## Monday, April 23, 2018

### Activity: Vectors (Day 150)

Today is the midpoint of the third trimester -- at least, it would be if the blog calendar wasn't based on a high school district that lacks trimesters. In previous posts, I've referred to this day as the end of the fifth "hexter." (It's uncertain whether the "hex-" in "hexter" means six weeks of 1/6 of a school year.)

Meanwhile, I subbed in a middle school special ed math classroom -- so obviously it's not in the high school only district. And in this district it's only Day 143, so it's not the end of the hexter yet, but it's likely close.

Actually, I've subbed in the class before -- back on March 13th. Thus this is my first repeat visit to a math class since I was hired in this district. You might wish to compare this "Day in the Life" below to my March 13th post.

8:15 -- Officially, this is the start of homeroom, but today's schedule is different. Back in my Pi Day post, I mentioned how there was supposed to be a diagnostic exam for high school placement. Well apparently, that test isn't given until today, over a month later! I suspect the reason for the delay is that a special block schedule had to be created for the day.

Here's how it works -- first, second, and third period are 70 minutes each, and the test is given to eighth graders during those periods. After lunch, the remaining classes are 35 minutes each. Then tomorrow, the schedule reverses so that fourth through sixth periods are 70 minutes, allowing the students who have math then to take the test.

Technically speaking, there is still a 10-minute homeroom before the 70 minutes of first period. (As I wrote in that March 13th post, the kids are always the same for homeroom and first period.) But the special aide and I choose to forego homeroom announcements and go straight to the test so that the students have an extra ten minutes to finish it.

The test is given on Chromebooks. Students are not allowed to use a calculator for this test. So we figure that many special ed students will struggle with it. Low-scoring students will likely be placed into a supplemental math class as freshmen -- or possibly into Algebra 1A (as explained in last Friday's post).

9:35 -- First period ends. As I explained in my March 13th post, this special ed teacher has two periods of co-teaching -- second and sixth periods. Second period is another eighth grade math class, and so they're just taking the placement exam as well.

The resident teacher has me help her set up the test. I notice that she passes out multiplication tables for the students to use on the test, and I tell her that maybe my special ed classes would have found these tables handy as well.

When the test begins, she informs me that she no longer needs me in the room. And so I end up having an unexpected break -- one that lasts until snack.

11:05 -- Third period is a seventh grade class, so there's no high school placement exam. To cover the 70-minute period, the aide begins by showing them a short video on decimals and percents. Then she hands the students a worksheet where they must perform the simple conversion. As you would expect, the difficult problems are decimals greater than 1 (100%) and numbers with just one decimal (multiples of 10%). Students spend the rest of the time using ALEKS software.

12:15 -- Many students in third period are loud, and so we must keep them in for three minutes of their lunch. A trio of troublemakers must stay in longer to discuss their behavior with the aide. For me, the lunch break lasts all the way through the teacher's conference period.

1:40 -- Fifth period is another eighth grade class, but now these are the periods that don't take the test until tomorrow. So instead, the students spend the time finishing and making up work. Keep in mind what I said at the top of this post about fifth hexter progress reports.

The main assignment for the students is the pizza project -- the same functions project that the gen ed eighth grade class I co-taught on March 13th was about to start. I just realized that I never mentioned this project on the blog until now -- the students are supposed to create linear functions where the slope is the price per pizza and the y-intercept is the delivery fee.

Some students are also working on a graphing activity. Unlike the March 13th graphing activity, this one really is in the same vein as "Cartesian Cartoons," although it's from another publisher.

2:15 -- The fifth period students leave, and so I head to the co-teaching classroom. Again, these students aren't testing, and so the resident teacher begins a new lesson on translations. It's obvious that these are the first transformations that she's teaching them -- and indeed, last year I began my eighth grade class with translations. As I wrote last year, at the eighth grade level it's best to start with translations as these are the easiest to understand. It's only in high school Geometry where we can't start with translations -- especially if we're following the U of Chicago text, where a translation is defined as the composite of two reflections (in parallel lines).

2:55 -- Sixth period leaves -- except for three students whom the resident teacher keeps because they try to pack up early rather than begin the homework.

Let's look at today's focus resolution:

2. Keep a calm voice instead of yelling at students.

Notice that I keep returning to the first three resolutions instead of following my original plan to rotate through the resolutions in order. Well, some of the later ones are irrelevant -- I guess you can argue that I upheld the seventh resolution, which relates to computer testing (but it's not quite the same, as it's not the SBAC):

7. If there is an official assignment to review for state testing, then implement it fully.

Anyway, I want to look at the second resolution, especially considering that the first time I subbed in this room, I raised my voice.

The interaction that may have crossed the line into arguing/yelling is at the end of third period. One of the three students whom the aide tells to stay tries to sneak out of the room. And so I call out to him loudly in order to get him to stay. All throughout the period, this student has been goofing off. He also asks for a restroom pass, even though a) this is, technically, the period after snack (the lone period between snack and lunch) and b) according to the aide (who travels to other special ed classes), this same student had asked for a pass during the previous period, a science class. Notice that once again, this student could have used the restroom twice without missing a single second of class (during passing to second period, and then snack). Fortunately, the regular teacher doesn't have a strict restroom policy, and I'm not as strict on testing block days (especially semi-unexpected testing block days). Anyway, today's argument is about skipping lunch detention, not using the restroom.

Last year at my old school, I also remember yelling to students who try to skip out on detention -- in this case, it was a group of seventh graders who wanted to escape an after school detention assigned by the English teacher (which I had to enforce as I had seventh grade last that day). I feel like yelling in these situations -- probably because I fear I'm letting another teacher down if I let the student escape (either the English teacher last year or the special aide today).

If I'm teaching in my own classroom, the best thing to do if a student evades detention is just to make a parent phone call. We know that the ideal classroom manager avoids parent phone calls, but the ideal manager avoids yelling even more. If I assign a detention and the student evades it, then it's the student's fault, not mine. I don't want to make excessive phone calls, but in this case there is a level of punishment between a warning and a phone call (namely the detention).

Seeing the other teacher teach eighth grade math also makes me wonder what could have been possible in my eighth grade class last year. In the sixth period class, she hands back the previous tests, now graded. This test covers many geometry-related standards, including the Pythagorean Theorem and the angle relations (corresponding, alternate interior, etc.). She congratulates the class for doing so well on the test, especially one girl who earns a score of 108%.

On the other hand, my eighth graders fared poorly on these standards last year. The reason wasn't because these are difficult topics -- again, the high scores in this class tell us that they aren't. The real problem is that the students had neutered the "no talking" rule, so that by the time I reached the geometry-related standards, the kids were ignoring me completely. Maybe my students wouldn't have scored 108% on my test (especially if there's no bonus question), but many more should have at least passed those tests, rather than stare blankly at them not knowing how to do Question #1.

Also, notice that this teacher covered the angle relations before transformations. I still wonder whether it's helpful to teach transformations first, so that students can visualize corresponding angles as an angle and its translation image, and so on. But I can't be sure, since management issues dominated the class.

The teacher also returned the pizza projects -- the special ed kids are wrapping up their projects, while the gen ed kids are getting them back graded. These grades aren't as good -- and as we know, the significance of such projects is that they prepare students for the SBAC Performance Tasks. I know that traditionalists complain about such tasks -- smart students can do well on the traditional tests but struggle on the Performance Tasks. (Then again, the girl who scored 108% also earned a perfect score on the pizza project, so go figure.)

Speaking of traditionalists -- no, I'm not trying to establish the habit of posting traditionalists on both the Friday test day and the Monday activity day. But once again, Barry Garelick has posted over the weekend, and this post has drawn eight comments:

I get fan mail from time to time and invitations to speak that most of the time never come to fruition.  One such invitation came from the treasurer of a Catholic school in the Los Angeles area.  He had read my book “Math Education in the US: Still Crazy After All These Years” and liked it so much that he ordered ten copies for various teachers in the school.

One time, however, he said that he thought teachers are born not made, and wondered what my opinion was on the matter.  I said I disagreed and that I had learned a lot about teaching techniques from articles I’ve read from reliable sources, (and including talks I heard at a researchED conference that I attended). One can always improve one’s teaching if one has the inclination–there is always something to be learned.  He apparently didn’t like this, and I never heard from him again.

Even though SteveH posted his usual comment, I wish to focus on another commenter first:

Chester Draws:
I think that many people are unsuited to be teachers, and no amount of experience will make them good teachers. I’ve seen good meaning, hard working people fail utterly because basic skills were absent — an inability to command a room, not enough intelligence, poor temper control are not something you can wish away.

That last one -- poor temper control -- goes right back to the second resolution. I wrote that resolution because my yelling isn't something I can wish away. As Draws informs us, no amount of experience will make me a good teacher unless I gain this important skill.

Chester Draws:
I’ve seen good teachers give it up because they can’t cope. Often progressives make this worse though, because demanding that you “care” is a strain on someone who has a life outside the classroom they competes for their attention. (Lots of educationalists who push really intense workloads, such as Dan Meyer, have very short spans actually teaching, so don’t see the cost first hand.)

Hmm, it's rare to see the former King of the MTBoS, Dan Meyer, mentioned in any traditionalist debate thread. Of course Meyer is a progressive -- and it's been a while since I've seen traditionalists call their opponents by the term "educationalists." I'm not quite sure exactly what criticism Draws is making of Meyer -- what does it mean to say that Meyer and those like him "have very short spans actually teaching"?

SteveH:
You can have highly competent content specialists who are not good teachers, but it’s much tougher to have process-only teachers be good at content and skills – especially in K-6 math. Our state requires subject certification and student academic class separation starting in 7th grade, and that’s the beginning of the transition to the real world reality of high school, college, and career. For too many, however, it’s tough to recover from K-6. I had bright 7th grade students in my opt-in after school SSAT class who told me that they were just stupid. My son told me in 8th grade that he really knew very little about history and how now the subject was taught completely differently – not thematically.

And of course, today's pizza project merely adds fuel to SteveH's fire. He implies that these seventh graders feel they're "stupid" because they struggled in project-based elementary math, yet would have succeeded had they been taught traditional math. Likewise, today's eighth graders didn't do as well on the Performance Tasks but excelled on the traditional test.

Then again, no one claimed that Performance Tasks are easy. The real debate is whether such difficult projects are worth doing, or they should be dropped in favor of a traditionalist curriculum.

Chapter 6 of Wayne Wickelgren's How to Solve Problems is called "Subgoals." Here's how it begins:

"A problem-solving method that is important but difficult to master is that of defining subgoals in order to facilitate solving the original problem. This method is sometimes called 'analyzing a problem into subproblems,' or 'breaking up a problem into parts.'"

Wickelgren explains why subgoals are important. He tells us to consider a state-action tree for a problem that requires n steps to solve, with m possible actions at each node:

"By systematic trial and error there are m^n alternative paths (action sequences) to be investigated in the original problem."

But, as he tells us, if we divide the problem into two subgoals, there would be two smaller problems each requiring only m^(n/2) steps to solve. If m and n are big, this is a huge improvement. On the other hand, some problems require clever insight instead of subgoals:

"These include problems in which one must represent the components of the problem in some suitable way, guess the correct set of givens (where there are multiple given states), or choose a solution approach that violates hill climbing but that requires choosing from among only a small number of action sequences, once the insight has been achieved."

Let's focus on subgoal problems -- so what is a good subgoal anyway?

"Although there is no method of defining plausible subgoals that is mathematically precise and that applies to every problem, you can take the first step by defining an evaluation function over different problem states, as was done as a necessary precondition in applying the hill-climbing method."

Wicklegren warns us:

"When you have defined two or more subgoals to be achieved in getting from the given state to the goal, you can make a logical distinction as to whether the subgoals bust necessarily be achieved in a certain order or whether they can be achieved in any order."

The author continues along this line:

"Being aware of the distinction between ordered and unordered subgoals permits greater flexibility in the solution of problems involving unordered subgoals. If you have defined n subgoals (whether ordered or unordered), you have automatically defined n + 1 subproblems to be solved -- namely, getting from one of the givens to one of the subgoals, getting from the first subgoal to the second subgoal, and so on, from the nth subgoal to the goal."

Wicklegren's first example of a subgoal problem involving making a trip, where we must pass through several towns along the way:

"Obviously, the subgoals are ordered in these trip-planning problems. As an example of a problem that is extremely easy to solve using the subgoal method (with unordered subgoals), consider the following."

http://mathcentral.uregina.ca/QQ/database/QQ.09.04/barb1.html

Nine men and two boys, trekking through the jungle, need to cross a river. They have a small inflatable boat and it's easy enough to row it across the river. The boat, however, can hold no more than one man or the two boys. How can they all get across?

"Stop reading and try to solve the problem, using the subgoal method."

Notice that at the above link, a hint is given in the question. The hint is, in fact, a subgoal:

(Hint. suppose there was only one man and two boys)

Wickelgren tells us that this subgoal requires four crossings (both boys, one boy, man, other boy), so repeating the subgoal nine times takes 36 crossings. Then the 37th crossing is so that the both boys can cross the river one last time.

This post is already bloated, and I want to hurry up and get to the Geometry problems in this chapter, but there's one non-Geometry problem that I just can't resist giving -- the Tower of Hanoi, which is discussed at the Brian Harvey Logo website. I definitely want to provide this link, since it's a highly visual problem:

http://people.eecs.berkeley.edu/~bh/v1ch8/recur2.html

One of the most famous recursive problems is a puzzle called the Tower of Hanoi. You can find this puzzle in toy stores; look for a set of three posts and five or six disks.

"Stop reading and try to solve the problem."

Wickelgren's example has six disks, but Harvey only uses five. But Harvey is trying to program Logo to find the solution, so of course he divides the task into subtasks:

We want to end up with all five disks on post B. To do that, at some point we have to move disk 5 from post A to post B. To do that, we first have to get the other four disks out of the way. Specifically, "out of the way" must mean onto post C.

The first part of the solution is to move disks 1 through 4 from post A to post C. The second part is a single step, moving disk 5 from post A to post B. The third part, like the first, involves several steps, to move disks 1 through 4 from post C to post B.

If you've developed the proper recursive spirit, you'll now say, "Aha! The first part and the third part are just like the entire puzzle, only with four disks instead of five!"

OK, it's finally time for Geometry:

"Given the parallelogram ABCD illustrated in [the figure], prove that the perpendiculars AE and CF drawn to the diagonal BD are equal. Stop reading and try to solve the problem by defining a relevant subgoal."

Wickelgren tells us that one way to prove segments equal it to prove that they are corresponding parts of congruent triangles -- in other words, CPCTC. So our first subgoal is to prove that Triangle ABE is congruent to Triangle CDF. (We also could have chosen AEB and CFB.) "Stop reading and attempt to solve the problem, possibly by defining a further subgoal.)

And the author does define a further subgoal -- prove Triangles ABD and CDB are congruent. "Stop reading and attempt to solve the problem, using this sequence of two goals."

Here is Wickelgren's proof (and no, I won't convert it to two columns since this post is already running so long): Triangles ABD and CDB are congruent by SSS. Angles ABD and CBD are congruent by CPCTC. Triangles ABE and CDF are congruent by AAS. So AE = CF by CPCTC, and the problem is solved.

Let's try another Geometry problem:

"Given the circle illustrated in [the figure], proceed from the circumference along a diameter of the circle for an arbitrary unknown distance, to point A, then turn perpendicular to the radius and draw a line connecting the radius to the circumference, point B. Then erect another perpendicular at B until it intersects, at point C, the diameter perpendicular to the original diameter. The diameter of the circle is 100 feet. Determine the length of the line AC. Stop reading and try to solve the problem by defining relevant subgoals."

It might be tricky to visualize this problem unless we draw the picture. Wickelgren tells us that a subgoal would be to find a connection between AC and a segment whose length is known -- either a diameter or a radius. The ultimate trick is that ABCO is a rectangle with diagonals AC and OB -- and the diagonals of a rectangle are congruent (easily proved using triangle congruence). So AC = OB, and OB is a radius of length 50. Therefore AC = 50.

Wickelgren's final example in the this chapter is the proof of the theorem that the sum of the first n integers is n(n + 1)/2. His proof uses an ingenious use of the subgoal method -- mathematical induction, which I've discussed in many posts:

http://www.9math.com/book/sum-first-n-natural-numbers

Skip directly to Proof 3, since this is the induction proof.

"Step (1) establishes that the theorem is true for n = 1, and step (2) establishes that if the theorem is true for n, it is true for n + 1." (Note: I changed this line from Wicklegren to match the proof given at the above link -- in the book, the author actually shows that if it's true for n - 1, it's true for n.)

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

Today I finally post the vector activity that I've been planning this week.

There is also a page to cut out with 36 given vectors -- since as I mentioned earlier, I don't want the students to choose the vectors. Of course, cutting out the vectors is time consuming -- even if the teacher does it before the class -- and those tiny slips of paper are easily lost. Then teachers will have to cut them out several times throughout the day.

Another way to have the students choose vectors would be to have something larger represent the vectors, such as playing cards. The playing cards can be converted to vectors, as follows:

For the horizontal component:
Ace through 10 -- valued 1 through 10
Jack -- valued -2
Queen -- valued -1
King -- valued 0

For the vertical component, use the suit:
Clubs -- valued -1
Diamonds -- valued 0
Hearts -- valued 1

Examples:
Eight of Diamonds -- (8, 0)
Ace of Clubs -- (1, -1)
Jack of Hearts -- (-2, 1)

Because they are larger, playing cards are less likely to be lost than the little slips of paper that I provide for this activity. But there are problems with using playing cards. First of all, the conversion from playing card to vector is another step that the first partner can get wrong -- and once again this may frustrate the second partner. (I've heard that some people don't even know the difference between clubs and spades!) Furthermore, vectors with large components, such as (8, 0) for the eight of diamonds above, become (30, -2) after performing the steps in Task Three -- and then they are asked to graph that vector (30, -2) in Task Four. So I leave it up to individual teachers whether to use playing cards or the slips of paper that I provide.