Palm Sunday Post

Table of Contents

1. Introduction
2. An April Fool's Day Problem?
3. Traditionalists: The Math Zombie Excuse
4. The Path Plan Revisited
5. Paths, Wayne Bishop, and JK Brown
6. The Tracking Debate
7. An Analogy
8. Cosmos Episode 8, "The Sacrifice of Cassini"
9. An April Metamath Proof
10. Conclusion

Introduction

Today is the Sunday before Easter -- also known as Palm Sunday. The fact that I'm posting on a Sunday and not a Monday -- and the fact that this post isn't titled "Lesson 14-2" -- gives it away. As expected, neither of my districts is going to open for students tomorrow.

In fact, both of my districts held board meetings the day after my last post to discuss the closure. My old district (whose calendar the blog still follows) is scheduled to open on Tuesday, May 5th (that is, Cinco de Mayo). Meanwhile, now the California State Superintendent, Tony Thurmond, is echoing Governor Gavin Newsom and his prediction that schools won't reopen this school year. My new district has listened to his guidance, and so in that district, the school year is already over (as far as we subs are concerned).

The scheduled reopen date for my old district is now Day 161. According to the digit pattern, all three remaining chapters of the U of Chicago text -- Chapters 13-15 -- will be skipped.

Indeed, Day 161 is usually when I switch my focus from the U of Chicago to the SBAC. By now, Governor Newsom has declared the SBAC exam to be canceled. I haven't quite decided what I'll do on the blog for the last twenty days of the school year. It's possible that I'll post last year's SBAC review pages anyway -- but if a question includes Chapter 13-15 material, then I'll also post the worksheet for that lesson. The other questions might also serve as review for the final (which is why I continued to post SBAC review worksheets last year even after most schools had given the SBAC).

Another possibility are just to post Chapters 13-15 during the last twenty days -- or more likely, just two of those three chapters, if we continue the digit pattern of one chapter per ten days. It's also possible to post something new -- perhaps probability (part of the California Common Core Geometry curriculum) or something interesting like spherical geometry. We'll cross that bridge when we come to it, since there still might not be a last twenty days of school.

By the way, in my new district, spring break would have been this upcoming week (Holy Week). I wrote last year that this district takes off the week before Easter in years when the holiday is late -- so apparently, April 12th counts as a "late" Easter. Of course, this year that district is closed the rest of the year, so the week of spring break is now a moot point.

The AP has already announced that the tests will be given online, so students can take them at home:

https://apcentral.collegeboard.org/about-ap/news-changes/coronavirus-update

Also, with SAT and ACT exams canceled, UCLA and the other UC's have announced that the Class of 2021 does not need to submit test scores with the application. This is the Class of 2021, not 2020, since 2020 should have already taken the test before the virus broke out. Notice that the eighth graders from the old charter school are now juniors. As I mentioned in our last post, our dean visited the eighth grade class in the first week of school and pushed the importance of high SAT scores for working towards high-paying Silicon Beach jobs -- and now that class won't need to take the SAT exam at all. (Of course, how was he to know that a virus would wipe out the tests entirely.)

This is my fourth spring break/coronavirus break post. During the closure, I've continued to spend my time watching Numberphile videos. Indeed, two more virus-related videos have been posted:

The first ten-minute video is a short version of the second 28-minute podcast. The speaker here, Kit Yates, is a Mathematical Biologist. (Add that to my "21 Years a Bruin" post from last time -- if I'd chosen to take courses in Mathematical Biology at UCLA, I'd definitely have a paying job now!)

By the way, just before I made my last post, someone commented on an old post from June 2016. It was all the answers to an old Chapter 15/SBAC Prep worksheet. In replying to that student, it was the first time I'd helped someone with a Geometry question in weeks.

As teachers switch to online learning, let me say that this blog is, yes, an online resource. If you wish, you can refer to my worksheets from earlier this year for Chapters 1-12 of the U of Chicago text. If you need any worksheets for Chapters 13-15 (Logic & Indirect Reasoning, Trigonometry & Vectors, and Further Work with Circles), then you can go back to earlier years on this blog. For a start, let me link here to the 2017 worksheets -- the year that I posted only Chapters 13-15 (after I left the old charter school):

http://commoncoregeometry.blogspot.com/2017/04/lesson-13-1-logic-of-making-conclusions.html

(By the way, on the Andrew Usher calendar, today is Easter. On that calendar, the holiday is always a Sunday between April 5th and 11th.)

An April Fools' Day Problem?

For April Fools' Day on her Daily Epsilon of Math 2020, Rebecca Rapoport wrote:

(1 * infinity)/infinity

At a quick glance, you might imagine that the infinities cancel to leave 1, since this was the problem for the first of the month. But then you might think, "Wait -- infinity/infinity is indeterminate!"

This is the second of three special days on the Rapoport calendar with a joke instead of a legitimate math problem. The first special day was Pi Day, while the third is later this year.

Of course, Rapoport wrote her April Fools' Day joke before the year started. For the most part, practical jokes have become a bit subdued this year due to the virus. For example, one of my favorite game shows, The Price Is Right, has had an April Fools' Day episode every year (except in years the holiday falls on a Saturday or Sunday) -- but not this year. The episode didn't tape until March 9th, by which time many were already preparing to close or cancel events.

The Metamath website hasn't had an April Fools' Day joke theorem since just before I started this blog -- so of course there wouldn't be one this year.

Now that I'm on a vacation schedule and posting only once or twice per week, it's likely that most Geometry problems on the Rapoport calendar will land on non-posting days. There was a question yesterday on how many triangle centers there are in classical Greek Geometry. The answer is four: the centroid, circumcenter, incenter, and orthocenter -- and of course, yesterday's date was the fourth.

Today's problem isn't Geometry, but on something else:

What is the Frobenius number of the numerical semigroup generated by 3, 7, and 8?

This sounds complicated -- you might recall from last week that a "group" is one of the three main objects of Abstract Algebra, and yes, a "semigroup" is related to this. But as it turns out, the concept isn't that complicated at all. Indeed, according to the following Numberphile video (from 7 1/2 years ago), the Frobenius number of the numerical semigroup generated by 6, 9, and 20 is 43:

The set Z of all integers is a group, but a semigroup is allowed to have some integers missing -- including zero and all negative integers. The Frobenius number is the largest missing number -- and in the case of Chicken McNuggets, that number is 43. (Note: In this country, we now have ten-packs rather than nine-packs, so now no odd number of nuggets is possible.)

Thus for today's Rapoport problem, we need the Frobenius number for 3, 7, and 8. Since we have 3, this means that all multiples of three are generated. And we can add any multiple of 3 to either 7 or 8 to obtain any greater non-multiple of 3. Therefore the desired Frobenius number is 5 -- and of course, today's date is the fifth.

Notice that the Frobenius number for football scores is 1. But in practice, safeties are rare -- and surely two safeties without any other points would be almost unprecedented. Thus while scores like 2, 4, or 5 are possible, they aren't common at all.

If we count only field goals (3), touchdowns with an extra kick (7), and touchdowns with an extra conversion (8), then we obtain today's Rapoport problem again. Of course, we should at least include touchdowns where the extra points are missed (especially for failed conversions). But this would be equal to two field goals, thus keeping the Frobenius number for realistic (non-safety) football at 5.

The following link discusses Frobenius numbers for football scores. It also mentions that it might now be possible to score a 1-point safety after all:

https://sports.stackexchange.com/questions/3208/how-does-one-get-5-points-in-a-football-game

Traditionalists: The Math Zombie Excuse

This is labeled as a traditionalists' post. Indeed, the main purpose of today's post is catching up on the traditionalists and their recent ideas.

Had it not been for the coronavirus, the Chapter 13 Test -- and hence my scheduled traditionalists' post -- would have been on April 2nd. Anyway, our main traditionalist, Barry Garelick, posted on exactly that date, April 2nd:

https://traditionalmath.wordpress.com/2020/04/02/the-math-zombie-excuse/

From an article about the Russian Math School:

“Meanwhile, Hilary Kreisberg, director of the Center for Math Achievement at Lesley University and a former fifth-grade teacher turned math coach, says her experiences with RSM students have led her to question the claim that Russian Math focuses more on developing a deep understanding of math instead of memorization. In fact, she has seen the opposite. “From what I’ve seen, they come in well above their grade-level standards in terms of memorization, but not in terms of content understanding,” she says. “Many of them very quickly get to an answer or can compute in a fast way, but they can’t necessarily explain to me what they’re doing or why they’re doing it.” And explanations, she says, are a critical component of mathematics. “In public school teaching, we are very strictly taught that the goal is not to accelerate,” Kreisberg says. “The goal is not to extend their thinking into another grade level, but to go deeper with the current grade-level standards because there’s always more you could learn about a topic.” “

OK, there are several things going on in this post. First of all, this article refers to something called "Russian School of Mathematics" (RSM). I read both Garelick's post and the Boston Magazine article to which he links to figure out exactly what this RSM is. It appears that RSM is some sort of tutoring center -- you know, the tutoring that our other traditionalist, SteveH, likes to talk about.

It's hard for me to tell what Garelick is saying about this RSM tutoring. I believe the gist of it is this -- like most tutoring centers that Garelick and his allies approve of, leans more traditionalist, with a focus more on memorization than understanding. But traditionalist memorization has been criticized so much that even RSM itself claims to focus more on understanding!

At the start of the original Boston Magazine article, I read the following:

On a January afternoon at the Russian School of Mathematics headquarters, a 6,500-square-foot bilevel brick building in a Newton corporate office park, Inessa Rifkin leads 15 excitable third and fourth graders through a lesson on graphs. She is in her sixties and chic in a blue-gray tunic over a crisp white button-down, skinny black pants, and a Tiffany-style silver pearl necklace. When she asks a question, her Russian accent unflinching, more than half of the students put their hands in the air. Not a single one has a cell phone on his or her desk. No one’s staring into space.

About halfway through the two-hour class she asks, “Quiz or break?” The group is unanimous in its decision: Quiz it is!

One thing I've often said in the traditionalist posts is that admittedly, traditionalism does teach math more effectively than reform methods -- but students are more likely to find it boring and will just leave traditionalist p-sets blank or find other excuses not to do the work. But one-on-one tutors, on the other hand, can get their students to work because they're sitting inches away from their student, so there's no way for the student to escape. (Recall that five years ago I was a math tutor, and so this was what I considered tutoring to look like.) Thus tutors can get away with traditionalist math, but not classroom teachers.

Yet in this article, the tutor is standing in front of the students as if she were a classroom teacher -- and there are 15 students (about as many as my eighth grade class at the old charter school). And the author makes it a point that the students aren't leaving the assignment blank or playing with phones -- and they even choose to take a quiz rather than a break.

Of course, we can't quite compare these kids in Grades 3-4 to my eighth graders (or, for that matter, my sixth graders). I've often named Grades 3-4 as right around the border, between the younger years when students are still eager to learn and thus accept traditionalist lessons, and the older years when they question everything and ask "Why do we have to learn this?" and "When will we use this math in real life?"

But as usual, the traditionalists like to make it seem as though when math is taught traditionally, then students are eagerly volunteering and they learn more, and when it's taught non-traditionally, then they lose interest. And we know what the traditionalists really want in this thread -- for all math classes taught during the school day to be just as it is at this RSM after school tutoring center.

Acceleration is mentioned several times in the article. The part I quoted above mentions that while students accelerate to material above grade level RSM, public schools prefer to go deeper within material at grade level.

I like to think of this from a Geometry perspective, as this is a Geometry blog. Going deeper within grade level means placing a sophomore into Honors Geometry rather than Geometry. In the honors class, students might get to prove some theorems which are essentially assumed as postulates in the regular class (for example, some of the area and volume formulas). Traditionalists, of course, would prefer accelerating this sophomore into Algebra II, since their goal is senior-year Calculus.

Let's see what the commenters have to say here. As usual, we begin with SteveH:

SteveH:

““Many of them very quickly get to an answer or can compute in a fast way, but they can’t necessarily explain to me what they’re doing or why they’re doing it.””

Compared to which other kids?

Where are all of these better understanding kids? Show me the college STEM programs they are in. None of these educators EVER answer these questions. 

The original article mentions RSM graduates going on to various colleges. Thus SteveH is asking for a similar list that shows non-traditionalist "better understanding" students getting into colleges -- and since such a list is lacking, he asserts that only traditionalist math works.

My usual retort is that at least there are fewer papers left blank in non-traditionalist classes. But that won't work in this thread -- all SteveH and the other traditionalists have to do is point to RSM, where the students are eagerly working on traditionalist memorization.

SteveH:

“It’s because we’re not doing it right”.

I’ve been waiting for more than 20 years!!! MathLand, then EverydayMath, but they aren’t complaining about them. All they do is haul out the arguments against traditional approaches that haven’t been seen in public K-6 schools in decades. And once again, they don’t account for all of the mastery work that now has to be done at home. As one of those parents, my hand is raised, but those educators are ignoring me. That’s because my son provides cover for their Everyday Math repeated partial learning.

When will they realize that they really are, fundamentally, not doing it right?

As usual, SteveH criticizes "Everyday Math" -- the U of Chicago elementary math texts. And as usual, when he says that "they" are not doing it right, he only counts traditionalism as "doing it right."

Another big traditionalist in this post is Ze'ev Wurman:

Zeev Wurman:

I believe that RSM is, in some sense, consciously deprecating memorization for marketing reasons. It can afford this, since the kids come ready with it anyway, so they pretend that memorization is not important.

A year back I published a report on RSM and Kumon. https://pioneerinstitute.org/download/axioms-of-excellence-kumon-and-the-russian-school-of-mathematics/

Most of this is what I already surmised earlier -- RSM only claims that it isn't based on memorization only because anti-traditionalists say that memorization is bad. In reality, RSM relies on memorization and is successful with it.

It's interesting that in his report that Wurman links to above, he's quick to defend the Russian School of Mathematics, while in previous posts, he criticized Transformational Geometry because it was first used and developed by Russians.

(Note: In that article, there is a section called "Math vs. Applied Math." In my last post, I also had a section about Math vs. Applied Math -- but that was all about majoring in Math vs. Applied Math at UCLA, and has nothing to do with young students at Kumon Tutoring Centers, which is what Wurman is discussing here. Please do not take the "Math vs. Applied Math" section in my last post as any sort of endorsement for traditionalist math.)

There is one more commenter in this thread, Wayne Bishop:

Wayne Bishop:
Former 5th grade teachers offer such a rich experience in mathematics understanding and application. It is unlikely that she has ever had a real math course for solved a the real math problem and yet she is responsible for the mathematics education of hundreds, maybe thousands, of students. Sin has many forms.

Here Bishop is mainly criticizing the specific fifth grade teacher mentioned in the article (Hilary Kreisberg) because she doesn't defend RSM or traditionalist math. But he extends his criticism to most fifth grade teachers for having never taken a "real math course" (by which he probably means something like an upper-division college course). Fifth grade teachers, after all, usually have Multiple Subject credentials.

And so this takes us right back to whether there should be specialist math teachers in elementary school, and something that I mentioned on the blog in the past but not recently -- the Path Plan.

The Path Plan Revisited

It's been about two years since I've written about the Path Plan. Indeed, my last discussion of it was in my Palm Sunday 2018 post. Well, since this is once again a Palm Sunday post, and since Wayne Bishop is criticizing how elementary school, let me cut-and-paste the Path Plan from that old post:

In past posts, I wrote about the system that my old elementary school introduced -- the path plan. In this system, students are officially divided into "paths," not "grade levels." Here are rough correspondences between paths and grades:

Early Learning Path: Headstart and Kindergarten
Primary Path: Grades 1-2
Transition Path: Grades 3-4
Preparatory Path: Grades 5-6

When I was a young student in the Preparatory Path, our classes stayed in homeroom (for English and history) until lunch. Then after lunch, we attended two more classes. One was math, and the other was an exploratory wheel that switched every trimester (art, health, and science).

Under this plan, only half of the Grade 5-6 teachers actually taught math -- so of course, the half that was stronger in math would teach the subject. My fifth grade teacher was strong in math, so I remained in her class for math. My sixth grade teacher taught science instead, so I went to another teacher's class for math that year.

A simpler schedule was used in the Transition Path. All Grade 3-4 teachers taught math at the same time, but some students moved to other classrooms during math time. I can easy see how the stronger math teachers could be assigned fourth grade math while the others taught third grade math. Thus the best 50% of teachers would cover math in Grades 4-6.

I once posted a more ambitious version of the Path Plan. We push everything down one path, so that Primary Path students leave homeroom only for math, Transition Path students leave for two classes, and Preparatory Path students leave for three classes. Then this ensures that the best 50% of teachers teach math in Grades 2-6, leaving only K-1 for weaker math teachers. Also, in the Preparatory Path, science can be a whole year, not a trimester. Back when I was a fifth grader, we could get away with less than a year of science. But now with state and NGSS science testing, a full year of fifth grade science is crucial. (When I was a fifth grader, our school actually tried having three classes after lunch, but then it switched to two classes with full implementation of the Path Plan.)

The following chart shows how many teachers each student would have at one time under the more ambitious Path Plan, along with which subjects aren't taught by homeroom teachers:

Path                    Teachers  Subjects not in HR
Early Learning   1
Primary              2              math
Transition           3              math, elective
Preparatory        4              math, elective, science

Here's how I'd make these classes fit a bell schedule -- all grades stay in HR until all grades have completed recess. Then after recess, all grades attend the first extra class. Afterward, Primary Path goes to lunch, and the other two paths attend the second extra class. Afterward, Transition Path goes to lunch, and Preparatory Path attends the third extra class, followed by lunch. After lunch, all classes return to homeroom. The length of each class therefore is the same as the length of lunch -- 40 minutes might be a good length of time for both lunch and math.

Notice how this path pattern can be continued beyond elementary school, to a Middle School Path (Grades 7-8) and Early High School Path (Grades 9-10):

Path                    Teachers  Subjects not in HR
Middle               5               math, elective, science, P.E.
Early High         6               typical high school schedule

So in middle school, students have the same teacher for English and history -- which is what my old middle school referred to as "Core class." After Grades 9-10, the pattern continues, but the number of classes goes down every two years:

Path                    Teachers  Comments
Late High           5              only 22 year credits often needed to graduate
AA/Lower Div.  4              (typical college schedule)
BA/Upper Div.   3
MA                     2
PhD                    1              dissertation adviser

Here's what we do to address SteveH's [2020 update: or Wayne Bishop's] ideas about teacher training - there could be two types of Multiple Subject Credential, namely math and no-math. Elementary schools should strive to hire about half math and half no-math for Grades 1-6. All students in Grades 2-6 would have a math credential for their math classes, while no-math teaches either K-1 math or no math at all. (Under the less ambitious Path Plan, it's half math for Grades 3-6, and Grades 4-6 are guaranteed to have a math credential for their math classes. No-math teaches either K-3 math or no math at all.)

Returning to 2020, there are a few tweaks to this system that I might suggest here. I was inspired by going back to the website of my old elementary school. Even though the Path Plan no longer exists, I did notice the new PE schedule, which is different from when I attended there. And so I wondered, is it possible to make this Path Plan fit the new PE schedule?

Here's what the new schedule looks like -- instead of everyday, PE meets only half the days. It meets Mondays and Thursdays (45 minutes each), and half the Wednesdays (30 minutes each). Each of the seven grades has its own PE time -- the middle grades (the ones our Path Plan places in Transition Path) are before recess, the upper grades (Preparatory Path) are between recess and lunch, and the lower grades (Primary Path) are after lunch.

Our plan has the students in homeroom before recess as well as after lunch, and so only the upper grades (Preparatory Path) would be in a class other than homeroom.

So here are the changes to make -- each class between recess and lunch is 45 minutes, not 40, so that it matches the length of PE. And let's switch science with the elective -- now both Transition and Preparatory Paths have science out of homeroom, while only Preparatory Path has an elective out of homeroom class. This elective can meet only Tuesdays, Fridays, and non-PE Wednesdays, so that depending on the day, students either go to elective or PE. (We don't need to do what my old charter did and have a completely different class order for M/Th, T/F, and Wednesday -- the only classes that change from day to day are elective and PE.)

This is better than having elective everyday and letting science be the class that only meets every other day when there's no PE. (Of course, if we did this, these sixth graders would still get more science than what my sixth graders got at the old charter school.)

Paths, Wayne Bishop, and JK Brown

I posted the Path Plan again because of Wayne Bishop's comment that I quoted above. He criticized multiple subject teachers for not knowing enough math to teach the young students.

Of course, we know Bishop's real motive -- he assumes that everyone who's taken higher math in college agrees with the traditionalists, so that anyone who doesn't agree with the traditionalists must not have taken higher math. He wants to rid elementary schools of those teachers and replace them with those who are more open to traditionalism. He and the other traditionalists imagine that with more math-trained teachers, every regular math class will be like that RSM class described above.

But there's a better reason to have math specialists in elementary school besides traditionalism. It's ideal to have a fifth grade teacher who is comfortable with fractions, for example. A teacher who, as a young fifth grader, struggled with fractions herself might exude that same fear in front of her fifth grade students. We don't need a fifth grade teacher to teach traditionalist methods, but we do need a fifth grade teacher to teach fractions.

(By the way, I'm not quite sure how strong the elementary teachers at the old charter school were when it came to math. I recall helping the fifth grade teacher out with some area problems, but she also came up with the "PRime = I and ME" mnemonic -- that is, 1 and itself -- that had never before occurred to me. Of the elementary teachers, the strongest at math might have been the kindergarten teacher, since she became my successor.)

The Path Plan was designed to have the strongest teachers teach math. For example, my own fifth grade teacher was strong at math, so she taught the math classes. My sixth grade teacher would teach science instead.

In the new version of the Path Plan described above:

• More mathy Primary Path teachers teach Math 2. Less mathy teachers teach Math 1.
• More mathy Transition Path teachers teach Math 3-4. Less mathy teachers teach science.
• More mathy Preparatory Path teachers teach Math 5-6. Less mathy teachers teach either science or an elective.

When I was a young student, every teacher had an elective at the end of the day, since this class wouldn't meet on Wednesdays (which were Common Planning, just like at the old charter). The new one has Grades 5-6 elective on Tuesdays, Fridays, and some Wednesdays, to match the corresponding PE time for that grade on the other days.

I wonder whether this version Path Plan would satisfy Wayne Bishop, since math is now being taught by stronger math teachers. Most likely, it all depends on whether those stronger math teachers are using traditionalist methods.

But there's another blog commenter who was complaining about math class recently -- JK Brown, whom I named in my last post. In this case, Brown wasn't complaining about math teachers and traditionalism, but about math classes and acceleration.

Recall Brown's vision of the future -- the schools are closed not only for the rest of this academic year, but into next year as well. Distance and online learning are extended -- and stronger students find that they can more easily accelerate into higher math courses online. Then when the schools finally open again, these advanced students choose not to return to school, because they've seen that they can learn more effectively at their own level online.

In Brown's imagined future, brick-and-mortar schools have become obsolete, and all learning is distance or online learning. Substitute teachers like me are no longer needed -- and moreover, not as many regular teachers are needed either, because a load of 300 or more students becomes possible if all the classes are online.

I, of course, want to save my job. And so I wonder, would a Path Plan discourage advanced students from wanting to take all classes online (as Brown describes) and keep them in the schools?

This the other major part of the Path Plan. Based on their reading level, advanced second graders are placed on the Transition Path, while advanced fourth graders are placed on the Preparatory Path. On the other hand, below basic third graders remain on the Primary Path and below basic fifth graders remain on the Transition Path. Thus students find that their classmates in their homerooms are closer to their own reading levels.

The reason I keep bringing up PE and bell schedules is that the bell schedules are supposed to line up, so that a student on one path for homeroom (ELA) can be on another path for math. Let's look at what a possible bell schedule might look like. (Recall that this schedule is similar -- but not identical -- to the bell schedule currently used at the elementary school I attended as a young student.)

8:20-8:30 -- Pledge of Allegiance
8:30-9:15 -- Third Grade PE, all others in HR/ELA
9:15-10:00 -- Fourth Grade PE, all others in HR/ELA
10:00-10:15 -- Latest possible AM recess, all other grades have earlier AM recess
10:15-11:00 -- Sixth Grade PE, all others in first non-HR class
11:00-11:45 -- Fifth Grade PE, Primary lunch, Transition/Preparatory in second non-HR class
11:45-12:30 -- Primary HR, Transition lunch, Preparatory in third non-HR class
12:30-1:15 -- Second Grade PE, Preparatory lunch, all others in HR
1:15-2:00 -- Kindergarten PE, all others in HR/PM recess
2:00-2:45 -- First Grade PE, all others in HR

Notice that kindergartners stay in homeroom all day except for PE -- so "all others" in the schedule really refers to non-kindergartners. And of course, "PE" only refers to PE days (Tuesdays and Fridays) -- on Mondays and Thursdays, this is replaced by homeroom (or Preparatory elective).

Let's now try some actual examples. Consider a hypothetical student who is at the fifth grade level for ELA and fourth grade for math. The student's chronological grade here is irrelevant, but let's assume that he's a fifth grader (so he's at grade level for ELA and below grade level for math). This student is placed on the Preparatory Path based on his ELA level. But he goes to the Transition Path when it's time for math. Since he's a fifth grader, from 11:00-11:45 he has either PE or his elective (say it's art) depending on the day of the week. This leaves either the 10:15-11:00 slot or the 11:45-12:30 slot for his Math 4 in the Transition Path -- and since the Transition Path teachers have lunch at 11:45, this leaves only 10:15-11:00 for Math 4. His last class, Science 5, is from 11:45-12:30.

What if this student in ELA 5 and Math 4 were a chronological fourth grader (that is, she's above grade level for ELA and on grade level for math)? Then she'd still be placed on the Preparatory Path (as an advanced fourth grader) and have the same schedule as above. In other words, she'd still have PE 5/elective and Science 5, since these are the classes available to her on the Preparatory Path. In other words, whether a student gets above grade level science depends on the ELA level, since ELA determines the path.

Our next example is a student who is first grade for ELA and third grade for math. Since ELA determines path, he is placed on the Primary Path. All Primary Path students have one class out of homeroom, namely math from 10:15-11:00. Thus this must be when he takes his Math 3 class.

For our final example, let's reverse the ELA and math grades from our first example -- we have a student who is in ELA 4 and Math 5, and let's say she's a chronological fifth grader. Since her ELA grade is fourth, she's placed on the Transition Path and thus has PE with the fourth grade. But if we also place her in Science 4, then she's missing out on valuable Science 5 material that she'll see on the California Science Test, which all fifth graders must take.

This is a fundamental problem with the Path Plan as written -- in theory, we should be ignoring chronological grades, but state testing mandates are based on chronological grades. (A second grader doesn't need to take SBAC, even if she's on the Transition Path. Meanwhile, a third grader must take SBAC, even if he's on the Primary Path.)

We might decide to have a rule that all chronological fifth graders must have Science 5 to prepare for the CAST, even those on the Transition Path. So our fifth grade example student would have her two non-HR classes be Math 5 and Science 5 -- which is possible, since she has PE with fourth grade. But one of these classes would have to be during fifth grade PE time -- but that's OK. There might be a special Science 5 class for Transition Path fifth graders -- and it might be taught by either a Transition or Preparatory Path teacher (depending on exactly how many students are in this situation).

The CAST, while taken in fifth grade, covers some Grade 3-4 material (just as the eighth grade CAST covers some Grade 6-7 material). So it might be desirable to require that students in the Transition Path be on grade level for science as well. This only leaves out chronological third graders who are on the Primary Path, who might not get the Science 3 material they need for CAST.  On the other hand, Transition Path second graders might miss out on Science 2 material -- but this is irrelevant as Science 2 isn't tested on the CAST.

Notice that Primary Path students get both Science 1 and Science 2 standards. Indeed, this is what happens with other subjects such as history -- the Preparatory Path students get half of the US History 5 standards and half of the World History 6 standards. These alternate each year, so that students who follow the normative paths get all the standards (and those who follow above or below grade level paths get most of them).

All of what I've described so far is based on my memory of the Path Plan as it was used back when I was a young elementary school student. But the truly ambitious Path Plan described in this post covers all of Grades K-12, not just K-6. This goes well beyond my experience as a young student.

For example, should an advanced sixth grade student be allowed to attend the 7-8 middle school (or, if we take my actual district, the 7-12 high school)? Back then, the answer was no, because the Path Plan ended at sixth grade. But under the 7-12 plan, it might be desirable.

We also have to figure out what happens to students who test at sixth grade for ELA and seventh for math, or vice versa. Of course, by this point this becomes common even at non-path schools -- a student who tests for eighth grade ELA and ninth grade math (which means Algebra I under the Common Core) is simply called an "eighth grader in Algebra I."

By the time we reach high school, grade levels and graduation is based on credits anyway. All the Path Plan really does is allow chronological eighth graders to earn credits that count towards high school graduation (and chronological freshmen to retake middle school classes that don't count towards graduation).

JK Brown wants us to focus on the students who are above grade level, and the Path Plan works for these students. It allows second graders to move up to Transition Path, fourth graders to move up to Preparatory Path, sixth graders to move up to the 7-8 middle school, and eighth graders to take high school classes for credit.

The Tracking Debate

In the most recent Barry Garelick post, the traditionalist teacher mentioned tracking. And where there's tracking, there's race. I wrote a little about race in that post, but as you already know, I save long, extensive discussions about tracking and race for the bottom of vacation posts, like this one.

By the way, recall that the Path Plan is also a mild form of tracking. Students weren't blindly assigned to tracks based on their grade level -- instead, students above grade level might be assigned to a higher path, while those below grade level would be on a lower path. So advanced fourth graders were placed in Preparatory Path while below basic fifth graders were placed in Transition Path. And likewise, advanced second graders were placed in Transition Path, while below basic third graders were placed in Primary Path. (I'm not sure whether the K-1 boundary was crossed this way -- at the time there was only half-day kindergarten, so crossing this line was awkward. Indeed, the K classes were almost always called "kindergarten" -- the name "Early Learning Path" was seldom used.)

Oh, and notice that students are assigned to paths based on their reading ability -- since after all, English (and history) is taught during homeroom. The students' math ability is taken into account when assigning the students to tracks. I know that fifth graders in the Preparatory Path might be assigned to sixth grade math, and so it's possible for fourth graders to be assigned to fifth grade math, even if their reading level keeps them in the Transition Path. I set up the bell schedule earlier so that it's easy to move to a different path during math time.

If it wasn't for race, tracking might still exist today. We might have something similar to a nationwide Path Plan -- and even the Common Core Standards could be set up to assign standards to paths, not grade levels.

But as long as tracking has a racial problem, it'll never be completely brought back.

I still consider my Path Plan to be a compromise -- a soft form of tracking. If done correctly, we can avoid the biggest racial problems with tracking. First of all, the Transition Path, for example, has both advanced second graders and below basic fifth graders. It's not the case that we have advanced students on one side of the campus and below basic students on the other -- segregation that often evolves into racial segregration.

The other racial problem in tracking occurs when members of one race are repeated held back while members of another race are allowed to proceed quickly through the grades. In practice, our Path Plan was only for Grades K-6, so all students began at kindergarten and left at sixth grade, even if they spend an extra year in Primary or Transition Paths. Hence it's not the same as retention.

It might get tricky though if we allow advanced sixth graders to move up to middle school -- would below basic seventh graders have to stay at the elementary school, and if they do, would those seventh graders be members of a certain race? A possible solution would be to allow students to move up to the next level school but not down.

(By the way, you might point out that most districts now have 6-8 middle schools -- just because my district as a young middle school student and the new district where I sub both count sixth grade as elementary, I shouldn't ignore the 6-8 middle schools. This is tricky -- it might be possible to divide the paths as K-1, 2-3, 4-5, but then that forces kindergartners onto a path. It also results in a 2-3 path where the second graders don't take state tests while the third graders do -- of course, this is also possible with our 1-2, 3-4 paths, but a 2-3 path makes it the norm. As of now, I don't have an easy path solution for K-5 elementary schools. On the other hand, some states and local charters have 5-8 middle schools, which are compatible with my Path Plan.)

An Analogy

This is an analogy that I wrote two years ago about the racial tracking debate. And yes, this analogy is still relevant today:

Here's another analogy -- a thought experiment, mind you -- to discuss the tracking debate further.

Imagine that there is a magic red button. Here's how it works -- as soon as the magic red button is pressed, the income of every black person immediately doubles. Actually, let me make this precise -- I already know how economists might say, "Sure, income doubles, but prices double as well, so no one is better off." If the red button is pressed, then the purchasing power of every black person doubles -- anything a black person can buy now, he or she can buy two of now.

So far, this sounds good. You might think that if we repeated the NY Times graphs in a world where the red button exists, then they'd be more favorable for blacks. More blue squares (representing blacks) would land on the paths to higher income and fewer would be on the paths to lower income.

There's just one problem -- I didn't say what effect the red button has on whites yet. And so let me do so now -- if the red button is pressed, then the income of every white person triples. Again, here I mean that the purchasing power of every white person triples -- anything a white person can buy now, he or she can buy three of now.

And as for Asians, Hispanics, Native Americans, and mixed-race individuals, let's say that the red button increasing the purchasing power by a factor between 2 and 3. Lighter-skinned individuals have a multiplier closer to three, while darker-skinned individuals have a multiplier closer to two. So if the red button is pressed, everyone will have greater purchasing power.

So what effect would the red button have on the NY Times graphs? Notice that the five income paths on the graphs are based on percentiles, so the highest path is the richest 20%. The red button triples whites' income while only doubling blacks' income. So the red button increases gaps -- there would be even more yellow squares on the highest path and blue squares on the lowest path.

But someone who is pro-red button could easily argue the following -- each black person has twice the purchasing power he or she would have without the red button. And so even with more blacks on the lowest path, they are better off with the red button than without it. And so if you oppose the red button, you are actually anti-black.

Skeptics can counter that relative wealth matters, and so blacks aren't truly better off in the world of the red button even if they have twice the purchasing power. Again, we can declare it's built into the magic of the red button that blacks are automatically twice as well-off if the button is pressed. In the end, it doesn't really matter because this is just a thought experiment -- there is no magic red button.

So far, I've written much about race and income, but didn't I say this analogy would have something to do with tracking?

Well, here's the kicker. According to tracking advocates, tracking is the magic red button. They concede that if tracking were brought back, more whites would find themselves on the higher track and blacks would wind up on the lower track. Yet somehow, everyone, regardless of race, would be better off with tracking.

Years ago, I recall quoting a traditionalist who defended tracking. This is what he wrote -- an Asian girl is placed on the highest track. She can take higher-level courses and learn the material at a faster rate than if her teachers had to slow down for the other students. When she grows up, she broader knowledge ultimately allows her to discover a cure for cancer. She's able to sell the medicine and become rich -- so her purchasing power triples, just as I claimed the red button would do.

[2020 update: Let's change "cancer" to "coronavirus" -- a disease we'd definitely like to find a cure for right now!]

In the same analogy, suppose a black boy is placed on the lowest track. From this track, he's qualified only for low-paying, blue-collar jobs. One day while working hard on the job, he catches coronavirus -- exactly the disease that the Asian woman has found a cure for. The money he saves on hospital bills can be spent on other things -- so his purchasing power doubles, just as I claimed the button does.

And so everyone is better off with the red button, even though the gap between races increases -- and even though the NY Times graph looks worse. Ironically, it's another NY Times article that argues that tracking benefits everyone:

https://www.nytimes.com/roomfordebate/2014/06/03/are-new-york-citys-gifted-classrooms-useful-or-harmful/tracking-students-by-ability-produces-academic-results

Notice that the study cited in the article takes place in Kenya, where presumably most of the students involved are black.

The reason that I can't wholeheartedly embrace tracking is that I'm skeptical that it will benefit everyone of all races as much as its advocates claim it will. I'm all for pressing the red button and helping everyone, but I'm not completely sold that tracking is the red button. For example, the black man placed on the lower track might not be able to get any job at all. He might not be able to afford the cure created by the Asian woman at all. And so he dies of the coronavirus. In the end, there's no point developing a cure for coronavirus if those stricken with disease can't afford the cure.

Many of the commenters in the old 2014 NY Times article above seem to agree with tracking. Let's look at some of these comments:

C:
I don't understand why everything is designed to force faux equality. When did it become wrong to recognize and encourage people for their talents? There is no racial or income requirement to get into one of these programs. Some people are just smarter than others. It's equally unfair to take someone who, can simply "get" 3rd, 4th, 5th, 6th, 7th, 8th or whatever math by looking at it to sit through endless lectures on how a fraction works.

It's obvious that placement on higher tracks benefit those who are so placed. What I don't believe yet is that those placed on lower tracks benefit. Well, here's another commenter:

Canis Scot:
For years I taught special needs students at their ability, they flourished and grew.

Then the "equality" fascists required that my students be distributed in the mainstream classrooms and I watched in horror as they failed. They retreated into themselves. They were being punished because they were not "normal."

OK, then, so let's imagine what would have happened if the equality "fascists" hadn't intervened. The students would continue to flourish and grow, and feel comfortable in their classes -- that is, until they turn 18. Then when it's time to be hired for a job, the lower-track students still don't have as many skills as those on higher tracks. And when they get their first paychecks, they see a lower number written after the dollar sign than those placed on the higher tracks. These students have become just another blue square falling to the lowest income level.

And also, consider what this placement looks like from the parents' perspective. When they see their children being placed on a lower track, they tend to object to the placement. They might wonder, is their child being placed low due to their race, as the NY Times graphs seem to suggest? Is the school condemning their child to a lifetime of low earnings? I suspect that a major reason that tracking is parental complaining.

Canis Scot's students feel that they are being punished in the gen ed class because they aren't "normal" -- but if tracking remained, they would have felt punished years later with a lower paycheck because they aren't "normal." Am I a "fascist" just because I want the students to have a job that pays well enough for them to have a roof over their head and food on their table?

Later on, another commenter writes that gifted students should be placed on higher tracks because we "need kids who master higher mathematics and design airplanes." But then the students placed on lower tracks are headed for low-paying jobs, where the pay is so low that they can't afford going on vacation or buying plane tickets. So who's going to ride on all those airplanes being built by the higher-track students?

Cosmos Episode 8, "The Sacrifice of Cassini"

And now suddenly we jump from our low-income income black boy who's stuck on the lowest track to Neil DeGrasse Tyson -- likely the most well-known black (and part Hispanic) astrophysicist. His biography reveals that in the 1970's he attended the Bronx High School of Science -- which some would count as the type of tracking described in the NY Times article listed above.

Note that Tyson mentioned the workings of the brain and the pseudoscience of "phrenology" in Episode 7, but I briefly discussed that episode in my last post. Today we review Episode 8 only.

Here is a summary of Cosmos Episode 8, "The Sacrifice of Cassini":

• The Cassini spacecraft explored Saturn for 20 years before making its final plunge.
• The planet J1407b is the first known ringed planet beyond the gas giants of our solar system.
• The rings of Neptune, Uranus, and Jupiter weren't discovered until recently by Voyager.
• Asteroid collisions may have first knocked Uranus on its ass.
• Scientists at NASA's Jet Propulsion Laboratory (JPL) used Cassini to study Saturn's rings.
• Galileo was the first to view Saturn through a telescope, so it wasn't just a distant point of light.
• A little later, Christian Huygens through a more powerful telescope and saw Saturn's moons.
• Giovanni Cassini, born in 17th century Italy, was first an astrologer before being a scientist.
• Through his telescope, Cassini discovered that the rings of Saturn are not solid.
• The Cassini-Huygens spacecraft was the size of a bus, and was powered mostly by gravity.
• Alexander Swergei/Kondrotiev* was an unknown Russian scientist who wrote a book about interplanetary space travel.
• He met Sergei Korolev, father of the Soviet space program, and told him his ideas.
• In the 19th century, Edouard Roche thought that Saturn's rings were the debris of an old moon.
• Each planet has its own Roche limit, which a moon can't approach without becoming rings.
• Cassini came to Saturn during its winter, survived spring storms, and was destroyed in summer.
• The "ball of yarn" refers to the 13 orbits of Saturn that the spacecraft made during its lifetime.
• The missionary scientists feared that if Cassini crashed, it might endanger life on its moons.
• Therefore Cassini had to make a controlled crash into Saturn at the end of its journey.

Once again, I wish to compare the science mentioned on the episode to the science that I could have taught in my middle school classes.

Based on the California-to-NGSS transition, I mange to avoid astronomy and space sciences. So I wouldn't teach the properties of Saturn that Cassini discovered.

Actually, there's one slight exception to this -- Standard MS-PS2-4 is all about physics, specifically gravitational interactions between objects. This includes the planets and how much objects weigh there, as well as how many moons each planet has. Notice that it was Saturn's gravity that kept Cassini in its orbit for so long.

*Notice that Tyson mentions the name of a Russian scientist whose name I can't find online, and hence I can't spell. Indeed, Tyson tells us that he went by two different names, but I have no idea how to spell either one. He meets Sergei Korolev, but no biography of Korolev that I can find easy mentions the other scientist's name. According to Tyson, this other scientist's name has been unknown for so long until he was honored recently during the Cassini mission, which explains why I find info about him online. But all that does is make it more difficult for me to spell his name or honor him properly on this blog.

An April Metamath Proof

Since Metamath doesn't give April Fool's Day jokes anymore, I'd like to attempt my own April Metamath-inspired proof. And it's based on Rebecca Rapoport's April Fool's Day problem:

(1 * infinity) / infinity

I was thinking back to the last April Fool's Day Metamath proof, from 2014:

http://us.metamath.org/mpegif/1div0apr.html

1 / 0 = empty set

And so if something is undefined, Metamath returns the empty set. Thus the theorem that we'll attempt to prove is:

(1 * infinity) / infinity = empty set

On Metamath we have a set called the "extended reals," symbolized as R*. Plus infinity and minus infinity are elements of the extended reals. We assume that Rappoport intends to have plus infinity in our proof, so let's write it:

(1 * +infinity) / +infinity = empty set

Now here's the thing -- Metamath defines addition, subtraction, and multiplication for extended reals, but not division! The website distinguishes between real operations (symbolized with a simple +, -, or dot for multiplication, but I'll write as * in ASCII) and extended real operations, which have a little subscript _e (that is, +_e, -_e, and *_e). We can use *_e in our proof:

(1 *_e +infinity) / +infinity = empty set

By the way, as we'd expect, 1 *_e +infinity is +infinity:

http://us.metamath.org/mpegif/xmulid2.html

Since there is no /_e, +infinity is already out of the domain of /. Thus we could just use the same proof as for 1/0. The only thing missing is that instead of 0 is not an element of C \ {0}, we need that +infinity is not an element of C \ {0} -- that is, +infinity is not a complex number. This step probably is a theorem somewhere on Metamath, but I can't find it.

This completes our proof, but it's not elegant. I'd much rather define /_e for extended reals and then exclude +infinity /_e +infinity from the domain of /_e.

One thing I notice is that both infinity -_e infinity and 0 *_e infinity are defined as 0, even though infinity - infinity and 0 * infinity are usually undefined (or indeterminate). In the Metamath notes, it's stated that proofs are simplified if the functions are total (that is, no elements are excluded from the domain of the functions).

If we were to have a /_e function, we'd probably keep infinity / infinity (for either sign) as well as anything divided by 0 (including 0 / 0) undefined. But 1 / infinity (for either sign) can be safely defined as 0.

Conclusion

Yesterday was the first Saturday in April. This means that yesterday was the biannual library book sale -- at least, it would have been had it not been for the coronavirus.

I still have one book from last October's book sale that I haven't read yet -- the computer book on the Visual Basic language. As I wrote in late February, the CD got stuck in my computer -- not only can I not load Visual Basic, but I can't get the disc out either.

I might still wish to discuss Visual Basic on the blog during the closure. I'll make a final decision after I finish the Cosmos series.