Friday, April 20, 2018

Chapter 14 Test (Day 149)

Today I subbed in a high school special ed class. This teacher covers both Algebra I and English, and I believe I taught enough math to justify a "Day in the Life" for today.

7:55 -- There is no first (zero) period today, so we begin with second period. This class is Algebra 1B, the second year of a two-year Algebra I course for special ed sophomores. The students are working in Chapter 8 of the Glencoe Algebra I text, which is on polynomials. Today was supposed to be the Chapter 8 Test, but probably because of the subbing day, the teacher chooses to postpone the test until Monday.

The biggest classroom management management mistake I make today involves attendance. I'm supposed to take attendance on the computer, but the password supplied by the office fails. This means that I should contact the office to print out paper rosters. But rather than send someone to get them right away, I wait until midway during the period to get them. My decision is influenced by Harry Wong's book on classroom management (mentioned in old posts from last year) -- he writes that it's better to get the students working on an assignment than waste time with attendance.

Perhaps Wong's advice doesn't work in a special ed class. The lesson plan states that the students should work on a review worksheet first (similar to the Chapter 14 Review worksheet that I posted here yesterday) followed by some whiteboard review (similar to what I did with my middle school classes last year). I help some students individually during the worksheet phase of the class and wish to lead the whiteboard portion, but can't because now I'm worried about attendance. Wong's advice probably succeeds in a general ed class where the students can work independently (and I can try the password one last time before giving up and contacting the office).

A special ed aide arrives at 8:15 -- who is herself a substitute. She ends up walking to the office to obtain the attendance rosters and taking attendance herself, and then the whiteboard review begins.

As far as math is concerned, I think my one-on-one help to two students helped them out greatly on their questions on adding, subtracting, multiplying, and factoring polynomials. But the students may have been confused by my explanation on the whiteboard problems. Out of habit, I skip too many steps for the special ed kids to understand. The aide takes over and writes out extra steps -- especially those involving subtraction, which she explicitly writes as addition of a negative, even if we're subtracting a monomial.

8:50 -- Second period ends. Like all special ed teachers in this district, this regular teacher co-teaches another class. For third period, I travel to a junior English class. The main teacher in this class is showing her class a movie, The Great Gatsby.

9:45 -- It is now time for tutorial. I return to my classroom. Some students come in for help on the math review worksheet, and the aide and I guide them. I also help some general ed students out with their assignments, including some Algebra I students who also appeared to be factoring. But one guy who needs math help starts working on science instead, in preparation for a test later on today. And another girl also works on science.

During tutorial, the special ed teacher -- who's out due to IEP meetings -- comes in. I inform her that some of the students are confused during the whiteboard lesson, and that I'm unable to access the computer for attendance.

10:25 -- Tutorial ends and the students leave for snack. This is followed by fourth period, which is the teacher's conference period.

11:40 -- Fifth period begins. This is a junior English class. They are also reading The Great Gatsby, but instead of watching a video, they are working on a project where they research how much it costs to hold a party similar to Gatsby's. They are surprised how many thousands it will cost -- and keep in mind that in the novel, Gatsby holds parties every week!

12:30 -- It is time for lunch. But something is wrong with the clock, and the bells start ringing about 30 seconds late. So some students decide to leave before the bell rings without permission, as they check their phones to see what the actual time is.

1:15 -- It is now sixth period and the second Algebra 1B class. One student arrives about 10 or 20 seconds late -- and I mark him tardy and tell him that he's really 40 or 50 seconds late, since the bells are ringing late.

This time, in helping the students, I write out the steps that the aide wrote earlier. As it turns out, there's supposed to be a second aide for this class -- but he's also out, and there's no sub for him. I hope that the students are able to understand more in this class than in second period.

And here is my second attendance-related error. You see, when I leave names for the teacher, I often write down the names of absent students (even if the teacher doesn't request it). Since it's the aide taking attendance in second period, I never see the names of the absent kids, so I don't write them -- and because I miss second period, I don't write the names of absent students for any class.

For some reason, I mistakenly believe that one girl on the roster is independent study rather than in Algebra 1B, and so I never hand her a whiteboard. It doesn't help that she sits next to the student TA rather than in the rows with the other students. It turns out that the actual independent study student is a male who happens to be absent. What leads me astray is that the previous period has a female independent study student. It's a mistake I'm less likely to make if I've been writing down the names of absent students as I usually do -- in this case, knowing who's absent helps me learn the names of the kids who are present (especially in a small special ed class). Instead, I've already sent the completed rosters back to the office and I can't remember who's absent -- and so I mistake present students for absent students.

And this indirectly leads to the third error. During the last thirty seconds of class -- when the bell should have rung but didn't -- the students line up by the door and then pull a prank. They open the door and then claim that another student has left early. A few seconds later, the bell rings, and then I ask the TA about the "missing" student. It turns out that the "missing" student is actually absent, and so no student has actually left early. Again, I wouldn't have fallen for the prank if I'd written down the names of absent students. (Notice that this lining up and "leaving early" problem wasn't much of an issue last year. I ended my classes with Exit Passes, and there are no bells, hence there are no bells that are 30 seconds late.)

2:10 -- It is now seventh period. A few things are going on here. Officially, the regular teacher is the tennis coach, and the team is actually playing today, hosting another school. One student comes to the classroom, but leaves quickly when he remembers the game. Most likely, the teacher doesn't schedule any IEP meetings for this period, and proceeds directly to the tennis courts after her last meeting.

Instead, she wants me to stay in the classroom and wait for two girls who are required to take their science test in a quiet room such as this one. One of the girls is the one who was studying for her science test during tutorial.

But suddenly, a sports team arrives in the classroom -- the soccer team. This sport isn't in season, but for some reason this is the classroom they go to during the offseason. Eventually, their coach sends one player a text to inform them that they don't have to stay.

In the end, the girl finally shows up to take her test. The second girl is absent today.

2:35 -- The student completes her science test, and so I complete the active part of my subbing day.

Let's look at one of my subbing resolutions:

3. Move on from past incidents instead of bringing them up with students.

It's hard to say whether I truly followed this resolution or not. It can be argued that I must have brought up the incident about fifth period kids leaving early during sixth period. It didn't lead to any arguments, but it did set up the prank about the "missing" absent student.

I prefer to err on the side of being too hard on myself. And so I tell myself that I shouldn't have mentioned the fifth period incident in sixth period. Instead, find a way somehow to tell the sixth period kids not to leave early without mentioning fifth period.

Today is a test day, and so it's a traditionalists day -- as if I haven't written too much about traditionalists in recent posts. Well, the traditionalists have been quieter lately. Here's what Barry Garelick writes in his last post -- excuse me, his last missive:

https://traditionalmath.wordpress.com/2018/04/19/clarification-and-amplification-dept/

In my last missive (a pretentious word, I admit, but I dislike the word “post” and I absolutely detest the phrase “smart and thoughtful post”, so please tell people who use such phrase to shut the hell up) I said the following:
“It’s a brave new educational world we live in. I want no part of it, nor any of the damned PD that comes with it either.”
Someone applied an interpretation I didn’t intend and tweeted: “These are the teachers who are retiring in droves.”
Garelick's April 19th missive doesn't draw any SteveH comments, but the original one from April 17th (linked to in the quote) does:

SteveH:
They hate the basic algorithms of math, but they luuuuv coding. Why don’t they try writing a program that simulates multi-digit multiplication. Better yet, do it for any base.

By "they," SteveH most likely means the reformist pedagogists. But to me, it's the students who hate the basic algorithms of math. My students from last year enjoyed the Monday coding classes because it gave them something to do other than learning the basic algorithms of math. Students learn more from coding even simple tasks than from math p-sets that they throw in the trash without attempting any of the problems.

SteveH:
Also, for a computer science career, you have to be able to get through Calculus II in college along with linear algebra or matrices – at least. PBL is vocational education that can never make up JIT for missing subject unit knowledge and skills.

Well, I still don't know his opinion of Statistics, but here SteveH plainly states that Calculus is required for a Comp Sci career. And yet there are some students who might wish to have a Comp Sci career, yet they sit in their AP Calculus track classes wondering, "When will we use this in real life?" insinuating that Algebra and Calc are 100% irrelevant to their future Comp Sci career.

SteveH:
Those who can, do. Later, they teach high school AP Calculus track courses. (Most of my son’s high school math teachers were from industry.)

I've seen other traditionalists say something similar in their posts -- they imply that the best teachers are on their second career. Does this mean that schools should exclusively hire candidates who are over 40 with 15+ years experience in industry (at least for math), and all others need not apply? I'd like to hear more from SteveH and the traditionalists about that.

OK, it's time to jump into Wickelgren.

Chapter 5 of Wayne Wickelgren's How to Solve Problems is "State Evaluation and Hill Climbing." It begins with:

"In the last chapter we reduced the amount of trial-and-error search in a problem by constructing equivalent state-action trees of reduced size. In this chapter, we discuss a very different way of reducing the number of state-action sequences that have to be searched before achieving the solution."

And this new way of solving problems is called hill climbing:

"This example arbitrarily uses an integer-valued evaluation function, with the beginning state having value 0, the goal state having value 10, and nongoal states having values intermediate between 0 and 10."

Wicklegren now draws a tree filled with these evaluation states. At each branch of the tree, we select the branch with the higher value. This is what he calls "hill climbing." In his example, the first decision point leads to branches labeled 3 and 4, so we continue along the branch labeled 4. Now eventually, the final leaf we land on is 9 -- the 10 goal leaf is on the 3 branch. The author tells us that more effective methods can help us reach the 10 leaf, such as looking ahead two levels, not one:

"This two-step hill climbing method would produce the goal the first time in the problem shown in [the figure]. Finally, you could question the evaluation function you had defined over the states in the problem."

For example, perhaps a single number isn't sufficient. Instead, we might use a pair of numbers to describe each state -- in other words, a vector (which fits here in Chapter 14).

"When the first hill-climbing path through the state-action tree fails to produce the solution, these nodes where you had good alternately choices are the obvious places to back up to and start new paths. Another approach to multidimensional evaluation functions is to combine the values on the separate dimensions into a single overall value for each state."

But this might take us right back where we started, depending on the particular problem.

Wicklegren's first example is a road trip, where the evaluation function is distance from the goal:

"Of course, choosing the road at the beginning of a trip that goes closest to the right direction may prove to be a bad choice. This road may eventually lead to a dead end or require you to go far out of the way to reach the goal."

Many of the examples in this chapter come from earlier in the book. Indeed, Wicklegren now repeats the one-heavy-coin problem:

https://www.algebra.com/algebra/homework/word/numbers/Numbers_Word_Problems.faq.question.324693.html

The author tells us that there are several possible evaluation functions. One might be the number of coins whose weight is known after each weighing -- we start with 0 and the goal with 24. The possible branches from the first step are "weigh 1 coin on each side," "weigh 2 coins on each side," all the way up to "weigh 12 coins on each side." The hill climbing node is "weigh 8 coins on each side," where the evaluation function is 16 known coins.

Wickelgren's next example is from Algebra I: solve 9x + 7 = 5x + 15. Here the author imagines that there's a four-valued vector evaluation function, with (9, 7, 5, 15) as the initial state. The goal state is (1, 0, 0, ???), where ??? is the final solution.

Wickelgren now returns to the six-arrow problem:

You are given six arrows in a row, the left three pointing up, and the right three pointing down. The goal is to transform these arrows into an alternating sequence such that the leftmost arrow points up, the next arrow to it points down, the next up, then down, then up, and then down. The actions allowed are to simultaneously invert (turn upside down) any two adjacent arrows. Note that you cannot invert one arrow at a time but must invert two arrows at a time, and the two arrows must be adjacent. Achieve the solution using the minimum number of actions (inversion of adjacent pairs.

"Stop reading and try to define three different evaluation functions that might be relevant to solving this problem by hill climbing, then read on."

  • The most obvious evaluation function is probably the number of arrows that are in the same position as in the goal state. This evaluation function starts out at four in the given state and ends at six in the goal state.
  • A somewhat different evaluation function that is considerably more useful in solving the problem is to count the number of runs of arrows (consecutive arrows with identical orientation). This evaluation function starts out at two runs for the beginning state and ends at six runs for the goal state.
  • However, the evaluation function that is optimal in conjunction with the hill-climbing approach to this problem is to consider the distance between the two incorrectly placed arrows and attempt to reduce that distance. Note that, in the given state, the value of this evaluation function is 3, and the successive actions in a correct solution to the problem can reduce this to 2 and then to 1, from which the final action is obvious.
Wickelgren calls his next example the discrimination reversal problem:

In the one-dimensional world of Lineland, there are two races of "people," whites and blacks. As in our three-dimensional world, the whites for a very long time discriminated against the blacks....

Hmm, I've unexpectedly mentioned race in this school-year post. Well, I'm protected by the "traditionalists" label, which is coming in handy yet again.

Actually, the description of this problem is much too long for me to type. The following problem I found on line is similar, not identical to Wickelgren's problem:


Notice that at this link, the problem is stated without invoking Rosa Parks, one-dimensional Lineland, or other elements from Wickelgren's problem. Since I can't find the exact problem online, and this post is growing in length, let's just skip this problem altogether.

(Hmm, I unwittingly stumbled upon someone's blog at the above link. I noticed that one of the featured posts there is relevant to the traditionalist debate -- the author agrees with the traditionalists that memorization is a good strategy for learning mathematics, especially at the elementary level.)


OK, let's get back to Wickelgren. His next example involves a simple chess endgame, which I can easily describe:

White: king at e6
Black: king at e2, rook at a1
Goal: Black to move and mate in as few moves as possible

"Stop reading and try to think of at least one evaluation function that is relevant to this objective and that might dictate the choice for black's first move in the present instance."
  • The most obvious objective of black is to minimize the number of squares to which the white king can move without being in check.
  • It is more useful to consider the evaluating function to refer to minimizing the number of squares to which the white king might ever be able to move (that is, minimizing the number of squares reachable by a sequence of several moves).
According to the second evaluation function, 1...Ra5 is the proper move.

The author returns to another previously given example, Instant Insanity:

https://www.jaapsch.net/puzzles/insanity.htm

"One natural four-dimensional evaluation function might be the number of different colors you achieved on each side of the tower. Hence, we may consider the beginning state to have the evaluation vector (0, 0, 0, 0) and the goal to have the evaluation vector (4, 4, 4, 4)."

But unfortunately, many nodes that have evaluation (3, 3, 3, 3) have no path that leads to the final goal mode (4, 4, 4, 4). It's likely that you'll be "close" to a solution and yet nowhere near actually reaching it. It's enough to drive players insane -- hence the name of the game.

The final example in this chapter is the famous missionaries-and-cannibals problem:

http://www.aiai.ed.ac.uk/~gwickler/missionaries.html

On one bank of a river are three missionaries and three cannibals. There is one boat available that can hold up to two people and that they would like to use to cross the river. If the cannibals ever outnumber the missionaries on either of the river’s banks, the missionaries will get eaten.
How can the boat be used to safely carry all the missionaries and cannibals across the river?

"Stop reading and try to solve the problem by explicitly defining some evaluation function and using a hill climbing approach, then see [the main figure at the link above] for a sequence of states that solves the problem."

As it turns out, if we choose the evaluation function "the number of people on the other side of the river," hill-climbing doesn't work! A detour step is necessary -- this is the step shown on the bottom of the diagram. One missionary and one cannibal already on the goal side of the river must sail back to the initial side.

Wicklegren concludes the chapter as follows:

"Some examples of this combined use of hill climbing and subgoal methods will be discussed in the mathematics, science, and engineering problems of Chapter 11."

Today I'm posting the Chapter 14 Test. According to the digit pattern, I should wait until Monday to post the test, but I wanted to avoid a Monday exam. Ironically, today I subbed for a teacher who, due to her absence, delayed her Algebra 1B test from today to Monday! (I wonder with all the help the sub aide and I gave the students today, how many of them will forget what we taught them at some point over the weekend!)

This is what I wrote last year about today's test:

Today is approximately the end of the fifth hexter -- the midpoint of the third trimester -- so it's a good test day.

Today is the Chapter 14 Test. Here are the answers to my posted test:

1. DE = 32, EF = 16sqrt(3).

2. TU = 16, US = 8sqrt(3), SK = 8, TK = 8sqrt(2).

3. 3/4

4. 3/5

5. 0.309

6. 0.625

7. 1/2

8. sqrt(3) (Some people may consider this question unfair, since the above question and both corresponding questions on the practice had rational answers, leading students to believe that they can just use a calculator to find the exact value rather than use 30-60-90 triangles.)

9-10. These are vectors that I can't reproduce easily here.

11. BC/AC (or a/b, if the students learned it that way).

12. AB and AD

13. ACD and CBD

14. This is a vector that I can't reproduce easily here.

15. (9, 6)

16. 115 feet, to the nearest foot.

17. (1, 4)

18. (3, -3)

19. (3, 2). (I hope students don't get confused here and solve these three backwards!)

20. This is a vector that I can't reproduce easily here.

Thus concludes Chapter 14. Stay tuned -- we're starting Chapter 15 on Tuesday, after an activity day on Monday!



No comments:

Post a Comment