Kwanzaa Post (Yule Blog Challenge #6)
Introduction
Today is the third day of Kwanzaa, a holiday celebrated by African-Americans. Five years ago, I worked at a majority-black charter school, and one girl there participated in a Kwanzaa pageant. In her honor, I've continued to label one of my posts between Christmas and New Year's as the "Kwanzaa post" in every year since I met her.
But hold on -- now I'm working at a majority-Hispanic high school, and as the Ethnostats teacher, I need to honor the students in my current classroom. Perhaps it's time for me to retire the name "Kwanzaa post" and focus on a holiday that my students celebrate.
Most Hispanics are Catholic, and on the Catholic Calendar, today is "Holy Innocents Day," (Dia de los Santos Inocentes), referring to the babies killed by King Herod around the time that Jesus was born:
https://www.timeanddate.com/holidays/mexico/day-of-the-holy-innocents
According to the above link, it is Mexico's equivalent to April Fool's Day.
Yule Blog Prompt #8: A Tool or Strategy from 2021 That I Will Continue to Use in the Future
One tool that I used in 2021 is the interactive notebook. It's something that I read about on the MTBoS for several years, most notably by Sarah Carter. Indeed, I was considering using it at that aforementioned charter school, but I decided against it. Then over the next few years, I subbed for several teachers who used interactive notebooks.
Then this year, Carter introduced something else in her Stats class, a "Stats scrapbook." And so not only did I decide to use a Stats scrapbook in my Ethnostats classes, but I even adopted some of her activities, including an NFL activity where each student selects a different team, chooses players on that team at random, and records their stats.
Notice that Carter never specified whether her "Stats scrapbook" is an interactive notebook (that is, one where worksheets are cut and pasted into the notebook) -- and indeed, I wasn't sure whether doing interactive notebooks is even a good idea during the pandemic (since scissors, glue/tape, and so on must be shared). But I noticed that the department chair (my partner teacher) uses interactive notebooks in her own math classes. So then I thought that if she can use interactive notebooks, so can I.
One thing I've always feared with interactive notebooks is that students might lose them, then say that they can't do any work since they don't have their notebooks. And the projects that I have them do in their scrapbooks are worth a large portion of their grade. So I've decided to keep the notebooks in my cabinet, passing them out only when doing projects. Only once did I let the kids take them home (for the data collection project), and then I warned them that they'd get 0 out of 100 points if they lose them.
While "interactive notebooks" are one possible answer to today's prompt, I know what sort of answer Shelli (and any other Yule Blogger) is expecting as a response -- something to do with technology. And so let me discuss what educational software I've been using in my class. Even though the Calculus teacher from the main high school (the one I'm supposed to be emulating) doesn't use tech (at least not on the day I observed him), my kids are used to using online software, so I'll continue to do so.
One website that I've been using is DeltaMath. Some of my quizzes and tests so far are on DeltaMath, and so leading up to such quizzes, I've give some review assignments on the website as well. And last month I was introduced to AP Classroom, another source of the type of questions that students are likely to see on the exam. This is, of course, in addition to the standard written assignments that come from the textbook.
In the upcoming second semester, I'd like to incorporate all three sources -- the text, AP Classroom, and DeltaMath -- into my assignments. I'm not quite sure how yet -- once again, it might depend on the lesson and what sorts of questions the students need to understand. Not all AP-like questions appear on DeltaMath, while the AP-like questions that do appear in Classroom might be too tricky for my students until they're ready for them. So I'll likely end up going back and forth among all three sources.
I also used Edpuzzle a few times in both Calculus and general Stats. You might also be wondering, what about Desmos -- the quintessential math website? While I haven't officially used Desmos, some of my Calculus students used it last year, and sometimes they refer to it this year when graphing. I inform them that while it's great that they can draw graphs on the website, they won't be able to use Desmos on the AP exam. So they must know how to graph on the TI calculators before the exam.
Calendar Reform: 7-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. It's easy to find a six-day calendar with three days of school -- the Liberalia Triday Calendar:
https://www.hermetic.ch/cal_stud/ltc/ltc.htm
This is the first calendar that supports a "hybrid" schedule -- indeed, scrolling down we see a schedule where "Team 1" and "Team 2" work on opposite tridays.
But of course, there's no real purpose in discussing calendars that will never be adopted (as interesting as these thought experiments might be). There are many religious adherents who will never drop the seven-day week. So if I do come up with an interesting schedule on one of these reform calendars, we must find a way to transfer it to the seven-day week if we expect it to be adopted.
Here's one way we can implement a triday schedule on the standard 7-day calendar -- we set aside one day, namely the Sabbath of the majority religion in each country. So here in the West we set aside Sunday, in the Middle East we set aside Friday, and in Israel we designate Saturday. Then the rest of the week can be divided into two tridays. In the USA, the first triday is Monday-Wednesday and the second triday is Thursday-Saturday.
Of course, after experiencing a hybrid teaching schedule last year, you might wonder why I'm so eager to write about another hybrid schedule. Well, one thing I like about hybrid schedules is that they eliminate weekends -- and the ability of companies like Disneyland to raise ticket prices on weekends.
Indeed, I was recently looking at the tiers for Disneyland tickets. Naturally, of the five tiers, most days that teachers are off are in the highest priced Tiers 4 and 5. Again, this makes sense, as demand rises when parents can easily take their kids to Disneyland, so prices rise as well. Parents can then take their annual two-week vacation from work to go to Disneyland. It's only a problem when your job is with kids (that is, teaching), so your schedule is the same as theirs. In fact, the only days below Tier 4 when teachers are off are Veteran's Day (Tier 2) and MLK Day (Tier 3).
Some people complain when teachers take days off when they work only 180 days per year. Their concerns are valid -- office workers get only two weeks of vacation per year while teachers get many more weeks off. But in exchange for having a longer break, we get less flexibility -- office workers can choose to take two weeks off when Disneyland prices are low, while teachers only get the weeks when prices are high.
The solution, of course, is to eliminate weekends and implement tridays instead. With different groups having different days off, it would be harder for Disneyland to raise prices on weekends.
There are several reasons that teachers disliked the hybrid schedule. One is that we couldn't really go to Disneyland (and not just because it was closed) -- when one group of students was in the classroom, the other was at home online. Students had extra days off, but teachers didn't. A true triday schedule would have two sets of teachers as well as two cohorts of students.
We might also wonder, is it possible to have a full 180-day school year with tridays? It's fairly obvious that we can't -- there are 52 weeks (of seven days) in a year, and 52 * 3 is only 156. And that's going to school year-round, with no vacation days. Then again, vacation weeks have already disappeared in this calendar along with weekends.
On the old blog, I once suggested a triday-like hybrid schedule with four school days per week. To accomplish this, there were three cohorts rather than two -- the first cohort was Monday-Thursday, the second cohort Wednesday-Saturday, and the third cohort Mon./Tues./Fri./Sat. With four school days per week, only 45 weeks are needed to reach 180 days, so that leaves seven weeks of vacation.
Perhaps we can do the same with our three-day week -- 45 weeks of three days each, even though this gives us only 135 school days. This might be acceptable -- we could label the missing 45 days as "asynchronous days" (as sometimes occurred on the real hybrid schedules) and assign 180 days' worth of work over 135 days of school. It can be argued that if we eliminate summer break (that is, spread the seven vacation weeks over the entire year), then there's less summer learning loss, so students can learn and remember as much over 135 spread-out days as they do over 180 days with a summer break.
And I found a way to distribute the seven weeks that respects established federal holidays. For those holidays that fall on Mondays (MLK Day, President's Day, Memorial Day, Labor Day, Columbus Day), we take off both Monday and Tuesday. (Tuesday is taken off so that the Mon./Tues./Fri./Sat. cohort doesn't have a single day on Tuesday surrounded by off days.) The Wednesday-Saturday cohort can then take Wednesday/Thursday off.
Thanksgiving fits into this plan as well -- Wed./Thurs. are the off days, except for the MTFS cohort, which takes off Friday and Saturday instead. Then these six holidays result in taking three weeks (that is, six half-weeks that are two days instead of four) off, leaving four more weeks of vacation. These can be the four federal holidays of Christmas, New Years, Fourth of July, and Veteran's Day -- these can fall on any day of the week and don't fit into the hybrid scheme, so it's easier to take the whole week off.
Of course, this plan isn't perfect. During Thanksgiving week, the W-S cohort must attend school on (Black) Friday and Saturday -- and indeed, many people would rather have the entire week off for Thanksgiving than Veteran's Day. Also, I came up with this schedule before the newest federal holiday was established for African-Americans -- no, not Kwanzaa, but Juneteenth. And Juneteenth is one that can fall on any day of the week, just like Veteran's Day.
(Speaking of Kwanzaa, notice that this holiday is one of the ones placed on the Tyerian 5-day calendar that I mentioned in my last post.)
There are ways to solve these problems on the triday version of the hybrid calendar. We can make it so that the entire week of Thanksgiving is taken off. Juneteenth and Fourth of July are about two weeks apart, so we can take the week in between off as well and make a short three-week summer break. Then along with the established two-week winter break, that gives us six weeks of vacation.
For the remaining six federal holidays, we only need to take one day off for each (say Monday for the M-W cohort and Saturday for the Thurs./Fri./Sat. cohort). But since each week has only three days, these six holidays add up to two weeks, so now there are only 44 weeks (that is, 132 days) of school instead of 45 if we take all six days off.
Cheng's Art of Logic in an Illogical World, Chapter 8
Chapter 8 of Eugenia Cheng's The Art of Logic in an Illogical World is "Truth and Humans." It's the first chapter of Part II, "The Limits of Logic." Here's how it begins:
"We have seen the power of logic in producing rigorous unambiguous justifications. Now we are going to address the limits of that power."
Cheng warns us logic has its limits -- it works well in the mathematical world, but it doesn't always work in the real world. In some ways, logic is a social construct. She writes:
"This might seem to fly in the face of everything I've said about mathematics being completely rooted in logic, but the situation is more subtle than that."
The author compares this to a trial by jury. Sometimes we lack time to make a logical argument:
"But another way in which logic isn't powerful enough is when we need to convince someone else of our argument. Logic turns out to be a good way to verify truth, but this is not the same as convincing others of truth."
Cheng writes that one way to make an argument is divide it into parts, and then subdivide each part of your argument into parts. She compares this to something we've seen before -- fractals:
"A fractal is a mathematical object that resembles itself at all scales, so that if you zoom in on a small part of it, that small part looks like the whole thing."
On the old blog, one of our side-along reading books was Benoit Mandelbrot's book on fractals -- and of course, Mandelbrot was the creator of fractal theory. And another side-along reading author, Wickelgren, also compared problem-solving to a fractal-like tree. Cheng draws such a tree in her book -- back in April, I wrote about how to program a computer to draw the tree. She explains:
"This tree represents how finding a proof works, in my head. The base is the thing you're trying to show is true, a but like in our diagrams of causation in Chapter 5. The two branches going into it are the main factors that logically imply it."
Fractal trees are infinite, but proofs are finite. So when do proofs stop? Cheng answers:
"In arguments in real life, we should keep going until the other person is convinced, or until we realize that our fundamental starting points are so different that we will never be able to convince them unless we can change their fundamental beliefs."
According to Cheng, even mathematical proofs are subject to a trial by jury, except that this trial by jury is called "peer review." We've discussed peer review back during another side-along reading book (on Perelman's proof of the Poincare conjecture). And just as we found out, submitting a paper anonymously does little to avoid its acceptance being swayed by human emotions, because the world of mathematicians is a small world after all:
"It's a bit like marking exams anonymously if your class has only three students and you've been working with them closely all year -- whether or not they write their names on the exam paper, you will know exactly which student is which."
The author tells us that to appeal to the human emotional side, research papers include material that isn't strictly logical:
"The help comes in the form of analogies, ideas, informal explanations, pictures, background discussion, test-case examples, and more. None of this is part of the formal proof, but is part of the process of helping mathematicians get their intuition to match the logic of the proof."
Cheng now leaves the world of mathematical papers and enters the world of politics:
"For politicians, the 'peer review' is the election -- it doesn't matter if they're right or not, and it doesn't matter if their if their arguments are sound or not, it just matters if people vote for them or not. Voters do not have to justify their vote either."
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.
Politics is all about raising skepticism. She tells us that reasonable skepticism about a mathematical proof can arise in two ways:
- Someone might think there's a gap or error in your logic.
- Your conclusion might contradict someone's intuition.
The statement "they eat dogs in China" is true, but it isn't very illuminating, especially if the current conversation is all about, say, a dog walking app.
Cheng explains that the only equations that are true in first-order logic are the equalities x = x:
1 = 1
2 = 2
3 = 3
4 = 4
...
and so on. These are true, but they aren't illuminating. As for all other equations, Cheng flatly states:
All equations that are illuminating are lies.
For example, she writes:
10 + 1 = 1 + 10
As far as first-order logic is concerned, this statement is a "lie," in the sense that it's not necessarily true on its own. It's true only in a system that has the Commutative Property of Addition -- without that property, the statement could be false. But Cheng explains why this statement is illuminating:
"One children work this out, they can use it to help them add up by counting on, knowing that it will always be easier to start with the larger number in their head and count on by the smaller number."
Notice that this is going to be a traditionalists-labeled post. And here's Cheng writing about things that traditionalists don't like -- adding by "counting on," or any method other than the standard algorithm of addition.
Anyway, Cheng tells us that the equations that really have nothing different about the two sides are the ones of the form:
x = x
and these ones are never useful.
At this point, Cheng repeats what she says about research papers containing stuff other than logic:
"Often in a research paper a logical proof will be accompanied by a description of what 'the idea is,' which is something more informal, not rigorous, but invokes ideas and imagery that might help us to understand the logic."
This leads back to the traditionalists' debate. Cheng tells us that some people -- traditionalists -- believe that rote memorization is necessary:
"But other people, often professional research mathematicians themselves (including me), are convinced that they have never really memorized anything in math. In fact, one of the main reasons I always loved math was exactly the fact that it didn't require memorization, only understanding."
And in fact, Cheng continues:
"However, I am perfectly fine at basic arithmetic and certainly above average compared with the general population, and yet I have never memorized my times tables. I know my times tables by some other, more subtle route that does not involve memorizing."
According to traditionalists, for those who don't memorize their times tables, "it's all over" and "doors are closed to STEM careers." Yet Cheng is a living, breathing counterexample -- she has a PhD in math yet never memorized her times tables.
Cheng wraps up the chapter with some Internet memes. Just like equations, memes can often be illuminating without being true. For example, here is such a meme:
HOW IT SHOULD BE:
Scientists (experts): There's a problem.
Politicians (non-experts): Let's debate a solution.
HOW IT ACTUALLY IS:
Scientists (experts): There's a problem.
Politicians (non-experts): Let's debate if there's a problem.
Cheng herself wants to edit this meme, as follows:
HOW IT SHOULD BE:
Scientists (experts): We think there's a problem. Here's a solution.
Politicians (non-experts): Let's debate funding their solution.
HOW IT ACTUALLY IS:
Scientists (experts): We think there's a problem.
Politicians (non-experts): Let's debate if there's a problem.
And here's another meme:
Funny how no country has ever tried to repeal universal healthcare.
It's almost like it works.
But the author responds:
"However, I'm not sure if the meme is true -- arguably some people have been trying to destroy and privatize the National Health Service in the UK."
Cheng concludes her chapter with another warning:
"We should particularly not pit emotions against logic. They are not opposites, but can work together to make things that are both defensible and believable."
Conclusion
I wish you a happy Kwanzaa, for those who celebrate it, and a blessed Holy Innocents Day, for those who observe it.
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