Ordinarily, I don't write a full "Day in the Life" for a high school class unless it's math. And so I won't -- and neither will I add the "subbing" label today. But I will give an overview of how my classroom management goes today.
First of all, I've covered this teacher before last year -- indeed, it's the same teacher who once had me show the movie McFarland USA to her class. Due to construction in the foreign language building, she's now in a science classroom with a DVD player. But there's no chance to try it out, since the students have worksheets to do.
The two Spanish I classes fill in blanks where they write their name, as well as the special Spanish nickname by which they want to be known in the class. (This is a foreign language class tradition -- as a young student, my high school French teacher did the same.) Meanwhile, the three Spanish II classes work on a crossword puzzle with 42 different verb conjugations.
Notice that this teacher didn't have Spanish I last year. Thus most of the Spanish II students don't recognize me at all. Indeed, I had more connections with the Spanish I freshmen who remember me from covering middle school classes last year. And I suspect it's because of my lack of connection to the second-year students that I have more behavior problems with these older kids.
The first of three classes go by smoothly. All of the students complete the puzzle -- the last student doing so with five minutes to spare. This gives me enough time to sing the reward song -- and of course, it's "Sign of the Times" since I'm in another Spanish class.
But in the other two classes, I must place five students each on my bad list, and I end up not singing the song in those classes at all. This is tricky -- you might argue that 42 verb conjugations in a single 50-minute period is a bit too much for me to require. Then again, the first Spanish II class is able to rise to the challenge, so it's reasonable to expect the others to do so -- but this is exactly what leads to an argument.
Back at the old charter school, I often compared the students to my young self -- when they claimed that my rules were impossible to follow, they replied "That's for you -- that's not for me." The reason for this is that even as a young student, I highly valued learning and following the rules, while these students value friendship and fun more than education. So their claim that rules were "impossible" to follow shouldn't have been taken at face value -- they really meant something like "It's hard to follow the rules," or "I don't want to follow the rules."
But at the time, I warned myself that while comparisons to myself are ineffective, comparisons to other students are equally pointless. Instead of "That's for you -- that's not for me," they say "That's for them -- that's not for me," when I point out today, during the second class, that the first class is able to finish the puzzle.
Yet it's difficult for me to avoid making that comparison. This is because when I entered the class, I didn't know how much work to expect of the students. Perhaps 42 conjugations in 50 minutes really are too much for the students. It's only because the first class is able to complete them that I feel justified in requiring the other classes to do so. Thus it's difficult to avoid making comparisons in explaining why I set the bar so high, when I use a comparison to determine where to set the bar.
In the second class, one girl plays a fairly common trick -- she's summoned to the office until about 10-15 minutes left in class, but when she returns, she does no work until there's about 2-3 minutes left, then claims that there's not enough time to do any work. But in the other class, a group of guys pulls the same trick without being called to the office at all. At the start of the 50-minute period, they say "I don't know how to do this," and "I'll do it at home," and then play with Tech Deck miniature skateboards until there are about 20 minutes, then claim that there isn't enough time to do the puzzle.
Even though in this third class I stop myself from comparing them to the first class, I end up making another useless comparison -- I tell the guys that the students who make the good list in this same (third) class finish the puzzle in about 20 minutes or less, so there's still time. Of course, this leads only to another "That's for them -- that's not for me." Afterwards, they ask how many points the assignment is worth, as well as what the punishment is for being placed on the bad list. I don't know the answer to either question, because the regular teacher doesn't specify.
Is there any way I could have avoided the comparisons, to either myself or other students? Well, here is what I could have done. I could say something like: "I know you can finish the assignment if you only work hard, without playing with Tech Decks." If they ask how I know this, or why they can't do the assignment at home, I resort to the old standby, "Because I said so."
Sometimes students don't want to work the second they enter the classroom -- preferring a few minutes to unwind from lunch or passing period. Maybe they are in the middle of texting on their phone, but as soon as the text is complete, they're willing to put the phone away and work. "OK, I'll do it soon -- just stop pressing me!" The problem occurs when they don't start working in a few minutes and keep playing with Tech Decks, as these guys do.
There are 42 conjugations for the students to complete in 50 minutes. For simple calculation, assume that it takes eight minutes for the students to unwind, leaving one minute per conjugation. Then I start checking work multiple times during the period based on this estimate. For example, 15 minutes after the start of class, that's eight minutes to unwind and seven conjugations to complete. So then I circulate around the class and write anyone with fewer than seven completed on the bad list. Let's say I check again at the midpoint of class. I erase the name of anyone with 17 conjugations and add anyone with fewer. I continue adding and erasing until the end of class. (This also solves the problem with the late-arriving girl -- I tell her that in ten minutes, she must do ten conjugations or be placed on the bad list.)
Notice that the final list that the regular teacher sees is anyone who fails the final check -- that is, anyone who doesn't finish the worksheet. But this is the exact same list that she'd see if I do only a single check at the end of class (which I actually do today). So why should I do multiple checks?
It's to remind the students that they must do the work throughout the period. It avoids the situation "There's only 20 minutes left, so there's no reason even to begin," when by that point in class I've already completed two or three checks and written their names down. They know that they must put the Tech Decks away and work the entire period.
There's also room for the song incentive here as well. Earlier this week, I told the middle school students from the start of the period that I'd sing them a song -- and whetted their appetite by singing one verse of that song. Sometimes I wonder whether I should have done the same today -- it seems weird to play that game with mature seniors.
But today's Spanish II sophomores are in between -- the first class is willing to work even though I don't mention a song until the last five minutes of class. Yet in the other two classes, some students tell themselves, "It's Friday, it's late, and there's a sub. I'm determined not to work this period!" I've already decided to make singing a part of my classroom repertoire -- and singing is probably the only thing in my repertoire that would have gotten those particular students to work today. And besides -- if I'm moving my mouth to sing, then I'm not moving my mouth in making comparisons that fail to motivate.
Oh, and speaking of comparisons, there is one time when making comparisons to either myself or other students can actually work. It's when I'm making a favorable comparison (that is, when the students standing in front of me are compared favorably to either me or the others). Indeed, back at the old charter school, one girl had the following to say about my comparisons -- "I respect my dad, who told me about all the bad things he did when he was young, much more than my teacher, who claims that he was a perfect kid." In other words, "I was bad. Don't do what I did!" is much more effective than "I was good. Do what I did!"
Here's what a better comparison would have looked like in my class today: "I get it -- it's Friday afternoon, and you'd just want to play and not do any work. When I was your age, I didn't want to work then either. But then I'd be stuck with extra work to do over the weekend -- and I often didn't do it then either, and I'd get in trouble. I found out the hard way that it's much better to finish the work in class on Friday so that you have no homework over the weekend -- and besides, you don't want to be the ones to cost the class the 'Sign of the Times' song, do you?"
(Likewise, it's only appropriate to compare the students to earlier classes if I'm saying "You guys are much better than the other period," and not "The other period was much better than you guys.")
Lesson 1-7 of the U of Chicago text is called "Postulates." (It appears as Lesson 1-5 in the modern edition of the text.) I've decided to restore the original inclusive definition of parallel (where a line can be parallel to itself). Therefore, this is what I wrote last year about this lesson:
Section 1-7 of the U of Chicago text introduces postulates. In the last section, the undefined terms -- the primitive notions -- point, line, and plane were introduced. Since these are undefined, we don't really know what they are unless we have postulates -- also known as axioms -- to describe them.
I reproduce the main postulate of this section, the Point-Line-Plane Postulate:
Point-Line-Plane Postulate:
(a) Unique line assumption: Through any two points, there is exactly one line.
(b) Dimension assumption: Given a line in a plane, there exists a point in the plane not on the line. Given a plane in space, there exists a point in space not on the plane.
(c) Number line assumption: Every line is a set of points that can be put into a one-to-one correspondence with the real numbers, with any point on it corresponding to 0 and any other point corresponding to 1.
(d) Distance assumption: On a number line, there is a unique distance between two points. If the points have coordinates x and y, we define this distance to be |x - y|.
Let's look at each of these four assumptions -- since postulates really are assumptions, or statements that are obviously true -- in detail. The first assumption, that two points determine a line, goes all the way back to Euclid's First Postulate. In Hilbert's formulation of Euclidean geometry, this is Axioms I.1 and I.2.
The second assumption, about dimensions, are often different in other texts. Some texts, for example, emphasize that three noncollinear points determine a plane -- and give the example of a tripod standing on its three legs, the ends of which are the three points determining the plane of the floor. Hilbert's Axioms I.3 through I.8 roughly correspond to this assumption.
Assumptions (c) and (d) often appear in geometry textbooks as the "Ruler Postulate." The Ruler Postulate was first formulated by the American mathematician George David Birkhoff, about eighty years ago. The Ruler Postulate basically states that rulers work -- that is, we can measure line segments.
The section continues with its first theorem, the Line Intersection Theorem:
Line Intersection Theorem:
Two different lines intersect in at most one point.
And then we have the definition of parallel lines:
Definition
Two coplanar lines are parallel lines if and only if they have no points in common, or they are identical.
I've discussed these in some of my introductory posts in July. The last four words of this definition are controversial: "or they are identical." But as I pointed out, using this definition often simplifies later proofs -- in particular, it often allows one to replace an indirect proof with a direct proof. And technically speaking, the proof of the Line Intersection Theorem is actually an indirect proof -- but it's so simple that the text includes an informal argument here while delaying other indirect proofs until Chapter 13.
So far, in the introductory posts, we wanted to prove that two (coplanar) lines are parallel -- using our definition that they are either non-intersecting or identical. We were able to do this two ways -- we could prove that if they have at least one point in common, then they must have every point in common -- or we could prove that if at least one point on one line fails to lie on the second line, then every point on the first line fails to lie on the second line. These are hypotheses and conclusions that can easily fit into the Given and Prove sections of a proof.
But as teachers, our priority is to make geometry easy for the students to understand. So which will confuse the students less: a definition of parallel containing those four extra words "or they are identical," or many indirect proofs? We can't be sure until actually teaching this in a classroom.
So for now, I will stick to the U of Chicago definition of parallel, with those words "or they are identical," and delay indirect proofs as long as possible. But on the following images, I'll just leave a space for "parallel" in the vocabulary section and leave it up to individual teachers whether or not to include those four extra words in the definition. (Notice that in my exercises derived from the U of Chicago text, I preserved the true or false question "a line is parallel to itself." Of course, the answer will depend on which definition the teacher decides to use.)
The section concludes with some Postulates from Arithmetic and Algebra. As I mentioned yesterday, I want to avoid mentioning "algebra" -- the subject that causes many students to hate math -- yet these are important properties that show up in proofs (for example, the Reflexive Property of Equality). And so I'll just call them "Properties from Arithmetic" and just leave "algebra" out of it.
As an aside, let me point out that I was once a teenager studying geometry. Like those of many students at that age, my thoughts turned toward girlfriends and boyfriends -- except that I was the geometry nerd who wasn't a part of all of that. Like most nerds, I could make the following assumptions:
David Walker's First Postulate:
I am physically attracted to no other person.
David Walker's Second Postulate:
I am physically attractive to no other person.
These are statements that were obviously true -- in other words, they're postulates.
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