Today I subbed in a seventh grade math class -- my first math subbing of the year. And so I definitely will do "A Day in the Life" today.
8:30 -- Let me begin by describing the distance learning schedule at middle schools, which is slightly different from the one at high schools. Instead of odd periods on Tuesdays/Thursdays, it's periods 1-3. Oh, and there's no period rotation or anything like that -- it's just 1-3 on Tuesdays/Thursdays and 4-6 on Wednesdays and Fridays.
First period is the only honors class of the day. But in this case, there's not much difference between honors and regular because both classes are taking a test on adding/subtracting integers. The test is given through the "Go Formative" website. I don't have access to this website, nor do I need it -- all that matters is that the students can see it.
I was wondering whether schools will attempt to give tests during distance learning, especially since hybrid is on its way soon (so any tests could be delayed to that day). Actually, several math teachers are out today, and most of them give a test today (since the assumption is that there won't be that many subs available who are good at math).
On one hand, this makes my decision to post a Chapter 1 Test here on the blog not look that bad. On the other hand, I'm still worried about test security. Some of the students have their cameras turned on but some of them don't. As far as I know, the students with their cameras off are asking their parents or someone else to cheat. (I hope that these honors kids would have no need to cheat.)
9:35 -- First period ends and second period begins. This is the first of two regular Math 7 classes.
While these students are taking the test, I check out the attendance rosters to check out which students are in Cohort A and B for the upcoming hybrid schedule. Unlike the high school students, the middle school kids are not divided alphabetically into cohorts. And of course, since these students are younger, many more of them are opting in to hybrid.
Speaking of hybrid, I read that somewhere in this country, there's a district that will indeed have special ed students attend four days a week -- this group is labeled Cohort C. (In that district, those who opt out of hybrid are considered Cohort D -- think D for distance.) Otherwise, that district will be implementing a concurrent hybrid plan just like mine.
10:40 -- Second period ends for nutrition/snack break.
11:00 -- Third period begins. This is the second of two regular math classes -- and unfortunately, this is when the trouble starts.
After having no problems with Zoom on Monday and Tuesday, today ends up more like last Friday. In fact, let me describe in full detail what Zoom was like for me both last Friday and today:
Indeed, whenever I try to log in, one of two things always happens:
Case 1: It works!
Case 2: I get stuck in the following loop:
Step 1: "Waiting for the host to start this meeting. This is a recurring meeting. If you are the host, sign in to start this meeting."
Step 2: I try to log in. I click "Sign in with Google" in order to enter my log in using the proper account.
Step 3: The computer notes that I'm already logged in -- and then it immediately returns to Step 1.
In Case 1, no message appears -- Zoom launches and there's no problem. But once I see the "Waiting for host..." message, I'm stuck in Case 2 and there's no way to get out. I don't know why sometimes I'm in Case 1 and others in Case 2.
And unfortunately, when I asked others both last Friday and today, and no one seems to know exactly why this keeps happening. The text of the error message makes it seem as if it's the fault of the regular teacher who forgets to let those who aren't the host (including me, since I must log in first before I can make myself host -- that is, making myself host is Step 4 but I can't get past Step 3) log in.
But that doesn't explain what happens to me today. After all, it's doubtful that both regular teachers would make that same mistake. Also, it's doubtful that today's regular teacher would let it work during first and second periods and then suddenly decide to disable it between 9:35 and 11:00. (After 11:00, the same second period link that worked at 9:35 no longer works either.)
There are only two possibilities left. One is that I'm doing something slightly, subtly different between my successful (Case 1) and unsuccessful (Case 2) logins, but I can't place a finger on exactly what I did Monday, Tuesday, and early this morning to make it work. The other is that there really is something wrong with Zoom, at least on our district's end.
When I first arrive on campus, a tech guy helps me log into all the websites. One thing I recall is that when I successfully sign in through Google (Step 2), it asks for my district email and password. But during the unsuccessful attempts, it doesn't ask (presumably because it's already logged in) -- it just goes back to Step 1. Even when I restart the computer later on, it doesn't ask for a username/password.
What ends up happening is during third period is that students attempt to log in, are unable to, and then complain on Google Classroom. And so I take the names I see in Classroom and use these for attendance, just as I did last Friday. The Go Formative link for the test is right there, and so I direct students to start the test right then and there.
Also, after Zoom fails and I return to Step 1, it allows me to start my own session. And so I do this and place a link to the temporary session on Google. I end up capturing a few more students who don't see the earlier messages. Still, there are many more absences in this period than the others, suggesting that some students still miss all messages (and possibly skip the test).
12:00 -- Third period ends for lunch. During lunch, I find out that one girl who was unable to log in during third period has called the school to find out why.
12:45 -- It's time for academic support -- organized differently here as opposed to high school. Rather than by period, students attend academic support by subject. Math is scheduled for Wednesdays and Fridays, and so there is no academic support for me today. Then again, it's not as if Zoom is working well enough for me to do the academic support anyway.
The song for today is "Same Sign, Add and Keep" since it's an integers test. I sing the song just before they start the test. Since there's lots of time left after the test -- after all, there are 25 questions and it doesn't take more than one minute to add or subtract two integers -- I add another song. This time, I choose "The Mousetrap Car Song" -- I associate this song, like "Meet Me in Pomona," with September.
Actually, "Mousetrap Car" reminds me of the seventh grade class at the old charter school. We already know that of all seventh grade standards, the ones on integer arithmetic are likely the most important, since fluency with integers correlates with math success in eighth grade in high school. At this school, this important topic is taught right at the start of the year, in September. But when did I teach integers?
That's right -- I hardly taught integers at all before I left. The reason for this major omission is how I interpreted the Illinois State text. I divided the year into two-week units -- one unit for each project in the STEM text. Each STEM project is associated with one or more of the Common Core Standards -- and that was the standard that I'd teach during that unit.
The first four projects -- including all three "Mousetrap Car" projects -- weren't associated with any grade-level standards at all. So I just went in the naive order that they're mentioned in Common Core, starting with RP, Ratios and Proportion. (This makes sense as mousetrap cars have a speed, and speed is a ratio.) But no project early in the STEM text correspond to integer operations (NS, Number System).
As we know by now (and as I mentioned in my August 31st post), if I'd followed the curriculum as intended by Illinois State, I was correct to follow the naive order of the standards, but I should have devoted only one week to each standard. If I started with the RP standards, then I would have reached NS and integer addition/subtraction in October. I should have ignored the STEM text and its order.
But even though I never figured this out and insisted on following the STEM text, I could have assigned NS standards with the first four projects. Integers are the meat and potatoes of seventh grade math. (I likewise could have started the sixth graders with NS, which in that grade focuses on the two big D's -- division and decimals -- during the mousetrap car projects.)
Lesson 2-2 of the U of Chicago text is called "If-then Statements." This is what I wrote two years ago about today's lesson:
Lesson 2-2 of the U of Chicago text continues the study of logic by focusing on "if-then" statements. I certainly agree with the text when it writes:
"The small word 'if' is among the most important words in the language of logic and reasoning."
There are a few changes that I will make to the text. First of all, the text refers to the two parts of a conditional statement as the antecedent and the consequent -- although it does mention hypothesis and conclusion as acceptable alternatives. I'm going to follow what the majority of texts do and just use the words hypothesis and conclusion. Actually, Dr. Franklin Mason doesn't even use the word hypothesis -- he simply uses the word given -- since after all, the hypothesis of a theorem corresponds to the "given" statement in a two-column proof.
When I teach or tutor students in geometry, one of my favorite examples is "if a pencil is in my right hand, then it is yellow." So I pick up three yellow pencils, and we observe that the conditional is true. But let's suppose that I pick up a blue pencil in addition to the three yellow pencils. Now the conditional is false, since we can find a counterexample -- the blue pencil, since that's a pencil in my right hand yet isn't yellow.
Notice that I decided to replace the word instance with the word example -- so that the connection between examples and counterexamples becomes evident.
The text has to go back to an example from that dreaded algebra again. Of course, it's an important example, since students often forget that 9 has two square roots, 3 and -3. But I decided to include it anyway since it's simple -- it's not as if I'm making students use the quadratic formula or anything like that.
Then the book moves on to a famous mathematical statement: Goldbach's conjecture, named after the German mathematician Christian Goldbach who lived 300 years ago:
If n is an even number greater than 2, then there are always two primes whose sum is n.
At the time the book was written, the conjecture had been verified up to 100 million, but the conjecture had yet to be proved. But what about now -- has anyone proved Goldbach's conjecture yet? As it turns out, the answer is still no -- but now the conjecture has been verified up to four quintillion -- that is, the number 4 followed by 18 zeros.
But there has been work on a similar statement, called Goldbach's weak conjecture:
If n is an odd number greater than 5, then there are always three primes whose sum is n.
This is called weak because if the better-known (or strong) conjecture is true, the weak is automatically true because we can always let the third prime just be 3. Ironically, when Goldbach himself actually stated his conjecture, he stated the weak version of the conjecture. It was a letter from Euler -- you know, the same Euler who solved the bridge problem that we discussed as an Opening Activity -- that convinced Goldbach to state the strong conjecture instead.
Now as it turns out, someone has claimed a proof of Goldbach's weak conjecture -- namely the Peruvian mathematician Harald Helfgott. Recently, Helfgott's proof was still being peer-reviewed -- that is, checked by other mathematicians to find out whether the proof is correct. By now, Helfgott's proof has finally been verified. Yes, mathematicians are still proving new theorems everyday.
Dr. M also mentions Goldbach's conjecture, on a worksheet for his Lesson 2-1. Often students are fascinated when they hear about conjectures that take centuries to prove, such as Goldbach's conjecture or Fermat's Last Theorem. I often use these examples to motivate students to be persistent when trying to come up with proofs in geometry -- if mathematicians Helfgott and Wiles didn't give up even after centuries of trying to prove these conjectures, then why should they give up after minutes? [2020 update: This year, this includes logging out of Zoom as soon as the lesson gets tough.]
If you thought that there would be no more "spilled milk" in this post, guess again. That's because it's become a tradition to write about the LA County Fair on Day 22. Back at the old charter school, the field trip was usually on Day 22.
Of course, that school is now closed, so I can't say "Today is the day I would have gone to the fair had I not left the school" anymore. No one from my old school is at the fair today, because my old school no longer exists [2020 update: and there's no fair due to the coronavirus]. I will still cut-and-paste from my old LA County Fair posts today. Yesterday, I've been singing the LA County Fair song in class.
Anyway, this is what I wrote four years ago about the field trip:
1) Teachers make a lot of decisions throughout the day. Sometimes we make so many it feels overwhelming. When you think about today, what is a decision/teacher move you made that you are proud of? What is one you are worried wasn’t ideal?
I think that the best decision I made during the first 22 days of school was to include a music break as part of my daily lesson. As I wrote in my First Day of School (August 16th) and August monthly posts, I try to sing a math related song three times a week. This motivates the students to want to sing along -- and by learning the words, they are learning math without realizing it. One of my most popular songs is the one I mentioned in my August monthly post, Count on It. Music break is ten minutes out of an 80-minute block -- but as an incentive, I extend the break to 15 minutes if the students are singing along.
As for the worst decision I made -- well, the field trip to the LA County Fair was two days ago, and so it's still fresh on my mind. There were a number of poor decisions I made on that trip. I know that this isn't supposed to be a Day in the Life post, but here is a brief overview of my field trip:
10:00 -- We arrived at the fair. All groups -- including mine of half a dozen sixth graders, five boys, one girl -- walked through the Jurassic Planet exhibit. My students were hungry and wanted to eat their lunch, but I tell them that all groups would eat near Mojo's Wild and Crazy Island.
12:00 -- The students eventually spent all of their money on the Extreme Thrills tickets. Since all of the other rides were now open, we walked towards the Carnival section -- only to find out that all of the rides require purchasing tickets. The kids kept walking hoping to find a free ride, but we didn't.
2:00 -- As we get ready to board the bus to leave, I met my Support Staff aide, who had a small group of sixth graders of her own. She told me that her group had taken a tram to the farm area, rode a few extreme rides, and still had money left over for the carnival rides!
At that point, one of my group members proceeded to blame me for giving them such a miserable day at the fair -- even though I wasn't the one who wouldn't let them ride. (That would be the carnies who told them that they needed tickets to ride.) On the other hand, he had a point, as there actually were a few things that I could have done to improve my group's experience at the fair.
Until I arrived, I didn't even know that there was a tram. That was something I should have looked into ahead of time -- when I was doing research for my song "Meet Me in Pomona, Mona." Finally, I should have found out that all of the rides require tickets -- perhaps if I'd told my students this, they would have saved money for the Carnival section.
In addition to today's worksheet, I'm restoring my old pattern of posting a weekly activity. In a way, nearly all of Chapters 0 and 1 are activities, so I resume this tradition here in Chapter 2. Our first activity from last year is a list of logic puzzles, to go along with the logic that we learn here in this chapter.
[2020 update: As I switch to the new district calendar, the activities that I posted last year no longer land on Fridays. I've decided to keep last year's activities on whatever day they happen to land on this year until we reach the start of hybrid. Then I will change the activities in order to fit a concurrent hybrid schedule.]
Here's a little of what I wrote last year about the logic puzzles:
As it turns out, I've seen a version of this puzzle before last spring. It is a similar brain teaser known as the "Sum and Product Puzzle." The next link contains a statement and solution of the puzzle:
http://www.qbyte.org/puzzles/p003s.html
Notice that in describing the solution, the author actually uses Goldbach's conjecture -- the unproved conjecture that I mentioned earlier in this post. Of course, the numbers involved in this problem are much too small to be counterexamples to Goldbach.
I'll repeat the same activity worksheet from last year, although it might be interesting to replace the old Puzzle #10 with Cheryl's birthday problem. The sum and product version of this puzzle might be suitable in an algebra class, especially near the lesson on factoring quadratic polynomials.
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