Boxing Day Post (Yule Blog Challenge #5)

Table of Contents

1. Introduction

2. Yule Blog Prompt #7: Start/Stop/Continue in 2022

3. Calendar Reform: 5-Day Calendars

4. Cheng's Art of Logic in an Illogical World, Chapter 7

5. Conclusion

Introduction

Today is Boxing Day, the day after Christmas. Indeed, now that Christmas has passed, it's time to look ahead to the new year. And today's Yule Blog prompt does exactly that.

Yule Blog Prompt #7: Start/Stop/Continue in 2022

What should I start in 2022? Well once again, I think back to the visit to the main high school and what I observed the veteran Calculus teacher at the main high school. And the big one is allowing time for the students to ask questions from the homework, and for me to answer those questions.

Here's what the veteran teacher did -- he insisted that each student give him a problem number. He didn't take "no" or "none" for an answer. That way, he was guaranteed to have a list of problems to work on with the class.

While I won't give lengthy assignments or extended time to do them, I will make sure that I answer the students' questions. There are six students in my Calculus class. If each student names a problem and I spend five minutes on each problem, then that's a half hour right there. (Note: I keep saying that there are six kids in that class, but I hear there might be a seventh student in Semester 2 -- a girl who studied independently during the first semester.) And the other classes are smaller -- only in my fourth period Ethnostats class will I be unable to take a question from every student.

This post asks me what I will start in 2022 -- and to me, that means that calendar year 2022, including August-December 2022. By that point, my magnet school will have moved over to the campus of the main high school. So another thing I'll start is paying attention to the bell schedule at that flagship high school, which will likely become our bell schedule next year.

I briefly discussed this schedule on the blog. Four periods meet on each block day. Thus it takes three block days (Tuesday-Thursday) for each of the six periods to meet twice. The other two days (Monday and Friday) are non-block days when all classes meet. So each class meets four times per week.

What should I stop in 2022? Well, on days when a student is absent, I want to stop tying Warm-Up/Exit Pass points to whether or not the absence is cleared -- that is, denying them points if the absence is unexcused, and giving them points for nothing if it's excused. (I wrote in an earlier Yule Blog post about the problems I had with certain students clearing their absences.)

It's much better just to have the student answer the Warm-Up and Exit Pass questions themselves, regardless of absence clearance. I keep all the notes in a folder anyway, so I can give returning students the folder for them to make up both the Warm-Up and notes.

What should I continue in 2022? Well, in most classes, I will continue to give homework assignments everyday and quiz or test every two weeks or so. By continuing to make each Calculus assignment be worth three points, the quizzes 75 points, and the tests 100 points, I will be able to keep the same grade percentages that I had first semester (that is, tests 40%, quizzes 30%, homework 20%, classwork 10%).

In Calculus class, I'll keep the early quizzes short, then lengthen them as we approach the AP exam. I will continue to integrate the textbook, AP Classroom, and DeltaMath into lessons and assessments.

As for allowing quiz corrections, I've decided that I'll continue to do so only for certain assessments. In particular, the department chair (my partner teacher) tells me that she sometimes does corrections on minimum days, as there's not much time to do anything else. So here's my decision -- I'll do corrections only if the Monday after the quiz/test is a monthly minimum day. I'll try to give as many quizzes as possible on the Thursday/Friday before a minimum day, and if the test happens to land on such a day, then I'll do test corrections as well. All other assessments will have no corrections.

Calendar Reform: 5-Day Calendars

One of my favorite topics on my old blog is Calendar Reform. I admit that this is one of my more esoteric interests -- the idea of replacing our current calendar (the Gregorian Calendar) with a new calendar, one with perhaps a different number of days in a week, or months in a year, and so on. On the old blog, I regularly blogged about Calendar Reform during the week leading up to New Year's (the time of the year when we think about calendars the most).

But I don't wish to invent a new calendar just for the sake of inventing a calendar. I want my calendars to fit events happening in my world, to see how things would be different on the new calendar. In particular, as a teacher, I want to see what the school year would look like on another calendar.

Last year, Calendar Reform happened in real life -- at least for the school year. Instead of having five school days per week, we had distance learning and hybrid schedules where students attended in-person fewer than five days per week. For example, one of my old districts declared four difference stages depending on the severity of the coronavirus:

Stage 1: Students attend zero days per week in person (pure distance learning).

Stage 2: Students attend one day per week in person.

Stage 3: Students attend two days per week in person.

Stage 4: Students attend four days per week in person.

Stage 5: Students attend five days per week in person (normal schedule).

All California schools began the school year in Stage 1. My old district spent most of April in Stage 2 and May in Stage 3. Note that Stage 4 was never reached (though I read that some districts outside of my state, such as Sarah Carter's school in Oklahoma, is currently in the equivalent of Stage 4). Most California schools are currently in Stage 5, with five days of in-person learning. (I've heard that some colleges, such as UCLA, are returning to distance learning, at least for a few weeks in January. With omicron on the rise, I wonder whether distance learning is around the corner in K-12 schools as well, as much as all parties involved wish to avoid it.)

Hmm, looking at the above list, we can't help but wonder whether something is missing. After all, we see stages for 0, 1, 2, 4, and 5 days of in-person learning -- but what about three days in person?

Of course, it's easy to see why three is missing in this list. The idea of hybrid is to divide the school into two groups and have each group attend half the week -- but unfortunately, the standard number of school days in a week is odd, so we can't take half of it easily. Instead, we have each cohort attend two days in person. The remaining day -- Monday, in my districts -- remains distance only for everyone.

This is for the main hybrid stage -- Stage 3. In Stage 4, both cohorts attend four days in in person, so that Monday can remain distance only. In Stage 2, we divide each cohort in half, leaving each of the four half-cohorts with its own day per week for in-person, and again leaving Monday as all-distance. So it's impossible to fit exactly three in-person days into this schedule.

Let me take that back -- it's impossible to have a three-day school week without Calendar Reform. And so this is the goal for my annual Calendar Reform week this year.

The simplest Calendar Reform with a three-day school week has four days per week -- three days of school and one day off. But I assume that no one wants to work three out of four days, so I consider this to be a non-starter. So we'll start with five-day calendars -- three days of school, two days off. We get the same weekend with a much shorter week.

The following is a link to a five-day calendar, the Quinta Calendar:

https://calendars.fandom.com/wiki/Quinta_Calendar

The calendar year has 12 months (January to December) each of 30 days equal to 6 quintas. The first month (January) is preceeded by a single quinta called Yule. Leap years have a leap quinta after the last month (December)

There's another link to a more detailed description of the Quinta Calendar, but that link is dead. I do recall (from years ago, back when that link was working) that the five named days of the week are Monday, Tuesday, Wednesday, Friday, and Saturday. The goal was to avoid having two days starting with T in the week, as well as two S's. Then we'd no longer have a problem with abbreviations. (Does MT mean every Monday/Tuesday, or every Monday/Thursday? On the Quinta Calendar, MT definitely means every Monday/Tuesday.)

Notice that this calendar as written has four days of school followed by one day off. If we're going to have just one day off, we might as well have chosen a four-day week with three on, one off -- having four on, one off is even worse. And there are other five-day calendars on the Calendar Wiki, but all of them divide the week as four on, one off. So we'll have to create our own three on, two off calendar.

This will not be a hybrid school calendar -- all students attend the same three days and have the same two days as the weekend. With an odd number of days, a block schedule could have one all-classes day and two block days. Or we could have all three days be block, if we follow the schedule established at the flagship school in my district with four periods meeting per day.

In fact, since that flagship schedule will probably be my own schedule next year, I might as well use it in this Calendar Reform. So the five days are Tuesday, Wednesday, Thursday, Saturday, Sunday. (Yes, I know it means we'll have two T's and two S's in our five-day week.)

Now let's set up the school year, In order to reach 180 school days, we need 60 weeks of school if each week has three days. One way of doing it on the Quinta Calendar is let June and July be summer break, and then each of the other ten months will have six weeks of school, giving us 60 weeks. But then there will be no long breaks other than the two-month summer break, and possibly the Yule Quinta which can serve as a short winter break.

On calendars of this type, I prefer to have just one month of summer break. Then the other six weeks can be distributed throughout the year to serve as holidays. One interesting way to do so is to let August be the month of summer break, and then the six vacation weeks plus the Yule Quinta divide the year into eighths ("quavers") alternating between seven and eight weeks in length. Here are the calculated vacation weeks:

Week of October 6th

Week of November 21st

Yule Quinta

Week of February 11th

Week of March 21st

Week of May 6th

Week of June 16th

Notice that here the Yule Quinta divides the third and fourth quavers -- in particular, it doesn't divide the first and second semesters. Then again, with all breaks the same length, it's not so important that the first semester end before the Yule Quinta. We might instead let some breaks (like the one near the Yule Quinta) be longer. Then we should make the first semester end before Yule (perhaps by making the summer break month be July instead of August).

Of course, I just placed the holidays at arbitrary weeks instead of celebrated holidays. Unfortunately, the Quinta Calendar link above doesn't provide us with any dates for holidays. There's a similar calendar called the Tyerian Calendar that does list some holidays:

https://calendars.fandom.com/wiki/Tyerian_Calendar

This calendar is a more bit complicated than the Quinta Calendar. That's because it's a "geosolar calendar" -- one day on this calendar isn't a solar day, but a sidereal day (23 hrs. 56 min., not 24 hrs.) I only wish to scroll down to where the holidays are, so we can ignore this "geosolar" stuff.

None of the listed weeks correspond with any major holidays (except the semester break, which falls close to "Groundhog Day" on the Tyerian Calendar). I leave setting holidays on our five-day calendar as an exercise.

Cheng's Art of Logic in an Illogical World, Chapter 7

Chapter 7 of Eugenia Cheng's The Art of Logic in an Illogical World is called "How to Be Right." It begins as follows:

"In the lovely stop motion film Chicken Run, the smooth-talking American rooster Rocky does a deal with the sly salesmen rats saying he will play them 'all the eggs he lays that month.' The rats are sly but not very knowledgeable about chickens, so they don't realize that roosters don't lay eggs, and that the total number of eggs Rocky lays that month is therefore going to be zero."

This chapter is "all" about quantifiers -- those little words like "all." Cheng tells us that technically, Rocky isn't lying -- he does give the rats "all" zero of his eggs.

Here is Cheng's next example:

"However, to express that fed-up-ness it may be tempting to say 'All men are sexist pigs!' but this might provoke someone to argue that not every man is a sexist pig, say, perhaps, Justin Trudeau [the Canadian prime minister]. In this case you've expressed true emotions but with inaccurate logic, and in doing so you're tempted certain types of people to argue with your logic instead of soothe your emotions."

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 two weeks and skip all posts that have the "Eugenia Cheng" label.

The author explains that this is how we find the negation of statements quantified by "all" (or another related word):

Statement: You never do the washing up!
Negation: I did the washing up one time.

Statement: You always leave a mess in the kitchen.
Negation: There was one time I did not leave a mess in the kitchen.

But as Cheng points out, people are prone to making sweeping statements. In fact, that last sentence is itself a sweeping statement. Is everyone prone to making sweeping statements? Cheng answers:

"I think everyone I know is, but I've met only a tiny proportion of people, so really I should just say 'Everyone I know is prone to making sweeping statements.' I have now refined my statement and made it less ambiguous, hence more defensible using logic."

Cheng tells us, though:

"Looking for the truth in someone's statement can be much more productive than pedantically demonstrating that they are wrong. I think this is an instance of the principle of charity, where you try to think the best of everyone all the time."

According to the author, there are two logically unambiguous ways to qualify a statement:

1. The statement is true of everything in your world. Perhaps "all mathematicians are awkward."
2. The statement is true of at least one thing in your world. Perhaps "there is at least one mathematician who is friendly." (Cheng hopes she counts as this. Don't worry, Eugenia -- I, at least, think you're friendly. Thus there exists at least one blogger who thinks you're friendly.)

Compare these two statements:

Everyone in the US is obese.
Someone in the US is obese.

Formally these two types of statement would be rendered using "for all" and "there exists" like this:

For all people X in the US, X is obese.
There exists a person X in the US such that X is obese.

As Cheng explains:

"'For all' and 'there exists' are called quantifiers in mathematics; they quantify the scope of our statement."

Now consider this statement:

All the elephants in the room have two heads.

According to the author, this statement is vacuously true. She explains why:

"There are no elephants in the room, and all zero of them have two heads. This is related to the fact that a falsehood implies anything, logically."

On the old blog, I attempted to sneak spherical geometry into high school Euclidean geometry by taking advantage of vacuous truth. For example, the statement "if two parallel lines are cut by a transversal, then corresponding angles are congruent" is obviously true in Euclidean geometry -- and it's vacuously satisfied in spherical geometry as well, because there are no parallel lines! Since then, I decided that this example is confusing rather than illuminating, and so I no longer recommend this as something to teach high school students.

Let's return to Cheng. Her earlier example "All men are sexist pigs!" can be rewritten as:

For all X in the set of men, X is a sexist pig.

And this is the negation of the above statement:

There exists X in the set of men, such that X is not a sexist pig.

According to the author, this statement is true, because her friend Greg is not a sexist pig. So the original statement must be false.

Cheng now describes one of her favorite mathematical jokes:

Three logicians walk into a bar. The bartender says "Would everyone like a beer?" The first logician says "I don't know." The second logician says "I don't know." The third logician says "Yes."

Let's explain the joke. Either:

• A: All three logicians want a beer.
• not A: There exists a logician who does not want a beer.

The first logician wants a beer. That's not enough to conclude A (since we don't know what the other two logicians want), but there's no logician for whom we can conclude not A either.

The second logician wants a beer. That's not enough to conclude A (since we don't know what the other third logician want), but there's still no logician for whom we can conclude not A either.

The third logician wants a beer. Finally, that's enough to conclude A, and so the answer is "Yes."

Here is Cheng's next example -- "all mathematicians are awkward." That is:

For all X in the set of mathematicians, X is awkward.

But Cheng doesn't consider herself to be awkward. Written as an implication, this statement becomes:

Being a mathematician implies being awkward.

In other words, either Cheng is awkward, or she isn't a mathematician. She tells us that she takes offense at both of these claims.

Cheng next writes, "Every female science student has been hit on by their supervisor." This is easy to refute -- a counterexample is any female science who has never been hit on by their supervisor. The author tells us that she herself is a counterexample. She proceeds:

"Whereas if I say 'Some female science students have been hit on by their supervisor' and you want to say that this isn't true, you have to do something much harder -- you have to check every single female science student, and make sure that nobody has been hit on by their supervisor. Unfortunately this won't be possible."

We have the following negations:

World: All female science students in time.

Original statement: For all female science students X, X has been hit on by their supervisor.

Negation: There exists a female science student X such that X was not hit on by their supervisor.

World: All female science students in time.

Original statement: There exists a female science student X such that X was hit on by their supervisor.

Negation: For all female science students X, X has not been hit on by their supervisor.

Cheng tells us that by using quantifiers, we now have predicate logic, or first-order logic. The opposite of predicate logic is propositional logic (no quantifiers). She explains "first-order":

"'First-order' is to distinguish it from higher-order versions of logic, which are more complicated in the way that the quantifiers work."

The author tells us that if we want to be right more often, we can be more careful with sweeping generalizations by using the following phrases:

In my opinion...

In my experience...

Maybe...

Sometimes...

Apparently...

It seems to me...

Cheng's PhD supervisor, Martin Hyland, often uses the phrase "There is a sense in which":

"'There is a sense in which Mozart is more boring than Brahms' is another way of correcting my sweeping statement 'Mozart is more boring than Brahms.' 'There is a sense in which teaching math can be a waste of time and money.'"

The author tells us that the last statement above could have been helpful earlier when she was in a Twitter debate with a traditionalist.

Cheng concludes the chapter with some advice:

"Someone might say something that is untrue in strictly logical terms, but perhaps they were really trying to say something else, perhaps something with strong emotional content that we should listen to if we are intelligent humans rather than intelligent emotionless robots."

Conclusion

I hope you enjoyed your Christmas holiday yesterday. And so let's enjoy the last few days in 2021 and start making plans for 2022.