(And no, I'm not talking about the hurricanes in Houston, although they did cause the cancellation of baseball games involving our local Angels.)
I'm speaking, of course, about last weekend's police shooting in Wisconsin. Even though Jacob Blake survived the shooting, it led to the cancellation of first the Bucks game (as that team comes from that state), and then all NBA games, including our local Lakers. And in baseball, while some games are still played, the three NL teams from California aren't playing tonight, while the A's are playing. (The Angels, once again, aren't playing due to the Houston hurricane.)
I want this blog to remain politically neutral, especially during the school year. I make efforts to avoid certain topics depending on the labels used for the post -- for example, it's OK for me to mention politics during "traditionalists" posts, since traditionalism is inherently political (and the traditionalists themselves often bring up politics). On the other hand, in posts labeled "Geometry" (and without an accompanying "traditionalists" label), politics is supposed to be off-topic.
When the George Floyd protests began, the schools were closed, and even distance learning was winding down for the summer. And so I felt it was OK for me to mention protests on the blog. I felt that even if the protests lasted a solid two months, they would be over before the basketball and baseball seasons were set to resume, and before the new school year.
And I was right -- that is, until yet another police shooting occurred.
Now NBA has been put on hold, and I don't know when it will resume. As a Southern Californian, I still hope that our local teams, the Lakers and Clippers, can meet in the Western Conference Finals, but I'm not sure whether the players on those teams are willing to compete any more.
I want to be politically neutral on this blog. But I am definitely not neutral when it comes to our students -- as teachers, it's our job to be on our students' side. And until I am hired once again as a regular teacher, I consider the students at my old charter middle school to be my students -- even though I no longer work there, even though the school longer exists, and even though they're high school students now. What I wish I knew is, what are my students -- the majority of whom are black, the majority of the rest Hispanic -- thinking about now, as they try to struggle through distance learning, as they hear about the events in Wisconsin?
I recall that right after the passing of George Floyd, some teacher-bloggers suggested changing their lesson plans in order to accommodate those students whose minds may be elsewhere -- thinking about world events too much to concentrate on math. Once again, I was hoping that the protests would have settled down by the first day of school so that we can focus just on the math. But thanks to what happened in Wisconsin, world events are taking over yet again.
Indeed, today's Geometry lesson is called "Perspective." But instead of Geometry, suddenly I wish to see the world from other people's perspectives. What happened that fateful Sunday night from Jacob Blake's perspective -- was he only trying to protect his three young sons in his car? What happened that fateful Sunday night from the officer's perspective -- did he truly feel threatened enough by Blake that he needed to shoot?
And again, what does this incident look like from my former students' perspective? Are they able to concentrate on math and their other distance lessons, or do they need someone to talk with, to make them feel safer in an increasingly uncertain world?
So far, I haven't received any subbing calls yet, but subbing calls are nevertheless possible. What should I do if I suddenly get a subbing call for tomorrow? Will those students be able to concentrate on whatever lesson I'm supposed to teach, or will Blake's injury weigh heavily on their minds? What does the world look like from their perspective?
I've admitted that I'm not good at talking about such incidents. On Pi Day Eve, the day before the coronavirus closure began, I just wanted to eat pie and sing Pi Day songs, but some senior girls in the last class I subbed that day were talking about how worried they were about possibly losing their prom and graduation. I could have spoken with them about the virus and what it means, but instead I mostly ignored their concerns.
And now the possibility exists -- not a high probability, but still, a possibility -- that I'll be called upon to sub when my students are thinking about world events.
I'm neutral in that I won't comment on whether Blake or the officer was in the wrong on Sunday. I wasn't there, and I can't see the world from either of their perspectives. But I want to make an effort -- certainly more of an effort than in the past -- to see the world from my students' perspectives. After all, if I'm called upon tomorrow, then it's my job to see the world from their perspective.
This is what I wrote last year about today's lesson:
Lesson 0.8 of Serra's Discovering Geometry is called "Perspective." This is the second of two sections appearing in the Second Edition but not in the modern editions. Serra begins:
"Many of the paintings created by European artists during the Middle Ages were commissioned by the Roman Catholic Church. The art was symbolic; that is, people and objects in the paintings were symbols representing religious ideas."
Unlike Lesson 0.5 on mandalas, which we choose to include on the blog even though it's "missing" from the modern editions, Lesson 0.8 can just be left out altogether. This is because we'll be starting the U of Chicago text next week, and that text already has a lesson on perspective (Lesson 1-5), so Day 15 would just be a repeat of Day 8.
Then again, we recall that in my class last year, the students in all grades had trouble drawing cubes even though those were on isometric paper rather than in true perspective). So we might wish to teach perspective on both Day 8 and Day 15. True perspective drawings should most likely be completed on plain unlined white paper, with a straightedge to draw lines toward the vanishing point. Lined notebook paper for one-point perspective drawing may also be acceptable -- but not for two-point perspective (the subject of today's worksheet).
The worksheet below comes from "marcandersonarts" and "Daisuke Motogi."
Here is the Blaugust prompt for today:
- Fav [math][ed] book read and take aways for this year (and beyond?)
Before I answer this question, let me announce that today would have been the 102nd birthday of Katherine Johnson -- the NASA mathematician who was the main subject of the movie Hidden Figures. I can't help but think back to our field trip to the movie theater during my year at the charter middle school.
[2020 update: Most of this is a reblog from last year's August 26th Blaugust post. But let's look at Katherine Johnson's achievements from another, um, perspective. When I mention race in the classroom, I'd much rather talk about Katherine Johnson than Jacob Blake. But again, Blake is the one who's on the minds of our black students now, not Johnson.]
Today's Blaugust prompt partly repeats an earlier one about my favorite blogs and books, except that this one focuses only on books. Of course, in honor of Katherine Johnson's birthday, I should begin with the actual book Hidden Figures.
I've actually only glanced through the book and never read it cover-to-cover. I do know that the film really only corresponds to the last few chapters of the book. Much of the book focuses on some of the other women besides Johnson -- and some of them don't even appear in the movie. The author, Margot Lee Shetterly, also describes their work for NACA, the predecessor agency to NASA. Who knows -- perhaps one of these days, I'll finally read Shetterly's book in full.
I decided to revisit old blog posts to find books that I've written about. My problem is that most of the books I read come from libraries -- including the books that I purchase from their book sales. Many of the math books that I described and blogged about are old textbooks -- and I doubt those are what Shelli had in mind when she came up with this Blaugust prompt.
During the first two years of this blog, I wrote about some books that I'd acquired not from my local library, but from my high school library back when I was still a young student attending there. One of these books was a textbook on coding in Pascal, which I no longer own. The other I still have to this day -- I'm glancing through it as I type. It's Introduction to the Theory of Sets by Joseph Brewer. This is technically a textbook (from 1958, during the "golden age" of textbooks when they were much smaller than they are today). I still remember being fascinated by Georg Cantor's theory that there are different types of infinite sets, "denumerable" and "non-denumerable."
(I mentioned these books together because there's a connection between them. In the coding text, the author was explaining algorithms. If someone is thinking of an integer, an algorithm to find the number is to guess 0, 1, -1, 2, -2, 3, -3, and so on -- eventually we'll guess the right integer. Then in Brewer's book, this exact same sequence is used to prove that the set of integers is denumerable.)
Of course, I can't post about books and not mention Glen Van Brummelen's Heavenly Mathematics. I did spend two summers (2017 and 2018) discussing this book on spherical trigonometry. But even that book is, at its heart, just another textbook.
So you might wonder, do I read any books that aren't textbooks? Once again, I do have to mention Ottaviani's series of graphic novels on famous scientists.
Once again, we notice that Shelli's prompt mentions "favorite [math][ed] book." Most of the books read by Blaugust participants are probably "ed" books. This includes Geoff Krall's Necessary Conditions (foreword by Fawn Nguyen). It's the type of book that I probably need to read more often if I wish to improve my teaching skills. I know -- by now you're likely tired of reading how I wish to read this book someday. My problem is that I don't often spend full price on books -- instead, I like to get my books cheaply at library book sales. It'll probably be years, if ever, before I ever find Krall's book at the local library.
Have I ever read any ed books? In the past, I did allude to Harry Wong's The First Days of School, along with another ed book by Fred Jones. Indeed, I blogged a little about Wong's book during the summer before I taught at the old charter school.
Did reading Wong's book help me at all during the year I taught there? Perhaps it helped a little -- but I might have focused more on what he wrote if I had, say, let his book be the side-along reading book that summer. Instead, I wrote more about spherical geometry (not from Van Brummelen, but from Legendre's old writings I found online). Of course, spherical geometry was completely irrelevant to this class I was about to teach.
Of course, even if I had read and blogged more about Wong's book, there's a difference between reading about Wong's principles and following them when I'm standing in front of students. For example, today I reread old posts and noticed what I wrote about my Warm-Ups. I wanted to make sure that I strictly enforced the rule that students must show their work on Warm-Ups, especially Pappas-style problems where the answer was the date (which should have been Exit Passes). But that failed miserably -- one day early in the year, most students wrote only the answer, and I didn't want to give lots of zeros to more than half of the class. And the rest is misery -- the "you must show work" rule was soon neutered.
Wong stresses the importance of enforcing the rules during "the first days of school." This time of year I even call the "Wong unit" (or "Willis unit" after Paul Willis, another author). But once again, it's easy to forget everything I read when I'm actually standing in front of students. Since I never made Wong's principles (enforcing rules and procedures early in the year) into habits, I promptly failed in the classroom. It's not enough to read or even blog about Wong -- I needed to figure out how to convert his principles into action.
Only one of our main Blaugust participants posted today -- Denise Gaskins. Fortunately, her post fits today's prompt, as she also writes about her favorite books -- including books that she's written:
https://denisegaskins.com/2020/08/26/its-almost-gone/
Five of her books, including Geometric Coloring Designs, are mentioned in this post. I assume that some of these designs fit with the Serra activities that we've discussed so far this week.
While Minnesota -- where George Floyd was shot -- is also the home of famous math teacher blogger Sara VanDerWerf, I don't know of many Wisconsin teacher bloggers who might discuss the Jacob Blake incident from a local, um, perspective. One blog I found is from Paige Sheehan:
https://mathmomentswithmrssheehan.blogspot.com/
She hasn't posted since July, so she isn't a Blaugust participant. Even though she hasn't blogged about the Jacob Blake incident yet, she did acknowledge the shooting in a tweet earlier this week.
Another interesting Twitter account I've seen recently is "Great Women of Mathematics." This site has been celebrating Katherine Johnson's birthday all month. But there are also threads there discussing problems on the Rebecca Rapoport calendar.
Yesterday, Dean Chung -- a co-author of the Rapoport calendar -- posted a solution to yesterday's fraction problem himself. Yes, the solution was that c can be any number equivalent to 2 mod 4.
And this solution generalizes. Chung proves that since 18 is the denominator given in the problem, any number that is singly even like 18 can be a winning value for c. If the given denominator were 20, then any doubly even value of c is a winner. If the given denominator were 24, then any triply even value of c is a winner. If the given denominator were 19, then any odd value of c wins. In short, both denominators must have the same number of factors of two in order to guarantee a winner.
I won't post today's Rapoport problem, but it is currently being discussed on Twitter as well. Today's date is being written using eleven 4's. (Four 4's is often used in math classes as an opening activity.)
You know what -- I'll post it anyway, because it does contain an error. Its parentheses aren't balanced:
[4 * (4 - 4/4) + 4/4) * (4/(4/4 + 4/4)]
At the point of the middle *, we opened two [( brackets but closed two )) brackets. Let's rewrite it so that matching brackets have matching shapes, and then we'll solve it:
[4 * (4 - 4/4) + 4/4] * [4/(4/4 + 4/4)]
= [4 * (4 - 1) + 1] * [4/(1 + 1)]
= [4 * 3 + 1] * [4/2]
= [12 + 1] * 2
= 13 * 2
= 26
So the answer is 26 -- and of course, today's date is the 26th. Some participants in the Twitter thread obtained 32 or 14 due to the parenthesis confusion
The four 4's (and similar) project is often used in math classes as an opening activity.
Earlier this week, I mentioned the late Wendy Menard and her final post. Even though she posted it back in 2018, this post -- on race and the "binary" -- has suddenly become more relevant this week:
https://hermathness.com/2018/09/03/the-binary/
Here the "binary" refers to the dichotomy between prejudiced and non-prejudiced -- but as Menard showed us in her post, this binary is an oversimplification.
I could simply add a "traditionalists" label to today's post and thus declare all this racial discussion to be on-topic, but I'm not really posting about traditionalists today. (The next traditionalists' post is scheduled for this Friday.) The racial discussion is on-topic because it might be on the mind of many of our students who are attending classes online or in-person.
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