First of all, whereas the other district high schools have late days on Mondays, this high school has early days for Common Planning (or "Collaboration"). With school dismissed at 12:55, these are almost like minimum days as there is no school lunch. The only break is snack, after fourth period.
There are four levels of art here. The highest level is AP Studio Art and the second highest is Advanced Painting and Drawing. There are so few students at these levels that they are combined into a single second period class. The next highest level is (regular) Painting and Drawing, which is third period. The bottom level is Fundamentals of Art, and there are three sections of this class, taught Periods 4-6. Officially, seventh period is Varsity Soccer -- but just like Varsity Basketball last week, coaches are in charge of this class and I, as the sub, have no responsibilities this period. The students in Fundamentals can be in any grade, but those in AP Studio Art presumably took art all four years, so these are all seniors.
As for the gender ratio, what's striking is that this is the exact opposite of the weight training classes from last week. Males are in the minority, especially in the higher levels. Indeed, in the AP class, there is only one guy in the entire class. I actually ask him today what it's like to be the only male in this class. He tells me about the first day of school two weeks ago -- he arrived to the art room early and watched more and more girls enter the classroom.
Some people might wonder why this is. After all, there are many famous male artists in history, from Leonardo da Vinci all the way up to Pablo Picasso while the only female artist who really comes to mind is Frida Kahlo.
But we must note that for most of human history, women were housewives and little else. When it comes to gender differences, it's often said that men prefer "action" or "doing things." Back when women were housewives, art counted as "action," but not in modern times. On the other hand, weightlifting is definitely considered action, so when it's time to choose electives, the guys tend towards weightlifting while the girls tend more towards art. (Also, notice that many young boys enjoy drawing -- so it's probably painting that they want to avoid.)
Notice that last week's weightlifters were all juniors and seniors -- that is, these are students who have already completed the two-year P.E. requirement for high school graduation in California. Likewise, only one year of art is required to satisfy the A-G requirement for college entry here -- and of course, that one year would be Fundamentals. Thus anything beyond two years of P.E. or one year of art is considered elective -- and that's where the gender differences are most noticeable.
Recently, it's pointed out that young girls tend to like another subject more than boys -- reading. I always think back to Disney's Beauty and the Beast, where Gaston tells Belle that it's wrong for a girl to read -- in the 19th century when the film takes place. (I don't know whether this line still appears in the live-action version of the movie, since I didn't watch it.) On the other hand, we compare this to The Simpsons, where the writers chose to make Lisa the smart one who likes to read much more than Bart does. Again, Gaston/Belle takes place when women were housewives and reading was seen as a masculine "action," while Bart/Lisa occurs when boys no longer consider reading masculine.
As a math teacher, I like to think about gender differences for two reasons. First, we know that there are fewer females than males in high-paying STEM careers, and so I wonder whether there's anything that we math teachers can do about it. The second reason is more personal. I wonder whether I, as a male teacher, am displaying any subtle bias in favor of male students in my class. Whenever I see the girls in any of my classes not perform as well as the boys, I fear that it's my fault.
Therefore during this week that I'm spending in this art classroom, I want to make sure that I'm interacting positively with the female students -- and there's no better opportunity than these art classes where girls are in the majority. Since everyone is working independently on art projects, I can walk around the classroom and ask some of the girls to describe their art. Then I can reply by saying something positive. By doing this, I'm working on positive interactions with girls so that I can more readily connect to female students the next time I'm in a math class.
Recall that the Illinois State text contains an art component. Since girls tend to enjoy art more, these art projects might help them do better at math (or science, as most of the art projects for middle school are actually for science). And so when I reflect on my year at the charter school and recall that I didn't connect with the girls very well, I wonder whether the Illinois State art projects could have made a difference.
Lesson 1-8 of the U of Chicago text is called "One-Dimensional Figures." (It appears as Lesson 1-6 in the modern edition of the text.)
This is what I wrote last year about today's lesson:
Lesson 1-8 of the U of Chicago text deals with segments and rays. The text begins by introducing the simple idea of betweenness. In Common Core Geometry, betweenness is an important concept, because it's one of the four properties preserved by isometries (the "B" of "A-B-C-D").
As I mentioned a few days ago, for Hilbert, betweenness is a primitive notion -- an undefined term, just as point, line, and plane are undefined. Yet the U of Chicago goes on to define it! It begins by defining betweenness for real numbers:
"A number is between two others if it is greater than one of them and less than the other."
Then the text can define betweenness for points:
"A point is between two other points on the same line if its coordinate is between their coordinates."
But Hilbert couldn't do this, because his points don't have coordinates. Recall that it was Birkhoff, not Hilbert, who came up with the Ruler Postulate assigning real numbers to points. Instead, Hilbert's axioms contain statements about order (Axioms II.1 through II.4), such as:
"II.2. If A and C are two points of a line, then there exists at least one point B lying between A and C."
Since we have a Ruler Postulate (part of the Point-Line-Plane Postulate), this statement is obvious, since points have coordinates and the same is true for real numbers -- between reals a and c is another real b.
I've seen some modern geometry texts mention a Ruler Postulate, but nonetheless leave the term betweenness undefined. Now as we mentioned earlier with point, line, and plane, if a term such as betweenness is undefined, then we need a postulate to describe what betweenness is. This postulate is often called the Segment Addition Postulate:
"If B is between A and C, then AB + BC = AC."
Notice that this statement does appear in the U of Chicago text. But the text doesn't call it the Segment Addition Postulate, but rather the Betweenness Theorem. As a theorem, we should be able to prove it -- and since after all, the text defines betweenness in terms of real numbers, we should be able to use real numbers to prove the theorem. Indeed, the text states that we can use algebra to prove the theorem, but the proof is omitted.
Following David Joyce's admonition that we avoid stating a theorem without giving its proof, let's attempt a proof of the Betweenness Theorem. We are given that B is between A and C. Now let us assign coordinates to these points. To make it easy to remember, we simply use lowercase letters, so point A has coordinate a, point B has coordinate b, and point C has coordinate c.
We are given that B is between A and C, so by definition of betweenness, we have either a < b < c, or the reverse of this, a > b > c. Without loss of generality, let us assume a < b < c (especially since the example in the book has a < b < c). Now by the Ruler Postulate (the Distance Assumption in the Point-Line-Plane Postulate), the distance between A and B (in other words, AB) is |a - b|. Since a < b, a - b must be negative, and so its absolute value is its opposite b - a. (To avoid confusing students, we emphasize that to find AB, we just subtract the right coordinate minus the left coordinate, so that AB isn't negative. This helps us to avoid mentioning absolute value.) Similarly BC = c - b and AC =c - a. And so we calculate:
AB + BC = (b - a) + (c - b) (Substitution Property of Equality)
= c - a (simplification -- cancelling terms b and -b)
= AC
The case where a > b > c is similar, except that AB is now a - b rather than b - a. All the signs are reversed and the same result AB + BC = AC appears. QED
Don't forget that I want to avoid torturing geometry students with algebra. And so I simply give the example with numerical values, with the variables off to the side for those who wish to see the proof.
The text proceeds to define segments, rays, and opposite rays in terms of betweenness. Notice that these definition are somewhat more formal than those given in other texts. A typical text, for example, might define a segment as "a portion of a line from one endpoint to another." But the U of Chicago text writes:
"The segment (or line segment) with endpoints A and B is the set consisting of the distinct points A and B and all points between A and B."
The definitions of ray and opposite ray are similarly defined in terms of betweenness.
The section concludes with the notation for line AB, ray AB, segment AB, and distance AB. But although every textbook distinguishes between segment AB and distance AB, many students -- and admittedly, many teachers as well -- do not. The former has an overline, but the latter doesn't. Unfortunately, Blogger allows me to underline AB and strikethrough
Now if
To avoid confusion, in the following images I threw out Question 8 from the text, a multiple choice question which states that
Returning to 2018, I've been thinking over the weekend about "drens" (beyond what I wrote in my Thursday and Friday posts). Back in my March 2nd post, I wrote about how perhaps there should be three levels, "smart," "almost smart," and "dren." Once again, the idea is that a "dren" is someone who is too lazy to do basic math. Student who are putting in the effort are on their way to becoming smart, and so they are "almost smart."
Anyway, "smart" / "almost smart" / "dren" reminds me of something the famous math blogger Sarah Carter once wrote on her blog -- A/B/Not Yet:
https://mathequalslove.blogspot.com/2015/06/students-speak-out-about-abnot-yet.html
It's surprising that I've linked to the Carter blog many times, but never A/B/Not Yet. But then again, A/B/Not Yet refers to a grading scale that she no longer uses.
Anyway, Carter used to give her students only three grades -- A, B, and Not Yet. And in this scale, Not Yets count as zeros -- only A's and B's are passing grades. In this class, students are given many opportunities to redo Not Yet assignments until they earn at least a B. Thus I now wonder whether I, back in my old class from two years ago, should have used a scale similar to Carter's, except that the three levels are "smart," "almost smart," and "dren."
Actually, "smart" is a dangerous name for a grade. It reinforces the idea that some students are destined to be "smart," and so the non-smart students shouldn't even try. Instead, I should use the word that I originally planned on pairing with "dren" -- hero. Notice that both "dren" and "hero" derive from "nerd" -- "dren" of course is "nerd" backwards. Both "nerd" and "hero" have the two middle letters in common, and we can think of cutting off the right side of "d" to make "o" and pass it to the left side of "n" to make "h."
The idea is that students shouldn't think of those with math smarts as nerds. Instead, they are heroes -- they are the ones who make possible many of the things we enjoy (such as the video game Fortnite, as I mentioned in Thursday's post). So the three grades would be "hero," "almost hero," and "dren."
According to Carter, A/B/Not Yet is a form of standards-based grading, or SBG. Back when I was tutoring, one of my students attended a school that used SBG. Her school used a four-point scale, which is fairly common for SBG. The four levels roughly correspond to the four SBAC scores -- and so SBG teachers in PARCC states might prefer to use a five-point scale.
Notice that SBG is actually compatible with Illinois State. The name "standards-based grading" refers to the fact that students receive grades for standards, not assignments. When setting up assessments on the Illinois State website, we first select a standard, and then questions from a bank appear. And so students can be graded directly on the chosen standards.
On the other hand, SBG is not compatible with PowerSchool, our grading software. PowerSchool divides all grades into categories. It distinguishes between "quizzes" and "tests" (which many teachers and texts do as well, but not Illinois State). Also, 10% of the grade is "participation," while 15% is "homework." Most SBG teachers, including Carter, don't grade homework at all.
Thus, "hero" / "almost hero" / "dren" probably wouldn't have worked at all in that class. Still, it is interesting to imagine how I could use it in a future class if possible.
For one thing, recall that on Dren Quizzes, the only possible scores are A and dren. Likewise, the "almost hero" score wouldn't be available for Dren Quizzes as these are for basic skills (such as single digit multiplication). For regular assessments, "almost hero" would be available.
Even though Carter's old scale didn't have a C grade, it might be advisable to allow C's only for special ed students. At my old charter, allowable grades were A, B, C, and F, and so the special ed C's would at least match up with this scale. (Likewise, some charters -- though not my old one -- use the A, B, C, F scale but allow D's for special ed students.)
In my old class, students earned Red Vines licorice for A's only -- I would give them one Red Vine for an A on a Dren Quiz and four Red Vines for an A on a regular quiz or test. Perhaps it might be advisable to keep the "hero" grade on a Dren Quiz at one Red Vine, but then give two Red Vines for "almost hero" and three for "hero" on a regular quiz or test. Yes, the A's would now get only three candies instead of four, but this is so that I have enough to give two to each of the B's.
What happened to Carter's old A/B/Not Yet scale. Actually, she dropped it the same year that I taught at the old charter:
https://mathequalslove.blogspot.com/2016/08/new-sbg-posters.html
This year Carter is teaching at a new school. She hasn't posted anything about her grading scale, so I don't know whether she's currently using A/B/Not Yet, 4/3.5/3/2/1, or traditional (non-SBG) grades.
But this week, I'm in an art class, and so math grades and SBG are irrelevant. The current goal is for me to get through this week of subbing, and to interact more with the female students.
As far as classroom management is concerned, I've already warned fourth and sixth periods about my pet peeve -- students who ask for restroom passes right after snack and lunch. Many of these students are freshmen who are observing their first early-out Monday at their new school (as two weeks ago was a special first-week schedule, and the Monday schedule was blocked by Labor Day last week), and so I reminded them about how the schedule works. On Tuesday through Thursday, these are the classes right after snack and lunch, and so my restroom policy will be strictly enforced then.
Of course, one behavior issue that might come up is "playing with paintbrushes." This isn't one of my pet peeves, since I'm not normally in a class with paintbrushes. Another thing that might come up is "playing with the compass" -- yes, in art, just like Geometry, students may need to draw circles.
Also, the lesson plans from the regular teacher are just "students continue working independently on their projects." As the week progresses, more and more students will finish these projects. So I must be prepared on Thursday to have many students say "I'm done with everything" -- and try to come up with something else for them to do.
In sixth period, I use teacher look when some students try to talk during attendance. So far, teacher look seems to work -- the class becomes quiet, and in fact sixth period is the "best class of the day" in the notes that I leave for the regular teacher.
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