Twelfth Night Post (Yule Blog Challenge #10)

Table of Contents

1. Introduction

2. Yule Blog Prompt #14: Read 3 Books, Blog Posts, Tweets, or Podcasts That Inspired Me in 2021

3. Cheng's Art of Logic in an Illogical World, Chapter 12

4. Conclusion

Introduction

Today is Twelfth Night, the end of the famous Twelve Days of Christmas. The reason, after all, that we're suppose to blog twelve times during winter break is due to the Twelve Days of Christmas. But it's difficult -- even Shelli, the leader of this challenge, had to settle for ten posts. I'm catching up to her with this post -- but I arguably did so by cheating, since my winter break is one week longer than hers.

Also, tomorrow is Epiphany -- also known as Dia de los Reyes, Day of the Three Kings, to my majority Hispanic students.

Yule Blog Prompt #14: Read 3 Books, Blog Posts, Tweets, or Podcasts That Inspired Me in 2021

Shelli, the leader of the Yule Blog Challenge, gave us bloggers a choice for this one -- we can write about posts, tweets, or even podcasts. But of course I'm going to write about books -- especially since I've already been writing about a certain book during winter break.

It was a few years ago when I first discovered Eugenia Cheng's first book How to Bake Pi. I just happened to be in the library and it was sitting on the hold shelf for another patron. So I decided to read it. At the time, I only had a passing familiarity with category theory (when some of my UCLA professors briefly alluded to it). But now I know a little more about it thanks to Cheng's explanation.

And so since then, I've checked out every book that she's written. As I've said countless times these past few weeks, in her third book, Art of Logic in an Illogical World, the author applies logic to race and politics -- the main topics of my current Ethnostats classes as well.

Perhaps I should have instead spent winter break discussing her fourth book -- x + y, on gender. The logic book is longer, and we won't have finished it on the blog by the end of winter break. But gender issues are definitely on my mind -- why, I ask myself, are my female students having so much trouble in my fourth period Ethnostats class?

In her book x + y, Cheng suggests one reason for girls' struggle in math class is that our math classes are too ingressive, or competitive -- and that satisfies many boys' interests. Girls, on the other hand, prefer to be more congressive, or cooperative. Thus according to the author, we can improve our girls' chances for success in our math classes by making them more congressive.

Indeed, there has been much discussion lately regarding the TV show Jeopardy! and gender. It has been noted that most of the top players of all time are male -- including Ken Jennings (now a co-host), Brad Rutter, and James Holzhauer, the participants of the Greatest of All Time tournament. But of course, Jeopardy! is a competition, hence ingressive -- and that's likely why men dominate. (And no, having a woman, Mayim Bialik, as the other co-host won't make the show any less ingressive.)

OK, let's count Cheng's x + y as the first of the three books to satisfy Shelli's challenge. I purchased the second book at last week's half-price book sale at Barnes and Noble. Another math author I'm familiar with is Ian Stewart, and so I found his latest book on sale -- What's the Use? How Mathematics Shapes Everyday Life. The title, of course, refers to a question often asked in math classes, "When will we use math in the real world?"

For previous Ian Stewart books, I would discuss the books on my old blog -- one chapter per day, just as we're doing over winter break with the Eugenia Cheng books. But once again, that was back when I was a substitute teacher with plenty of time on my hands. Now that I'm a full-time teacher, I don't have as much time for side-along reading.

As for the third book -- well, I have several books to mention in connection with Ethnostats. Recall that last month, the department chair (my partner teacher) sent me the real syllabus for this class. And now that I have the syllabus, I no longer have to piece together lessons from my two predecessors -- and now I know what the required reading materials are supposed to be.

The Ethnostats course is divided into four units. These units correspond to the four units for Statistics and Probability according to the Common Core Standards:

http://www.corestandards.org/Math/Content/HSS/introduction/

1. Interpreting Categorical & Quantitative Data
2. Making Inferences & Justifying Conclusions
3. Conditional Probability & the Rules of Probability
4. Using Probability to Make Decisions

Unfortunately, based on my pacing so far, I've just barely started the second unit. That's because the text is divided into five parts, but Unit 1 of the curriculum corresponds to Parts I-II of the text. Only because I squeezed Chapter 10 into first semester did our class reach Unit 2 -- indeed, each of the remaining Parts III-V is equivalent to one of the Units 2-4.

(The same thing often happens in Geometry classes, as we saw on the old blog -- these are six units, but Unit 1 on Congruence can take up most, if not all, of the first semester, leaving Units 2-6 for spring.)

The current Unit 2/Part III includes Chapters 11-12. It might take me about four weeks to cover the chapters, taking us through January into the first few days of February.

As I wrote above, the syllabus lists the course materials for this unit:

1. Pedagogy of the Oppressed
2. Money, Race, and Success: How Your District Compares
3. They Got Me Trapped: Structural Inequality and Racism in Space and Place Within Urban School System Design
4. Culture as a Disability
5. Transforming Deficit Myths about Learning, Language, and Culture

The first item above is a book by Paulo Friere. Hmm, perhaps I should have looked for that book at the Barnes and Noble sale -- then again, they likely didn't have a copy. So instead, I ended up requesting the book from the library. On the other hand, the third item is also a book, but there doesn't appear to be a copy at my library. That's OK -- technically I already started Unit 2 in December, so I'm not expected to cover all of the materials during the remainder of the unit.

The last two items on the list are articles. Articles are often difficult to access -- there are websites like JSTOR that have articles, but those sites cost money. I do remember someone talking about JSTOR at a teachers meeting -- I'm hoping that it's our history teacher (and not, say, a teacher from my long-term school last year, since I don't remember how long ago it was). If so, then maybe I could ask her for advice regarding how to access the articles.

Here's what I plan on doing in my Ethnostats class. I'll discuss the Friere book depending on when my copy arrives at the library -- hopefully it's next week. Then the next two weeks can be for the two articles on JSTOR.

This I'll intersperse with the material from Chapter 11. Notice that although there's no real reason for me to give quizzes in Calculus and Ethnostats simultaneously, it's been a habit to do so. (With larger classes, I'd try to avoid giving quizzes at the same time, but for my small classes, I've developed the habit of quizzing all classes at once.) Thus the timing of the Chapter 11 quiz ultimately depends on when the monthly minimum day is scheduled. I might spend either two to three weeks on Chapter 11.

For Chapter 12, the week when Calculus is taking its chapter test, Ethnostats will get a project instead -- and this project will be based on the second item from the list above. It's a New York Times article -- and notice that "How Your District Compares" refers to our district. That is, we can type in the name of our district and get data about our community.

Some of the other Unit 2 stuff on the syllabus include the following: "Students must demonstrate proficiency on a computerized unit exam that requires students to look at and understand relationships between variables. Students must be able to come to conclusions about correlation, association, and causation in scatter plots and explain their findings." Recall that I received this syllabus a few days before the final exam, and so I already included this on the final.

The goal for second semester is at least to start Unit 4/Part V, which would be Chapter 17. If I follow the current pacing, then I should reach Chapter 18 and perhaps even start Chapter 19. Since 19 is the longest chapter of the text, I'm unlikely to complete it.

So to answer Shelli's question, my three books are Paulo Friere's Pedagogy of the Oppressed, Ian Stewart's What's the Use? and Eugenia Cheng's x + y.

And just as x + y suggests, throughout the second semester, I'll try to make the class a little bit more congressive so that my fourth period girls can be more successful. And speaking of Eugenia Cheng --

Cheng's Art of Logic in an Illogical World, Chapter 12

Chapter 12 of Eugenia Cheng's The Art of Logic in an Illogical World, "Fine Lines and Gray Areas," begins as follows:

"One night during my first term at university another fresher was found in the kitchen just before midnight eating a bowl of cereal. He explained that, according to the best-before date, his milk was about to go off at midnight."

The point of this story, of choice, is that there's no fine line or exact moment when the milk changes from unspoiled to spoiled. Likewise, there's no single moment when one person falls in love. Cheng quotes from Jane Austen's Pride and Prejudice as Mr. Darcy says to Elizabeth:

"I cannot fix on the hour, or the spot, or the look, or the words, which laid the foundation. It is too long ago. I was in the middle before I knew that I had begun."

Cheng reminds us that although logic may be black and white, the world rarely is. There are often gray areas:

"But I worry about the world turning into certainties that are almost certainly flawed. We should understand different ways of dealing with gray areas and become better at working with their nuance, instead of longing for the false promise of black and white clarity."

As with many of her examples, the author likes to return to cake. She writes:

• It won't hurt to eat one small piece of cake.
• And however much cake I've already eaten, it can't hurt to eat one more mouthful.

As Cheng points out, we can then prove that it's okay to eat arbitrarily much cake. This is an example where the logic of the situation pushes us to one of two extreme positions, either:

• it is not okay to eat any cake at all, or
• it is okay to eat infinite amounts of cake.

The trouble is the gray area. She tells us that this is what happens when we try to eat one more bite of cake, or when a child tries to stay up an extra two minutes:

"But really, the bedtime itself is spurious -- it is an arbitrary line that has been set in a gray area between 'sensible bed time' and 'much too late.' One way to get round this logic is just to shrug and say just because something is logically implied, that doesn't mean I'm going to believe it."

At this point Cheng refers to the "deductive closure" -- the deductive closure of a set of axioms is the set of all statements that can be proved from those axioms. She tells us that having a deductively closed set of beliefs is part of being a logical human being -- but gray areas are problematic.

The author asks, so where do we draw the line? When it comes to eating cake, she writes that she should draw the line on the safe side:

"This is especially true because I am liable to stray over my line a but, so I should put a little buffer zone in to be on the safe side."

Now Cheng proceeds to write about grade boundaries -- something that's important to us teachers:

"In the UK system, students graduate from university with a degree that is classed as first class, 'upper second' class, 'lower second' class, or third class, otherwise known as first, 2:1, 2:2, or third. But where should the boundaries be drawn?"

Try changing this to "US Common Core exams" (like SBAC or PARCC) and suddenly we wonder where to draw lines -- the cut scores between 1, 2, 3, and 4 (and 5 for PARCC). She points out:

"No matter where you place it, someone will argue that it's unfair to the person just below it, and as a result the line tends to shift further and further down. There is no logical place to put that line. I think the only logical thing to do is get rid of the lines and publish either averages on a fully sliding scale or percentiles instead."

She also writes about Obama. How could he be the first black president if his mother is white? Now according to Cheng, "black" here is being used to mean "non-white""

"At least talking about white people and non-white people is a genuine dichotomy, where black and white is a false one."

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.

This is a great time for me to put in that disclaimer, since Cheng now goes into sexual harassment:

"It can be especially hard to draw lines when dealing with people who do not respect boundaries. This can happen if you are the kind of person who likes being generous and being able to help people."

And of course, all the hearings and FBI investigations are all about whether a certain man crossed such a line with a certain woman 36 years ago. Again, the author reminds us that it's difficult to know exactly when a line has been crossed:

"Mr. Darcy's inability to draw the line for love also applies to hurt -- we don't necessarily know where is the exact moment that something is going to start hurting us. We only know it will definitely hurt us a lot, like if inappropriate physical contact becomes rape."

And so Cheng draws her line closer to the safe side, thus leaving a buffer zone where's it's not necessarily dangerous yet, but for extra protection. (Note that this refers to when the contact is actually happening and not necessarily in a hearing 36 years after the fact.)

The author now proceeds to discuss Body Mass Index, defined as mass over height squared:

"The first thing that people object to about this is that it doesn't take into account how muscular you are, and so very strong athletes are liable to have a high BMI as muscle is very dense."

For women like Cheng, the cutoff to remain healthy is about 25. But to remain on the safe side, Cheng tries to remain around a BMI of 24:

"One interpretation of what I'm doing is treating the line itself as something hazy, so I try to stay far enough on the good side of the line that I'm out of its hazy range."

The author's next topic is mathematical induction. I've mentioned induction on the blog several times before, including last year during another side-along reading book (Stanley Ogilvy). Cheng returns to her favorite example, baked goods:

• It's fine to eat 1 cookie.
• If it was fine to eat some number of cookies, it's okay to eat 1 more.

Therefore it's fine to eat any number of cookies.

Cheng also writes this by letting P(n) be the statement "It is fine to eat n cookies":

• P(n) is true.
• P(n) => P(n + 1).

Then by the principle of mathematical induction, P(n) is true for all whole numbers n. She proceeds:

"This is fine for whole numbers, but it gets tricky if you're trying to deal with a sliding scale that includes all the numbers in between, or even just all possible fractions."

And indeed, Cheng once passed out cookies to 20 students. The cookies she handed to adjacent students are almost the same size, yet the last cookie is twice the size of the first cookie!

The author now discusses fuzzy logic, where there are truth values between "true" and "false":

"It also might be true in the case of probability, where we can't be certain what the truth is, we can only be a certain percentage sure, with the rest being in some doubt."

This applies to weather forecasts (chance of rain), as well as percentages given on a test (as opposed to just pass/fail):

"Fuzzy logic is currently used more in applied engineering than in math, to deal with gray areas in control of digital devices."

Examples include rice cookers and heating/air conditioning units.

Cheng now returns to cookie sizes. She shows us a picture of her batch of cookies -- the one where each cookie has a slightly different size:

"This way I can meet my own needs and other people's at the same time, without actually having to know what anyone else's perfect size of cookie is -- I can be sure it's in there somewhere. This is an application of the intermediate value theorem, a theorem in rigorous calculus that math students usually study as undergraduates. It says that if you have a continuous function that starts at 0 and goes up to some number a, it must take every value in between."

Last year, Cheng talked to an art student in Chicago who created visual illusions and tried to see if viewers would believe if they were real or digitally manipulated:

"The question was to find the sweet spot where people would be really unsure which it was. I realized that she could invoke the intermediate value theorem: make a series of pieces starting with one that was obviously a physical construction, gradually becoming less obvious until she ended with one that was obviously physically impossible so must be a digital manipulation."

And of course her "sweet spot" was somewhere in between.

Cheng returns to races. What should we call someone who is mixed Asian and white, depending on whether that person is more Asian or more white?

"As described throughout this chapter there are various possibilities, each of which has advantages and disadvantages."

The goal, Cheng tells us, is to bridge the gap between the two sides, since in a way, we are all living on a gray bridge between black and white:

"If we all acknowledge that, and build even more bridges, I think we will achieve better understanding."

Her final example of such bridge-building is on social services:

"The person who believes that everyone should take responsibility for themselves might be able to acknowledge that some particularly 'worthy' people need help, perhaps members of the military who have been injured during active service."

Cheng concludes by telling us how we might draw such a person onto our bridge:

"The first step is to think about understanding a difficult argument by comparing it with a more understandable one that has something in common, that is, it is analogous in some way. This is the subject of the next chapter."

Conclusion

Tomorrow isn't just Epiphany -- it's also Phi Day. January 6th was chosen because the the digits of Phi begin with 1.6... -- 1.618 is a bit more accurate. With a three-week winter break, there's much less likely to be school on Phi Day. (On the LAUSD calendar, there can never be school between December 21st and January 6th inclusive -- I suspect that it's the same in my current district.)