But what, exactly, is a mandala? Serra explains:
"A mandala is a circular design arranged in layers radiating from the center. The word mandala comes from Hindu Sanskrit, the classical language of India, and means 'circle' or 'center.'"
As Serra points out, other cultures had mandalas, not just the Hindus. The Aztec calendar, for example, was constructed as a mandala.
Many mandalas exhibit threefold or sixfold symmetry. They are related to the regular hexagon, and so the compass and straightedge can be used to construct them. At any rate, the compass should at least be used to draw the circle that is the base of any mandala.
All the mandalas on these pages come from a Google image search. There is no project in this section, but of course "draw your own mandala" is a natural question for this section.
Here is the Blaugust prompt for today:
How do you handle Parent Communication? What has been successful for you?
That's one thing I don't write about much on this blog -- parent communication. During the one year I worked at a charter middle school, I wrote about parent communication during the first week of school:
9:55 -- The previous day we discussed the classroom rules. Today I pass out Behavior Contracts for all of the students. Each student writes down the rules we agreed to, and then the students take them home for the parents to sign. In this way, each student is to be made accountable for his or her own behavior throughout the year -- and if a parent requests a conference, I can just take out the contract and let the parents know that their child has violated that contract.
I also wrote once about my main form of home communication -- Parent Conferences Week. Let me cut and paste some of what I wrote that day:
7:50 -- Our first parent arrives -- yes, parents can choose to come before school if they desire. It's the mother of one of our sixth graders. The mother and her daughter arrive at the history room, where all three middle school teachers -- the history teacher, the English teacher, and me -- are sitting. We've agreed that it's easiest to do it this way, so parents can speak to all three of us together.
As it turns out, this girl is earning straight A's and is one of the quietest students our classes. So naturally, all of us have only positive things to tell her mother. We three teachers wish that all of our students were like this girl.
1:30 -- Our first afternoon conference begins. It's the father of one of our most troublesome seventh graders -- he just barely scraped through my class with a C, but he failed history and English, and even received a C in music -- a class in which almost everyone gets an A. I tell his father that he can do much better, but he hangs out with the wrong crowd. Just yesterday he or one of his friends pulled a classic prank -- putting a tack in my chair.
2:55 -- After a lull without any parents, the mother of a sixth grader arrives. The girl is getting B's in both my class and history, yet is failing English. The mother is a Spanish speaker, so our history teacher must translate for the English teacher, who explains that the girl has trouble writing. My colleague tells her mother about an time last month where I had the whole class write standards when they were too loud (during IXL time, of course), and the poor girl cried as she was unable to finish.
3:00 -- In the middle of the English teacher's exposition, the mother of an eighth grader arrives. I go over to talk to this mother as the English teacher is still talking to the sixth grader's mom. The eighth grade girl is earning C's in all three classes -- mine, history, and English. I tell the mother that even though her daughter passed some of her tests, she failed some others. Still, I let the mother know that the girl is very well-behaved and likes to help us out after school, especially the English teacher.
3:40 -- The parents of a sixth grader arrive -- but they are in a hurry due to a family emergency, so they just pick up their son's report card (I gave him a C) and leave.
3:55 -- No more parents show up. The history teacher counts out the remaining reports and figures that about two-thirds of the parents showed up this week.
But six months before the first day of school that year, I wrote the following statement on my blog:
I know that I can be a better teacher by showing common courtesy to students, parents, and staff members.
So this is the guiding principle when it comes to communication with anyone, including parents. I'm not quite sure whether I adhered to this principle fully -- was I completely courteous to parents?
There is one form of parent communication that I used that year -- yet I never wrote about it in any blog entry. At our school, I was required to file all completed assignments in folders -- the students had nothing to take home. Some kids wanted to show something to their parents, and so one day, I decided to type up notes to parents of students who earned A's on tests. This way, parents can receive positive information about their children, not just negative info. Unfortunately, I also had to make many phone calls to the parents of misbehaving students.
Today's Blaugust link is Jenna Laib, a Massachusetts math coach:
https://jennalaib.wordpress.com/
https://jennalaib.wordpress.com/2019/08/20/listening-to-early-understandings-of-division-12-%c3%b7-5-is-either-3-or-0/
Laib is a math coach, not a math teacher -- and she works mostly with elementary students. So what Laib describes here will not be relevant to any class I've taught or will teach. She writes:
I wrote this post while engaged in a residency in a third grade classroom. We had recently launched a unit on division.
I think of third grade as the year when division is first introduced -- most of the heavy lifting of division is taught in fourth grade. Anyway, she continues:
Enter: 12 ÷ 5.
I presented this problem to the students in 3F. There was some uncharacteristic silence in this typically chatty classroom. Then, Neil said:
“Oh. I get it. It’s impossible. You can’t do that.”
“I think it’s -3,” Yosef stated coolly.Perhaps emboldened by Yosef, Luuk decided to share, too. “I think it’s 0,” he shrugged, as if apologizing for taking up space with his body or ideas. Thankfully, other students heard him, and started to use American Sign Language for “same.” “Me, too!”
(Yeah, who'd have thought that I would blog an obelus two days in a row!) Actually, I find the answer -3 to be the most interesting, since I'm surprised that a third grader would know negative numbers.
Laib's post is intriguing, but I don't wish to dwell on a third grade division lesson. So instead, let me go back glance at a few Blaugust entries posted late last night.
First of all, Shelli -- the leader of the Blaugust challenge -- posted a Q-Bits puzzle and some forms in her weekly "Made for Math Monday" series:
http://statteacher.blogspot.com/2019/08/mtbosblaugust-puzzle-and-some-forms.html
At the left, you can see my Student Aide To-Do list, a "Why were you Tardy" note, and a "Please put away your cell phone" note. I realized after the fact that I should have made the cell phone note into a laminated paper so I could reuse them... Oh well. :)
One management problem that I had in my class at the old charter school is arguments -- I tell students that they are doing something wrong, and an argument ensues.
So we can easily see the wisdom of Shelli's tardy and cell phone notes. Students feel embarrassed when the teacher calls them out for wrongdoing -- and they respond to that embarrassment by arguing with the teacher. Most likely, Shelli hands students the tardy and phone notes without speaking. Thus students aren't embarrassed, and so they don't feel the need to argue.
In yesterday's post, I mentioned one of the famous Sara(h)s of math, Sarah Carter. Although she isn't really a Blaugust poster -- only once has she ever labeled a post "Blaugust" -- many participants of that challenge consider her to be an inspiration. So let's look at what she blogged late last night:
https://mathequalslove.blogspot.com/
https://mathequalslove.blogspot.com/2019/08/teaching-train-game.html
I guess I'll start by blogging about an activity I *should* have blogged about last year. For the past two years, the Train Game has been my go-to 2nd day of school activity. I've mentioned it on my blog before, but I've never written an entire blog post explaining it fully.
And this explains why Carter's posts are so popular during Blaugust -- she regularly posts "first day of school" and "second day of school" activities.
Carter spends much of this post explaining what constitutes a "train," since the object of the game is to make as long a "train" as possible. (In short, a "train" is a nondecreasing number sequence.) But I wish to focus on the following rules:
Let's see if I can summarize the rules of game play. 20 numbers will be drawn. After each number is drawn, the number must be written in one of the train cars on the game board. Numbers cannot be moved or erased once they are written down. Each number must be written down before the next number is called.
There's no way I could have played this game in my old classroom, since Carter didn't post this game until last night. (Technically, the Train Game is based on a real board game that she received in 2014, so it's possible I could have acquired and played the game in class back then. But in reality, most teachers have never heard of a game until a star teacher like Carter blogs about it.)
But here's the thing -- there's a similar game that I definitely would've known of back then -- "But Who's Counting?" which comes from Square One TV. In fact, I still remember how the host reminds the players, "Once you place a number on the board, you can't move it!" And we see that Carter's Train Game has essentially the same rule -- "Numbers cannot be moved or erased once they are written down."
In "But Who's Counting," the object isn't to make long "trains." Instead, the goal differs by episode -- something it's to make the smallest number, or largest number, or smallest sum, and so on.
Here's a sample playing. The goal is to make the smallest number. The wheel contains digits 0-9, and the host spins the wheel five times:
__ __ __ __ __ (first spin: 3)
__ 3 __ __ __ (second spin: 7)
__ 3 __ 7 __ (third spin: 9)
__ 3 __ 7 9 (fourth spin: 7)
__ 3 7 7 9 (fifth spin: 0)
0 3 7 7 9
In this case we obtained the smallest possible number, 03779. Notice that the smallest number is also a "train," and indeed, strategies for making trains and the smallest number are similar.
Some strategies immediately jump out -- if a 0 is spun, place it as far to the left as possible, and if a 9 is spun, place it as far to the right as possible. Here's another playing:
__ __ __ __ __ (first spin: 6)
__ __ 6 __ __ (second spin: 5)
__ 5 6 __ __ (third spin: 4)
4 5 6 __ __ (fourth spin: 3)
4 5 6 3 __ (fifth spin: 9)
4 5 6 3 9
In this case we get 45639, but if our opponent placed the 6 in the fourth position, then we lose, since the opponent likely reaches 34569.
This game might have worked in my classes at the old school. What's bad is that it never even crossed my mind to play Square One TV games in my class -- despite my singing songs from that show all the time.
END
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