"We have heard of body language created by the various messages we give off by the way we move or posture our bodies. But what's body music?"
This is a one-page section in Pappas, simply titled "Body Music." The subject of this page is DNA -- apparently, it's possible to make music out of its strands. Pappas explains:
"A musical link becomes apparent, when one considers the recurring sequences [of bases] as recurring melodies of a song. In fact, many of them have been put to music, both in an octave and in other intervals."
Here it appears that just as the digits of pi and the Fibonacci numbers mod seven are used to make songs (as I mentioned back in my Pi Approximation Day post), it's possible to take the bases of DNA to create body music.
I can only imagine how this works. Recall that the four bases are adenine (A), cytosine (C), guanine (G), and thymine (T). We notice that A, C, and G are already musical notes, so all we have to do is change T to a valid note and we'll have a song. An interesting choice is F, because then F-G-A-C are the four playable notes on the Fischinger Google Doodle from two months ago (if we play the scale in F major rather than C major).
Moreover, DNA bases appear in pairs, with A paired with T, and C paired with G. If we play two-part music to represent the two strands, then all the intervals are consonant, with F-A as a major third (and its inversion the minor sixth), and C-G as a perfect fifth (and its inversion the perfect fourth). If we still had the Fischinger player available, we might choose random numbers from 1 to 4 (F-G-A-C), create a "base bass line" using the complementary bases, and play some "body music."
So far in these Pappas science posts, I've been writing about how I should have taught the subject in my own class. Well, I already said that I should have taught DNA to my seventh graders. Making "body music" probably isn't a suitable project in a school, as students would suddenly have to learn more music than science. The only way I'd do it is if every student had a Fischinger player to make the notes, so all the students have to do is pair the bases correctly to create the song.
But even though I wouldn't play body music in my class, I did play music in my class. Just yesterday I wrote about two of my songs, "The Dren Song" and "Earth, Moon, and Sun." I regret many things about my first year of teaching, but the idea of having a music break is not one of them. Yet often the students didn't know the purpose of music break -- and sometimes neither did I.
Recall that according to Fawn Nguyen, it helps to have a vision of the classroom you want. So what exactly was my "vision" for music break? To me, music break served two purposes:
-- To break up the monotony of 80-minute blocks
-- To give the students something to help them remember difficult topics
For example, I wondered whether I should let the students simply ignore my song and talk all the way throughout the break. According to the first purpose, this is okay -- why not let the students talk if I'm giving them a break?
But according to the second purpose, this is no good. When I first came up with music break, I thought about the famous "Pop Goes the Weasel" parody for the Quadratic Formula. If students sing the song, it might enter their heads when it's time for the Quadratic Formula test. But if they're not paying attention, then the song would be of no use come test time.
Too often, my students talked right through music break. But then again, they had already neutered the "no talking" rule -- if they were talking throughout the lesson, why would they suddenly stop when it's break time? It would only be for a particularly funny line in the song -- and then they would stop to laugh at my singing, not to remember anything mathematical.
One of my most successful songs for jogging the memory was "Measures of Center" -- a "Row, Row, Row Your Boat" parody about the mean, median, and mode. One of my sixth graders who transferred from another school told me how her former teacher had sung her this song. Later on, the coding teacher asked the seventh graders about mean, median, and mode -- and they couldn't remember what they were until I started singing this song.
But there was a blown opportunity for me to sing a song to jog student memories. During SBAC Prep time, I set up a practice test on the computers, and the first few questions were on exponents. Many of the eighth graders didn't remember the laws of exponents -- or possibly, they never learned them because they were talking throughout the lesson. I'd written a song about these laws -- a parody of the UCLA fight song -- and I could have sung it just before they began their test, but I didn't. The students were then became frustrated when they couldn't answer a single question on the practice test.
Why didn't I sing the song then? Well, when I first covered exponents in October, I'd posted the lyrics on the wall, but then I took them down and packed them away in the closet when it was time for a new lesson and new song. Indeed, I'd forgotten that I even had an exponent song -- recall that this wasn't really a song about exponents, but a tack-on verse of the song "Unit Rate," which was geared towards my sixth and seventh graders. It's easy to remember "U-N-I-T Rate! Rate! Rate!" but more difficult to recall the second verse on exponents.
Since I left my school, I decided to collect all my songs into one notebook. Then I could whip out the notebook and sing the songs whenever I needed them -- especially in preparation for cumulative exams such as the SBAC.
Some of my songs were more popular than others. Parodies were often more popular, since it's much easier for me to change the lyrics than invent a tune from scratch. Of course, Square One TV songs were even more enjoyable, since I didn't have to create the tune or the lyrics.
One song that was surprisingly popular, especially with the seventh graders, was "Fraction Fever." I point out that, just like a parody, I didn't have to create a tune -- the music ostensibly came from the old computer game from 30 years ago. But in practice, I actually did ad-lib a few notes, since I couldn't remember all the notes from so long in the past.
Many seventh graders enjoyed singing "Fraction Fever." The problem was that the lyrics of the song were all about playing Fraction Fever. They weren't about anything that could help them on a test, such as adding, subtracting, multiplying, or dividing fractions.
I should have taken advantage of the song's popularity and adding verses about fraction arithmetic. I could even keep some of the lines about game rules and combined them with arithmetic lines -- perhaps rhyming "elevator" (to get to the next level in the game) with (common) "denominator."
Indeed, I could even have sung this song during Learning Centers. One of the DIDAX manipulatives involved fractions, and so whenever we used them I could sing the song. Students who tend to groan whenever they see fractions might now look forward to seeing them, knowing that I'd be about to sing "Fraction Fever." Unfortunately, the class that liked the song the most -- seventh grade -- was also the class I saw the least, and so I never tried Learning Centers with them.
But in general, during music break the students should either be silent or sing along with me. Music break really is a break from writing, but it's still class time, so they should be listening to me.
Pappas concludes:
"Therefore, it is not unusual for scientists and mathematicians to seek out the music of the body."
...and the same is true for science and math teachers.
I didn't want this to turn into another classroom management post -- I was planning to finish with management during the summer posts. But the success of my "music break" ultimately depended on classroom management to keep the kids quiet and listening to my songs. (Again, I can never have too much management after what happened last year.)
I've said that management is a rarely blogged topic. But another MTBoS challenge has started -- not Blaugust, but something called "Sunday Funday." Julie Reulbach, a North Carolina high school teacher, is the leader of this challenge. Each Sunday, she posts a topic, and participants write about the challenge during the week.
Last week's Sunday Funday topic was classroom management. I assume that she was strongly influenced by that Fawn Nguyen post when she chose this topic -- indeed, Nguyen is listed as the first post, and several other participants quote Nguyen in their own entries.
https://ispeakmath.org/2017/08/20/classroom-management-help/
Sunday Funday is another challenge in which I'm not worthy to participate until I hopefully make my return to the classroom. That doesn't stop me from linking to other participants, though. Here are a few key entries that I found in Reulbach's list:
https://euclidsmusings.blogspot.com/2017/08/classroom-management.html
Peggy Bondurant (state not listed) is one of the newest MTBoS bloggers. After acknowledging Nguyen as the originator of this prompt, she writes:
I enlist the help of a time-watcher or two to help make sure there is time for clean-up at the end of class. I teach how to put away our laptops - log-off all websites, shut down (not sign-off), place in #'ed slot and plug-in power chord. Anytime there is an algorithm for a daily or weekly task, I teach it. My HS Ss do well with these, though the laptops and restroom procedures get re-taught often!
Of course, Bondurant's post reminds me of my own problems with IXL and laptop procedures. We see that her slots were "#'ed" (i.e., numbered). That's the main reason I had problems -- I should have numbered the slots before the first day of school. On the other hand, my class didn't have enough power chords, so that wouldn't have been part of my procedure.
http://busybeebe.blogspot.com/2017/08/classroom-management-sundayfunday.html
Brianne Beebe, a New York high school teacher, is already familiar as one of the participants of "Day in the Life." (Her daily posting day was the seventh.) She writes:
I rarely contact parents about student misbehavior. Most student misbehavior that I encounter can be handled in the classroom. The worst thing my students do most of the time is talk at the wrong time. (I hope I'm not jinxing myself for this year. *Knock on wood) My typical discipline hierarchy is as follows: first a warning, second a small consequence (taking phone for the period, moving seat, etc), third a referral with phone call home. I think I've only made it to the third step once in five years. Typically, when I call parents about student misbehavior it's because the student's behavior has been repeated for a few days and the consequences in the classroom have not deterred it.
Beebe is what I call the "ideal classroom manager." The ideal manager reaches the higher levels of the hierarchy very rarely -- she says that she only needed one referral in five years. She's also a "natural classroom manager," particularly if those are her first five years of teaching.
I, meanwhile, had to give many referrals and phone calls in just five months of teaching. And it's easy to see why -- look at that second step, where Beebe confiscates phones and changes seats. In my class, my students would either refuse to surrender their phones, or as we've seen, even claim that it's just a phone case and not the actual phone. And they'll refuse to change seats when I tell them to. So I'd have no choice but to make the referral or phone call with the reason "refusal to surrender phone or change seats after being asked to do so more than four times."
Since Beebe is a natural, I assume that she has a strong teacher tone. When she speaks in that teacher tone, the students surrender their phones and change seats when asked. Since I lack teacher tone, the students refused to do what I ask. Therefore I must reach higher levels on the hierarchy than a teacher with a strong tone like Beebe.
https://hazeleyedmathnut.blogspot.com/2017/08/classroom-management-be-pro-active-and.html
Tara Daas, a Georgia high school teacher, is also familiar from "Day in the Life." (Her daily posting day was the 26th.) She writes:
Though I have not really had a lot of issues with classroom management in the years past the start of my career, I feel that many methods I tried and used lacked so much purpose and meaning. In reflection, I feel most of it was driven by re-action to past problems rather than pro-active measures. Though I had become pro-active with parent communication, the pro-active switch on everything else was stalled. The pro-active parent communication helped behind the scenes of my classroom early in my teaching career, and it kept amount of communication less frequent and more positive. When I began to implement more pro-active and positive measures in the main stage of my classroom with students in later years, I never really gave classroom management much thought anymore – it managed itself.
The distinction between reactive and proactive management is also made by Lee Canter. I've never thought of it that way before -- proactive interactions with students (and parents, as Daas points out) tend to be more positive.
Okay, that's enough with music and management. Let's finally get to Geometry.
Lesson 0.5 of Michael Serra's Discovering Geometry is called "Mandalas." This is the first of two sections included in the old Second Edition yet omitted from the modern editions.
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.
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