Christmas Eve Post (Yule Blog Challenge #4)
Introduction
Today is Christmas Eve. Many people finally get today off, including schools in New York (which often stays open through the 23rd) as well as federal workers (today is Christmas Observed, since the actual holiday is on a Saturday).
Yule Blog Prompt #5: Something Unexpected That I Faced/Learned
Well, there's one thing I faced this year that was completely unexpected -- my Ethnostats class. I'd never even heard of Ethnostats until after I was hired. While waiting for my paperwork to be processed so that I could start working on the second day of school, I researched the district and read about Ethnostats. (I already described a little about this class in my first Yule Blog post.)
When I arrived at my school, I found that two of my classes are Ethnostats -- Periods 2 and 4. Of these two classes, second period is smaller, with four students (all guys). The fourth period class is my largest, with thirteen students (including five girls).
One assignment I gave was a data collection project. The idea of Ethnostats is to tie statistics to the students' own worlds -- not just their ethnic or gender identity, but themselves and relationships with others. So I assigned this the week before Thanksgiving -- echoing the fact that the teacher from two years ago likewise assigned this project in mid-November.
A major topic of Ethnostats is the achievement gap -- why do statistics show that students of color don't perform as well as white students? And why is there a gender gap in math as well? Of course, the last thing I want to do, as the Ethnostats teacher, is reinforce those gaps. But this is what worries me -- let's just look at the first semester grades. In particular, I wish to compare the highest-scoring girl to the guy with the lowest grades for most of the semester.
With lots of projects for the students to do, only one student was failing for most of the semester, and he barely managed to get a D by the date of the progress report. Soon after, he raised his grade to a C -- even so, he still had the lowest grade in the class, since there were no other D's or F's in the class.
Meanwhile, the highest scoring girl does well on most of her quizzes during the semester. But she does have several absences, leading her to miss several assignments. She took the final exam, but much to her dismay, her grade on the final exam was 60%. I mentioned earlier this week that too many students earned D's on their finals. Notice that D's in Stats aren't as terrible as they are in Calculus (where students expecting to pass the AP exam need to earn higher than a D on their final).
I offered to let her come in and do test corrections, but she didn't come in. And guess who did take my offer instead? That's right -- it's the lowest-scoring guy. He made up the missing study guide (and many other assignments as well) and did the test corrections. When all was said and done, he'd raised his grade all the way up to a B! That's right -- the lowest scoring guy who'd done little work for large parts of the semester somehow ends up with a B, while the highest scoring girl ends up with a C+. My interventions designed to help the girls end up helping the guys instead.
The second highest girl in the class also has a C+, but she's farther away from the grade border. Her test grades are higher, but she has many more missing assignments. I know that she aspires to attend my alma mater, UCLA, because she said so when she answered her own survey question for the data collection project. (And so I really, really want the admissions officers at UCLA to see a B- for this class instead of a C+.) But there's no way to justify raising her grade. The other girls also have C's, including one girl with a C- (who was absent and missed the data collection activity completely).
Once again, a major question in Ethnostats is, why do girls consistently perform lower than boys in math classes? But instead, I must ask myself, why are my girls consistently performing lower than boys in my math class? It's important for me to form relationships with all of my students, regardless of gender identity. While Eugenia Cheng's fourth book, x + y, deals with gender issues in more detail, we're still in the middle of her third book here on the blog. But she writes about gender and relationships in today's chapter as well.
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 Bonnie Basu, who's also a California high school math teacher. (I believe she teaches all three levels of Integrated Math.). She chose the same topic that I did for Christmas Eve, "Lessons Learned":
https://reflectreviserepeat.blogspot.com/2021/12/lessons-learned.html
Basu writes:
I started this year with such great hopes.
I was wrong. I made so many incorrect assumptions.
After reading Basu's blog, I realize that both of us learned the same lesson this year -- that it's going to be harder than we imagined to engage our students. While we both had trouble engaging our students this year, I had extra trouble keeping the girls in one class motivated.
I enjoyed reading the rest of Basu's blog, all the way back to her very first post, as well. She started her blog last year, during the pandemic, and so I just read her entire blog. As a Californian, she was also subject to Governor Newsom's strict pandemic rules. It appears that her school spent at least the first semester of the 2020-2021 school year (and possibly the entire year) in distance learning.
She ends her blog entry with:
I have learned that I need to take a step back before I help my students step forward.
And I've definitely learned this lesson too.
Cheng's Art of Logic in an Illogical World, Chapter 6
Chapter 6 of Eugenia Cheng's The Art of Logic in an Illogical World is called "Relationships." Here's how it begins:
"In the previous chapter we saw how crucial it is to consider whole systems of interactions rather than people or events in isolation. The idea of considering thins in relation to each other is one of the important basic principles of modern mathematics."
So this chapter is all about relationships. The first interaction between events that Cheng considers is the vicious cycle. She draws a diagram with arrows, but I can sum it up in words right here:
When I feel bad, then I overeat.
When I overeat, then I feel bad.
Cheng explains:
"Some people don't suffer from either arrow: emotions don't cause them to eat, but also, when they do overeat they don't feel bad about it. Suffering from one arrow is unfortunate but at least it doesn't cause the escalation that the two cause in conjunction."
She tells us that one arrow ("I overeat" -> "I feel bad") represents "feelings," while the other ("I feel bad" -> "I overeat") represents "action." Here's her next example:
When Alex feels disrespected, then Alex can't show love. (action)
When Alex can't show love, then Sam feels unloved. (feelings)
When Sam feels unloved, then Sam can't show respect. (action)
When Sam can't show respect, then Alex feels disrespected. (feelings)
Cheng's next example is very relevant to recent events in the news:
When police feel threatened by black people, police aggressively defend themselves against blacks. A
When police aggressively defend themselves against black people, blacks feel threatened by police. F
When black people feel threatened by police, blacks aggressively defend themselves against police. A
When black people aggressively defend themselves against police, police feel threatened by blacks. F
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.
How do we end such a vicious cycle? Cheng writes:
"Some people argue that black people should simply 'do what they're told' by the police. But tragically, there are well-documented examples of black people being shot by police even when they were doing what they were told."
She tells us that it's important to understand that this is a cycle, and claiming there is one root cause is an oversimplification, unless we acknowledge that the cycle itself is the root cause.
In the next section, the author writes about category theory. As we recall from her first book How to Bake Pi, Cheng is a category theorist. So it's understandable that she continues to write about category theory in her next two books. She explains her field as follows:
"Category theory is a field of modern mathematics that brings relationships to the forefront. In this approach, the framework for thinking starts with deciding what objects and relationships we're going to focus on."
In her first example, she writes down the factors of 30:
1, 2, 3, 5, 6, 10, 15, 30
Then she draws a diagram in which arrows join numbers to their factors. Again, I don't draw the diagram, so let's follow the author as she describes it:
"We now see that this has the structure of a cube -- a more interesting structure than just some numbers listed in a straight line. We can then think about the hierarchy of these numbers in the picture."
At the bottom of the factor hierarchy we have 1. Then Cheng writes:
"At the second level we have the factors 2, 3, and 5 because nothing goes into them except 1. That is, they are prime numbers."
The next level up has:
6 = 2 * 3
10 = 2 * 5
15 = 3 * 5
Finally at the top we have:
30 = 2 * 3 * 5
Cheng tells us that the cube structure appears because 30 has three different prime factors. Other such numbers will produce the same pattern, such as 42. (At the Dozens Online forum, we'd say that 30 and 42 have the same abstract prime factorization or APF, namely {1, 1, 1}. Another accepted name for APF is "prime signature.")
42 = 2 * 3 * 7
has the following factors, in order of size:
1, 2, 3, 6, 7, 14, 21, 42
When she draws the same cube diagram for 42 as she does for 30, she points out that 7 is on the second level and 6 on the third, even though 6 < 7. She explains:
"If we also represented the size hierarchy, we would have to skew the diagram to look like this, a cuboid rather than a cube."
Here Cheng uses "cuboid" to represent what Pappas calls a (rectangular) "prism" and what the U of Chicago text calls a "box." Officially, a cuboid need not even be a prism (much less a box) -- it's only necessary that its six faces be quadrilaterals.
Returning to 30, Cheng redraws the picture so that each number is represented by the set of its prime factors -- so that {2, 3, 5} is at the top, and then {2, 3}, {2, 5}, {3, 5} right below it. At the bottom of the diagram she draws the empty set, a symbol often rendered as 0 in ASCII:
"Here 0 is used to represent the prime factors of 1, because 1 does not have any prime factors."
The author draws a similar diagram for 42 ({2, 3, 7}), and then generalizes to {a, b, c} where these can be prime factors or any objects.
And of course, Cheng's next example is all about privilege. So the three objects that she considers for a, b, and c are "rich," "white," and "male." She draws this diagram twice -- first with "rich white male" at the top, and then subsets of this through the diagram with 0 at the bottom. Then she redraws this diagram with "non-male," "non-white," and "poor."
Cheng points out that perhaps a cuboid diagram (as for 42) may be appropriate for privilege:
"Many people would argue that rich white women have higher status then rich black men, for example, and that rich black men in turn are better off in society than poor white men (not just in terms of wealth). It turns out that money goes a long way towards mitigating other problems."
And so she redraws the privilege diagram yet again as a cuboid. "Rich white male" remains at the top, followed closely by "rich white non-male." Then "rich non-white male" is next, followed by "rich non-white non-male." Below this is "poor white male" and so on. As Cheng writes:
"In particular, this provides a logic-based account of why some poor white men are so angry in the current socio-political climate -- because they are considered to be privileged from the point of view of number of types of privilege (white and male) but they are in reality less advantaged than many people who count as having fewer types of privilege than them."
Cheng finally draws one more cube diagram -- this one is more relevant in the context of feminism (such as last year's Women's March). All the members of this diagram are women, but now the three dimensions of privilege are "rich," "white," and "cis." She explains:
"This helps us understand why there is a lot of anger towards rich white cis women, among women activists who feel excluded by mainstream feminism."
She summarizes these charts as follows:
"Animosity tends to occur when someone is prone to thinking of themselves in a context that makes them underprivileged (a "victim") while others tend to view them in a context that makes them overprivileged."
And Cheng concludes the chapter with some advice for all of us:
"If we all become more adept at seeing things from both a privileged and a non-privileged point of view, we will achieve greater understanding of disadvantaged people's struggles but also of the actions, whether malicious or ignorant, that cause bigotry and oppression."
Conclusion
This concludes my post. I hope you have a wonderful Christmas holiday -- or at least a semi-normal Christmas (after last year's was ruined by the pandemic, which is unfortunately still raging).
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