As is usually the case, the best class of the day was strings (third period). In her lesson plans, the teacher warns me to watch out for the two "music appreciation" classes, fourth and sixth periods. In the music appreciation classes, the students watch YouTube videos about two famous singers, Shawn Mendes and Camila Cabello. At the same time, they can prepare for tomorrow's partner performance. I figure that the performance will be played on keyboards, since they're given a keyboard worksheet for them to "practice" on. They are also allowed to take out cell phones and use an app that allows them to play their songs.
It goes without saying that one major issue is when students try to do something else on the phones other than music. But as has been the case lately, the real problem is at clean-up time. Students are supposed to put the chairs and music stands away. Instead, they play on the laptop and audio system that's set up for YouTube -- after the Camila Cabello video, they click on a random video (which happens to be about Shawn Mendes) and then turn it up loud. They also play with the piano, the lights, and the doors (if I guard the front door, they try to run out the back).
I try to correct this during sixth period. Doing so is crucial, since this happens to be the class that rotates into the after-lunch block (in other words, silent reading time). I repeat over and over again that students should be quiet during silent reading time, and I believe that this helps set the tone for the class. Not as many students try to practice for the performance as in fourth period -- which also means that not as many students play around on their phones than the other class. In the end, I name sixth period as the second-best class of the day.
The only other problem that I must report to the regular teacher is in fifth period band. The students are supposed to practice, but the percussionists can't because the teacher doesn't want them playing on the drums when there's a sub. The lesson plan directs the percussionists to use "practice drums" instead, but this pair -- a boy and a girl -- don't want to. The lesson plan also allows me to play another YouTube video (on the importance of listening to classical music) if the students have nothing else to do, and that appears to be the case in this class. Nonetheless, the two percussionists keep messing around with each other, and another boy who plays the clarinet also gets involved. So I have no choice but to report the percussion and clarinet sections to the teacher.
One thing that worries me is when I don't have access to the students' names. Obviously, this puts me in a position of weakness, since merely saying the students' names along with a teacher look might be enough to get them to change their behavior. In many classes I can guess the students' names by using the seating chart, but music doesn't have a traditional classroom with a seating chart. On the bad list I write only "percussion" and "clarinet." This does identify who the misbehaving students are (since the teacher knows which students are in which section), but still, it doesn't sound as strong to either the students or the regular teacher as having exact names would.
I need to be cleverer when it comes to figuring out student names. Obviously, misbehaving students won't tell me their own names, and students won't tell me others' names if they fear that they'll be labeled a "snitch." But perhaps I could have sat the clarinetist out of earshot of the percussionists and ask him to write down their names. Then a little later, I could ask the percussionists for the name of the clarinet player.
Writing down names or even the instruments of the students who misbehave in fourth period is even trickier today, because it all happens mostly near the end of the period, after they have already put away their instruments. And I don't see exactly who plays with the lights or piano. This continues to be a problem in my classes when I'm paying more attention to whether materials (Chromebooks in previous classes, chairs and stands today) are put away properly than to student behavior. So this is definitely something to watch out for in future classes.
At the start of the school year, I did say that on a day that I sub in music, I'll return to the Mocha computer music that we explored over the summer. I also wrote that I'll return to Mocha music around holidays -- and of course, Halloween is just around the corner. In fact, today I'm able to create an original song on Mocha and actually play this new song for students!
I'm also glad that I get to write about music while we're exploring Sue Teele's book. After all, one of the Multiple Intelligences is musical intelligence. And besides, we can also make a connection to visual intelligence via the rainbow metaphor. After all, Teele's book is Rainbows of Intelligence, and Kite's color notation describes the musical notes that I can play on Mocha.
But Teele doesn't actually reach the rainbow metaphor until Chapter 5 -- tomorrow's chapter. And so I'll save the Mocha music post until tomorrow's post. That will be the big day in which we both introduce Teele's rainbow of intelligence and revisit Kite's rainbow of music -- and you'll get to learn more about the actual song that I play in class today. But first, let's get to today's chapter.
Chapter 4 of Sue Teele's Rainbows of Intelligence, "Changing Our Thinking About Education," begins as follows:
"The recent release of the Third International Mathematics and Science Study, which reported on cross-national comparisons of student achievement in mathematics and science, curricula, textbooks, and teacher practices for a range of 20 to 50 countries, depending on the particular comparison, supports the need to expand our educational paradigm and to analyze, evaluate, and more effectively understand beliefs and practices that have been the foundation of the American educational system for over 100 years."
Recall that Teele wrote her book in the year 2000. By the way, the TIMSS test still exists, but now T stands for "trends," not "third." It's given every four years, including next year. Like many other tests, next year's test will be given digitally -- it will be known as eTIMSS. Teele writes:
"As our nation moves toward a more standard-based system, current information on the neurobiology of the brain and the ways individuals process information must be integrated into the new system if we are to ultimately improve student achievement."
Of course, that "standard-based system" is now known as Common Core. Anyway, the author highlights the following statement:
"We must provide the best education possible to prepare our students to be successful in the 21st century."
And she also writes about the future workforce:
"The future workforce will require students to think at higher levels and to envision new ideas and products that have not been previously designed."
Of course, traditionalists dislike any discussion of the future workforce, because they fear that "21st century learning" is nothing more than an excuse to abandon traditional teaching methods. I suspect that traditionalists won't like anything that appears in today chapter. Teele continues:
"Because many of our students will be working in their homes on their computers, they will need to possess additional personal qualities that require them to manage their time independently, have integrity, and be able to take individual responsibility for their work."
She explains that both intrapersonal and interpersonal intelligences will be necessary, since many employees will work independently at times and collaborate at other times.
"Education should not be looked upon as the mere acquisition of academic subject matter, but as a part of life itself." -- John Dewey
In quoting the famous educationist, Teele points out that the TIMSS report agrees with him -- countries that presented academic material realistically and practically scored higher. And so she makes a suggestion:
"Our educational system should be incorporating current knowledge of how students learn, recent research on the brain, and ways to effectively teach all students."
Moreover:
"To accommodate these diverse students, our educational system should be incorporating the current knowledge of how students learn, recent research on the brain, and ways to effectively teach all students."
The author now describes a test she created -- the Teele Inventory for Multiple Intelligences (TIMI):
"Individuals are asked to select one of the two choices that they feel is most like them. There are no right or wrong answers."
The test then reveals which of the multiple intelligences are strongest. I won't reproduce Teele's chart here, which shows us how students in four different grade spans (primary, upper elementary, middle school, and high school) tend to score. She points out that while students tend to score well in the linguistic and logical-mathematical intelligences in the lower grades, by the higher grades the interpersonal intelligence dominates, while all grades score highly in both the visual-spatial and bodily-kinesthetic intelligences.
Traditional education supports only the linguistic and logical-mathematical intelligences. This explains why, once again, I agree with traditionalism in the primary (which I assume here means K-2) grades where those two intelligences score well, but not in the higher grades where they don't.
Now Teele writes about gender differences among the intelligences. This is important to me, since I always fear that I'm not teaching all the girls in my classes effectively. Again, I won't post the whole chart, so let's just skip to her summary:
"These studies support the findings that females, on average, are more linguistic and interpersonal, and males, on average, are more logical-mathematical and spatial."
And of course, on the TIMSS scores reveal that boys scored higher than girls in both math and science (slightly so in math, significantly so in science). She writes:
"Perhaps a model should be considered that is based on variables that are both biological and social and, therefore, cannot be classified into one of these two categories. Learning is both a socially mediated event and a biological one."
She highlights the following statement:
"Psychological, biological, social, and environmental factors may all contribute to guiding the learning process."
And here is her suggestion:
"Students can be taught how to translate from their dominant intelligences to their less dominant intelligences."
Traditionalists should take the following statement to heart:
"Once students discover they can succeed in learning, they are more willing to try harder."
Teele concludes this chapter by highlighting some more suggestions:
"We need to personalize instruction to accommodate the different ways students learn."
The following statement is especially relevant to the grade-span I plan to teach. Again, tt's definitely something that traditionalists should keep in mind:
"The need for acceptance, that is, development of self-esteem during adolescence, is heavily influenced by peer groups, particularly same sex groups. This is why interpersonal intelligence is so dominant in both males and females at the secondary level."
But why is this the case?
"Biological and environmental factors are intertwined in ways that affect intelligence and learning."
Teele concludes the chapter as follows:
"The changes engendered by these discoveries will greatly influence both our educational system and our future workforce. We must redesign our educational system to allow all students to develop their intellectual abilities to their full potential."
OK, let's finally get to the U of Chicago text.
Lesson 4-7 of the U of Chicago text is called "Reflection-Symmetric Figures." (This corresponds to Lesson 6-1 in the modern Third Edition.) This is what I wrote last year about today's lesson:
Section 4-7 of the U of Chicago text deals with reflection-symmetric figures. A definition is in order:
A plane figure F is a reflection-symmetric figure if and only if there is a line m such that r(F)=F. The line m is a symmetry line for the figure.
In other words, it's what one usually means when one uses the word "symmetry." Some geometry texts use the term "line-symmetric" instead of "reflection-symmetric." Some geometry and algebra texts use the term "axis of symmetry" instead of "symmetry line" -- especially Algebra I texts referring to the axis of symmetry of a parabola. Some biology texts use the term "bilateral symmetry" instead of "reflection (or line) symmetry" - in particular, when referring to symmetry in animals. As animals are three-dimensional, instead of a symmetry line there's a sagittal plane.
Indeed, it is this last topic that makes symmetry most relevant and interesting. Most animals -- including humans -- have bilateral symmetry. I once read of a teacher who came up with an activity where the students look for the most symmetrical human face. The teacher blogged about how students who are normally indifferent to geometry suddenly came fascinated and engaged to learn about the relationship between symmetry and human beauty. Unfortunately, this was more than a year ago, and I can't remember or find what teacher did this activity -- otherwise I'd be posting a link to that teacher's blog right here!
In the Common Core Standards, symmetry is first introduced as a fourth grade topic:
CCSS.MATH.CONTENT.4.G.A.3
Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.
Later on, symmetry appears in the high school geometry standards:
CCSS.MATH.CONTENT.HSG.CO.A.3
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
Notice that if a reflection over a line carries a polygon to itself, then that line is a symmetry line. But symmetry lines for polygons formally appears in Chapter 5 of the U of Chicago text. Right here in Chapter 4, we only cover symmetry lines for simpler shapes -- segments and angles. The text reads:
"In the next chapter, certain polygons are examined for symmetry. All of their symmetries can be traced back to symmetries of angles or segments."
For segments, the text presents the Segment Symmetry Theorem:
A segment has exactly two symmetry lines:
1. its perpendicular bisector, and
2. the line containing the segment.
The text gives an informal proof of this -- as the mirror image of an endpoint, there can only be two possible reflections mapping a segment
But we also want to work with angles. The first theorem given is the Side-Switching Theorem:
If one side of an angle is reflected over the line containing the angle bisector, its image is the other side of the angle.
An informal proof: the angle bisector divides an angle into two angles of equal measure. The picture in the U of Chicago text divides angle ABC into smaller angles 1 and 2. Now the reflection must map ray AB onto a ray that's on the other side of the angle bisector BD, but forms the same angle with BD that AB does with BD. And there's already such a ray in the correct place -- ray BC. Notice that part b of the Angle Measure Postulate from Chapter 3 already hints at this -- the "Two sides of line assumption" gives two angles of the same measure, one on each side of a given ray. QED
The other theorem, the Angle Symmetry Theorem, follows from the Side-Switching Theorem:
Angle Symmetry Theorem:
The line containing the bisector of an angle is a symmetry line of the angle.
Earlier this week, I wrote that we'd be able to prove the Converse of the Perpendicular Bisector Theorem after this section. As it turns out, the Side-Switching Theorem is the theorem we need.
Converse of the Perpendicular Bisector Theorem:
If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
Given: PA = PB
Prove: P is on the perpendicular bisector m of segment
Now I was considering giving a two-column proof of this, but it ended up being a bit harder than I would like for the students. But as it turns out, even though the U of Chicago text doesn't prove this converse, in Section 5-1 it gives a paragraph proof of what it calls the "Isosceles Triangle Symmetry Theorem," and the proof of this one and the Converse of the Perpendicular Bisector Theorem are extremely similar. After all, we're given that PA = PB -- so PAB is in fact an isosceles triangle!
Proof:
Let m be the line containing the angle bisector of angle APB. First, since m is an angle bisector, because of the Side-Switching Theorem, when ray PA is reflected over m, its image is PB. Thus A', the reflection image of A, is on ray PB. Second, P is on the reflecting line m, so P' = P. Hence, since reflections preserve distance, PA' = PA. Third, it is given that PA = PB. Now put all of these conclusions together. By the Transitive Property of Equality, PA' = PB. So A' and B are points on ray PB at the same distance from P, and so A' = B. That is, the reflection image of A over m is B.
But, by definition of reflection, that makes m the perpendicular bisector of AB -- and we already know that P is on it. Therefore P is on the perpendicular bisector m of segment
Let's think about what we're trying to prove here. We want the Converse of the Perpendicular Bisector Theorem -- and consider what I wrote earlier about the proof of converses. The proof of the converse of a statement often involves the forward direction of the theorem and a uniqueness statement -- and even though we didn't use the forward direction of the theorem here, we did use a uniqueness statement here. As it turns out, given two distinct points A and B, there exists only one line m such that the mirror image of A over m is B -- and that line is the perpendicular bisector of the segment
In this section, we found symmetry lines for simple figures such as segments and angles. But can we find symmetry lines for the simplest figures? As it turns out, a point has infinitely many lines of symmetry -- any line passing through the point is a symmetry line. But a ray has only one line of symmetry -- the line containing the ray.
Finally, does a line have a line of symmetry? This is exactly the answer to Question 25 of this section, in the Exploration/Bonus Section. A line -- considered as a straight angle -- contains more than one symmetry line. This is because any point on the line can be taken as the vertex of that straight angle. Since straight angles measure 180, their angle bisectors must divide them into pairs of 90-degree angles. Therefore, any line perpendicular to a line (straight angle) is a symmetry line of the given line. This is what I called the Line Perpendicular to Mirror Theorem. It implies that a line (straight angle) has infinitely many symmetry lines. (Of course, the line has one more symmetry line that I didn't mention -- namely the line itself.)
I included Question 24, even though it appears to mention corresponding and same-side interior angles formed by two lines and a transversal. But nowhere in the question does it mention anything about the two lines being parallel.
I left out Questions 16 and 17, which give the construction of an angle bisector. I finally plan on going to constructions sometime next week. But here's another video from Square One TV, where doctors have to perform a "bisectomy" on an angle. (Unfortunately, only the entire 30-minute show is available on YouTube -- the "bisectomy" doesn't begin until the 11-and-a-half-minute mark.)
No comments:
Post a Comment