Palm Sunday Post: Miscellaneous Ideas from Traditionalists

Table of Contents:

1. Pappas Problem of the Day
2. The Tweeter "Common Core Math (even if your state renamed it)"
3. The Remaining Two-Year Gap
4. The Path Plan Revisited
5. Another Geometry Problem
6. The Tracking Debate
7. An Analogy
8. Conclusion

Pappas Problem of the Day

Today on her Mathematics Calendar 2018, Theoni Pappas writes:

Arc ABC is 320 degrees. Find the area of the shaded sector.

(The circle's radius is 15sqrt(pi)/pi. The shaded sector is bounded by minor arc AC, not a major arc.)

Oddly enough, Lesson 8-9 gives only the area of an entire circle, not a sector -- even though Lesson 8-8 does teach arc length. There are a few examples on sector area in the corresponding lesson of the modern Third Edition of the text. The idea, of course, is to multiply the area of the circle by 40/360 -- not 320/360, since the arc is bounded by minor arc AC = 360 - 320 = 40 degrees:

Area of the shaded sector = 40/360 * (area of Circle O)
                                          = 1/9 * pi(15sqrt(pi)/pi)^2
                                          = 225/9
                                          = 25

So the area is 25 square units -- and of course, today's date is the 25th. This is another one of those questions where we must start with a unnatural radius (15sqrt(pi)/pi) in order to make the answer come out to a natural number.

This is my first spring break post. As I often do during vacations -- but seldom do during the school year -- I post on the weekend. Today is Sunday the before Easter -- also known as Palm Sunday.

No, I didn't choose today to post because of Pappas. As it turns out, Pappas actually has a whopping five Geometry posts this week, and I won't post five times this week just to get them all. Indeed, five posts would make this feel more like a regular school week, not a vacation week.

And besides, I already skipped posting the Pappas question on a school day. Back on March 14th, Pappas had a Distance Formula question. To me, it's a borderline case whether to count a Distance Formula problem as Geometry. Of course, we do see this formula in Lesson 11-2 of the U of Chicago text, but then again, it's studied in several other courses, not just Geometry.

In the end, I chose not to post it because that day was Pi Day. My post was already jam-packed with so much other stuff (the Pi Day songs I played in class that day, the walkout that occurred on Pi Day, the regularly scheduled lesson). And so I didn't want to add something else to that post that wasn't directly related to pi. Had today's problem occurred on Pi Day instead (with the final answer changed from 25 to 14, of course), then I would have posted it that day after all. Thus I'll decide which Pappas problems to write about and which ones not to write about.

The Tweeter "Common Core Math (even if your state renamed it)"

Most Blogger and Twitter users who use the labels "MTBoS" or "I teach math" aren't traditionalists. I do know of exception -- "Common Core Math (even if your state renamed it)." This tweeter (whose gender isn't public) is highly critical of the standards -- and, as the username implies, that criticism begins with states who no longer use the name "Common Core," yet made only insignificant modifications to the standards:

https://twitter.com/CCSSIMath

It definitely appears that the user CCSSIMath (so I don't keep writing the entire username over and over, I'll just use the name in the URL instead) -- is a traditionalist. Let's look at CCSSIMath's pinned tweet, dated January 8th:

On the left, a "released" 2016 @PARCCPlace 10th grade geometry test question. On the right, two pages from a Japanese 5th grade #math textbook. Now, your best arguments how #CommonCore is: (1) internationally benchmarked (2) rigorous

In case you didn't click the Twitter link above, let me describe it. Both lessons are nearly identical -- they show a derivation of the circle area formula (another tie to today's Pappas question). A similar diagram appears in Lesson 8-9 of the U of Chicago text. The difference, according to CCSSIMath, is that the American question is for sophomores, while the Japanese question is for fifth graders.

Now, of course, we see what the intended answers for (1) and (2) are above. We're to conclude that the Common Core Standards, despite claims to the contrary, are neither internationally benchmarked nor rigorous. And if internationally benchmarked and rigorous standards are desired, then the first thing to do is immediately include derivation of the circle area formula as a fifth grade standard under the Common Core.

We already know about other standards which traditionalists believe should be taught in lower grades than specified under the Common Core. These include the standard algorithms for arithmetic. And we already know the solution for this -- replace all mention of "strategies involving place value" with "the standard algorithm" and skip the corresponding standard in the higher grade.

But this one is trickier, since it's Geometry, not arithmetic. In fact, just before January 8th, the tweeter CCSSIMath made several other posts comparing Common Core Math to Japanese math. In each case, the Asian nation teaches in upper elementary school what the Common Core teaches in high school -- and more often than not, this material is related to Geometry. Since this is a Geometry blog, I want to discuss this in more detail.

Let's look at the pinned tweet again. First of all, notice that the circle area formula isn't a high school standard, but a seventh grade standard:

CCSS.MATH.CONTENT.7.G.B.4
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

Indeed, the U of Chicago Transition Math text for seventh grade derives this in Lesson 12-5. So there's really only a two-year gap between the U.S. and Japan, not a five-year gap. So why then does the circle area derivation appear in high school Geometry?

(By way, notice that Pi Day was Day 128 on the blog calendar. If we were following the seventh grade text on the blog, then Lesson 12-5 would have been taught on Day 125, just in time for Pi Day.)

There are several things going on here. First of all, let's what David Joyce says about this:

Theorem 5-12 states that the area of a circle is pi times the square of the radius. A "work together" has students cutting pie-shaped pieces from a circle and arranging them alternately to form a rough rectangle. That idea is the best justification that can be given without using advanced techniques.

By "advanced techniques," Joyce means Calculus. So he has no problem with this informal proof being given in Geometry (and indeed the Common Core admits that it's an informal derivation). The problem CCSSIMath has is that the derivation is given several years too late.

I think the problem here is that Geometry is all about proofs. This means that, as David Joyce puts it, "Every theorem should be proved, or left as an exercise, or noted as having a proof beyond the scope of the course. " But he says nothing about results from before the scope of the course. If a Geometry text leaves out the derivation of the circle area formula -- or more to the point, the triangle area formula -- then the text isn't rigorous, even if the result was proved in an earlier grade. In fact, let's give this as the answer to CCSSIMath's second question:

Now, your best arguments how #CommonCore is: (2) rigorous

Answer: Common Core Geometry is rigorous because all theorems require proof. This includes proofs of results learned in earlier grades. The derivation shown on the PARCC is admittedly informal, but a true derivation requires ideas from Calculus and beyond. And so this proof is, as Joyce puts it, the best justification that can be given at the level of high school Geometry.

The Remaining Two-Year Gap

OK, so now we still have CCSSIMath's question (1) to answer. Even if we admit that the circle area derivation is a seventh grade, not a tenth grade, standard, then there's still a two-year gap between the American and Japanese lessons.

I think back to SteveH and his notions that K-6 math is a "fantasy land" based on a "low slope," while high school math makes the "nonlinear jump" to AP Calculus. Of course, there's some truth to this.

Arithmetic is generally taught by elementary school teachers with Multiple Subject credentials. Many of these teachers choose their profession because they like kids -- not because they like math. These teachers, back when they were young students, might have earned grades of C (or even lower) in their own math classes, and now they must teach rigorous math to their own students.

So the Common Core and other state standards try to take the burden off of elementary teachers by delaying rigorous topics such as the area of a circle until seventh grade, when they're more likely to be taught by a math specialist. But if we want to follow CCSSIMath's suggestion that the standards should be "internationally benchmarked" to Japan, then we must push more difficult math into elementary schools with their Multiple Subject teachers. And here I mean topics like deriving the area of a circle, not just teaching standard algorithms earlier.

Is there a solution to this? Let's double-check what SteveH says about this:

https://traditionalmath.wordpress.com/2016/11/15/whatever-gets-you-through-the-night-dept/

SteveH:
This is a systemic problem for K-8 educators and charter schools are not a complete solution. Most K-8 educators are cut from the same cloth. My son’s first grade teacher admonished my wife and I by saying that our son had a lot of “superficial knowledge.” All teachers should be required to get a degree in a content and skills-rich field, not one where they ironically get directly taught their turf – full inclusion, differentiated and natural learning, and a dislike for “mere” facts and “rote” skills – anything that will justify their social goals that end up having the opposite effect – hiding the skills and knowledge tracking at home and increasing the academic gap.

That doesn't provide a full solution for the elementary school problem. OK, so let's say that all elementary teachers are required to get a degree in a content field. Now suppose that this chosen field is English or history. How would this degree help them teach math any better?

I remember once reading that in Japan and other nations, students begin learning math from specialists somewhat earlier than American students do. In the elementary schools, the students stay in one classroom and the teachers move from class to class -- just like music and P.E. teachers do here at American elementary schools. I wanted to find a link to confirm this, but I can't -- surely some traditionalists have brought this up in the past. If this is true, then it explains why Japanese fifth grade teachers feel comfortable teaching the circle area formula, but not American fifth grade teachers.

The Path Plan Revisited

In past posts, I wrote about the system that my old elementary school introduced -- the path plan. In this system, students are officially divided into "paths," not "grade levels." Here are rough correspondences between paths and grades:

Early Learning Path: Headstart and Kindergarten
Primary Path: Grades 1-2
Transition Path: Grades 3-4
Preparatory Path: Grades 5-6

When I was a young student in the Preparatory Path, our classes stayed in homeroom (for English and history) until lunch. Then after lunch, we attended two more classes. One was math, and the other was an exploratory wheel that switched every trimester (art, health, and science).

Under this plan, only half of the Grade 5-6 teachers actually taught math -- so of course, the half that was stronger in math would teach the subject. My fifth grade teacher was strong in math, so I remained in her class for math. My sixth grade teacher taught science instead, so I went to another teacher's class for math that year.

A simpler schedule was used in the Transition Path. All Grade 3-4 teachers taught math at the same time, but some students moved to other classrooms during math time. I can easy see how the stronger math teachers could be assigned fourth grade math while the others taught third grade math. Thus the best 50% of teachers would cover math in Grades 4-6.

I once posted a more ambitious version of the Path Plan. We push everything down one path, so that Primary Path students leave homeroom only for math, Transition Path students leave for two classes, and Preparatory Path students leave for three classes. Then this ensures that the best 50% of teachers teach math in Grades 2-6, leaving only K-1 for weaker math teachers. Also, in the Preparatory Path, science can be a whole year, not a trimester. Back when I was a fifth grader, we could get away with less than a year of science. But now with state and NGSS science testing, a full year of fifth grade science is crucial. (When I was a fifth grader, our school actually tried having three classes after lunch, but then it switched to two classes with full implementation of the Path Plan.)

The following chart shows how many teachers each student would have at one time under the more ambitious Path Plan, along with which subjects aren't taught by homeroom teachers:

Path                    Teachers  Subjects not in HR
Early Learning   1
Primary              2              math
Transition           3              math, elective
Preparatory        4              math, elective, science

Here's how I'd make these classes fit a bell schedule -- all grades stay in HR until all grades have completed recess. Then after recess, all grades attend the first extra class. Afterward, Primary Path goes to lunch, and the other two paths attend the second extra class. Afterward, Transition Path goes to lunch, and Preparatory Path attends the third extra class, followed by lunch. After lunch, all classes return to homeroom. The length of each class therefore is the same as the length of lunch -- 40 minutes might be a good length of time for both lunch and math.

Notice how this path pattern can be continued beyond elementary school, to a Middle School Path (Grades 7-8) and Early High School Path (Grades 9-10):

Path                    Teachers  Subjects not in HR
Middle               5               math, elective, science, P.E.
Early High         6               typical high school schedule

So in middle school, students have the same teacher for English and history -- which is what my old middle school referred to as "Core class." After Grades 9-10, the pattern continues, but the number of classes goes down every two years:

Path                    Teachers  Comments
Late High           5              only 22 year credits often needed to graduate
AA/Lower Div.  4              (typical college schedule)
BA/Upper Div.   3
MA                     2
PhD                    1              dissertation adviser

Here's what we do to address SteveH's ideas about teacher training - there could be two types of Multiple Subject Credential, namely math and no-math. Elementary schools should strive to hire about half math and half no-math for Grades 1-6. All students in Grades 2-6 would have a math credential for their math classes, while no-math teaches either K-1 math or no math at all. (Under the less ambitious Path Plan, it's half math for Grades 3-6, and Grades 4-6 are guaranteed to have a math credential for their math classes. No-math teaches either K-3 math or no math at all.)

Another Geometry Problem

Here is another Geometry problem that CCSSIMath tweets about -- questions that are taught in high school under Common Core, but in elementary school in Japan.

March 6th tweet -- Given:

• Semicircle O of diameter AB = 12
• Arc AC = 135 degrees
• D midpoint of chord AC
• S: shaded area bounded by CD, BD, and arc CB

Find:

• S

Here's how I solved the problem:

First, let's do the easy part -- the area of the entire semicircle. The diameter is 12, so the radius is 6, so the area of the circle is 36pi, so the area of the semicircle is 18pi.

Now we must subtract the area of both Triangle ABD and the segment bounded by the chord AC to leave only Region S. Let's try the segment first -- the area of the segment itself found by subtracting the area of Triangle ACO from that of the sector bounded by Arc AC.

Area of sector: (135/360)(36pi) = 27pi/2
Area of triangle: (1/2)(6)(6)sin(135) = 9sqrt(2)
Area of segment: 27pi/2 - 9sqrt(2)

Now we need the area of Triangle ABD. The base of this triangle is easy to find -- AB is a diameter of the circle, so its length is 12. But the height isn't as easy to find. I imagined Circle O to lie on a coordinate plane with O at the origin. Then the coordinates of C are (6 cos 45, 6 sin 45), which we write as (3sqrt(2), 3sqrt(2)), and the coordinates of A are (-6, 0). The height of the triangle is the same as the y-coordinate of D, the midpoint of AC. So the height is 3sqrt(2)/2. (We don't even bother to find the x-coordinate of D.)

Area of triangle: (1/2)(12)(3sqrt(2)/2) = 9sqrt(2)

So the area of the segment and triangle together is 27pi/2 - 9sqrt(2) + 9sqrt(2) = 27pi/2. This we subtract from the area of the semicircle to find S:

Area of S = 18pi - 27pi/2 = 9pi/2

I admit that this question took me 15 minutes to figure out -- and even then I made a mistake. I didn't realize that the two 9sqrt(2) terms had opposite sign and so I had a more complicated answer.

Now here's what CCSSIMath writes about the question:

To compare: this is a fairly typical 6th grade problem in top #math nations' schools, but it would be done in a high school geometry class in the US, if it's covered at all.

So this is supposed to be a sixth grade problem? I had to use the sine function twice -- one in the formula A = (1/2)ab sin C, and the other to find the y-coordinate of C. And look at the mistake I made earlier -- a sign error. Sixth graders usually aren't proficient with signed numbers yet, so if I made this mistake, surely sixth graders would make the same error.

At the link, a tweeter named "eylem gercek boss" had a different solution. This tweeter added in two extra segments -- OC and OD. The region bounded by AB, AC, and Arc BC is divided into five smaller regions:

Region A: Triangle ADO
Region B: Triangle DFO (where F is the intersection of BD and CO -- eylem gercek boss never gives this point a name)
Region C: Triangle BFO
Region D: Triangle CDF
Region E: "Triangle" BCF (actually, BC is an arc, not a segment, but you know what I mean)

Now eylem -- uh, let's just say "boss" -- writes:

A = B + D = B + C

I see what the boss is thinking here -- the median of a triangle divides the triangle into two smaller triangles of equal area. This is because the two smaller triangles each have the same base (half that of the large triangle) and height (same as that of the large triangle).

D = C

OK, so the boss subtracted B from both sides.

==> S = D + E = C + E.

OK, so S is given as the sum of D and E, and we substitute to get S = C + E. But notice that C + E is just the sector of the circle! This is easy to find -- it's just like the Pappas problem I wrote about in today's post:

Area of the shaded sector = 45/360 * (area of Circle O)
                                          = 1/8 * 36pi
                                          = 9pi/2

So now I see what's going on here -- S is indeed an oddly shaped region, but the boss here makes a clever transformation to show that S has the same area as some other much simpler region, in this case a sector. To those who spot the clever transformation, the problem is easy, but to those who don't, the problem is tricky. I never saw the clever transformation and so I struggled -- and I'm a trained math teacher. How much more unlikely then will it be for sixth graders to spot the clever move to make -- or even the sixth grade math teachers?

At the very least, a sixth grade teacher who wants his or her students to solve these problems would have at least pointed out that a median divides a triangle into two equal-area triangles. Not every teacher or textbook emphasizes this.

Also, when we push problems like this down into sixth grade, we expect complaints such as "Why do we have to learn this?" and "When will we use this in real life?" to appear. Also, here's another line I heard my sixth graders say last year -- "This problem takes too long." To many students, any problem that takes more than 30 seconds to solve isn't worth solving. Math isn't something to become proficient at -- it's a barrier to get past in order to reach more enjoyable, non-mathematical activities.

I could go on with CCSSIMath's Twitter page forever, but there are other things I wish to discuss in today's post.

The Tracking Debate

In the most recent Barry Garelick post, the traditionalist teacher mentioned tracking. And where there's tracking, there's race. I wrote a little about race in that post, but as you already know, I save long, extensive discussions about tracking and race for the bottom of vacation posts, like this one.

By the way, recall that the Path Plan is also a mild form of tracking. Students weren't blindly assigned to tracks based on their grade level -- instead, students above grade level might be assigned to a higher path, while those below grade level would be on a lower path. So advanced fourth graders were placed in Preparatory Path while below basic fifth graders were placed in Transition Path. And likewise, advanced second graders were placed in Transition Path, while below basic third graders were placed in Primary Path. (I'm not sure whether the K-1 boundary was crossed this way -- at the time there was only half-day kindergarten, so crossing this line was awkward. Indeed, the K classes were almost always called "kindergarten" -- the name "Early Learning Path" was seldom used.)

Oh, and notice that students are assigned to paths based on their reading ability -- since after all, English (and history) is taught during homeroom. The students' math ability is taken into account when assigning the students to tracks. I know that fifth graders in the Preparatory Path might be assigned to sixth grade math, and so it's possible for fourth graders to be assigned to fifth grade math, even if their reading level keeps them in the Transition Path. I set up the bell schedule earlier so that it's easy to move to a different path during math time.

As for race, notice that even tweeters like CCSSIMath mention race from time to time:

Extensive Data Shows Punishing Reach of Racism for Black Boys https://nyti.ms/2GFoJ7L  via @UpshotNYT

I'll only quote a small part of this New York Times article that CCSSIMath links to here:

The disparities that remain also can’t be explained by differences in cognitive ability, an argument made by people who cite racial gaps in test scores that appear for both black boys and girls. If such inherent differences existed by race, “you’ve got to explain to me why these putative ability differences aren’t handicapping women,” said David Grusky, a Stanford sociologist who has reviewed the research.

A more likely possibility, the authors suggest, is that test scores don’t accurately measure the abilities of black children in the first place.

I quote this part of the article because it mentions test scores -- which is the most relevant part for the tracking debate. It explains why tracking disappeared, and why many people are wary of trying to bring it back -- even without tracking, disparities in income exist. How much worse, then, would the gaps be if we tried to bring back tracking?

If it wasn't for race, tracking might still exist today. We might have something similar to a nationwide Path Plan -- and even the Common Core Standards could be set up to assign standards to paths, not grade levels.

But as long as tracking has a racial problem, it'll never be completely brought back.

An Analogy

Here's another analogy -- a thought experiment, mind you -- to discuss the tracking debate further.

Imagine that there is a magic red button. Here's how it works -- as soon as the magic red button is pressed, the income of every black person immediately doubles. Actually, let me make this precise -- I already know how economists might say, "Sure, income doubles, but prices double as well, so no one is better off." If the red button is pressed, then the purchasing power of every black person doubles -- anything a black person can buy now, he or she can buy two of now.

So far, this sounds good. You might think that if we repeated the NY Times graphs in a world where the red button exists, then they'd be more favorable for blacks. More blue squares (representing blacks) would land on the paths to higher income and fewer would be on the paths to lower income.

There's just one problem -- I didn't say what effect the red button has on whites yet. And so let me do so now -- if the red button is pressed, then the income of every white person triples. Again, here I mean that the purchasing power of every white person triples -- anything a white person can buy now, he or she can buy three of now.

And as for Asians, Hispanics, Native Americans, and mixed-race individuals, let's say that the red button increasing the purchasing power by a factor between 2 and 3. Lighter-skinned individuals have a multiplier closer to three, while darker-skinned individuals have a multiplier closer to two. So if the red button is pressed, everyone will have greater purchasing power.

So what effect would the red button have on the NY Times graphs? Notice that the five income paths on the graphs are based on percentiles, so the highest path is the richest 20%. The red button triples whites' income while only doubling blacks' income. So the red button increases gaps -- there would be even more yellow squares on the highest path and blue squares on the lowest path.

But someone who is pro-red button could easily argue the following -- each black person has twice the purchasing power he or she would have without the red button. And so even with more blacks on the lowest path, they are better off with the red button than without it. And so if you oppose the red button, you are actually anti-black.

Skeptics can counter that relative wealth matters, and so blacks aren't truly better off in the world of the red button even if they have twice the purchasing power. Again, we can declare it's built into the magic of the red button that blacks are automatically twice as well-off if the button is pressed. In the end, it doesn't really matter because this is just a thought experiment -- there is no magic red button.

So far, I've written much about race and income, but didn't I say this analogy would have something to do with tracking?

Well, here's the kicker. According to tracking advocates, tracking is the magic red button. They concede that if tracking were brought back, more whites would find themselves on the higher track and blacks would wind up on the lower track. Yet somehow, everyone, regardless of race, would be better off with tracking.

Years ago, I recall quoting a traditionalist who defended tracking. This is what he wrote -- an Asian girl is placed on the highest track. She can take higher-level courses and learn the material at a faster rate than if her teachers had to slow down for the other students. When she grows up, she broader knowledge ultimately allows her to discover a cure for cancer. She's able to sell the medicine and become rich -- so her purchasing power triples, just as I claimed the red button would do.

In the same analogy, suppose a black boy is placed on the lowest track. From this track, he's qualified only for low-paying, blue-collar jobs. One day while working hard on the job, he catches cancer -- exactly the cancer that the Asian woman has found a cure for. The money he saves on hospital bills can be spent on other things -- so his purchasing power doubles, just as I claimed the button does.

And so everyone is better off with the red button, even though the gap between races increases -- and even though the NY Times graph looks worse. Ironically, it's another NY Times article that argues that tracking benefits everyone:

https://www.nytimes.com/roomfordebate/2014/06/03/are-new-york-citys-gifted-classrooms-useful-or-harmful/tracking-students-by-ability-produces-academic-results

Notice that the study cited in the article takes place in Kenya, where presumably most of the students involved are black.

The reason that I can't wholeheartedly embrace tracking is that I'm skeptical that it will benefit everyone of all races as much as its advocates claim it will. I'm all for pressing the red button and helping everyone, but I'm not completely sold that tracking is the red button. For example, the black man placed on the lower track might not be able to get any job at all. He might not be able to afford the cure created by the Asian woman at all. And so he dies of the curable cancer. In the end, there's no point developing a cure for cancer if those stricken with cancer can't afford the cure.

Many of the commenters in the old 2014 NY Times article above seem to agree with tracking. Let's look at some of these comments:

C:
I don't understand why everything is designed to force faux equality. When did it become wrong to recognize and encourage people for their talents? There is no racial or income requirement to get into one of these programs. Some people are just smarter than others. It's equally unfair to take someone who, can simply "get" 3rd, 4th, 5th, 6th, 7th, 8th or whatever math by looking at it to sit through endless lectures on how a fraction works.

It's obvious that placement on higher tracks benefit those who are so placed. What I don't believe yet is that those placed on lower tracks benefit. Well, here's another commenter:

Canis Scot:
For years I taught special needs students at their ability, they flourished and grew.

Then the "equality" fascists required that my students be distributed in the mainstream classrooms and I watched in horror as they failed. They retreated into themselves. They were being punished because they were not "normal."

OK, then, so let's imagine what would have happened if the equality "fascists" hadn't intervened. The students would continue to flourish and grow, and feel comfortable in their classes -- that is, until they turn 18. Then when it's time to be hired for a job, the lower-track students still don't have as many skills as those on higher tracks. And when they get their first paychecks, they see a lower number written after the dollar sign than those placed on the higher tracks. These students have become just another blue square falling to the lowest income level.

And also, consider what this placement looks like from the parents' perspective. When they see their children being placed on a lower track, they tend to object to the placement. They might wonder, is their child being placed low due to their race, as the NY Times graphs seem to suggest? Is the school condemning their child to a lifetime of low earnings? I suspect that a major reason that tracking is parental complaining.

Canis Scot's students feel that they are being punished in the gen ed class because they aren't "normal" -- but if tracking remained, they would have felt punished years later with a lower paycheck because they aren't "normal." Am I a "fascist" just because I want the students to have a job that pays well enough for them to have a roof over their head and food on their table?

Later on, another commenter writes that gifted students should be placed on higher tracks because we "need kids who master higher mathematics and design airplanes." But then the students placed on lower tracks are headed for low-paying jobs, where the pay is so low that they can't afford going on vacation or buying plane tickets. So who's going to ride on all those airplanes being built by the higher-track students?

Conclusion

When I was in second grade, my teacher allowed me to study Pre-Algebra independently. I also spent much of my early years reading above grade level. And so I know firsthand the benefits of helping out the students who are above grade level. (On the other hand, I didn't really benefit from the Path Plan because I was already in sixth grade by the time it was fully implemented.)

But despite this, I can't fully support the tracking movement. I simply am afraid that the students -- and races -- who end up on the lowest tracks are set up for lower lifelong earnings.

What I want is a world where the graphs on the original NY Times article -- the one where yellow squares tend to float up to the highest level and blue squares tend to sink -- no longer exist. If it can be shown that tracking truly helps everyone regardless of race, I'd favor it. But until then, I must remain a skeptic.

Once again, I trust the parents who complain about their children being placed on the low track. Only in a different world -- one where I see parents, of all races, actually celebrating that their students are being placed on the low track to get the help they need to be successful -- will I ever accept that tracking works. Then we will have truly pressed the red button.

Lesson 13-5: Tangents to Circles and Spheres (Day 135)

Lesson 13-5 of the U of Chicago text is called "Tangents to Circles and Spheres." In the modern Third Edition of the text, tangents to circles and spheres appear in Lesson 14-4 -- the only lesson that's in Chapter 13 of the old text and Chapter 14 of the new text.

Spring break is definitely on many of our minds right now. As so often happens when I work in two districts, they end up taking different weeks off for spring break. In my new district, it's easy -- spring break is always the week after Easter (that is, Easter Week or Bright Week). Thus there is one week left until spring break in that district.

But the blog calendar follows my old district, not my new district. And in that district, spring break works a little differently. First of all, in this district, spring break has nothing to do with Easter. So
instead, spring break occurs during the same fixed week each year -- and that's next week. In other words, today is the last day before the vacation on the blog calendar.

We notice that today is Day 135 on the blog calendar, and 135 is 3/4 of 180. Despite this, spring break is not considered to divide the third and fourth quarters. This is because winter break, which really does divide the first and second semesters, occurred after Day 83, not Day 90. So the start of the fourth quarter is closer to the mathematical midpoint of the second semester, a few days ago.

I'm not sure why the district doesn't simply have spring break right after third quarter. Perhaps it's so that today can be the deadline for submitting the third quarter grades, or maybe today's the day the students actually take home the third quarter progress reports. I'm not sure, since of course I didn't sub in this district today.

But this district doesn't ignore Easter completely. The rule is that Good Friday and Easter Monday are always holidays in this district. Since the spring break week already contains Good Friday, an extra day is taken off for Easter Monday. And so the blog calendar will resume on Tuesday, April 3rd. In both 2017 and 2019, Easter falls on the third Sunday in April. So

Meanwhile, the LAUSD (and hence my charter school from last year) always takes off the week before Easter (that is, Holy Week), so they are closed next week as well. And just like the blog calendar, LAUSD is also closed an extra Monday -- not for Easter, but for Cesar Chavez Day. The labor leader's March 31st birthday ends up being part of spring break in years when it is close to Easter, as it obviously is this year.

Chavez Day is used to define spring break in the California State University system. Since the actual date falls on a Saturday, campuses are closed on Friday, March 30th instead. Spring break is thus considered to be all of next week -- there are no classes, but the campuses are open every day except Friday, Chavez Day Observed. In the University of California system (including my alma mater UCLA), Chavez Day is considered to be the last Friday in March, which is also March 30th. So once again, the UC spring break is next week as well.

Finally, I want to point out when spring break is in New York City -- the largest district in the nation and home to many MTBoS bloggers. In the Big Apple, spring break occurs during Passover, which is usually near Easter anyway. This year Passover starts on Saturday, March 31st. But New York, just like the blog calendar, has a rule that Good Friday is always a day off (though Easter Monday isn't part of this rule). And so New York takes off Good Friday and the week after Easter. (In some years, Passover falls a month after Easter -- in such years, Good Friday is a mere three-day weekend.) I am aware that there were closures in the city and other areas this week due to snow.

As usual, my plans for the blog are to post once or twice during "spring break" as observed by the official blog calendar. But I'll probably be subbing in my new district next week -- and I promised a "Day in the Life" if it's a math class. It'll look strange to make a "Day in the Life" post during "spring break," but it's possible.

Today, meanwhile, wasn't math, but the last day in the music class. Students all played the same songs as yesterday, except for seventh grade band. The student teacher explains that they're trying out different songs to play at a recital coming up in May. Today they played "John Williams: Movie Adventures," a medley of four Hollywood songs -- "Star Wars (Main Theme)," "Duel of the Fates," "Theme from Jurassic Park," and "Theme from E.T. (Extra Terrestrial)."

Being surrounded by all this music puts me in a mood for another music post -- and lately, I've been writing about music during vacations anyway. So expect a music post coming up for spring break. I am already thinking about ideas to include in this post:

• I'll have more songs based on digits of mathematical constants.
• For microtonality, I already have a 7-limit scale. Since I've been writing about the number 11 a lot lately (Eleven Calendar, Eleven Clock), I can't help but consider an 11-limit scale.
• And I want to compose a new song for Easter.

OK, let's get to the lesson. I posted Lesson 13-5 last year, but I didn't have much to say about it. But remember that this is still an important lesson -- tangents to circles definitely appear on the PARCC and SBAC exams!

On the other hand, today is an activity day -- and it's the last school day before Easter. Last year I omitted this, but this year I'm bringing it back -- Spring Spheres! And not only that, but it actually fits this final lesson on tangents to circles and spheres.

And so this is what I wrote two years ago about today's activity (including more comments about the Easter Date):

Here's my idea of an activity: we take the idea of dividing the surface of a sphere into figures that are nearly polygons and run with it. Now Dr. M divides his surface into triangles, but here we will use square Post-it notes instead. After all, we measure areas in square units, not "triangular units." The task directs students to estimate how many Post-it notes it takes to cover the surface of a sphere before they actually try it.

I have decided to name this activity "Spring Spheres." The name actually refers to an an incident a few years ago (it was 2011 -- the year when Easter was very late) where a volunteer in a classroom was not allowed to bring Easter eggs to school because they were religious. So she decided to bring the students "spring spheres" instead. Here I twist the use of that name around -- it's springtime and we're finding the surface area of a sphere -- hence the name "Spring Spheres."

http://mynorthwest.com/11/459668/Seattle-school-renames-Easter-eggs-Spring-Spheres

This activity is long and requires that there are several balls in the classroom -- and even if we divide the class into groups and ask some students to bring balls to school, they may simply play with the balls rather than complete the activity. So here are some other activities that I am posting today:

• From the U of Chicago text: I decided to change the activity from two years ago to one that actually fits Lesson 13-5. It's about three important spheres -- earth, moon, and sun. One question is about a lunar eclipse -- think back to the "super blue blood moon" two months ago.
• Let's balance out "Spring Spheres" with a question about Easter -- specifically the Easter date. Even though Easter is determined by a table, the table can be calculated using a formula. The following link gives a link to what is known as the Conway Doomsday Algorithm -- and that's Conway as in John Horton Conway, the mathematician who also argued for the inclusive definition of trapezoid. In fact, Doomsday is used to determine the day of the week -- and that's part of calculating Easter, since we need to know when Sunday is. The link also describes how to calculate the Jewish holidays of Passover and Rosh Hashanah, but these are more complicated than calculating Easter. A neat trick is to verify that the Easter calculation works this year, then calculate when it falls next year. Notice that this is a math lesson, but if your school is similar to the Washington state school where Easter eggs have to be called "Spring Spheres," then just stick to the Spring Spheres lesson in the first place.

http://quasar.as.utexas.edu/BillInfo/ReligiousCalendars.html

If you pay attention to the dates of Easter and Passover from year to year, you will notice that although they usually fall within a week or so of each other, on occasion Passover falls about a month after (Gregorian) Easter. At the present time, this happens in in the 3rd, 11th, and 14th years of the Metonoic Cycle (i.e., when the Golden Number equals 3, 11, or 14). The reason for this discrepancy is the fact that although the Metonic Cycle is very good, it is not perfect (as we've seen in this course). In particular, it is a little off if you use it to predict the length of the tropical year. So, over the centuries the date of the vernal equinox, as predicted by the Metonic Cycle, has been drifting to later and later dates. So, the rule for Passover, which was originally intended to track the vernal equinox, has gotten a few days off. In ancient times this was never a problem since Passover was set by actual observations of the Moon and of the vernal equinox. However, after Hillel II standardized the Hebrew calendar in the 4th century, actual observations of celestial events no longer played a part in the determination of the date of Passover. The Gregorian calendar reform of 1582 brought the Western Church back into conformity with astronomical events, hence the discrepancy. 

Similarly, you will notice that in many years Gregorian Easter (the one marked on all calendars) differs from Julian (Orthodox) Easter, sometimes by a week, sometimes by a month. Again, this is due to the different rules of calculation. A major difference is that Orthodox Easter uses the old Julian calendar for calculation, and the date of the Vernal Equinox is slipping later and later on the Julian calendar relative to the Gregorian calendar (and to astronomical fact). Also, the date of Paschal Full Moon for the Julian calculation is about 4 days later than that for the Gregorian calculation. At present, in 5 out of 19 years in the Metonic Cycle--the years when the Golden Number equals 3, 8, 11, 14 and 19--Orthodox Easter occurs a month after Gregorian Easter. In three of these years, Passover also falls a month after Gregorian Easter (see above).

Yes, every time I turn around, it's Christmas or Thanksgiving -- or Easter, that is. Yes, the Big March is over, and it's now spring break. My next regular school post will be on Tuesday, April 3rd.

And so this concludes my last post before spring break. Once again, I plan on making one or two posts next week, during spring break itself.

Lesson 13-4: Indirect Proof (Day 134)

Lesson 13-4 of the U of Chicago text is called "Indirect Proof." In the modern Third Edition of the text, indirect proof appears in Lesson 11-3.

Today I subbed in an instrumental music class. It's at the same middle school as the recent Digital Film class -- and in fact, the regular teacher is out for the exact same reason. Her class is on an out-of-state field trip -- this time to the Disney World Music Festival.

I mentioned Florida in recent posts to a recent hot topic -- of course, I'm speaking of the Year-Round Daylight Saving Time bill. Orlando, where Disney World is located, would be on Year-Round Eastern Daylight Time, equivalent to Atlantic Standard Time. (On the Eleven Clock, all of Florida would be united in the Eastern Time Zone.)

Last week, I did a "Day in the Life" for the first day of the Digital Film class even though it's not math, since I wanted to focus on classroom management on a multi-day subbing assignment. But this time, there's a student teacher who takes charge of all classes. Normally, he teaches only half a day at this school and spends the rest of the day at a high school in the same district. But once he found out that a music specialist sub wasn't available, he rearranged his schedule to cover all classes today.

Thus there's no reason for a "Day in the Life" today, as classroom management isn't an issue. But I do wish to write a little about the music classes for today -- Day 126 in the new district.

There are three eighth grade classes and two seventh grade classes. Within each grade, there is one band class (typical band instruments) and a strings class (mostly violins and cellos, of course, with a few violas and one guy playing the bass in each class). That's only four classes -- well, the fifth class is jazz band. These eighth graders play the same instruments as regular band, including saxes (three types), trombones, and trumpets. (There would have been a drummer, but he's in Florida.)

In previous posts, I mention the musical notes Bb-A-Bb in connection with band classes. Let me remind you what these notes are -- a common "Warm-Up" for band is to play the notes Bb-A-Bb, then Bb-Ab-Bb, then Bb-G-Bb, and so on. I brought it up in connection with the traditionalist debate, where traditionalists lament that musicians know how important repetitive practice (Bb-A-Bb) is to being proficient musicians, yet progressive reformers oppose repetitive practice in math.

I actually never subbed for band on a Bb-A-Bb day before -- I only know about it because I would sometimes overhear Bb-A-Bb while covering another class next door. Now that I'm actually in a band class today, you might be wondering, do I hear Bb-A-Bb today?

Well, the answer is sort of. I do hear repeated notes as a "Warm-Up," but it's not Bb-A-Bb. Instead, the "Warm-Up" begins with F, not Bb. So the eighth graders play F-E-F, F-Eb-F, F-D-F, and so on all the way down to F-F-F (an octave down and up), all in half notes. The seventh graders have a slightly easier "Warm-Up" -- F-E-F-Eb-F-D-F, all the way down to the octave, in whole notes (or longer).

All classes have breathing exercises before F-E-F -- even the string players. And after F-E-F, all classes move on to scales. The seventh grade string players begin with C major and proceed via the circle of fifths until they reach B major. On most band instruments, it's easier to play flat scales than sharp scales, so they play the other side of the circle of the fifths -- from F major to Gb major. The eighth grade classes only play a single scale, but it's one of the more difficult scales -- B major for strings and Db major for band. The jazz class plays a completely different scale -- the "blues scale" commonly used in their genre. They play an F blues scale -- F-Ab-Bb-B-C-Eb-F.

Again, traditionalists wonder why math classes can't promote mastery of arithmetic and p-sets -- the mathematical equivalent of breathing exercises, F-E-F, and scales. In the past, I point out that students are willing to practice for something that's easy, fun, or high-status. So F-E-F and scales are neither easy nor fun, but they lead to the high status of being a musician. On the other hand, being a mathematician isn't high status for many teens. They view mastering arithmetic and completing their p-sets as "not worth it," whereas F-E-F and scales are "worth it."

And so here are the songs the students play after their scales -- both grade level strings work on a song called "Ancient Ritual." Eighth grade band plays "The Great Locomotive Chase," while seventh grade band is beginning a brand new song today. Titled "Best of the Beatles," it's a medley of a trio of Fab Four songs -- "Ticket to Ride," "Hey Jude," and "Get Back." The jazz students don't play a new song, but instead practice jazz standards from their textbook, including "Birdland."

I'll have more to say about music after my second and final day in this class tomorrow.

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

But Section 13-4 is the big one. This section is on indirect proof. I've delayed indirect proofs long enough -- now is the time for me to cover them. Actually, indirect proofs aren't emphasized in the Common Core Standards, but they were in the old California State Standards, where they were known as "proofs by contradiction."

What, exactly, is an indirect proof or proof by contradiction, anyway? The classic example in geometry is to prove that a triangle has at most one right angle. How do we know that a triangle can't have more than one right angle? It's because if a triangle were to have two right angles, the third angle would have to have 0 degrees -- since the angles of a triangle add up to 180 degrees -- and we can't have a zero angle in a triangle. Therefore a triangle has at most one right angle.

And voila -- that was an indirect proof! Notice what we did here -- we assumed that a triangle could have two angles -- the opposite (negation) of what we wanted to prove. Then we saw that this assumption would lead to a contradiction -- a triangle containing a zero angle. Therefore the original assumption must be false, and so the statement that we wanted to prove must be true. QED

Indirect proofs are often difficult for students to understand. One way I have my students think about it is to imagine that they are having a dream. Normally, when one is dreaming, one can't tell that they are having a dream, unless something impossible happens, such as a pig flying in the background, or the dreamer is suddenly a young child again. I recently had a dream where I was suddenly younger again, and I was flying off the ground! Naturally, as soon as those impossible events happened, I knew that I was in a dream.

And so a proof by contradiction works the same way. We begin by assuming that there is a triangle with two right angles, and then we see our flying pig -- a triangle with a zero angle. And as soon as we see that flying pig, we know that we were only dreaming that there was a triangle with two right angles, because there's no such thing! And so all triangles really have at most one right angle. So an indirect proof is really just a dream.

We saw how an indirect proof was needed when we were trying to prove that there exists a circle through any three noncollinear points A, B, and C. The proof that such a circle exists requires an indirect proof to show that the perpendicular bisectors m of AB and n of BC actually intersect. The indirect proof goes as follows: assume that they don't intersect -- that is, that they are parallel. Then because, m is perpendicular to AB and parallel to n, by our version of the Fifth Postulate, AB must be perpendicular to n. Then, now that n is perpendicular to both AB and BC, by the Two Perpendiculars Theorem, AB and BC are parallel. But B is on both lines, so we must have, by our definition of parallel, that a line is parallel to itself -- that is, AB and BC are on the same line. But this contradicts the assumption that A, B, and C are noncollinear. Therefore the perpendicular bisectors m and n aren't parallel -- so that they actually exist.

Returning to 2018, let me post my worksheets. I begin with the second side of the worksheet that I posted yesterday. Then, since this lesson naturally leads itself to activity, I also include some old logic problems that I did post last year.

By the way, it's back-to-back scientists featured in the Google Doodle. Today's Doodle features Japanese geochemist Katsuko Saruhashi. Her specialty is measuring pollution in water, in particular radioactive pollution.

Lesson 13-3: Ruling Out Possibilities (Day 133)

Lesson 13-3 of the U of Chicago text is called "Ruling Out Possibilities." In the modern Third Edition of the text, ruling out possibilities appears in Lesson 11-1.

This is what I wrote about Lesson 13-3 last year:

Here are a few things that I want to point out. First of all, some texts refer to the Law of Ruling Out Possibilities in Section 13-3 by another Latin name, modus tollens. Here is a link to the Metamath reference to modus tollens.

http://us.metamath.org/mpegif/mt4.html

As we can observe in the proof at the above link, modus tollens is essentially modus ponens (The Law of Detachment) applied to the contrapositive (Law of the Contrapositive, or contraposition.)

Section 13-3 is another section that lends itself to an activity, since many of its questions are actually logic problems, like the ones that often appear in puzzle books.

I don't have much else to say today. After so many lengthy posts about circle clocks and circle constants, the Queen of the MTBoS and the King of Traditionalism, and so much subbing, today is a much-needed short post.

(OK, I can at least mention today's Google Doodle since it honors a scientist -- Guillermo Haro, the first Mexican member of the Royal Astronomical Society.)

Lesson 13-2: Negations (Day 132)

Lesson 13-2 of the U of Chicago text is called "Negations." In the modern Third Edition of the text, negations appear as part of Lesson 11-2 (which is the old Lessons 13-1 and 13-2 combined).

Today on her Mathematics Calendar 2018, Theoni Pappas writes:

Find AB^2.

(We're given Triangle ABC with right angle at B, altitude to the hypotenuse BD, AD = 2, DC = 8.)

This is clearly a case of the Right Triangle Altitude Theorem, Lesson 14-2:

Right Triangle Altitude Theorem:
In a right triangle,
b. each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg.

So this tells us that AB is the geometric mean of the hypotenuse AC = 10 (= 2 + 8) and AD = 2. As we're asking for AB^2 rather than AB, we don't even need to use square roots:

AB^2 = AC * AD
AB^2 = 10 * 2
AB^2 = 20.

So the desired value is 20 -- and of course, today's date is the twentieth. It's the spring equinox -- and by definition, the moment of the equinox (9:15 AM Pacific Time) is Nowruz, Persian New Year.

By the way, our students haven't seen Chapter 14 yet. But our students might be able to figure it out using the similar triangles of Chapter 12 -- though Triangle ABC ~ ADB is a bit tricky.

Today I subbed in a middle school English class. So there is no "Day in the Life" today, but I do want to follow yesterday's Fawn Nguyen post with another issue that comes up today -- restroom passes.

The teacher I'm covering today has a strict restroom pass policy -- one pass per trimester. (It is Day 124 in the new district, hence it's near the start of the third trimester.) The rule is that the kid must produce a student planner, which the teacher signs. Presumably, the teacher uses the planner to determine whether the student's single pass has been used up that trimester.

Let's approximate how many passes this teacher gives in a typical day. She has six classes (five English and a yearbook elective) with about 30 students in each class, so 180 students. Each student gets one pass per trimester, so 180 passes per trimester. Finally, a trimester is about 60 days, so we estimate that she gives out about three passes per day. Hence I would have considered it good if only three students ask to go to the restroom today.

So how many students ask to go the restroom today? No, it wasn't three -- try ten. As a comparison, if the regular teacher gave out three passes per day, every student would have used up his or her pass by the time the SBAC is given. And with those daily two-hour blocks for testing, the students will wish that they still have their restroom passes!

Here's the breakdown by period -- one student each in fourth and fifth periods ask for a pass. And two students in each of the other classes ask to go. And for comparison, the ten students who ask to go today is more than the number who go during the four days of the digital film class combined -- which is not what I'd expect in a class with such a restrictive restroom rule.

It's obvious that some students ask because they assume that I, as a sub, don't even know about the regular teacher's restroom rule. The two students in first period -- the rotation actually starts with first today -- find out that I read the pink lesson plan where she lists rules when I ask for their planner. In second period, one of the two students who ask doesn't have his planner, and so he doesn't go. The other student tells me that the pass page has fallen out so that I should sign elsewhere in his planner. I wouldn't be surprised if he's hidden the signature page intentionally so that he can ask the regular teacher to sign for a second pass later on.

In third period, I begin to get upset with the fifth student -- two beyond my goal of three -- asks for a restroom pass. I start warning him about using up his pass so early in the trimester and not having a pass for later on. This warning is enough to convince the sixth student to change his mind about the restroom -- as does the lone student who asks when fourth period begins.

In fifth period, a student asks to go just as the class is about to start. This would aggravate me even if the teacher didn't have a one-pass-per-trimester rule -- because he's asking right after lunch. If he had gone to the restroom ten minutes earlier, he wouldn't have needed to ask a teacher -- and his one pass for the trimester would still be intact.

Of course, we all know what's going on here. The student would rather miss class time than lunch time with his friends. To him, sacrificing his pass is worth it if it means he doesn't have to lose a precious second with his friends.

When I was a young student, I was the opposite. I'd rather spend my lunch in the restroom -- sometimes going twice during break -- rather than miss even one second of class. My eighth grade English class was always after lunch, and I didn't miss a second of class time. Yes, English wasn't my favorite subject, and yes I sometimes found it boring, and yes I sometimes thought it was irrelevant to my future career -- but I didn't want to miss a single second. I didn't always care about spending time with friends.

Given a choice between being popular with my fellow teenagers and being popular with adults, I chose the latter -- and it wasn't even close. Also, I wasn't trying to be a "teacher's pet." The name "teacher's pet" implies that teachers judge students by feelings rather than by merit. I wanted teachers to like me for meritocratic reasons. I tried my best in English class, even if I didn't always get the highest grades. For example, I earned a "C" grade in the third quarter -- the quarter when our class read The Diary of Anne Frank (the play this class is reading today). I'd much rather be reading the story then asking my teacher for a pass.

And if I'd been a student in a class with a one-pass-per-trimester policy, I definitely would have saved the third trimester pass for SBAC two-hour testing block season, not blow it well before testing -- especially if I'm an eighth grader who should remember two-hour testing from seventh grade.

Here's what I do when the fifth period student asks for a pass today -- I tell him that since lunch begins at 12:45, he should have gone to the restroom at 12:46, or 12:47, or 12:48, or 12:49, and so on for all forty minutes of lunch, until 1:25. Yes, I name every single minute out loud. Some of the students start to laugh -- but it's effective, since the student changes his mind about going.

The reason I name all the minutes is because of a new classroom management idea I thought of. If I ever have my own class again and a student asks to go to the restroom right after break, I'll let the student go, but then the student owes me standards -- one line for each minute of the break during which he or she should have gone:

At 12:46, I could've gone to the restroom, but I didn't.
At 12:47, I could've gone to the restroom, but I didn't.
At 12:48, I could've gone to the restroom, but I didn't.
At 12:49, I could've gone to the restroom, but I didn't.
...
At 1:25, I could've gone to the restroom, but I didn't.

Again, this is management that is based on how students actually think. Unlike my young self, many students value friendship time over class time and are willing to miss class for friendship time, so no argument about favoring adults' opinions over their friends' (or what I used to do, tell them about how I went to the restroom thousands of times over three years of school without missing single a second of class) can be effective. Now it's, I'll go to the restroom during lunch so that I won't have to write 40 lines in class.

Recall what Fawn Nguyen writes about restroom passes, in her big classroom management post:

http://fawnnguyen.com/all-i-got-on-classroom-management/

A few years ago I was sitting in the lodge at CMC-North in Asilomar when someone recognized me from my Ignite talk and came over to chit chat. He was lamenting the frequency in which his students were asking to use the restroom pass. I asked some related questions to learn more before I realized that he wasn’t lacking classroom management skills as much as just lacking a good lesson.

This explains why I had eight restroom passes during five periods of English when I didn't have as many passes in sixteen periods of digital film -- film is more interesting than English. Students rarely ask for passes if they find the lesson interesting or fun -- which unfortunately, Anne Frank isn't to many students.

That takes us to sixth period Yearbook class. Yearbook, like Digital Film, is an elective, so perhaps the stream of kids of students asking for passes would end here.

Well, there are a few other issues going on here. That same pink lesson plan also mentions that the students must be in their assigned seats, and that (going back to yesterday's post) cell phones may be used only for listening to music, not texting or taking photos. (In particular, the phones must be either face down on the desk or in a pocket.) But the students complain that their assignment is to get together in groups (thus violating the seating chart) and upload the photos they took on phones for the yearbook assignment.

I concede that maybe the teacher doesn't have Yearbook in mind when she lists these rules. But there's one rule that she explicitly applies to Yearbook class -- "zero level of talking." Apparently, there are four levels of acceptable noise, and of course "zero level" is silence. I can't make those words disappear from the pink lesson plan, and so I'm obligated to enforce it. The students continue to object, since "zero level" would interfere with the group projects to which they are assigned.

And this is when a student asks for a restroom pass. She has her planner ready for me to sign, and it's not the period right after lunch. The only problem I have is that she's the ninth student to ask for a pass today when my goal is three passes or fewer. Again -- students tend to ask for passes during boring classes, and a class with "zero level of talking" is boring.

So this is when the argument begins. I tell her that she's the ninth student asking for a pass when there should have been only three -- and I suspect that six students have asked me, the sub, for a pass when they wouldn't have asked the regular teacher. She counters, "How can you 'calculate' when someone needs to go to the restroom?"

In the end, I finally sign the planner, but I continue to argue along the way. The argument ends when security, in the form of an academic coach, finds out what's going on here. The students tell her their side of the story and I tell her mine. In the end, she decides to make it "level one of talking" and allows one more girl -- the tenth student -- to use the restroom.

I admit that I don't really want to enforce the "zero level" rule today anyway. Students who have completed their projects are allowed to use the class as a study hall. Two students are working on math -- an eighth grader is beginning rotations, while a seventh grader is finding area. The older girl doesn't draw her 180 degree rotation correctly and the younger girl calculates the perimeter instead of the area -- and yet I can't help either one! For not only would helping them make it harder to enforce the "zero level" rule, I'd be violating it -- I'd want to ask the students questions about their homework and they'd answer, which then wouldn't be "zero level" of talking.

It's notable that the pink lesson plan is actually the same sub plan she uses everyday -- the actual lesson for the day (at least for English) is posted in the Google Classroom. And so the Yearbook lesson plan doesn't change -- perhaps some days they're working on individual assignments with "zero level," and today they're working on a group assignment.

I keep thinking that the Yearbook class would have been the best class I subbed for in a while had it not been for having to enforce the pink rules -- or if the teacher had specified different rules for the Yearbook class (such as "The students are seated in their assigned groups. It's OK for them to talk quietly among their group -- Level 1. It's OK for them to upload photos from their phones"). Perhaps at least one of the two girls wouldn't have been bored enough to want a restroom pass. And I could have helped one or both of the other girls with their math assignments.

It's ironic that I'd have trouble keeping students away from restroom passes on a day when the students are reading Anne Frank. After all, the young Jewish girl was only allowed to use the "WC," or water closet (bathroom), at night. Our students think it's oppressive when we make them wait until snack or lunch to use the restroom, but that's nothing compared to what Anne Frank had to endure.

But this doesn't mean that my strategy for keeping the kids away from the restroom should have been to compare them to Anne Frank (or to the guard at the Tomb of the Unknowns, or to myself and my thousands of restroom trips at school, as I have in previous posts). As a regular teacher, I should have had the students write standards as mentioned above. And as a sub -- well, the students, I assume, already have some penalty if they make a second restroom visit in a trimester, and that wasn't enough to stop the demand for passes today.

So instead, I could have an incentive for the students, to be implemented from now on whenever the regular teacher mentions a strict restroom policy in the lesson plan. At the start of the day, I tell the students that according to the teacher, she usually gives out only three passes a day (which is technically true, albeit indirectly inferred from the calculation). So our goal is for there to be no more than three total passes during the entire day.

Now within each period, there's a reward if the class goes the whole period without a pass. So what would be a suitable reward for the entire class?

Well, last year I sang songs in class as part of music break. So here's the idea -- as the students enter, I sing (without instruments, since I don't carry them around when subbing) the first few lines of one of the more popular songs that I played last year. I know that these are math songs, but I suspect that kids in any class will enjoy some of them. Then I tell them that I'll sing the rest of the song during the last few minutes of class if all of them can avoid using a restroom pass the entire period.

Of course, suppose one student takes the pass at the start of class. This means that they fail to earn my reward -- so they think they might as well all go to the restroom that period. Well, here's the sneaky part of the reward -- as long as the student returns before singing time, I'll keep the reward alive but just sing half of the song. The goal is now to avoid a second student asking for the pass the rest of the period.

Now suddenly, the students have a real reason to skip using the restroom pass -- one that should be more effective. Just maybe, the Yearbook girls who insisted on using the restroom would have been the second and third students to go, not the ninth and tenth. And the earlier classes might have earned a song at the end of their Anne Frank reading. This is definitely something I want to consider, especially as we head into SBAC two-hour testing block season.

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

For example, the statement:

All unicorns are white.

is actually true -- after all, we have never seen a unicorn that isn't white (precisely because there exists no unicorns at all, much less ones that aren't white). Another way of thinking about this is that there are zero unicorns in this world, and all zero of them are white! In if-then form this statement becomes:

If an animal is a unicorn, then it is white.

The hypothesis is false (since there are no unicorns), so the entire conditional is true. This statement has no counterexamples (unicorns that aren't white), and conditionals without counterexamples are normally called true.

The book then derives, from the statement 1=2, the statement 131=177. There is a famous example of a derivation of a false conclusion from a false hypothesis, often attributed to the British mathematician Bertrand Russell, about a hundred years ago. From the statement 1=2, Russell proved that he was the Pope:

The Pope and I are two, therefore the Pope and I are one.

that is, he used the the Substitution Property of Equality from the hypothesis 1=2.

In today's lesson, the U of Chicago text introduces the symbol not-p for the negation of p. In other texts, the notation ~p is used, but I have no reason to deviate from the U of Chicago here.

Before leaving this site, let me point out that this [Metamath] site gives yet a third way of writing the "not" symbol used in negations:

http://us.metamath.org/mpegif/wn.html