(Of course this is in my new district, where today is only Day 18. In my old district, it's Day 27.)
7:55 -- This is second period. (As usual, "first period" = zero period.) It's a special ed "Academic Improvement" class -- similar to the "Academic Enrichment" that I covered on Monday in the other district.
Students are supposed to show me their weekly planners. (This is similar to the Monday Enrichment class, except today's planners are weekly, not everyday the class meets.) One girl refuses to show me her planner, and so I leave her name for the regular teacher.
8:50 -- Second period ends and third period begins. Third period is Algebra 1B, the second of two years that special ed students spend in Algebra I. The students are now working in Chapter 6 of the Glencoe text, which is on systems of equations. Specifically, the students are learning how to solve systems using the elimination method with multiplication.
(I explained back during my failed attempt to land a summer school position that the second semester of Algebra I -- and hence Algebra 1B -- begins with systems of equations. In other districts, systems of equations make up the last unit of the first semester.)
One guy refuses to work on his assignment -- actually, he lost the worksheet from yesterday, and even though the aide makes another copy for him, he doesn't do as much as copy down the two problems I do on the board. I start to argue with him -- and stop myself, since I know that arguing doesn't work. The aide tries to reason with him, but he still doesn't work and ultimately returns the second copy. After class, she informs me that even though he never works today, at least he is quiet. For me, this marks improvement from June -- three months ago, I would have continued to argue, and then the student wouldn't have remained quiet.
Meanwhile, I help another student in this class solve one of his algebra problems. The final step of one problem is to solve "2x = 8," and so he must divide both sides by 2. He tells me that he's not good at division, until I ask him to figure out what half of 8 is. Then he realizes that x = 4.
9:45 -- It is now time for tutorial. Some students come in for help on the systems worksheet, and the aide and I help them.
10:25 -- Tutorial ends and it's time for snack. There's a pep rally for tonight's football game. As it turns out, the opponent just happens to be the school that I attended as a young freshman (before I moved to a new district near the end of football season). Normally, I would cheer on the school where I'm subbing, but it's difficult this time.
10:45 -- Snack ends and fourth period begins. As it turns out, this is a new level of math that this teacher didn't teach back in June -- and it's our favorite math class, Geometry! Officially, this is "Basic Geometry" (and not "Geometry A" for some reason). I presume that some of her same Algebra 1B sophomores from last year are now Basic Geometry juniors.
The students are learning about angles in Lesson 1-4 of the Glencoe text. This just goes to remind us how pathetic the U of Chicago text is when it comes to angles. We compare this class I sub in to my own Geometry course on the blog, based on the U of Chicago text. The class I sub in is a slower-moving special ed class, while the class I blog on is based on a district calendar that started a week and a half before this special ed class. And yet the special ed class makes it to angles before the U of Chicago text does!
No other Geometry text besides the U of Chicago waits until Chapter 3 to introduce angles. And we're still only in Chapter 2 -- albeit the final lesson of the chapter. But still, it means that we haven't quite gotten out of Chapter 2 yet. Lesson 1-4 of the Glencoe text is akin to Lesson 3-2 of the U of Chicago text. There are more equations to solve in Glencoe 1-4 than U of Chicago 3-2, despite Glencoe only teaching bisectors in this lesson (along with acute, obtuse, right) and not the complements and supplements of U of Chicago 3-2.
This is the worst-behaved class of the day. Many of the students talk during the entire lesson, but fortunately (unlike June) there isn't much throwing of objects (except maybe a few pencils in the first few minutes of the period). I grow frustrated because I can see that some of the students need extra help with setting up the equations. In the end, I leave the names of three students -- an entire group sitting together -- for the regular teacher. One of them is the girl from earlier who doesn't fill out her weekly planner, along with two guys.
11:35 -- Fourth period finally ends. Like all special ed teachers in this district, this regular teacher co-teaches another class. For fifth period, I travel to a junior English class. The main teacher in this class is having her students watch a video on The Crucible.
12:30 -- It is time for lunch. After working through much of the day, I enjoy a well-needed break during the teacher's conference period.
2:10 -- It is now seventh period. As I explained in June, the regular teacher is the tennis coach, and now it's currently the girls' tennis season. (The team competed in a contest yesterday.) As often happens when I sub for a coach, the assistant coaches run the practice, and so I don't need to stay.
I return to the classroom to pick up my things. I notice that the aide is posting some worksheets on the board. It looks like an opening week activity. The title of the worksheet is "I wish my teacher knew...my math mindset." (This worksheet deserves a separate post, but I don't want to squeeze a discussion of it into today's post.)
As I reflect upon this class, I wonder how I could have taught the class better. In June, my focus resolution for this class was:
3. Move on from past incidents instead of bringing them up with students.
I believe that I follow this resolution well enough today -- as tempting as it might have been to mention the students' June behavior again today.
Moreover, I'm tempted to tell the misbehaving girl in fourth period that I don't trust her because she doesn't fill out the second period weekly planner. But fortunately I avoid the temptation -- her second period assignment has nothing to do with her fourth period behavior, and so there's no reason to bring it up again.
For that matter, her fourth period assignment has nothing to do with her fourth period behavior. She quickly copies down the three problems that I do in class, but that doesn't stop her from talking loudly throughout my presentation.
So why was fourth period much worse than third? I suspect it's because the aide helps out with management more in third period. In fourth period, there are two students who are actually in Algebra 1B, not Geometry. The aide usually works with these two separately, so she can't help me with management.
And indeed, one of these two students is a handful. First, he actually shows up earlier during tutorial, yet he has no intention of working -- he just plays with phones the whole time, and so the aide just kicks him out of tutorial. Then when he arrives for fourth period, he continues refusing to work. The aide tells me that most likely, soon he'll be expelled from the school!
I wish to compare the three main classes where I needed to manage the most effectively (because it's not an honors class and there's no aide or other teacher to help manage) -- these are yesterday's sixth and second period Science 7 classes and today's Basic Geometry class. And so I'll name what I think are my management problems in those classes.
As you already know, I lack a true teacher tone. One of the most effective ways for a teacher to keep and maintain control of the class is to use the teacher tone. Since I don't have a teacher tone, it means that I must be perfect with all other aspects of management. But unfortunately, I'm nowhere near perfect in certain aspects.
For example, let's think about line-crossing. At what point should I say that misbehaving students have crossed the (proverbial) line? We think of this in terms of our three archetypal teachers (from Harry Potter), Snape the hostile teacher, Flitwick the nonassertive teacher, and McGonagall the assertive teacher. Before crossing the line, it would be Snapelike to punish minor misbehavior -- but after crossing the line, it becomes Flitwicklike to ignore major misbehavior.
A rule of thumb is, "the line is crossed" about ten minutes before I lose control of the class. But this means that by the time I realize it's time punish misbehavior, it's too late! It reminds me of what my old distance running coach used to say -- drink ten minutes before you're thirsty! This means that runners should always be mindful to keep hydrated -- and ideal classroom managers are always mindful to be aware of how much the students are misbehaving.
Yesterday in second period, one boy kept doing a bunch of little things -- first moving to the wrong seat, and then glancing and whispering during silent reading. But I knew the line had been crossed when someone put glue in his seat -- that shouldn't happen in a well-controlled class. So I realized that his name should be on the list of troublemakers -- but from his perspective, I was writing the name of a pranking victim on the list. I really should have written his name on the list ten minutes earlier -- this would have shown that I was in control of the class. Then maybe the incident with the glue would never occur.
Today at lunch, the aide tells me that perhaps I should have separated the group of juniors who are talking loudly and disrespectfully. It reminds me that I do indeed have one more way to establish control of the class -- changing students' seats (if only for that one day).
But I'm not sure whether seat changing would have worked yesterday, since the three seventh graders were already in different groups. The one time it might have worked is after the glue incident -- with glue in his seat, the boy might have been all too willing to move to another seat! Then I could have placed him in a group that's even farther away from the other two troublemakers -- and hopefully from the originally glue prankster.
I blamed several students for causing problems in class these past two days, but the responsibility lies with me to improve my classroom management skills. This is a segue into Eugenia Cheng:
Chapter 5 of Eugenia Cheng's The Art of Logic in an Illogical World, "Blame and Responsibility," begins as follows:
"On 9 April 2017 United Express Flight 3411 was overbooked. The airline bumped a particular passenger off the flight, but he didn't go voluntarily and was dragged off by security officers, sustaining injuries along the way."
Cheng asks us, whose fault was it? There are two opposing viewpoints:
- It was United's fault for their unreasonable use of force.
- It was the passenger's fault for refusing to leave his seat when asked.
Her next example is about a certain comic strip. Often appearing in traditionalists' debates, it's about students who fail classes.
"In the panel for the good old days, a parent and child are in a teacher's office, and the teacher is scolding the student for their bad grades. In the panel for today, the image is the same except now it's the parent scolding the teacher for the student's bad grades."
Cheng's point is that the student's contribution and the teacher's contribution combine to cause the outcome -- as she shows us:
the student didn't work hard enough}
and } => the student failed the exam
the teacher didn't teach well enough}
The main idea of this chapter are logical connectives. Cheng writes:
"For example, 'The student didn't work hard enough and the teacher didn't teach well enough.' The connecting word here is 'and.' How could the student have passed? Perhaps if 'The student worked harder or the teacher taught better.' The connecting word is 'or.' These two words are the two basic connectives in logic."
The general situation is that given two statements A and B, we get two new statements by using "and" and "or":
- A and B are both true.
- A or B is true.
Cheng tells us that "and" works the same in logic and English -- although in English we might leave the word "and" out in certain situations. But the word "or" could be exclusive or, which means "one or the other but not both," or it could be inclusive "or." Here's her example:
"Similarly you have higher social status if you are rich or male (or both). You are an LGBTQI person if you identify as lesbian, gay, bisexual, transgender, queer or intersex, or more than one (for example, transgender and lesbian)."
Again, I remind you that Cheng writes about race and politics throughout her book. If you prefer not to read this, then I suggest that you avoid this blog for the next three weeks and skip all posts that have the "Eugenia Cheng" label.
In logic, we tend to use the inclusive "or," even though we might use exclusive "or" more often in speaking English.
At this point Cheng shows us some Venn diagrams. She illustrates the set of black women in the region where the circles for "black people" and "women" overlap:
"In the language of sets and Venn diagrams this is called the intersection, and in this case it consists of black women."
(By the way, lately we've heard the word "intersectionality" in recent news articles pertaining to the social sciences. I believe that the word "intersectionality" here goes back to exactly the sort of intersection that Cheng is describing here, as in "black women.")
In her next diagram, Cheng shows us the set of people who are either black or women:
"This gives us the region shown in the following diagram, which in the language of Venn diagrams is called the union of the two sets."
Unions and intersections -- hey, this is exactly what yesterday's Lesson 2-6 was all about! For the second time this week, Cheng and the U of Chicago are out of sync by one day. (If this occurs a third time next week, I'm seriously considering combining Chapters 1-2 and editing the posts so that the Cheng topics line up with U of Chicago lessons.)
Now Cheng shows us how to find the negation of a statement containing "and" or "or." In her first example, the set of white men (that is, those who are "white" and "men") is the intersection of the sets of white people and men. Its negation consists of three regions:
The author's next example is all about the education system. Once again, many of her ideas have been mentioned earlier in some traditionalists' debates. Cheng writes:
"The education system is full of problems, in my opinion. There are problems to do with funding, expectations, objectives, standards, and so on. Problems with the education system is where I believe math phobia comes from."
She points out that one source of this math phobia is the pressure placed on teachers -- pressure caused by standards and tests (such as Common Core and SBAC). But she adds:
"This causes children to lose interest in it, often towards the end of elementary school, when the math has become to hard for general elementary teachers to be comfortable with, but specialist math teachers have not yet been invoked."
Cheng now returns to the example of the United passenger. She repeats the simplest argument:
Lesson 2-7 of the U of Chicago text deals with polygons. Notice that this lesson consists almost entirely of definitions and examples. But this chapter was setting up for this lesson, since a polygon is defined (Lesson 2-5, Definitions) in terms of unions (Lesson 2-6, Unions and Intersections) of segments:
A polygon is the union of three or more segments in the same plane such that each segment intersects exactly two others, one at each of its endpoints.
It follows that this section will be very tough on -- but very important for -- English learners. I made sure that there is plenty of room for the students to include both examples and non-examples of polygons. The names of n-gons for various values of n -- given as a list in the text -- will be given in a chart on my worksheet.
The text moves on to define a polygonal region. Many people -- students and teachers alike -- often abuse the term polygon by using it to refer to both the polygon and the polygonal region (which contains both the polygon and its interior). Indeed, even this book does it -- when we reach the chapter on area. Technically, triangles don't have areas -- triangular regions have areas -- but nearly every textbook refers to the "area of a triangle," not the "area of a triangular region." Our text mentions polygonal regions to define the convexity of a polygon -- in particular, if the polygonal region is convex (that is, if any segment whose endpoints lie in the region lies entirely in the region), then the polygon itself is convex.
The text then proceeds to define equilateral, isosceles, and scalene triangles. A triangle hierarchy is shown -- probably to prepare students for the more complicated quadrilateral hierarchy in a later chapter.
Many math teachers who write blogs say that they sometimes show YouTube videos in class. Here is one that gives a song about the three types of triangle. It comes from a TV show from my youth -- a PBS show called "Square One TV." This show contains several songs that may be appropriate for various levels of math, but I don't believe that I've ever seen any teacher recommend them for the classroom. I suspect it's because a teacher has to be exactly the correct age to have been in the target demographic when the show first aired and therefore have fond memories of the show. So let me be the first to recommend this link:
Another song from Square One TV that's relevant to this lesson is "Shape Up." Notice that many geometric figures appear on the singer's head -- though not every shape appearing on her head is a polygon:
[2018 update: Today in subbing I taught angles, so "Angle Dance" is a Square One TV song that would have been relevant. I'll wait until Chapter 3 before posting that song. Hmm, I wonder whether I could have sung "Angle Dance" in class today as an incentive to get the students quiet, such as I used "Quadratic Weasel" in Algebra 1B back on June 5th. But I would have needed to be careful about how I used the song in order for it to be successful.]
Today is an activity day. Last year right after Lesson 2-7, I posted activities for the Daffynition Game and Jeopardy, and so I repeat those activities today. This is what I wrote two years ago about these activities:
And now I present my worksheet for the Daffynition Game. Remember that only one of these worksheets need to be given to each group -- in particular, to the scorekeeper in each group. The students write their guesses for Rounds 1-4 (or 5) on their own separate sheet of paper. I recommend that it be torn into strips so that they are harder to recognize. And the teacher provides the index cards, one for each student. Make sure that the students give back the index cards so you can reuse them for the next period. The students may keep their "guess cards," so there should be one for every student in every period.
The second page is for the Jeopardy game -- just as with the Daffynition game, there should be index cards, with the number of points on one side and the question (um, the answer, since the response is the question) on the other. In my class the questions were taped to the front board. In this version of the game, the categories correspond to the four lessons covered earlier this week. Of course, some lessons, such as Lessons 3-1 and 3-2 on angles, are tailor-made for Jeopardy, but unfortunately we haven't quite covered the lesson. Of course, we'll get there next week. I didn't include a Final Jeopardy Question, but here's a tricky one:
Final Jeopardy Category: Types of Polygons
If two points lie in the interior of this type of polygon, then the segment joining them lies in the interior.
In logic, we tend to use the inclusive "or," even though we might use exclusive "or" more often in speaking English.
At this point Cheng shows us some Venn diagrams. She illustrates the set of black women in the region where the circles for "black people" and "women" overlap:
"In the language of sets and Venn diagrams this is called the intersection, and in this case it consists of black women."
(By the way, lately we've heard the word "intersectionality" in recent news articles pertaining to the social sciences. I believe that the word "intersectionality" here goes back to exactly the sort of intersection that Cheng is describing here, as in "black women.")
In her next diagram, Cheng shows us the set of people who are either black or women:
"This gives us the region shown in the following diagram, which in the language of Venn diagrams is called the union of the two sets."
Unions and intersections -- hey, this is exactly what yesterday's Lesson 2-6 was all about! For the second time this week, Cheng and the U of Chicago are out of sync by one day. (If this occurs a third time next week, I'm seriously considering combining Chapters 1-2 and editing the posts so that the Cheng topics line up with U of Chicago lessons.)
Now Cheng shows us how to find the negation of a statement containing "and" or "or." In her first example, the set of white men (that is, those who are "white" and "men") is the intersection of the sets of white people and men. Its negation consists of three regions:
- white people who are not men
- men who are not white, and
- people who are neither white nor men.
In general:
(A and B) is false means A is false or B is false (or both).
She also negates her earlier example involving "or" -- the negation of the set "black or women" consists of those who are both non-black and non-women:
(A or B) is false means A is false and B is false.
To sum up:
Original statement: You are black and female.
Negation: You are not "black and female." So you could be black but not female, female but not black, or neither.
Original statement: You are black or female.
Negation: You are not "black or female." So you are not black and you are not female. In more natural language you are neither black nor female.
The author now moves on show how to assign blame. Cheng writes that if she breaks a glass she might say there were two contributing factors:
A: I dropped the glass.
B: The floor was hard.
The glass broke because of A and B. So to prevent the glass from breaking, we must invoke the negation -- A is false or B is false. This extends to more "ands":
(A and B and C and D) is false means A is false or B is false or C is false or D is false.
Cheng expands upon this idea:
"I heard an interesting talk by software developer Jessica Kerr who summed this up as understanding the system rather than blaming the individual. So instead of arguing about trying to attribute blame individually, it is more productive to understand how the system makes all those factors interact with each other to cause that outcome."
This applies, for example, when considering who's to blame for a divorce:
A: Alex had an affair.
B: Sam does not forgive Alex.
X: Alex and Sam split up.
Cheng points out that often we forget to consider other contributing factors:
C: they both refuse to go to couples therapy, or
D: they went but the couples therapist wasn't very good.
The author's next example is all about the education system. Once again, many of her ideas have been mentioned earlier in some traditionalists' debates. Cheng writes:
"The education system is full of problems, in my opinion. There are problems to do with funding, expectations, objectives, standards, and so on. Problems with the education system is where I believe math phobia comes from."
She points out that one source of this math phobia is the pressure placed on teachers -- pressure caused by standards and tests (such as Common Core and SBAC). But she adds:
"This causes children to lose interest in it, often towards the end of elementary school, when the math has become to hard for general elementary teachers to be comfortable with, but specialist math teachers have not yet been invoked."
Cheng now returns to the example of the United passenger. She repeats the simplest argument:
- It was United's fault for their unreasonable use of force.
- It was the passenger's fault for refusing to leave his seat when asked.
- The flight was unexpectedly overbooked.
- Some crew needed to get to Louisville to work on another flight, so the airline decided to remove people from the flight.
- Nobody accepted the offer of money to get off the flight.
- United decided not to offer more money.
- United chose a particular passenger to be asked to leave the flight.
- The passenger decline.
- United staff called security officers.
- The security officers used excessive force to remove the passenger.
She draws a fairly complicated diagram containing these and even more possible factors. Of course I won't include this diagram, but she does write:
"I think identifying important factors is an aspect of powerful rational thinking, related to knowing when to stop asking or explaining 'why' in a given situation."
And then the author adds in more complex diagrams explaining the result of the 2016 US presidential election (on the bottom of course is "Trump won the electoral college vote" => "Trump was elected") and why she gained weight ("take in more energy than I burn" => "gain weight").
"So who is to blame? It is possible futile even trying to answer that question. A better question is: who is going to take responsibility for changing it?"
And so Cheng concludes this chapter as follows:
"It is tempting to point the finger of blame at one factor or person, especially if that exonerates ourselves, but I believe it is much more productive to understand the connections of the system. Outcomes are always caused by whole systems, but we can still as individuals take responsibility for change."
Lesson 2-7 of the U of Chicago text is called "Terms Associated with Polygons." (It appears as Lesson 2-6 in the modern edition of the text.)
This is what I wrote last year about today's lesson:
This is what I wrote last year about today's lesson:
Lesson 2-7 of the U of Chicago text deals with polygons. Notice that this lesson consists almost entirely of definitions and examples. But this chapter was setting up for this lesson, since a polygon is defined (Lesson 2-5, Definitions) in terms of unions (Lesson 2-6, Unions and Intersections) of segments:
A polygon is the union of three or more segments in the same plane such that each segment intersects exactly two others, one at each of its endpoints.
It follows that this section will be very tough on -- but very important for -- English learners. I made sure that there is plenty of room for the students to include both examples and non-examples of polygons. The names of n-gons for various values of n -- given as a list in the text -- will be given in a chart on my worksheet.
The text moves on to define a polygonal region. Many people -- students and teachers alike -- often abuse the term polygon by using it to refer to both the polygon and the polygonal region (which contains both the polygon and its interior). Indeed, even this book does it -- when we reach the chapter on area. Technically, triangles don't have areas -- triangular regions have areas -- but nearly every textbook refers to the "area of a triangle," not the "area of a triangular region." Our text mentions polygonal regions to define the convexity of a polygon -- in particular, if the polygonal region is convex (that is, if any segment whose endpoints lie in the region lies entirely in the region), then the polygon itself is convex.
The text then proceeds to define equilateral, isosceles, and scalene triangles. A triangle hierarchy is shown -- probably to prepare students for the more complicated quadrilateral hierarchy in a later chapter.
Many math teachers who write blogs say that they sometimes show YouTube videos in class. Here is one that gives a song about the three types of triangle. It comes from a TV show from my youth -- a PBS show called "Square One TV." This show contains several songs that may be appropriate for various levels of math, but I don't believe that I've ever seen any teacher recommend them for the classroom. I suspect it's because a teacher has to be exactly the correct age to have been in the target demographic when the show first aired and therefore have fond memories of the show. So let me be the first to recommend this link:
Another song from Square One TV that's relevant to this lesson is "Shape Up." Notice that many geometric figures appear on the singer's head -- though not every shape appearing on her head is a polygon:
[2018 update: Today in subbing I taught angles, so "Angle Dance" is a Square One TV song that would have been relevant. I'll wait until Chapter 3 before posting that song. Hmm, I wonder whether I could have sung "Angle Dance" in class today as an incentive to get the students quiet, such as I used "Quadratic Weasel" in Algebra 1B back on June 5th. But I would have needed to be careful about how I used the song in order for it to be successful.]
Today is an activity day. Last year right after Lesson 2-7, I posted activities for the Daffynition Game and Jeopardy, and so I repeat those activities today. This is what I wrote two years ago about these activities:
And now I present my worksheet for the Daffynition Game. Remember that only one of these worksheets need to be given to each group -- in particular, to the scorekeeper in each group. The students write their guesses for Rounds 1-4 (or 5) on their own separate sheet of paper. I recommend that it be torn into strips so that they are harder to recognize. And the teacher provides the index cards, one for each student. Make sure that the students give back the index cards so you can reuse them for the next period. The students may keep their "guess cards," so there should be one for every student in every period.
The second page is for the Jeopardy game -- just as with the Daffynition game, there should be index cards, with the number of points on one side and the question (um, the answer, since the response is the question) on the other. In my class the questions were taped to the front board. In this version of the game, the categories correspond to the four lessons covered earlier this week. Of course, some lessons, such as Lessons 3-1 and 3-2 on angles, are tailor-made for Jeopardy, but unfortunately we haven't quite covered the lesson. Of course, we'll get there next week. I didn't include a Final Jeopardy Question, but here's a tricky one:
Final Jeopardy Category: Types of Polygons
If two points lie in the interior of this type of polygon, then the segment joining them lies in the interior.
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