricaturizing traditional math approaches as “rote” never gets old, though, as the writer of the article shows.
Naturally, Garelick takes more issue with the cons:

Then there’s this:
  • Less of a focus on applied mathematics than traditional U.S. math textbooks. For instance, the Everyday Mathematics program emphasizes data analysis using real-life, multiple step math problems, while Singapore Math’s approach is more ideological.
Less focus on applied math than traditional textbooks? Really?  Primary math is all about applied math, but the term “applied math” has a different meaning when talking about math education these days. It used to be we could talk about students learning to solve problems.  But “solving problems” in today’s edu-lexicon means the standard one-answer, non-open-ended type of problems (like “Johnny had 5 apples and gave some away. He has three apples left. How many did he give away?”) which is viewed as “dull, boring and deemed to not imbue students with a problem solving “schema”.  Furthermore, the problems are viewed as contrived, and not something they would use in “real life” (even though they are very applicable to real life).
First of all, notice that the article mentions the U of Chicago elementary texts -- also known as "Everyday Mathematics." I concede that Garelick does have a point about how primary math is "applied math" -- I often do agree with traditionalists regarding math in the lowest grades. To me, "applied math" is math for which the question "When will we use this in real life?" is not asked by students (because the "application" to real life should be obvious). Students don't generally ask this question until upper elementary school -- so that's when I'd start worrying about whether math can be "applied" to real life.

By the way, I observe that whenever the Singapore and U of Chicago texts make their way into the traditionalists' debates, it's always the elementary texts that are being mentioned. I seldom see any mention of the U of Chicago texts for any grade higher than sixth, even though my own blog is all about the U of Chicago high school texts.

Again, I anticipate a few issues that the traditionalists might have with our high school texts. Even though the U of Chicago text called "Algebra" is considered an eighth grade text, this doesn't mean that the U of Chicago supports the AP Calculus track. The reason for this is because a text called "Functions, Statistics, and Trigonometry" is the junior-year text, between Algebra II and Precalc. In fact, the Algebra I text moves a bit slowly, as does Algebra II (which could be the reason that an extra Functions, Stats, and Trig text is needed -- the Algebra II text doesn't adequately prepare the students for Precalc).  Finally, we know better than anyone that the Geometry text supports transformations, which traditionalists tend to hold in disdain.

As for the high school Singapore Math texts, I've blogged about them in years past. The only problem with them is that they're integrated -- after all, Singapore, like most countries, doesn't divide high school math into Algebra I, Geometry, and Algebra II. I once suggested that we could compromise between traditionalist and reform math by using the integrated math Singapore texts -- traditionalists would like the "Singapore" part, while reformers would like the "integrated" part.

But according to the traditionalists, integrated math has already "lost the battle." It's no wonder, then, that traditionalists never mention Singapore math for any grade higher than sixth -- it would be too embarrassing for them to see the integrated math they despise being published by the country whose elementary curriculum they love.

SteveH's Response to Garelick's Post

Garelick's post draws several comments, including the usual pair by SteveH:

SteveH (quoting the article):
“Materials are consumable and must be re-ordered for every classroom every year. This can put a huge financial burden on already strained school budgets.”
Has she looked at the cost of Everyday Math for consumables? Publishers just love modern K-6 math curricula. In high school for my son’s traditional AP math track classes, they used good ol’ student covered textbooks. I still have my son’s old consumable EM workbooks to review and once again be horrified.
Actually, I admit I don't quite understand the relationship between consumable texts and modern math classes -- I only know that many modern texts are paperback. Here both sides criticize the other for using paperback texts (the author attacking Singapore, SteveH attacking the U of Chicago).

Early elementary texts have always been consumable. I remember that as a young student, my elementary school had a paperback first grade text and and a hardback fourth grade text. So I don't recall exactly at which grade the hardback texts began.

But as both the author and SteveH point out, consumables at higher grades is a new phenomenon. For example, the Illinois State text from my charter school from last year used paperback texts. Officially, the word from the administration is that only the "traditional texts" (or Student Journals) -- that is, the ones with actual p-sets -- are treated as consumable (so that the students can write in the answers). So the students were not supposed to write in texts containing STEM projects, since these would be used again the following year.

As a sub, I've observed Common Core paperback math texts in high school. I know that Pearson has published such texts for Integrated Math, but I've also seen paperback Algebra I texts as well.

One thing I've noticed about many of these secondary paperback math texts is that they're divided between Volumes I and II -- presumably the first and second semester. Of course, hardback texts contain two semesters' worth of material.

SteveH:
“Less of a focus on applied mathematics than traditional U.S. math textbooks. For instance, the Everyday Mathematics program emphasizes data analysis using real-life, multiple step math problems, while Singapore Math’s approach is more ideological.”
What! Singapore Math teaches more deep “ideological” understanding in math than Everyday Math? This is K-6 and the goal of math is not to track students into a lower level career path that virtually closes all STEM education doors – IN K-6!
And here we go again as SteveH worries about "closing doors." A fourth or fifth grader might be too young to know whether he or she wants to attend college, much less major in STEM. On the other hand, the child isn't too young to know that he or she doesn't want to do endless p-sets -- and might begin to hate math because of all the p-sets that need to be completed.

SteveH's next comment is directed towards the homeschooling parent (of an eighth grader) who asks, "Is Singapore Math good or bad?"

SteveH:
After 6th grade, you need to move to proper math textbooks that lead to the AP Calculus track in high school. If you are still hoping for proper STEM career preparation, you really need to push to achieve proper Algebra I course material in 8th grade that leads to honors geometry as a freshman in high school. This is required to follow the Geometry-Algebra2-PreCalc (trig)-AP Calculus track.

Notice that SteveH suggests that the mother move (that is, away from Singapore Math) towards texts that prepare for Calculus. One would think that the Singapore high school texts prepare students for Calc, since after all Singapore twelfth grade math contains Calculus. The problem, of course, is that the Singapore high school texts contain Integrated Math, whereas according to SteveH, Integrated Math "already lost the battle."

SteveH:
If you are headed for engineering, math, physics or even chemistry, you need to get to AP Calculus. Other STEM career paths might be OK for reaching only Pre-Calc, but students have to be prepared to get through at least 2 semesters of calculus in college and perhaps courses like linear algebra. In my college math and CS teaching days, I saw many students who had to change their majors because they could not get through the math requirements. I saw this even with nursing students who couldn’t get through a course in statistics.

Here SteveH provides paths for math, engineering, and physics majors (AP Calc or bust!) and a path for other STEM majors such as bio (at least Pre-Calc). Then he mentions prospective nurses who end up failing Stats in college. I wonder what classes SteveH would recommend that future nurses take in high school to be able to pass college Stats. (AP Calc or bust? Pre-Calc? Presumably AP Stats should prepare students for college Stats, but then what should they take to prepare for AP Stats?)

SteveH:
When I was growing up, the top level in high school was called “College Prep”, but now that they think everyone should go to college, that has become the lowest level. In my son’s high school, the only kids not in College Prep are those who are about two years behind. This means that the new (and realistic) College Prep courses are Honors and AP Classes. They are traditionally taught and use old fashioned textbooks. This is where all of the best students go.

When SteveH writes that "they think everyone should go to college," he implies that "they" (whoever "they" are) are wrong to think that. So at least SteveH admits that not everyone should go to college, but notice how he assumes that the homeschooler (whose mother he's responding to) is not only college-bound but STEM bound. What class would SteveH have recommended for this eighth grader if the mother had said that her child is definitely not going to college or definitely not a STEM major?

SteveH:
I had to help out my son at home by pushing proper p-sets and mastery. By the time he got to high school, I did not have to help at all. Math is the most difficult to recover from after 6th grade because it’s cumulative and poor or missing skills are hard to diagnose and fix while charging ahead on new material.
Again, let's compare SteveH's son to Sikora's student. SteveH tells us that his son didn't need help in high school because he pushed p-sets on him in middle school. So again, that's probably what he thinks the girl's parents should have done -- they should have hired Sikora to tutor her in middle school by pushing p-sets and mastery on her. Then supposedly by now, the girl (like SteveH's son) wouldn't need any help at all. SteveH writes about how "math is the most difficult to recover from after 6th grade," and this is why Sikora's student is struggling so much.

Joanne Jacobs: Eighth Grade Math in SF

There have been many articles on several websites lately regarding the implementation of Common Core here in California -- specifically in San Francisco. Apparently San Francisco Unified has decided to teach Common Core 8 to all eighth graders, and not Algebra I. Naturally, traditionalists are strongly opposed to this policy. (Of course, my old charter school didn't teach eighth grade Algebra I either, but there's a huge difference between a small charter with 13 eighth graders without Algebra I and a big district with thousands of eighth graders without any of them in Algebra I.)

Once again, here's my own opinion of this issue -- while I don't believe that eighth graders should be forced to take Algebra I, the fact that AP Calc exists implies that at least some students should be allowed to take eighth grade Algebra I. But I wonder how much of SFUSD's reluctance to teach eighth grade Algebra I has to do with the existence of the SBAC test -- on which eighth graders are required to answer Common Core 8 questions. If there were a simple rule that eighth graders in Algebra I don't have to take the SBAC (just as freshmen and sophomores in Algebra I don't have to take the test), then would SFUSD be more willing to teach that class to them?

Let's keep this in mind as we read the following article from the Joanne Jacobs website. (Again, I often link to Joanne Jacobs because many traditionalists comment there.)

http://www.joannejacobs.com/2018/06/one-track-for-math-is-equity-unfair/

Nobody takes algebra in eighth grade in San Francisco’s district schools any more, reports Ed Week‘s Stephen Sawchuk. In the name of equity, district leaders decided four years ago to place all students in “math courses of equal rigor through geometry.”
That means no “honors” classes. No gifted track. No weighted GPAs until later in high school. No 8th grade Algebra 1.
The article explains that it's still possible to reach AP Calculus after all:

To enable achievers to prepare for AP Calculus in 12th grade, “the district permits students to accelerate after completing Algebra 1 in 9th grade — most notably through a compressed class combining Algebra 2 and precalculus,” writes Sawchuk.
In addition, he writes, “ambitious parents . . . shell out thousands of dollars for students to take non-district algebra classes over the summer in the hopes of getting their children into geometry early.”
OK, the traditionalists do have a point here. They might argue that Algebra II and Pre-Calc are difficult classes that are difficult to double up. Students might fail this class since it contains so much material and thus not make it to AP Calculus -- while these same students might have made it to AP Calc easily if only they'd been allowed to take Algebra I in eighth grade.

I recall how I once tutored over the summer. Many of the students were trying to get ahead -- not because they wanted to get to AP Calc as seniors, but because they wanted to get there as juniors. I remember how those students who took Algebra I or Geometry over the summer passed it, but the one who attempted Algebra II over the summer failed it. Part of the reason is that the teacher gave an overwhelming amount of homework (as in four HW assignments per day, with 10-20 questions on each assignment). But part of it is because Algebra II is a difficult course -- and so it's not easy to compress the assignment over the summer.

This means that if students want to accelerate, they should do so no later than Geometry. In other words, they must be enrolled in Algebra II as sophomores in order to make it comfortably to senior AP Calculus.

Barry Garelick analyzes the story here and explains what’s wrong with Common Core math.

Ah, so now Jacobs links to Garelick. I won't discuss those two Garelick articles here, since I already wrote about one of his articles in this post. (And besides, SteveH doesn't comment at either of those linked articles. The one Garelick article I chose is a two-fer, since it allowed me to catch SteveH's comments as well.)

Jacobs continues by discussing the other problem with Common Core math -- explaining answers:

A few miles south, Silicon Valley’s math-centric parents don’t understand why they can’t understand Common Core math, reports Karen D’Souza in the San Jose Mercury News.
“The idea is to promote critical thinking,” says Arun Ramanathan, a former teacher who now runs Pivot Learning, an Oakland-based nonprofit providing training and support to California schools. “It’s not as straightforward as it used to be. The idea is to have a conversation about how to solve the problem.”
But even Ramanathan, who has a doctorate from Harvard and a background in teaching, admits to struggling with his daughter’s seventh grade assignments.
I'm not sure what problems this Harvard Ph.D. is having with seventh grade math, since all the examples of Common Core horror given in the article are from third grade (including the infamous problem with 3 + 3 + 3 + 3 + 3 vs. 5 + 5 + 5). If I were to guess, perhaps it's the Double Number Line or Tape Diagrams that are used to solve proportions in sixth and seventh grade. These are used in order to avoid proportion equations, which students don't learn to solve until either Common Core 8 or Algebra I.

Traditionalists' Responses on the Jacobs Website

Typically, "Bill" is the main traditionalist I read at the Jacobs site. But Bill makes only one post among over 30 comments in this thread:

Bill:
I didn’t have algebra I until 9th grade back in 1977, but a college prep track back then looked like this:

The list that he provides here isn't significantly different from the a-g requirements for the University of California. In other words, it matters more to Bill that college-bound students have a strong high school curriculum overall than for them to have Algebra I in eighth grade.

There are other comments in this thread regarding tracking and middle school Algebra I. Momof4 is a traditionalist whose comments I've discussed a few weeks ago:

Momof4:
I cannot see this as anything other than a deliberate attempt to (hopefully, but not likely) decrease The [Achievement] Gap by holding back the top kids. The fact that the establishment is complaining about parents (=cheaters) who tutor their kids to enable them to get ahead, by whatever means, makes this pretty clear. The top URMs [underrepresented minorities] whose parents cannot provide tutoring will be casualties.

In many ways, this is a debate about tracking -- and where there's tracking, there's race. But ignoring race for the moment, notice that momof4's words echo SteveH's. The parents (whom she sarcastically calls "cheaters," to represent the district's opinion of them) provide tutoring for the students who want to take Algebra I before freshman year, and the ones who are left out are the lower income students who are smart enough to take eighth grade Algebra I but can't afford the tutoring.

Momof4:
My oldest grandkids are fortunate to be in a high-performing school district that believes in maximizing opportunities. I am pretty sure that my oldest grandson is finishing algebra I this year, as a 6th-grader and think his twin sister will take it next year.

So momof4's grandson is already taking Algebra I in sixth grade? Notice that there's another reason besides Common Core for schools not to offer early Algebra I -- there aren't enough other sixth graders taking Algebra I to fill the seats in the class.

There are two possibilities here. One is that the grandson is taking Algebra I with a class of mostly seventh and eighth graders. But that just kicks the can down the road -- he'll be ready for Algebra II in eighth grade, but his classmates are already in high school, on a separate campus. So there won't be 20+ students available to be in the same math class when he's an eighth grader. The other possibility is that in momof4's "high-performing school district," there really are 20+ other sixth graders in Algebra I with her grandson, ready to move on to Geometry and Algebra II together.

I also notice that momof4's granddaughter isn't in the same class as her grandson -- even though they are twins. One thing I'm curious about is the effect tracking has on twins. It might not be as bad when both twins are above grade level. But I know that if I had a twin and the twin were placed in a class at grade level while I was placed in a below-level class, I wouldn't like it. In fact, momof4 goes on to describe a pair of twins for which exactly this happens:

Momof4:
I am old enough to remember when it was not uncommon to offer kids working a year (or more) above grade level the chance to move to that grade, and it was common to hold back kids not functioning at grade level. One half of the set of twins who started school in my class was held back (cannot remember if it was in first or second).

Again, I wouldn't want to be the twin held back in this situation!

Continuing along the lines of tracking, let's look at wahoofive:

wahoofive:
Tracking is better for smart kids, who can race ahead together. Integrated classes are better for slow kids, who get more teacher and peer support because they’re not in a class of all slow kids. It’s just a matter of priorities.

Notice that here "integrated classes" means non-tracked. If tracking is better for smart kids and integrated classes are better for slow kids, then tracking becomes a zero-sum game. What benefits one group hurts the other, and vice versa.

But one commenter disagrees with wahoofive -- and it's none other than Ze'ev Wurman, a major traditionalist who posts at various websites:

Ze'ev Wurman:
Actually, even that is not true, although I agree that too many teachers religiously believe so.
Research indicates that slower kids *like* being in tracked classes because it doesn’t make them always feel “stupid” and always be the last to “get it” or to raise their hand.

According to Wurman, slower kids feel more comfortable in lower-track classes. But usually, the ones complaining about tracking are the parents, not the students. That's how all the controversy about tracking and race began -- the parents (not "educationists," not "social justice warriors," but the parents) complain that their child has been placed in too low of a track. And when they see that many of the other students on the low track are of the same race, the parents (not "educationists," not "social justice warriors," but the parents) grow skeptical that the district has the best interests of that race at heart. They fear that by placing their child on the low track, the child will not be prepared either for college or for any job that pays over minimum wage. The students might feel comfortable now that they aren't the last to 'get it,' but will they feel comfortable after they turn 18 and they aren't qualified for college or any high-paying job due to their low-track skills?

Here's another way of thinking about it -- when has any parent, of any race, ever said to school admins, "Please place my child in a lower track, so that he/she won't always be the last to 'get it' or raise their hand?"

By the way, here's what Jacobs wrote about race in the original post:

Compared to the previous cohort, fewer detracked students in all ethnic and racial groups are receiving D’s and F’s in algebra, data show. “About a third more students are ready for calculus, and that pool is more diverse than it’s ever been,” writes Sawchuk.
However, black students continue to lag in state test scores and enrollment in calculus.

Most of the remaining traditionalists in the thread are responding to the Common Core requirements that students explain their answers. This subthread begins with Darren Miller (as in Right on the Left Coast) -- the blogger who filled in for Jacobs during her recent vacation:

Darren:
Common Core fetishizes explanations over understanding. Do we really need to explain *why* 5×8=40, or can we just *know* it?
But on to the the thesis of the article. Doesn’t it seem like the height of hypocrisy for educators to refuse to allow students to learn as much as they can? And all because of someone’s pet political theory?

I'm not sure what "pet political theory" Miller is referring to here. But keep in mind that Miller is the author of Right [-Wing] on the Left Coast, while Common Core was first created under a Democratic administration, so that probably explains it. Commenter George Larson agrees with Miller here:

GEORGE LARSON:
I agree with you about rote memorization and mathematics, but I think rote memorization and numeracy/arithmetic are fine. But… I recall a statement from Norbert Weiner’s autobiography saying he memorized math before he truly understood it to get through his early years.

One of the posters in this subthread is BGarelick -- presumably, this is Barry Garelick, since Jacobs already quoted him in her article:

BGarelick:
As much sense as this makes, there were many blistering comments on this article which states what you’re saying. https://www.theatlantic.com/education/archive/2015/11/math-showing-work/414924/ and an extended debate here: http://blog.mrmeyer.com/2015/understanding-math-v-explaining-answers/

Here Garelick links to two of his other articles. One is that infamous Atlantic article from 2015 -- I wrote about it for a month after he first published it. The other is from the website of the former King of the MTBoS, Dan Meyer. This marks the only time that a traditionalist ventured onto the MTBoS.

One commenter here is Tara Houle -- a frequent commenter on the Garelick website, and so she follows him onto the Jacobs website:

Tara Houle:
Gawd here we go. A “visual” explanation goes soooooo much further than requiring kids learn through procedural steps, perfected over 2 millennia.

Houle is clearly being sarcastic here. She continues:

Tara Houle:
Get real. Why do you think so many kids are now enrolled in Kumon? Because they WANT to be there? If you’re only interested in throwing insulting comments from the curb, best to save it. Funny how math is the only subject which is under constant scrutiny for demanding more than a simple google click for an solution.

That's because math is the only subject that leads to "I'm not a math person" upon hearing that you teach the subject. That takes us back full circle to the hostess from Fawn Nguyen's post.

In fact, here's our final post from this thread:

SC Math Teacher:
Understanding that 5 sets of 8 = 8+8+8+8+8 = 8*5 = 40 is important when first learning multiplication. One must transition from that to automaticity as soon as possible, though. This emphasis on explanation is just a back door way to place equal value on trying and being correct. For my part, I want the people designing bridges to be correct; I don’t need an explanation.

Bridges -- hmm, there are many bridges in Philadelphia, such as the Ben Franklin Bridge. It's possible that the hostess in Philly crossed one of those bridges to get to work the night she met Nguyen. As SC Math Teacher points out, those bridges exist because their designers sat in math classes and were given A's only for correct answers and F's for incorrect answers. In other words, the hostess benefits from math's insistence on correct answers.

And yet the hostess informs Nguyen that she's "not a math person." She hates math because she was unable to find correct answers and got F's for it. If we asked her for her favorite subject, the hostess would probably say she's an English person, since English is a subject where explanations, not exact answers, lead to A's. The Ben Franklin Bridge doesn't make her hate math any less.

I assume that SC Math Teacher is a math teacher (from SC, South Carolina). So I assume that he or she has heard snide comments from adults who "aren't math people" or "hate math" upon hearing about the occupation. So wouldn't SC Math Teacher want to make math more palatable so that those tiresome "I'm not a math person" and "I hate math" will disappear?

Case Study: My Old Sixth Grade Class

As we've seen in today's traditionalists' debates, sixth grade is a critical year. The tone suddenly switches from "Singapore Math good, Everyday Math bad" to "get to the AP Calc track." SteveH himself tells us that after sixth grade, it's all over for those who didn't have traditional math (whether in class or via tutors) to prepare them for Calculus. But sixth grade is also around the time when students begin to question authority. They start asking "Why do we have to learn this?" In short, this is around the year when they begin to hate math or view themselves as "not a math person."

My own sixth grade class from my old charter middle school was anything but traditional. It was based on the Illinois State text, which was Project-Based Learning. In this section, I'll describe my process for planning a math unit, including my considerations and planning steps.

For this post, I choose the unit that I taught in late September to mid-October 2016. I believe that this unit is most representative of the lessons I taught that year. You can refer back to the blog posts that I made at this time, but recall that most of my posts were geared towards the eighth graders, and so I wrote little about the sixth graders at the time (until now).

The Illinois State text provides a pacing guide based on the Common Core Standards in order. Since this is the third unit of the year, the sixth graders are working on the third standard of the year:

CCSS.MATH.CONTENT.6.RP.A.3.A
Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

The Illinois State text suggests the following pacing guide for each unit:

  • Project, 1 day
  • Traditional Lesson, 1 day
  • Learning Centers, 1-2 days
  • Interactive Review, 0.5 day
  • Summative Assessment, 0.5 day
But to me, this pacing guide is impossible to follow for two reasons. First, I find that this provides too little time for the students to learn between the project/lesson and the assessment. Second, I'm required to teach other subjects in my class besides math. A coding teacher comes in every Monday to teach the students on computers. And since our small charter school has no science teacher, I must find time to teach science as well. I intentionally insert science lessons between the suggested math lessons while the students complete math for homework. Hopefully, this will allow extra time for the students to prepare themselves for the math assessment.

So this is the pacing guide that I actually followed:
  • Wed., Sept. 28th: Project
  • Thurs., Sept. 29th: Science
  • Fri., Sept. 30th: Project (Continued)
  • Mon., Oct. 3rd: Rosh Hashanah (No School)
  • Tues., Oct. 4th: Review for Science Quiz
  • Wed., Oct. 5th: Science Quiz
  • Thurs., Oct. 6th: Traditional Lesson
  • Fri., Oct. 7th: Traditional Lesson (Continued)
  • Mon., Oct. 10th: Coding
  • Tues., Oct. 11th: Interactive Review
  • Wed., Oct. 12th: Yom Kippur (No School)
  • Thurs., Oct. 13th: Learning Centers, Summative Assessment
Unfortunately, when I give the summative assessment, I find out that more than half the class has not mastered the main concepts. So let me describe my next steps.

First, the reason that some students struggle with the ratios assessment is that they have trouble with basic skills such as multiplication. So my first step is to review multiplication skills during the next unit (which is on Standard 6.RP 3b). Then on October 20th (one week after the original summative assessment) I give the students a Basic Skills Quiz (a "Dren Quiz"), on which the students must answer 50 questions on their 3's times tables.

Second, Illinois State allows teachers to creates assessment from a question bank. And so on October 27th (two weeks after the original summative assessment), I'm able to create a quiz containing both 6.RP 3a and 6.RP 3b questions. The students do much better on this quiz, showing that they've learned the material since the original assessment.

Let me now describe the role of data in my classroom. I implement "Learning Centers" (part of the suggested Illinois State pacing guide) with the assistance of a volunteer college student from UCLA ("Bruin Corps"). His first day in the classroom is the day of the original summative assessment. With his help, I'm able to divide the class into three groups (or "Learning Centers"), with an adult in charge of each group (either the Bruin Corps volunteer, the support staff aide, or myself).

A data wall is used to divide the students into three groups. At the beginning of the year, students complete Benchmark Tests based on the upcoming year's material, and twice a week (on Tuesday and Friday afternoons), the sixth graders answer questions on IXL based on the current standard. The scores on the Benchmarks and IXL are posted on the data wall in a corner of the classroom, so that both the students and I are aware of their growth in the class. And so I use the data wall to determine which students are to be placed in the high, middle, and low groups.

Part of the agreement with UCLA is that the Bruin Corps member must work with the slower students, and so I assign him the low group. Even though the first time these students work with him is the day of the original assessment, I believe that he's able to help some of them pass the test. And so I continue to have these students work with him as he begins to volunteer more regularly. A few students still do not improve as their skills are very deficient. For example, during the multiplication quiz, one student answers every question wrong because he adds instead of multiplies.

Let me now provide an example of Project-Based Learning (PBL) in my classroom. The Illinois State text includes a Student STEM Project Edition that contains all of the projects I use.

The project that I give on September 28th is called "What's the Best Advantage?" since this is the project that corresponds to standard 6.RP 3a. For this project, the students build mousetrap cars. Here is what the text directs the students to do in this project:

The Mousetrap's Lever:
  • Figure out the purpose of the mousetrap's lever. Pull the lever back and forth a few times to find out how it works.
  • Put the string on the lever.
  • Design a data table to record your findings.
  • Compare any patterns that you observed with your teammates. Discuss how changing the pulling location of the string on the lever affects the force.
The Wheels:
  • Predict what changes will occur when you change the wheel diameter on the mousetrap car.
  • Test your car over a four-meter distance. Record performance data for each trial as you complete test runs using both wheel diameters.
  • Compare any patterns that you observed with your teammates. Discuss how changing the whel diameter affects performance.
The Hub:
(replace "wheels" with "hub" in the steps above)

On the day of the project, many of the students struggle with creating the table, which is why a second day is needed to complete the project. For subsequent projects, I find that the projects go much more smoothly if I provide the student with a worksheets that give them hints, such as empty tables in which they should enter the data. (These, after all, are the tables that demonstrate ratios as required by Standard 6.RP 3a.) I must create these worksheets myself -- they aren't provided by Illinois State, nor does the text recommend that I give out such worksheets. It's my own experience which tells me that passing out such worksheets is a good idea.

Progress on Translation Proof

I've been continuing to work on the proof that I started last year and mentioned in my last post:

A line and its translation image are parallel.

We found out last time that this is equivalent to:

Given a composite of two reflections, if a line and its image are nonparallel, then so are the mirrors.

Recall that the proof is supposed to be "natural" -- that is, it should work in both Euclidean and spherical geometry without distinguishing between the two. Thus we can't write the following:

Proof:
Case I: Spherical Geometry
All lines intersect in spherical geometry, therefore the mirrors intersect. QED

Case II: Euclidean Geometry
etc.

We also found out that the theorem to prove is false in hyperbolic geometry, therefore the proof must require a statement that holds in both Euclidean and spherical, but not hyperbolic, geometry. As it turns out, Euclid's Fifth Postulate, depending on how it's written, actually holds in spherical geometry (for example, Playfair's Postulate mentions "...at most one parallel line..." and so on). Thus such a statement must appear in the proof.

Here's what our proof looks like so far -- l is the original line, l' is the image of l reflected over the first mirror m, and l" is the image of l' reflected over the second mirror m'. And so we're given that l and l" intersect, and we're to prove that m and m' intersect. We found out that we can draw l' at any position, and then we divide into cases:

Case I: l' is parallel to both l and l"
This case is impossible by Transitivity of Parallels, as we're given that l and l" intersect. This is our first use of a Parallel Postulate. (It holds vacuously in spherical geometry as there are no parallels.)

Case II: l' is parallel to exactly one of l and l"
Without loss of generality, assume that l' is parallel to l. This is where those lemmas that I mentioned last year come into play. If a line is parallel to its reflection image, then the mirror is parallel to both of them (and halfway between them), so l' | | m. But then m' can't be parallel to m, since then l' would be parallel to m' and thus l" as well, by Transitivity of Parallels. So m and m' intersect.

Case III. l, l', and l" are concurrent
The other part of the lemma tells us that if a line and its reflection image intersect, then the mirrors are the angle bisectors. So in this case, all five lines l, l', l", m, m' are concurrent.

Case IV: l, l', and l" form the sides of a triangle
This is the most general case -- all the others are considered special cases. But it's the case when the above proofs don't work.

Notice that m and m' could be two of the angle bisectors of this triangle. So it would seem that the proof that angle bisectors of a triangle are concurrent would work here. The problem is that the usual proof of this requires both AAS and HL, which fail in spherical geometry. It still might be possible to prove that the angle bisectors of a spherical triangle are concurrent using AAA instead, but this fails the requirement to avoid distinguishing between Euclidean and spherical cases.

Also, notice that the lemma tells us that there are two possible mirrors that map a line to another line that intersects it. In short, the second mirror could have bisected the exterior angles of the triangle rather than an interior angle. The concurrency of the (interior) angle bisectors tells us nothing about whether exterior angles intersect or not.

I'm actually thinking about the actual Fifth Postulate as given by Euclid. The two interior angles of a triangle are each less than 180 (even in spherical geometry), and so bisecting these give us two interior angles that are each less than 90. So these two interior angles are indeed less than two right angles -- which means that they intersect by Euclid's Fifth. If we take the exterior angles instead of the interior angles, then they form angles that are less than two right angles outside the triangle.

But there is a special case -- if we take one exterior angle and one interior angle, then the sum of these angles could be exactly 180. Yet it's possible to show that if these two angles add up to 180, then two angles of the original triangle must have been 180 as well -- which is possible if we're already in spherical geometry (where any two lines intersect anyway). So this is still not quite the proof that I'm seeking. I'll continue to work on it.

Conclusion: Three Californias?

As you know, I live in California. Some of you might have heard on the news about a proposition that is coming up on the November ballot -- to divide California into three states:

https://ballotpedia.org/California_Three_States_Initiative_(2018)

What is my opinion of this issue? Well, this is a tough one. I currently sub in two districts -- and these two districts would wind up in different states (California and South California) under the proposal. I must admit that because of this, it's doubtful that I'll vote for this proposition.

Meanwhile, another proposition is close to qualifying for the November ballot, one that I've discussed on the blog before -- Year-Round Daylight Saving Time in California:

https://ballotpedia.org/California_Daylight_Saving_Time_Measure_(2018)

I've stated before that I have no problem with the biannual clock change, but if I were to choose one clock to keep the whole year, it would be DST. So it's possible that I might vote for this proposition, but I admit that it would be more logical if Nevada would also convert to Year-Round DST. Then all three states California, Nevada, and Arizona would have the same time all year.

The timing of the election might help Year-Round DST pass -- Election Day is right after the clocks move back to Standard Time. Supporters of the proposition can point to the dark early evenings and point out that one could eliminate them by voting for the proposition.

Both of these propositions require congressional approval to have any effect.

The nightmare scenario for me is if the state divides -- and then only South California adopted the proposed Year-Round DST while the other two states keep the clock the way it is (which could happen, as only South California borders Arizona, while North California might fear late winter sunrises under Year-Round DST). Then not only would my two districts be in two different states, but two different time zones during a large part of the school year.

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