New Year's Adam Post (Yule Blog Challenge #7)

Table of Contents

1. Introduction

2. Yule Blog Prompt #10: A Look Back at My Professional Growth from 2021

3. Links to Other Challenge Participants

4. Calendar Reform: 9-Day Calendars

5. Cheng's Art of Logic in an Illogical World, Chapter 9

6. Conclusion

Introduction

Today is New Year's Adam -- the day before New Year's Eve. I often use the word "Adam" (the day before an Eve) when a Sunday holiday is celebrated at a school on a Friday. (For example, this year we had Pi Day Adam on Friday, March 12th and Halloween Adam on Friday, October 29th.)

Also, yesterday I received my booster shot for the Moderna vaccine. It was exactly seven months after I got my second shot of Moderna.

Yule Blog Prompt #10: A Look Back at My Professional Growth from 2021

How much have I grown as a teacher this year? Well, the answer is that I've grown plenty -- after all, I began 2021 having had only one year of experience as a teacher (and I didn't even complete that one year), and now I finally have a full-time job once again. Much of my focus is on making sure that I don't repeat the mistakes from five years ago again this year.

As most experienced teachers already know, the best way to run the class is to make sure that the students know what they are doing at all times. In my last post, I mentioned how I'm incorporating both interactive notebooks and educational websites into my lessons. Five years ago, at the old charter middle school, I struggled with both -- I didn't use notebooks because I was afraid the kids might lose them (and at the time, I hadn't heard of many other math teachers who used them). And I didn't incorporate technology effectively because, well, my mind was still stuck in the 1990's, when I was a student, before such websites were commonly used.

In my Ethnostats class, I've used several projects. Some of these I found on the Sarah Carter website, including a dotplot on how many states visited that I gave the class back in September. I combined the lesson with one my Ethnostats predecessor once taught -- a dotplot on privilege number. Students drew both of these dotplots in their interactive Stats scrapbooks.

I also did several projects with my middle school students at the old charter school. But many of these projects lacked direction, and sometimes the students didn't know what they were supposed to do. Too often I ended up just giving everyone 100%. I'm trying to do better with this year's projects -- giving the students clear directions and taking off points if they don't follow them. Even so, for the most recent project (which involved drawing some graphs and answering questions), I gave no one less than a score of 68/75 (the lowest possible A). I must continue of making sure that the students are really learning something from these projects and taking points off if they haven't learned what they're supposed to.

Classroom management is, of course, related to this -- students are more likely to behave if they know what is expected of them at all times. Most of my classes are tiny so classroom management isn't really much of a problem. I do want to make sure that the students in my two largest classes -- Advisory and fourth period Ethnostats -- are on task at all times, and I think I still have room for improvement there.

Links to Other Challenge Participants

Whenever I participate in the Yule Blog and other challenges, I like linking to other participants. Today I link to Cheryl Leung, a sixth grade math teacher. She also writes about her professional development in her New Year's Adam post:

https://matheasyaspi.wordpress.com/2021/12/30/pandemic-lessons-teaching-students-in-quarantine/

In this post, Leung worries about her students who miss class due to a COVID-19 quarantine. Notice that her school apparently never had a hybrid schedule (that is, in 2020-21 her district had pure online learning, just like my current district). Even so, she is currently running her class like a hybrid -- she has both in-person and distance (the quarantined) students. She explains:

Every week, I post the work for the upcoming week in my Google Classroom as assignments. I number the items to help kids keep track of what comes first, what comes next, and so on. I organize them under a topic with the dates for the week.

In my current class, I haven't had that many students who needed to quarantine. But one girl in my Ethnostats class missed the entire week before Thanksgiving (presumably to quarantine). She had trouble working on the data collection project that week, and fell behind her regular written work -- and ultimately, she ended up with a C-, the lowest grade in the class.

It never occurred to me to set up a Google Meet for this student and run the class like a hybrid -- and I have no idea whether she would (or could) have joined the meet even if I'd done so. Perhaps after reading Leung's post, I might consider doing the same the next time a student is out for a week.

Meanwhile, reading about Leung's sixth grade class reminds me of my old charter middle school. Yes, Leung writes that she uses technology in her own classroom, mainly out of necessity -- she uses Google Classroom, Meet, and Slides, as well as Desmos. In that class I ended only used IXL -- and even then, not all sixth graders got to use it because there weren't enough laptops in the class for all the kids. Once again, even though that class was before the pandemic, I could have used technology more often -- either in partners or by giving those without a computer something else to do. Then perhaps I might have been more successful with that class.

Calendar Reform: 9-Day Calendars

Let's continue looking at Calendar Reform -- again, this year we're focusing on calendars that can support a three-day school week. This time, let's move up to nine days per week, since a new 9-day calendar was posted on the Calendar Wiki this year:

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

There are three versions of A Calendar for Time to Come posted at this link. This first one represents the mildest reform -- it keeps the standard seven-day week. It contains 12 months per year, with each month having 30 days. Extra days (not blank days) appear at the equinoxes and solstices (with 2-3 such days at the spring equinox, including Leap Day). This version of the calendar is nearly identical to the Fixed Quarters Calendar:

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

(except that the extra days are labeled 0-1 in Time to Come and 31-32 in Fixed Quarters).

The second version introduces the nine-day week. There are eight months, with each month having 45 days (that is, five of these 9-day weeks). Actually, they aren't weeks but "nonads," and they aren't months, but "octants." The octants correspond to seasons, with the first octant starting on the day of the spring equinox. Then the second octant starts at "Beltane" (near May Day, a pagan cross-quarter holiday), the third octant starts at the summer solstice, and so on.

The third version has four quarters rather than eight octants, with the ten nonads in each quarter labeled with the numbers 0-9. The "0" nonads are for holidays -- in fact, all holidays are squeezed into the "0" nonads, which fall near the equinoxes and solstices. This is like the Fixed Festivity Week Calendar, which does the same thing except with standard seven-day weeks:

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

Thus, even though Time to Come doesn't name the holidays that fall during the zero nonads, we can use the Fixed Festivity Week holidays for ideas. For example, Fixed Festivity Week uses the winter and spring holiday weeks for religious (Christian) holidays and the summer and fall holidays weeks for secular holidays. It squeezes all of Lent into a single week, starting with Shrove Monday, Mardi Gras, Ash Wednesday, and ending with Good Friday, Holy Saturday, Easter Sunday. (The calendar author couldn't think of anything else for Thursday, so Valentine's Day was chosen.)

So we can do the same with the zero nonads in Time to Come. There are ten days (labeled 0-9, since the zero nonad also has a zero day) in the holiday period, so we can squeeze in a few extra holidays not appearing in Fixed Festivity Week.

So what does the school year look like in Time to Come? First of all, notice that with nine days per week, we can actually include two three-day periods of school in a nonad. Indeed, the author of Time to Come was clearly inspired by the Triday Calendar -- within each nonad, two tridays are for work and the last triday is for rest. A six-day school week accommodates both types of block schedule -- the A/B schedule used at my current magnet high school and the A/B/C schedule used at the main high school.

With six days of school per nonad, we need 30 nonads to reach 180 days, so that means there can by ten nonads of vacation. All of the holidays fall during the four zero nonads, so we can afford six additional nonads off. Perhaps we might take off Nonads 1-6 in the summer to make a summer break, or maybe Nonads 7-9 in spring and 1-3 in summer (so that the winter zero week divides the semesters evenly). Or we might just simply keep all 216 days in the school year. Notice that the office work year has been reduced to these 216 days, so this can keep the work year and school year aligned.

The link doesn't state which three days of the nonad are the days off. Since the author was inspired by the Triday Calendar, perhaps a hybrid schedule is intended -- divide everyone into three cohorts, and each cohort works two of three tridays (similar to the 4/7 schedule I mentioned in my last post). Once again, I like the way hybrid schedules avoid price gouging at Disneyland. (While amusement parks would naturally jack up the prices during the zero holiday nonads, the other nonads would remain at regular price, allowing one to attend during their off-tridays.)

Otherwise, we might avoid hybrid and simply designate a six-day week and three-day weekend (for example, the week is Days 1-6 and the weekend is 7-9, or maybe 2-7 is the week, etc.)

Cheng's Art of Logic in an Illogical World, Chapter 9

Chapter 9 of Eugenia Cheng's The Art of Logic in an Illogical World is called "Paradoxes." Here's how it begins:

"I am an avid writer of to do lists. I find it an excellent way to procrastinate in a mildly useful way. Sometimes if I'm feeling particularly tired or stressed I will put some very easy things on my to do list so that I can easily declare that I've achieved something."

This chapter is all about paradoxes. Recall that we've already looked at paradoxes during a previous side-along reading (or side-along DVD viewing) -- David Kung's lectures on paradoxes. This was back in January 2016. So I might be referring back to posts from that month in today's blog entry.

What does Cheng's to do list have to do with paradoxes? Well, Cheng wonders what would happen if one entry on her to do list is "Do something on this list." Can she then immediately cross it off? She also mention another real paradox in her life -- to apply for a visa application, she had to enter her name exactly as written on her passport, but the online application won't accept her hyphenated middle name. So of the two commands "fill in your name exactly as written on your passport" and "only alphabetic characters are allowed," she can obey one or the other, but not both. She writes:

"I think of these loops and contradictions as paradoxes of life. Paradoxes occur when logic contradicts itself or when logic contradicts intuition."

Cheng begins with the liar paradox -- "I'm lying!" It can also be written as:

1. The following statement is true.
2. The previous statement is false.

David Kung also mentions the liar paradox in his very first lecture. Back in my New Year's Day 2016 post, I wrote:

-- The Liar Paradox: "This sentence is false."

And indeed, the title of that first lecture is "Everything in this lecture is false." Here is another example given by Cheng:

Cette phrase en francais est difficile a traduire en anglais.

which can be translated literally as:

This sentence in French is difficult to translate into English.

but it no longer makes sense.

Cheng's next paradox is called Carroll's paradox, named after British author Lewis Carroll. I don't think that Kung ever mentions Carroll's paradox in his lectures -- I've mentioned Carroll on the blog before (most recently in my Thanksgiving post), but never in connection to Kung. So this is a new paradox for us.

Carroll's story is called "What the Tortoise Said to Achilles." The titular tortoise asks the ancient Greek hero to show that a triangle is isosceles by measuring his sides, and he does:

A: Both sides of the triangle equal the length of 5 cm.

Z: Both sides of the triangle equal each other.

The tortoise asks "Does Z follow from A?" Achilles replies "Yes," so the tortoise adds it to the list, since it's a new assumption that we're making:

A: Both sides of the triangle equal the length of 5 cm.

B: A implies Z.

Z: Both sides of the triangle equal each other.

The tortoise asks "Do A and B imply Z?" Achilles replies "Yes," so the tortoise adds it to the list, since it's a new assumption that we're making:

A: Both sides of the triangle equal the length of 5 cm.

B: A implies Z.

C: A and B imply Z.

Z: Both sides of the triangle equal each other.

The tortoise asks "Do A, B, and C imply Z?" Achilles replies "Yes," so the tortoise adds it to the list, since it's a new assumption that we're making:

A: Both sides of the triangle equal the length of 5 cm.

B: A implies Z.

C: A and B imply Z.

D: A, B, and C together imply Z.

Z: Both sides of the triangle equal each other.

And the tortoise tortures Achilles with these statements ad infinitum. Cheng tells us that the only real escape is to use the rule of inference, modus ponens. It's modus ponens that allows us to conclude Z from the earlier statements.

The author repeats her example from earlier about breaking glass:

A: I dropped the glass.

B: A implies Z because the glass was too fragile.

C: A and B imply Z because the floor was too hard.

D: A, B, and C together imply Z because gravity intervened.

E: A, B, C, and D together imply Z because I didn't catch the glass.

F: A, B, C, D, and E together imply Z because nobody else caught the glass.

G: A, B, C, D, E, and F together imply Z because...

Z: The glass broke.

Cheng tells us that Carroll's choice of a tortoise and Achilles as characters goes back to more famous paradoxes -- Zeno's paradoxes. Kung mentions Zeno's paradoxes in his fifth lecture. Cheng writes about the race between the tortoise and Achilles, in which the reptile gets a head start:

"But then Zeno argues like this: by the time Achilles gets to the place where the tortoise started, the tortoise will have moved forwards a bit, say to point B. By the time Achilles gets to point B, the tortoise will have moved forwards a bit, say to point C."

Hey, what am I doing typing out Zeno's paradoxes in full? This is what I wrote in January 2016 about Zeno's paradoxes -- including a link to all three paradoxes:

Zeno's Paradoxes are so well-known that it's easy to find links to them. The following link mentions the first two of them:

http://platonicrealms.com/encyclopedia/zenos-paradox-of-the-tortoise-and-achilles

When Kung gives the story of Achilles and the tortoise, he doesn't give any specific numbers, but the above link does. We say that Achilles is running at 10 m/s (about the same speed as an Olympian sprinter like Usain Bolt) and the tortoise can only walk at 1 m/s. So how long does it take for Achilles to catch up to the tortoise? At first the answer may appear to be one second since that's how long it takes for Achilles to run 10 meters, but in that second, the tortoise has moved up one meter. And then Achilles can cover that meter in 0.1 second -- but by then, the tortoise has moved another 10 cm. And so on -- and that is Zeno's first paradox.

Returning to Cheng, she writes that a falsidical paradox is one where a fault of logic has been hidden in the argument:

"Zeno's paradoxes are falsidical paradoxes: the error is in the logic, not in our intuition about the world. The error is very subtle though, and it took mathematicians a couple of thousand years to work out how to correct it."

And as Kung tells us in his lectures, those corrections are now known as calculus.

Cheng's next example is related to the infinite sum

1 + 2 + 3 + ...

She tells us that according to a Numberphile video, the sum is -1/12. I've linked to Numberphile myself on the blog in the past, and so it's no effort for me to bring up the relevant video:

Cheng explains:

"I hope you feel that the end result is absurd, not least because all the numbers we're adding get bigger and bigger to infinity. Indeed this is why the infinite sum cannot be said to have a sensible answer without substantial qualification."

And she adds:

"Unfortunately, the video fooled millions millions of people, partly because of the good reputation of Numberphile videos in general. It is perhaps a case in point about memes being popular and believable even if they contradict both logic and intuition."

Mathologer refutes the Numberphile video. I assume Cheng would approve of the contents of this video:

But again, Numberphile videos are usually spot on. Indeed, here's a reputable Numberphile video about Zeno's paradoxes:

Cheng's next paradox is Hilbert's paradox. I don't need to type this us again -- Kung mentions it in his sixth lecture, and Cheng herself wrote about it in her second book Beyond Infinity. Let me cut-and-paste from the Kung posts of January 2016, which in turn contains a link to the full paradox

Kung begins his lecture by discussing the Hotel Infinity -- a hotel with infinitely many rooms. He states that this is often called Hilbert's Hotel, named after David Hilbert -- the mathematician who first came up with this analogy. (Yes, this is the same Hilbert who came up with a rigorous formulation of Euclidean geometry.) Hilbert's Hotel is so famous that it's easy to find a link to a description of the hotel, such as this one:

https://nrich.maths.org/5788

Cheng writes:

"It warns us that we can't just extend our intuition about finite numbers to infinite numbers, because strange things start happening. Those things aren't wrong, they're just different."

The author tells us that this is related to the current issue of Internet piracy. How is it possible to steal digital media when there can be potentially infinitely many copies of a file? She explains:

"Indeed, the theory of infinitely developed following Hilbert's paradox tells us that subtracting one from infinitely still leaves infinite. The math can't tell us what to do about these moral issues, but it can give us clearer terms in which to discuss them."

Cheng's next paradox is Godel's paradox, which Kung mentions in his first lecture. She recommends another book, Godel, Escher, Bach by Douglas Hofstadter, who explains the paradox more elegantly than any of us can:

"In it Hofstadter elucidates not only the incompleteness theorem but all sorts of fascinating links between logical structures and abstract structures in the music of Bach and the prints of Escher, both of whose works are deeply mathematical while also being immensely artistically satisfying."

In a nutshell, Godel's paradox is the following statement:

This statement is unprovable.

This is what I wrote back in my New Year's Day 2016 post:

On the other hand, Godel's statement "This statement is not provable" can be written as a mathematical statement -- but you have to be as smart as Godel (the Austrian mathematician Kurt Godel) to figure out how. The conclusion is that there exists statements that are true, yet not provable in mathematics.

And Cheng adds:

"This is an example of the fact that even in the logical world of mathematics if a conclusion feels wrong there are mathematicians who refuse to believe it although they can't find anything logically wrong with the proof."

In the next section, Cheng writes about Russell's paradox. She begins:

"When I meet people and say I'm a mathematician I often get slightly strange responses. It's funny how some people immediately boast about how bad they are at math, but other people immediately try to show off how knowledgeable they are."

Replace "mathematician" with "math teacher," and recall that Fawn Nguyen basically wrote about the former response ("I'm not a math person!") in a summer post. Anyway, Cheng tells this story because one person replied with, "Doesn't Russell's paradox show that math is a failure?"

Kung describes Russell's paradox in his ninth lecture, but he also mentions a simpler version of it, the barber paradox, in his first lecture. It's about the barber who shave anyone who doesn't shave himself, which Cheng illustrates as follows:

• If person A shaves person A, then the barber doesn't shave person A.
• If person A does not shave person A, the barber shaves person A.

This results in a problem if person A is the barber:

• If the barber shaves the barber, then the barber doesn't shave the barber.
• If the barber does not shave the barber, then the barber shaves the barber.

Each of these statements produces a contradiction. This is Russell's paradox. Cheng writes about how set theorists avoid Russell's paradox:

"The idea is to say that we have different 'levels' of sets, a bit like how we have different 'levels' of logic. Russell's paradox is caused by statements involving sets that loop back on themselves."

Cheng's last example is about tolerance. First, she explains that two "nots" cancel out:

"If I am 'not not hungry' then I am hungry. If we add up the 'nots' we find that 1 'not' plus another 1 'not' makes zero 'nots.'"

Cheng compares this to the structure of the Battenberg cake -- one of her favorite baking analogies that goes all the way back to her first book How to Bake Pi. (Cheng writes that it's because she does love Battenberg.) It's analogous to addition modulo 2, or adding even/odd numbers, or multiplying positive/negative numbers.

She believes it also comes up if we think about tolerance and intolerance:

• if you're tolerant of tolerance then that is tolerance
• if you're intolerant of tolerance then that is intolerance
• if you're tolerant of intolerance then that is intolerance
• if you're intolerant of intolerance then that is tolerance

Cheng adds:

"For me this means that I feel no pressure to be tolerant of hateful, prejudiced, bigoted, or downright harmful people, and moreover, I feel an imperative to stand up to them and let them know that such behavior is unacceptable."

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.

(Look at how far we got in the chapter before we needed that disclaimer!)

The author explains how different levels of sets resolves Russell's paradox:

1. Collections of objects, carefully defined. These are called sets.
2. Collections of sets; these are sometimes called large sets.
3. Collections of large sets, which we might call super-large sets.
4. Collections of super-large sets, which we might call super-super-large sets.
5. ...and so on.

We could do this with tolerance as well. We could set up levels like this:

1. Things
2. Ideas about things
3. Ideas about ideas about things; we might call these meta-ideas.
4. Ideas about meta-ideas, which we might call meta-meta-ideas.
5. ...and so on.

So we can be tolerant of people's ideas, but not their meta-ideas. Intolerance is such a meta-idea.

Cheng also shows us that we can do the same thing with knowledge:

1. Things
2. Knowledge about things
3. Knowledge about knowledge about things; we might call this meta-knowledge.
4. Knowledge about meta-knowledge, which we might call meta-meta-knowledge.
5. ...and so on.

She tells us that this arises when allegations of sexual harassment emerge, especially when it's against a well-known figure. She writes:

"This is one of the reasons the aggressors try to prevent communication between victims, with threats and abuses of power, or even a settlement and non-disclosure clause, or other forms of payment."

Cheng concludes the chapter by clarifying this surprising link between logic and the real world:

"But to me this is just part of the fact that logical thinking helps us in all aspects of life, even in our personal interactions with illogical humans."

Conclusion

This concludes my New Year's Adam post. I wish you all a Happy New Year, and that our professional growth will continue in 2022.