"This wonderful magic square was created by Leonhard Euler in the 18th century. As in most magic squares its rows, columns, and diagonals total the same number, in this case 260."
This page is titled "Leonhard Euler & the Knight's Tour." It refers to yet another magic square, this time of order 8. As usual, here's the link:
https://www.johndcook.com/blog/2011/04/06/a-knights-magic-square/
The term "Knight's Tour" refers to a chess knight. Pappas says a knight "can land on every number of the entire square sequentially from 1 to 64, by just following the moves allowed of the knight."
If you read the comments at the above link, notice that the attribution to Euler is controversial. It's possible that the creator of this square is William Beverly, 65 years after Euler's death -- and in fact, the charge is that Euler never came up with a knight's tour at all, much less a magic square. Finally, it's pointed out that not only don't the diagonals add up to 260, it's provable that no knight's tour can be a magic square that includes the diagonals.
Well, we're done with magic squares in Pappas, but not with Euler. For the rest of this week she writes about something else attributed to Euler -- and hopefully this attribution is correct. One commenter, John V., says:
Regarding the attribution, people love appeals to authority and it is always easier to remember a famous name than one less famous. I think that is one reason for so many incorrect attributions.
Another related reason is hero-worship. People who admire a famous person are quite inclined to believe anything positive about them. I suspect many of the anecdotes about folks like Feyman, Erdos, [....]
Erdos -- well, let's not tell Hoffman that many of the Erdos stories in his book are false. Oh, and speaking of Erdos:The final chapter -- more of an epilogue -- of Paul Hoffman's The Man Who Only Loved Numbers is called "We Mathematicians Are All a Bit Crazy." This chapter isn't numbered eight, but rather a sideways eight -- Chapter Infinity. Hoffman doesn't begin with a quote, but instead tells us a little something about himself:
"I am not a mathematician. I have never proved a theorem, let alone a surprising conjecture. My Erdos number is infinity."
Well, that explains why the chapter is numbered infinity. The title, meanwhile, comes from a quote Erdos heard from fellow mathematician Edmund Landau. As Hoffman says, many mathematicians indeed suffered from mental illness, most notably John Nash, the main character of A Beautiful Mind.
Hoffman ends the book by telling us how he meets Erdos in 1986, as he prepares to write this biography about him. The author had the famous mathematician critique his book, and Erdos told him to leave out the part about Benzedrine, because "I don't want kids who are thinking about going into mathematics to think that they have to take drugs to success." And here's Hoffman's response to close out this most enjoyable book:
"That was Erdos, always thinking about the epsilons."
Well, I interacted with the epsilons today. You see, today I subbed in a high school history class. As I've done before, I want to write about this day of subbing in "Day in the Life" format:
6:55 -- Believe it or not, this teacher has a zero period class. This is the first of three sophomore World History classes. I expect there to be many tardies in this class, and there are (about ten).
The students are learning about the Industrial Revolution and answering questions on three pages in the text. In the middle of this class, the dean enters and gives a lecture on classroom and school behavior rules.
Five students fail to answer even the first question, which asks for seven definitions from the text. I inform the regular teacher in my sub notes for him, but unfortunately there is no seating chart or any way to identify the student names. It's also difficult because many students say that they are skipping around instead of doing the definitions first -- and I know that some of the students making this claim are really trying to pass off yesterday's work as today's. Some students also give the excuse that the dean's ten-minute lecture prevents them from doing work the rest of the hour. I do have the foresight to tell a tardy student who arrives 20 minutes before class ends that 20 minutes is nonetheless enough time for him to do the definitions. Of course, he still doesn't do the work.
8:00 -- First period arrives. This is the only junior U.S. History class of the day. Two of the students are wearing Dodger shirts, and so I ask them whether they are looking forward to watching Game 1 of the World Series tonight. Notice that this is the first time a Southern California team has qualified for the finals in one of the four major sports since I created this blog -- the Kings having won the Stanley Cup a few months before I began the blog. (Yes, some Northern California teams have played in the Finals, most notably the Golden State Warriors, but this is Southern California.) It's the first class of the day for many of these students, and so there continue to be many tardies in this class too (about five).
In this class, students are researching the Ku Klux Klan and completing a written group report. They are allowed to use Chromebooks for this assignment. This time, I catch two students not doing the work, with one of them having a phone out. Once again, I don't have the names, but I've decided to write in the sub notes where they are sitting as well as a description of their backpacks. You see, I figure that the students will wear different clothes but the same backpack tomorrow, so describing the backpack gives the teacher a hint who the student is.
8:55 -- The second World History class arrives. This time, I'm more prepared for the students as I let them know up front that they are to complete at least the first question with all seven definitions. One student fails to finish the seven definitions -- but this time I manage to catch his name. A few students start working on their Chemistry assignment. I ask them whether they know what holiday it was yesterday. One girl knows that it was Mole Day. (And since these are tenth graders in Chemistry, they are "sopho-moles," according to yesterday's joke page.)
9:50 -- The students leave for nutrition.
10:10 -- The third and final World History class arrives. Two students fail to finish their definitions, and I must describe their backpacks again for the teacher.
11:05 -- Fourth period is the teacher's conference period. As it happens, I must cover another class during this time -- a special ed English class. Most of the students are sophomores, but there are a few freshmen and juniors in the class. Many of these students have just arrived in the country and speak mostly Spanish or Arabic, with very little English.
The students are supposed to be working on a speech about animals, but the regular teacher knows that they would struggle on this without her. So it becomes a study hall instead. I see that some of the students are working on math (Algebra I), and so I try to help two of them (a brother and sister, I believe). The lesson is on linear functions and mathematical modeling. Since this is the only mathematical part of my day, let me at least describe this in a little more detail.
CCSS.MATH.CONTENT.HSF.BF.A.1.A
Determine an explicit expression, a recursive process, or steps for calculation from a context.
Determine an explicit expression, a recursive process, or steps for calculation from a context.
The example the students are given shows a pattern made out of tiles:
Figure 0: 3 tiles
Figure 1: 7 tiles
Figure 2: 11 tiles
Figure 3: 15 tiles
Figure 4: 19 tiles
I decide just to tell the students that the function is f(x) = 4x + 3, but I do show them why this function is correct by plugging in 0, 1, 2, and so on for x and pointing out that this gives the correct number of tiles. Students are then asked how many tiles Figure 20 will require. Of course, the answer is f(20) = 4(20) + 3 = 80 + 3 = 83 tiles. They are also asked to find which figure has 203 tiles. I tell them that they must set up an equation, 4x + 3 = 203, which they can solve easily -- 4x + 3 = 203, 4x = 200, x = 50, so it's Figure 50.
I realize that the language is the biggest barrier for these siblings, not the math. But as I've written on the blog before, I had many English learners in one of my student teaching classes, and I was able to help them. I only know a few words in Spanish (mostly numbers and a few other key math terms), but fortunately they already know the basics (in both English and math), and so I think they can understand most of what I say.
As you may remember from last year, I like to hand out pencils around the holidays. Since there are only 13 students plus a TA, I give everyone a Halloween pencil.
12:05 -- In this district, students have a sort of "Interventions" or tutorial class. At all of the other schools in this district the time is embedded into the two-hour block schedule. But the school I subbed at today clearly does not have a block schedule. So instead, Interventions is considered part of lunchtime. All freshmen must attend Interventions, but older students only go if they need extra help or are failing a class. For some reason, Interventions is labeled as an "eighth period" class in attendance, even though there is no seventh period.
Since there are no freshmen in the history classes, only nine sophomores show up today for tutorial, plus one junior, a peer assistant. I see that she's wearing a Dodger "Fly the Pennant" blouse, and so I ask her about tonight's game. She tells that -- believe it or not -- her mom is taking her to the game tonight! I ask her about the cost of the tickets, and she informs me that they are $600 each -- but it would have been $1000 were the Yankees playing instead of the Astros.
Two sophomores from zero period complete their seven definitions during this time.
12:30 -- Lunchtime proper begins.
1:05 -- During fifth period, this teacher coaches the JV football team. I enjoy working out and actually lifting weights with the team. Of course, I can't bench press nearly as much as what these guys can lift!
1:55 -- Fifth period ends. Most students and teachers with a zero period don't need to attend sixth period, and so this ends my day. I go home to type up this blog entry.
Teachers are always working on improving, and often have specific goals for things to work on throughout a year. What have you been doing to work toward your goal? How do you feel you are doing?
I've been thinking about using backpacks to identify students ever since I left my old school and returned to subbing. I believe that I developed some bad classroom management habits back when I was a sub. The students would misbehave, I wouldn't know the kid's name, and then I end up not punishing the student at all. (Of course I don't even bother to ask for the student's name -- why in the world would students who know they're in trouble reveal their names?) The problem would be that I then developed the habit of not punishing students effectively, which persisted even after I became a regular teacher (and thus knew the students' names). Identifying students by backpack allows me to focus on effective responses to misbehavior, rather than attempting to figure out their names.
That being said, I still need to work on emphatically calling out a misbehaving student's name in cases when I do know the name. Only once today do I catch a student's name -- but that one time, I still fail to call out the name. It's as if I'm afraid to call it out, thinking that the student would then begin to argue -- even though I'm working on a teacher look to avoid such arguments.
Oh, and I do use my teacher look effectively during one of the classes. A group of girls start laughing during attendance, and I use teacher look to quiet them down.
It's time to review for the Chapter 4 Test. Two years ago, I didn't write much about the review but went directly to the worksheet, so here it is. (Notice that in the past, I referred to it as a Chapter 3 and 4 test and included questions from both chapters.)
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