Orthodox Christmas Post (Yule Blog Challenge #11)
Introduction
Today is Christmas -- at least to Orthodox Christians. That's because the Orthodox Churches still use the old Julian Calendar, and December 25th Julian corresponds to January 7th Gregorian. (There was an official statement from the White House today to celebrate Orthodox Christmas.)
Many people confuse Orthodox Christmas with Gregorian Epiphany. But actually, the 12 days it took the Three Kings to reach the infant Jesus have nothing to do with 13-day difference between the Julian and Gregorian Calendars. Indeed, the Orthodox have their own Epiphany (or Theophany) coming up in 12 days, on January 19th.
Also, yesterday wasn't just Orthodox Christmas Eve or Gregorian Epiphany, but Phi Day. I celebrated Phi Day by reading some websites about whether January 6th is the best possible date for Phi Day -- after all, it's close enough to Christmas that some schools (such as mine) are still on winter break:
https://latkin.org/blog/2015/01/06/today-is-phi-day-at-least-it-ought-to-be/
Another suggestion is to divide the year in the Golden Ratio, and let August 14th be Phi Day. If there are 29 days in February, then August 13th is a better fit for the Golden Ratio.
Also, suppose we want a Phi Approximation Day, like Pi Approximation Day on July 22nd. We recall that the best rational approximations to Phi come from the Fibonacci sequence, with higher values giving better approximations. Then the best approximation that fits on the calendar is 8/13 -- August 13th, which, as we see above, just happens to divide the year in the Golden Ratio as well! So perhaps August 13th is the best choice for a Phi Day. The only problem, of course, is that it's too close to the first day of school (and many schools haven't started yet).
If our main criterion is to fit Phi Day on the school calendar, then the best date is probably May 19th -- listed at the above link as also dividing the year in the Golden Ratio, but in the other direction. A possible Phi Approximation Day can also fall in that month, on May 8th (for 5/8).
Yule Blog Prompt #16: What I'm Looking Forward to in 2022
There's one thing I'm definitely looking forward to in 2022 -- teaching my new Trigonometry class. As I've mentioned before, my general Stats class is one semester, with Trig second semester. But I have one huge problem with my upcoming Trig class -- I don't have the textbook. When my students show up to class on Monday, I'll have nothing to teach them -- and they might not have the book yet either. (Recall that only my general Stats class is one semester -- the Ethnostats classes are a full year.)
Let me explain what happened. First of all, I wasn't even informed that Stats and Trig are one-semester courses until November -- the day before my visit to the main high school and the Calculus class. So when I was at the flagship campus, I asked for a copy of the Trig text. They didn't have any teacher editions available, so they handed me a student text.
After the meeting, I arrived back at my own campus and informed the principal that I was now in possession of the Trig text. But as soon as she saw my student edition, she promptly took the book away from me and told me that she'd request the teacher edition (presumably from the same flagship school where I got the student text). She stated there was a possibility that a teacher edition would be found for me during winter break -- but now that it's Friday, the last weekday of the break, I must assume that the copy isn't coming.
This is what I call letting the perfect become the enemy of the good. While I'd rather have a teacher text than a student text, I prefer having a student text to having no text at all. At the time I saw the student text, it was November, and I didn't want to think too much about a January class in November. But now, of course, it's January, and I'm less than 72 hours away from facing the class, not knowing whether they or I will even have a textbook.
I've written much about all my other classes during the Yule Blog challenge, but I've avoided discussing my fifth period Trig class until now (mainly because I was still holding out that I'd get the textbook so I can write about my plans here). So I might as well start blogging about that class now.
The Trig class will be my smallest class of all. Indeed, I've only been seeing four students for fifth period -- and one of those students is a guy in special ed. I already have access to my second semester classes in Aeries, and I see that the special ed student will not be enrolled in the Trig class. (Indeed, his one-on-one aide was surprised that he was even enrolled for Stats.)
Of the three students staying for trig, I have two guys and one girl, all seniors. One of the guys is also in Ethnostats and he's definitely one of the top students. The other guy started out strong but started to taper off a little near the end of the semester. He ended up with an A- once I realized that there was an error in my final exam and I fixed the grades.
Unfortunately, the lone girl in the class is struggling the most. After grading the final exam, her grade dropped to a D+. But (unlike the girls in fourth period Ethnostats) she was willing to stay during sixth period for test corrections. I let her correct enough questions to raise her grade to a C-, but I warned her that she needs to keep up with the material better, since Trig is a much tougher class than Stats.
Once again, I'm keeping Eugenia Cheng's x + y in mind and wondering whether my classes are too ingressive, which would explain why so many girls are struggling in my classes. Then again, there are some strong female students in my Calculus class, so it's not as if I can't teach girls at all.
OK, now let's get to the Trig itself. As I wrote before, when I observed the Calculus class at the flagship high school, I noted that I didn't want make so many radical changes to my own Calc class, a class that I'm already in the middle of. But Trig is a different matter, since it's a brand new class. So I have no problem making changes as we move from Stats to Trig.
I do remember a little about the Trig text during that brief time between the flagship Stats/Trig teacher giving it to me and my principal taking it from me. The text has seven chapters, and the flagship teacher tells me that he expects to cover four full chapters, perhaps start Chapter 5 during the semester. And I do recall that the first few chapters have five sections each, and Section 3.2 is on radian measure -- and that's the section I want to reach by Pi Day, March 14th.
That's right -- Phi Day was yesterday, and now I'm already thinking about Pi Day. (It's a bit like retail stores promoting Christmas the day after Halloween.) And I shouldn't be planning an entire semester around one day in mid-March. But had it not been for Pi Day, I wouldn't have remembered anything about the new textbook at all. It was only because I was excited about Pi Day that I even opened the book in the few minutes that I held it, eager to see when the radian lesson was (the obvious link between Trig and pi) and whether I could reach it by the math holiday.
(And if I'd known that the principal would take the book from me, I likely would have written down the entire contents of the text on paper while still in my car, and then posted those contents on the blog during Thanksgiving break, before I had a chance to lose that paper.)
Recall that in Calculus class, I have to cover Chapters 4-7 before the May AP exam, so that each chapter will span four weeks. I can use a similar pacing guide in Trig -- spend four weeks on each chapter, so that I complete Chapters 1-4 likewise by early May. This means that I will complete four chapters and start Chapter 5 by the end of the semester (as suggested by the flagship Trig teacher) and that Chapter 3 will be taught around March (allowing me to reach radians by Pi Day).
Let's now look at a pacing plan for the first chapter. Once again, I don't remember exactly what's in Chapter 1, except that it has five sections. Following the advice of the Calculus teacher, I don't need to finish a section in just one day, so let's assign two days per section, based on the block schedule:
EDIT: This schedule is already outdated. Scroll to bottom of post for announcement.
Once again, I don't know when the minimum day Mondays are -- but that's irrelevant. Each section spans two days, at least one of which is a block day. So I'm guaranteed to spend at least one block period on each lesson.
Again following the veteran teacher, the first homework is assigned on Tuesday, the day when Section 1.1 is completed. And I take several days to go over the assignment -- discussing questions from Section 1.1 before teaching Section 1.2. I expect that HW not to be collected until perhaps the end of the second week -- and only then is the next HW assigned. With only three students in the class, I might even ask them to choose two questions each for me to go over each day.
Of course, there's one glaring problem with the above pacing guide -- on Monday, it shows me starting Section 1.1, but we don't know whether the students or I will even have a book. Most likely, on Monday I'll give an alternate assignment that doesn't require the text.
It won't be too terrible if the kids don't get the text until Tuesday. Since there won't be any old HW or quizzes to go over, I can conceivably cover all of Section 1.1 on Tuesday and assign the first HW of the semester that night. (Notice that all seniors will be switching from Government to Econ or vice versa -- these are also one-semester courses with their own texts. So when they switch their Government/Econ books, they can switch their Stats text to Trig as well.) The real problems occur if we get through the entire week and the students still don't have the text.
The veteran teacher gives short weekly quizzes. I will do the same, but notice that my quizzes likely won't be every week. If each chapter is four weeks, I can't give a quiz the first week -- the chapter just started and there's no material to assess. And I can't give a quiz the last week -- the chapter's almost over and I'm about to give the test anyway. So I expect to give two quizzes per chapter, during the second and third weeks of the chapter.
I can give the quizzes on Tuesdays and go over them on Thursdays. The first quiz will be on the 18th -- assuming that we get the books in time, that quiz will be on Section 1.1 (since we won't have finished 1.2 in time for the quiz). The second quiz will be on the 25th and cover up to Section 1.3. Notice that the quizzes won't line up with Calculus quizzes (which will be on Thursdays), but they will line up with the tests.
The Chapter 1 Test will be on February 3rd, and the Chapter 2 Test should be one month later, which works out to be March 3rd. So now we can look to see how Chapter 3 lines up with Pi Day (since as of now, this is the only lesson I know the content for):
Ah, so Pi Day will be on a Monday this year. (It marks the first time since 2019 that Pi Day is on a weekday rather than the weekend.) Notice that if I fall behind a little in Chapter 2, that test might be given on Monday the 7th instead (unless that's the monthly minimum day for March) and thus start the radian lesson on Pi Day. But I like the idea of starting the lesson before Pi Day -- during the lesson, I can inform them that Pi Day is coming up on Monday, and I plan on bringing pizza, so please tell me what your favorite toppings are.
Some might wonder whether it's a good idea to bring food to share during the pandemic. One of the other teachers had a club meeting in her room at lunch, and one day she bought them pizza -- but then again, that was before omicron. I'm hoping that omicron will have settled down by March, and that another wave won't have begun. (The next Greek letter after omicron is pi -- no, please, I don't want there to be a pi variant on Pi Day!) Also, to avoiding sharing, I can get each student a personal pizza -- this I'm willing to do because there are only three students.
The Pi Moment is 3/14 at 1:59 PM. Unfortunately, fifth period ends at 1:40 on Mondays -- 1:59 will be during my sixth period conference. (That, of course, is assuming that March 14th isn't itself the minimum day -- then of course school will be out well before 1:59.)
OK, that's enough about Pi Day -- I need to worry about how I'm going to teach Chapter 1 in January, not how to teach Chapter 3 in March. But again, it all depends on the availability of the book.
Of course, there are other ways to teach besides the book, such as DeltaMath. But for this class I promised to follow the veteran teacher, and he doesn't use DeltaMath in his class. Then again, I do recall that he used Desmos one day, so maybe I might be able to find some Trig lessons there.
Also, the veteran teacher regularly has group activities in his classes, but group activities won't work when there are just three students. Perhaps I can come up with something that will require the students to discuss and talk math with each other. It will definitely be helpful for the struggling girl if she can communicate with the stronger guys in the class -- indeed, that would be congressive. And that, according to Eugenia Cheng, is what girls need to be more successful in math. Speaking of Cheng --
Cheng's Art of Logic in an Illogical World, Chapter 13
Chapter 13 of Eugenia Cheng's The Art of Logic in an Illogical World is called "Analogies." Here is how it begins:
"We have seen that abstraction is how we get to the world where logic works. The abstract world is a world of ideas and concepts, removed from our concrete, messy world of objects."
And as the title implies, one major method of abstraction from the real world to the logical world is through the use of analogies. Cheng reminds us that there can be many different ways of finding an abstract version of the same situation:
"This doesn't mean that some are right and some are wrong, it means that different abstractions show us different things, and we should be aware of what we have lost and gained by doing it."
According to the author, we learn to abstract as soon as we learn to count. The number "two" is an abstract concept. Concrete instances of "two" include "two cookies" and "two bananas."
Cheng draws many diagrams in this chapter. Her first diagram looks something like this:
2 things
/ \
/ \
v v
2 cookies 2 bananas
She tells us that this diagram of abstraction looks a bit like a pivot:
"The number two enables us to pivot from a situation involving two cookies (perhaps one and then another one) to a situation involving two of something else."
Cheng now provides us a framework for analogies:
"The general situation is that we are making an analogy between concepts A and B, via an abstract principle X that is often implicit rather than explicit. The diagram looks like this:"
X
/ \
/ \
v v
A B
These diagrams don't look good in ASCII, and so I'm no longer drawing them on the blog. Instead, I'll describe each diagram to you.
For example, her next two diagrams have a + b and a * b on top, respectively. She writes:
"People often tell me that they lost [understanding of math] when 'numbers became letters.'"
But then she combines these two diagrams into a tree. At the top of this tree is a ( . ) b, where the symbol is actually a dot within a circle (that's even harder to draw in ASCII) that represents a binary operation that could be +, *, or something else. The author proceeds:
"One of the important lessons my PhD supervisor Martin Hyland taught me was the importance of finding the right level of abstraction for the situation."
For example, Cheng draws a tree with "2 apples" and "2 bananas" at the bottom. These are linked together by "2 fruits." Then this in turn, along with "2 chairs," are linked together by "2 things":
"If we only go up to the level of '2 fruits' we will get an analogy between 2 apples and 2 bananas, but will omit 2 chairs. In order to include '2 chairs' we need to go up further, to the level of '2 things.'"
Cheng's next chart is based on the factor "cubes" (trees) of Chapter 6. These trees demonstrate an analogy between the factors of 30 and the factors of 42. So she draws a new analogy tree with "cube of factors of 30" and "cube of factors of 42" on the bottom, linked by "cube of factors of a * b * c." In her next chart, this in turn, along with "cube of privilege" (as in rich, white, male), are linked together by "cube of subsets of {a, b, c}.
Again, I remind you that Cheng writes about race and politics throughout her book. If you prefer not to read this, then I suggest that you avoid this blog for the next week and skip all posts that have the "Eugenia Cheng" label.
The author continues:
"Much of my argument about math comes from my view that math is a little removed from normal life, so if we pick low pivots we will not get very far out of math, perhaps only a far as physics."
Cheng's next example is very real in life. In this tree, "black people" is linked by "visible minorities," which in turn, along with "gay people," are linked by "minorities." This in turn, along with "women," are linked by "oppressed people." This in turn, along with "straight white men," are linked at the top by "people."
Now she asks how long to draw each of the arrows -- that is, should all the classes of people be written on the same level at the bottom of the chart? This leads to divisive arguments because people pick the level of abstraction that best suits their own argument.
The author reminds us about axioms:
"In Chapter 11 we talked about finding axioms for our personal belief systems, that is, the fundamental beliefs from which all our beliefs stem."
Cheng's example here is also divisive:
"When arguing about affirmative action on grounds of race, some people are opposed to it on the grounds that there are people of color who come from well-off backgrounds who need help much less than some disadvantaged white people. I still believe that we should try to help all people of color, and all people from less privileged schools, even if some of them don't 'need' it.
"Being falsely accused of sexual misconduct is indeed a trauma that nobody should have to go through, but I believe we need to be concerned with the quantity of sexual misconduct going unstopped."
The author draws diagrams for these two examples. In the first tree, "social services" and "affirmative action" are joined together by "help people even though they might not need it." In the second tree, "cancer screening" and "sexual harassment" are joined by "take evidence seriously although it might cause unwarranted action."
"Opposite is the diagram of the different levels of abstraction producing different analogies:"
Then she combines the two previous trees, joining "help people even though..." and "take evidence seriously although..." with "avoiding false negatives is more important than avoiding false positives," which is one of her basic axioms.
[2022 update: Cheng wrote this book before the pandemic. Of course, many of these ideas apply to the pandemic as well, including false negative/false positive COVID tests, as well as other preventative actions such as masks, vaccines, and so on.]
In fact, she realizes that she favors compulsory voting, as in Australia, for the same reason -- she fears that without compulsory voting, there could be voter suppression:
"It is yet another situation where I care about preventing false negatives the most; I just hadn't realized that was the issue until someone pointed it out to me."
Two weeks ago, I mentioned how those who didn't support Obama were accused of racism. Now Cheng repeats this example along with the counterargument -- they didn't support him because "he was inexperienced." She also discusses the 2016 election, in which those who didn't support Clinton were accused of sexism. A counterargument is that "she was a liar."
"We can clarify this using diagrams. Someone might think they're applying a principle about inexperienced people in general, regardless of the fact that person A is a woman:
In her first chart, "an inexperienced woman A" is linked by "inexperienced people." In her second chart, "an inexperienced man B" is also linked by "inexperienced people." But in her third chart, sexism is at play, with "inexperienced women" placed as an intermediate level between "an inexperienced woman A" and "inexperienced people."
Cheng's next chart is the abstract version of this. "A person from group A doing X" is linked first by "people from group A doing X." This in turn, along with "a person from group B doing X," are now linked directly by "people doing X." She writes:
"If the person from group A is treated differently from the person in group B, it's a sign that the intermediate principle is at work, not the general one."
And she repeats her example from Chapter 3 about black people shot by police in the US. Now she moves on to the debate between science and religion.
In her first chart, "shouldn't believe religion" is linked by "shouldn't just believe books and teachers," and in her second chart, "shouldn't believe religion" is also linked by "shouldn't just believe books and teachers." But in her third chart, a "more nuanced principle" is placed as an intermediate level between "shouldn't believe religion" and "shouldn't just believe books and teachers." She adds:
"That more nuanced principle might be that we shouldn't just believe books and teachers unless they are backed up by reproducible evidence, but that still leaves the question of how we can tell if the evidence is reproducible."
For the next set of examples, Cheng introduces another symbol to denote the power relationship between a powerful group and an oppressed group. This is actually a triangle or delta symbol, but in ASCII let's represent it by the letter V.
So in her next diagram, "men V women" and "Oxbridge V non-Oxbridge" are linked by "privileged group V oppressed group." Notice that here "Oxbridge" refers to Oxford and Cambridge, the two most prestigious universities in England -- and indeed, Cheng admits that she has privilege as a Cambridge alumna. An American would likely replace this with "Ivy League V non-Ivy League."
Also, Cheng reminds us that she is an Asian person. Because of white privilege, she draws "white people V non-white people," but she also concedes that Asians are arguably more privileged than others among non-white people. Thus she draws "Asian people V black people."
So she uses this analogy to perform a pivot. "White people V Asian people" and "Asian people V black people" are linked by "privileged group V oppressed group."
And Cheng abstracts this even further. Everyone is less privileged than someone and more privileged than someone else. So group A: more privileged than you V you V group Z: less privileged than you.
So she uses this analogy to perform another pivot. "Group A V you" and "you V group Z" are linked by "privileged group V oppressed group."
In her next example, the author writes about the idea that everyone should take responsibility for themselves, therefore we should oppose universal healthcare. Again, she uses charts here.
In her first chart, "oppose universal healthcare" is linked by "everyone should take responsibility for themselves," and in her second chart, "oppose roads" is also linked by "everyone should take responsibility for themselves." But in her third chart, "everyone should take responsibility for their own optional extras" is placed as an intermediate level between "oppose universal healthcare" and "everyone should take responsibility for themselves." She adds:
"This actually explains the sense in which the healthcare denier thinks healthcare and roads are different."
Now Cheng tells us that we must pick the right level of analogy. She begins:
"My wise friend Gregory Peebles says analogies are like bridges that can take us anywhere -- so we'd better be careful what bridge we choose."
Here's how we usually use analogies in discussions:
A is analogous to B.
B is true.
Therefore A is true.
This is much less watertight than using an actual logical equivalence:
A is logically equivalent to B.
B is true.
Therefore A is true.
In other words, there is some implicit principle X:
A is true because of principle X.
B is also true because of principle X.
B is true.
Therefore A is true.
But as Cheng points out:
"There is now a logical flaw in the argument, which is that just because B is true it doesn't mean that principle X is true. In a sense we are trying to move backwards up the right-hand arrow."
X
/ \
/ \
v v
A B
For example, she draws a tree in which "straight marriage" and "gay marriage" are linked by "2 unrelated adults." But opponents of gay marriage disagree with this tree. For example, their erroneous tree might have "straight marriage," "gay marriage," and "incest" all linked by "2 adults." Or they might have "straight marriage" linked to "an unrelated man an woman."
Perhaps a correct way to draw this tree is to begin with "straight marriage" linked to "an unrelated man and woman." Then this in turn, along with "gay marriage," is linked to "2 unrelated adults." This in turn, along with "incest," is linked to "2 adults."
It's even possible to draw a tree incorporating various slippery-slope arguments. We continue by linking "2 adults" and "pedophilia" to "2 humans." Then this in turn, along with "bestiality," is linked to "2 creatures." Cheng points out that there was once a level below "an unrelated man and woman," namely "an unrelated man and woman of the same race." In all cases, the claim that going up one level necessarily involves going up more than one is an erroneous argument.
The author wraps up the chapter by discussing these implicit levels. She writes:
"Disagreements over analogies basically take two forms, as in the argument about gay marriage. It starts by someone invoking an analogy of this form:"
X
/ \
/ \
v v
A B
However, typically X is not explicitly stated. Now someone objects, either because they see a more specific principle W at work (between A and X), or they see a more general principle Y at work (that links X along with some objectionable thing C).
For example, is racism by whites against blacks the same as racism by blacks against whites? She answers with another diagram. "Racism of white people against black people" is linked to "prejudice of privileged people against oppressed people." This in turn, along with "racism of black people against white people," is linked to "prejudice of people against people." The argument is really about which level of principle we should go up to.
Cheng concludes the chapter, as usual, with a summary and a preview:
"In the end the whole aim is to reach greater understanding of in what way situations are equivalent and in what way they are not. This is the subject of the next chapter."
EDIT: School Reopening and COVID-19
My district has just made an announcement regarding the school reopening and COVID-19. The announcement was made a few minutes after the timestamp of this post, and so I'm including it here as an EDIT.
Due to the ongoing omicron surge, all students and staff must now be tested for COVID-19 before returning to campus. In order to give everyone sufficient time for the testing, the first day of school has been delayed to Wednesday. (I hear that LAUSD is doing the same thing.)
This means that the pacing plan I made earlier in this post is already outdated. My Trig students definitely won't have the textbook on Monday since they won't even be in class that day. What this means is that I'll definitely have to compress some lessons in Chapter 1 in order to be finish the first chapter by February 3rd. The quiz on January 18th might not happen at all -- if it does, it may be very short (as in just one question from Section 1.1).
It hasn't been announced yet what the bell schedule for Wednesday will be. The most logical schedule would be for Wednesday to be like an all-classes Monday, and then Thursday and Friday to be regular block days (odd and even respectively) as usual.
As frustrating as it is not to have the Trig text yet, it's understandable. Indeed, as we see right here in this EDIT, the focus is on reopening safely. The Trig text is the very last thing on anyone's mind now.
It's more important to make sure everyone's protected than for me to get the Trig book. Indeed, that's Cheng's philosophy as she states in her book -- she's much rather do something unnecessary than something unsafe. She fears false negatives more than false positives -- and so do I, especially when it comes to COVID.
As a Californian, I must congratulate the newest Jeopardy! millionaire -- Amy Schneider, who's also from the Golden State!
If there are any Orthodox Christians reading here, I wish you a Merry Christmas. And if you aren't Orthodox, I hope you enjoyed your Phi Day celebration yesterday.
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