Saturday, July 22, 2017

Pi Approximation Day 2017

Today is Pi Approximation Day, so-named because July 22nd -- written internationally as 22/7 -- is approximately equal to pi.

It also marks three full years since my very first post here on the blog. Yes, I've declared Pi Day of the Century -- March 14th, 2015, to be the ceremonial start date of this blog. But the chronological start date was July 22nd, 2014. I like celebrating both Pi Day and Pi Approximation Day -- including eating pie both days -- and so I celebrate both of them as launch dates for the blog.

Last year, I used the anniversary to create a FAQ about my blog and myself. I do the same this year -- and this year's FAQ will reflect the changes that have happened to my career since last year. As usual, let me include a table of contents for this FAQ:

1. Who am I? Am I a math teacher?
2. Who is Theoni Pappas?
3. How did I approximate pi in my classroom?
4. What is the U of Chicago text?
5. Who is Fawn Nguyen?
6. Who are the traditionalists?
7. What are number bases?
8. Why wasn't my first year of teaching as successful as I'd like?
9. What did I do about cell phones in the classroom?
10. How should have I stated my most important classroom rule?

1. Who am I? Am I a math teacher?

I am David Walker. I'm trying to start a career as a math teacher, but I must admit that my career isn't off to as good a start as I'd like.

Last year was my first as a teacher at a charter middle school, but I left that classroom in March. And this year, I was just about to start at a charter high school, but I was told that enrollment at the school is declining, and so they decided not to hire me as a teacher.

By now, I fear that I won't be hired to teach this year at all. If I'm not hired, then I'll try to return to substitute teaching. But this will make the launch of my teaching career that much more difficult.

So the answer to this question is no, I'm not currently a math teacher, but I'm hoping that I'll be one again soon.

2. Who is Theoni Pappas?

Theoni Pappas is the author of The Mathematics Calendar. In most years, Pappas produces a calendar that provides a math problem each day. The answer to each question is the date. I enjoyed, but only rarely wrote of, the Pappas calendar during the first three years of this blog, 2014, 2015, and 2016.

Last year, as I was about to begin my first year of teaching, I came up with the idea of starting each class with my own Pappas question -- a Warm-Up question whose answer is the date. But this quickly fell apart when I found out that our school curriculum provided a "daily assessment" question that I was required to give as the Warm-Up. Of course, the answers to these questions weren't just the date.

I was disappointed when Pappas didn't publish her calendar for 2017. So in January and February, I had neither my own Pappas calendar in the classroom Warm-Up, nor any questions directly written by Pappas for my home calendar.

In March, my local library held its biannual book sale. For fifty cents, I found a book actually written by Pappas, The Magic of Mathematics, published in 1994. And so I decided to blog about one page from her book everyday that I post for the rest of this year, until 2018, when it appears that she will publish her actual Mathematics Calendar once again.

I wanted to retain the tradition of incorporating the date in choosing a page to blog about. And so I decided that the page number will be the same as the (Julian) day count of the year. For example, today is July 22nd, the 203rd day of the year, so I'll blog about page 203 today.

This is what Theoni Pappas writes on page 203 of her Magic of Mathematics:

"Ever wonder, irritatedly, how it is you receive so much junk mail -- catalogues you never requested, offers for 'super' buys? How were your name and address accessed? Welcome to the electronic age and the loss of privacy."

This page is in Chapter 8, "The Computer Revolution." It is the only page of the section titled "Cryptography: Anarchy, Cyberpunks, & Remailers."

Keep in mind that Pappas wrote the above in 1994, when the Internet was in its infancy. Imagine how much easier it is to access personal information now that the Internet is in its prime.

It also reminds me of the old show Square One TV. In a Mathnet segment produced a few years before Pappas wrote her book, the villain was a woman named I.O. Privacy. She used personal information on a computer to send mailers to those who were most likely to spend a weekend on vacation at Pocono's Paradise. When the victims were out of town, she would send her henchmen out to their homes and rob them. Oh, and the culprit's initials, I.O., stood for "Invasion Of."

So Pappas warns us that living in the digital age has its drawbacks as well as its advantages. Here's how she ends this section:

"Many feel that if governments can use methods and devices to encode important information, an individual also has a right to use similar methods to insure personal privacy."

3. How did I approximate pi in my classroom?

Since today is Pi Approximation Day, I should write something about approximating pi. I won't let my disappointment -- of thinking I'd be hired at a new job only to be denied -- get in the way of celebrating the special day.

The dominant approximation of pi in my classroom was, of course, 3.14. I didn't quite reach the unit on pi in my seventh grade class -- Grade 7 being the year that pi first appears in the current Common Core Standards. But I did reach a unit on the volume of cylinders, cones, and spheres in my eighth grade class.

The main driver of my use of 3.14 as the only approximation of pi was IXL. I used this software to review the volume problems with my eighth graders. The software requires students to use 3.14 as pi, even though I provided them with scientific calculators with a pi key. For example, the volume of a cylinder of radius and height both 2 is 8pi cubic units. IXL expects students to enter 8(3.14) = 25.12, even though 8pi rounded to the nearest hundredth is 25.13. IXL will charge the students with an incorrect answer if they enter 25.13 instead of 25.12. And even on written tests, I didn't want to confuse the students by telling to do something different for written problems, and so I only used 3.14 for pi throughout the class.

But there was another problem with our use of calculators to find volumes. Some of the calculators were in a mode to convert all decimals into fractions. Thus 25.12 appeared as 628/25 -- and I couldn't figure out how to put them in decimal mode. This actually allowed me to catch cheaters -- only one of the calculators was in mixed number mode. Here was one of the problems from that actual test, where students had to find the volume of a cylinder of radius and height both 3:

V = pi r^2 h
V = (3.14)(3)(3)(3)
V = 84 39/50

so it displayed 84 39/50 instead of 84.78 or 4239/50. So any student who wrote 84 39/50 on the test -- other than the one I knew had the mixed number calculator -- must have been cheating.

Here's an interesting question -- suppose instead of catching cheaters, my goal was to make it as easy as possible on my students as well as my support staff member, who was grading the tests. That is, let's say I want to choose radii and heights carefully so that, by using 3.14 for pi, the volume would work out to be a whole number (which appears the same on all calculators regardless of mode). How could have I gone about this?

The main approximation of pi, 3.14, converts to 314/100 = 157/50. It's fortunate that 314 is even, so that the denominator reduces to 50 rather than 100. So our goal now is to choose a radius and height so that we can cancel the 50 remaining in the denominator.

We see that 50 factors as 2 * 5^2. The volume of a cylinder is V = pi r^2 h -- and since r is squared in this equation, making it a multiple of five cancels 5^2. It only remains to make h even to cancel out the last factor of two in the denominator. Let's check to see whether this works for r = 5, h = 2:

V = pi r^2 h
V = (3.14)(5)^2(2)
V = 157 cubic units

which is a whole number, so it works.

If we had a cone rather than a cylinder, then there is a factor of 1/3 to deal with. We can resolve the extra three in the denominator by making h a multiple of six (since it's already even). Let's check that this works for r = 5, h = 6:

V = (1/3) pi r^2 h
V = (1/3)(3.14)(5)^2(6)
V = 157 cubic units

Spheres, though, are the trickiest to make come out to be a whole number. This is because spheres, with the volume formula V = (4/3) pi r^3, have only r with no h. Thus r must have all of the factors necessary to cancel the denominator. Fortunately, the sphere formula contains the factor 4/3, and the four in the numerator already takes care of the factor of two in the denominator 50. And so r only needs to carry a factor of five (only one factor is needed since r is cubed) and three (in order to take care of the denominator of 4/3). So r must be a mutliple of 15. Let's check r = 15:

V = (4/3) pi r^3
V = (4/3)(3.14)(15)^3
V = 14130 cubic units

Notice that the volume of a sphere of radius 15 is actually 14137 to the nearest cubic unit -- that's how large the error gets by using 3.14 for pi. Nonetheless, 15 is the smallest radius for which we can get a whole number as the volume by using pi = 3.14.

Now today, Pi Approximation Day, is all about the approximation 22/7 for pi. Notice that if we were to use 22/7 as pi instead of 157/50, obtaining a whole number for the volume would be easier.

For cylinders, the only factor to worry about in the denominator is seven. Either the radius or the height can carry this factor. So let's try r = 1, h = 7:

V = pi r^2 h
V = (22/7)(1^2)(7)
V = 22 cubic units

For cones, we also have the three in 1/3 to resolve. Let's be different and let the radius carry the factor of three this time. For r = 3, h = 7, we have:

V = (1/3) pi r^2 h
V = (1/3)(22/7)(3^2)(7)
V = 66 cubic units

For spheres, unfortunately our radius must carry both three (for 4/3) and seven, and so the smallest possible whole number radius is 21:

V = (4/3) pi r^3
V = (4/3)(22/7)(21)^3
V = 38808 cubic units

This is larger than the radius of 15 we used for pi = 3.14. But notice that there are many extra factors of two around in the numerator -- 4/3 has two factors and 22/7 has one. This means that we can cut our radius of 21 in half. Even though 21/2 is not a whole number, the volume is nonetheless whole:

V = (4/3) pi r^3
V = (4/3)(22/7)(21/2)^3
V = 4851 cubic units

To make it easier on the students, I could present the radius as 10.5 instead of 21/2. (Recall that the students can easily enter decimals on the calculator -- they just can't display them.) There is still some error associated with the approximation pi = 22/7, as the volume of a sphere of radius 10.5 is actually 4849 to the nearest cubic unit, not 4851. Still, 22/7 is a slightly better approximation than 157/50 is, with a much smaller denominator to boot.

With such possibilities for integer volumes, I could have made a test that is easy for my students to take and easy for the grader to grade. I avoided multiple choice on my original test since I didn't want to give decimal choices for students with calculators in fraction mode (or vice versa). With whole number answers, multiple choice becomes more feasible. Of course, the wrong choices would play to common errors (confusing radius with diameter, forgetting 1/3 for cones).

4. What is the U of Chicago text?

In 1991, the University of Chicago School Mathematics Project published a series of secondary mathematics texts. I was able to purchase both the Algebra I and Geometry texts for two dollars each at a local public library. One thing I discovered was that the U of Chicago Geometry text is based on the same transformations that appear in the Common Core Standards -- even though the U of Chicago published the text nearly two decades before the advent of the Core.

The common denominator is NCTM, the National Council of Teachers of Mathematics. This group was behind both the U of Chicago text and the Common Core Standards. Still, this means that one can satisfy most of the Common Core Standards by teaching out of the U of Chicago text.

There are newer versions of the Geometry text than the one that I own. I've read that while my old version introduces reflections in Chapter 4 and the other isometries in Chapter 6, in the new version all isometries appear in Chapter 4.

The U of Chicago text is not the only book that I mention here on the blog, not by a long shot. Many posts here refer to the Illinois State text. This is mainly because I was a math teacher, and I used the Illinois State text when teaching at my old school. 

To supplement the Illinois State text (mainly when creating homework packets), I used copies of a Saxon Algebra 1/2 text and a Saxon 65 text for fifth and sixth graders.

To learn more about some of these texts, here are some links to a webpage by Cathy Duffy. Duffy's goal is to help homeschooling parents choose a textbook for their children. Here is her review of the U of Chicago text:

and the Saxon series:

5. Who is Fawn Nguyen?

For years, Fawn Nguyen was the only blogger I knew who was a middle school math teacher. Back when I was a middle school teacher, I enjoyed reading Nguyen's blog, but now that I've left, it's not as important for me to focus on her blog over all others.

Nonetheless, let's take a look at Nguyen's blog today. She hasn't posted in two months, and so the following link is to her most recent post:

This is a computer program that takes two whole numbers as input. Then the computer draws a rectangle whose dimensions are these two numbers.

Afterward, the computer performs the following algorithm -- inside the rectangle, it draws the largest possible square that fits inside the rectangle. The side length of this square will equal the smaller dimension of the rectangle. If there is a rectangle left over after cutting out the square, then the computer performs the same algorithm on the remaining rectangle. The algorithm terminates when the final rectangle is itself a square.

The name "Euclid's Algorithm" reveals its origins -- the Greek mathematician proved that if the sides of the original rectangle are whole (or rational), then the algorithm terminates. Euclid referred to the side length of the final square as the "common measure" of the dimensions of the original rectangle, and nowadays we can think of this length as the GCF of the original whole number dimensions.

But not all lengths have a common measure. Some are incommensurate (irrational). The most famous example of a rectangle with incommensurate dimensions is the golden rectangle. If a square is taken away from a golden rectangle, the result is a rectangle that is similar to the original figure. And if a square is taken away from the remainder, the result is again a golden rectangle -- thus Euclid's algorithm fails to halt for this rectangle. The ratio of the dimensions of a rectangle is (1+sqrt(5))/2, which I mentioned in another post about a month ago as the golden ratio, Phi.

But today is Pi Approximation Day, not Phi Approximation Day, so let's try Euclid's algorithm on a rectangle with length pi. For simplicity, the width of the rectangle will just be 1. (By now we've left Nguyen's original post long behind, since her class only dealt with whole number lengths.) We can approximate pi by pretending that there is no remaining rectangle, which would then make the dimensions rational.

-- Our starting rectangle has dimensions (pi, 1).
-- The first iteration allows us to cut three unit squares from the rectangle. Ignoring the remainder would then imply that pi = 3 -- and actually, Nguyen does do the (3, 1) case. And indeed, this crude approximation pi = 3 is implied in the Bible.
-- The second iteration takes the remaining rectangle of dimensions (pi - 3, 1). Seven small squares can be cut from this rectangle. Ignoring the remainder would then imply that pi = 3 + 1/7 (since the seven small squares would then have the same length as one of the large squares). This, of course, is equal to 22/7, which makes today Pi Approximation Day. The Greek mathematician Archimedes was the first to use this approximation.
-- The third iteration takes the remaining rectangle and cuts out one more smaller square. Ignoring the remainder would then imply that pi = 3 + 1/8 = 25/8 (since it's now as if eight small squares can fit on the side of the unit square). The ancient Babylonians used this approximation.
-- The fourth iteration takes the remaining rectangle and cuts out 15 smaller squares. Ignoring the remainder would then imply that pi = 333/106. This is less obvious, but if we were to take Nguyen's program and enter (333/106, 1) -- actually (333, 106) has the same effect -- we would see three large squares, then seven medium squares, then one square of its own size, and finally fifteen of the smallest squares. There is a theory that this approximation also appears in the Bible -- here's a link to a Reddit post (written exactly six years ago on Pi Approximation Day):

-- The fifth iteration takes the remaining rectangle and cuts out one more smaller square. Ignoring the remainder would then imply that pi = 355/113. Again, we can enter (355, 113) in Nguyen's program and see that there is one more smallest square than there was for (333, 106). This approximation was known to the ancient Chinese.
-- The sixth iteration takes the remaining rectangle -- which is very thin, because 355/113 is such a great approximation. Indeed, the remainder is 292 times as wide as it is long. And so we normally cut off the approximation at 355/113.

The rational approximations 3, 22/7, 25/8, 333/106, and 355/113 are called "continued fractions." We can use continued fractions whenever we need to find a rational approximation. (Calendar reform and microtonal music are two topics mentioned on the blog that can make use of continued fractions, though I didn't explain how in any of those posts.)

I hope Fawn Nguyen posts again on her blog soon!

6. Who are the traditionalists?

I use the phrase "traditionalists" to refer to those who oppose Common Core for mathematical -- not political -- reasons. The traditionalists tend to believe that many of the topics in the Common Core Math Standards are taught too late.

My own relationship with the traditionalists is quite complex. I tend to agree with the traditionalists regarding elementary school math and disagree with them regarding high school math.

For example, they believe that the standards in Grades 4-6 that require the use of the standard algorithm for arithmetic should move down at least one grade level. I tend to agree with the traditionalists on this point here -- most of the reasons for opposing direct instruction, rote memorization, and drill don't apply to the youngest students. I remember once learning about two nonstandard algorithms for addition -- the Left-Right Method and the Plus-Minus Method -- on an old 15-minute math show for sixth graders, "Solve It," that used to air on our local PBS station back when I was that age. For mental math, these nonstandard algorithms are great, but for a young elementary student first learning to add two-digit numbers I prefer the standard algorithm.

On the other hand, the traditionalists also prefer that high school seniors take AP Calculus in order to open the door to college STEM majors. In particular, some STEM colleges here in Southern California, such as Cal Tech and Harvey Mudd, require calculus for admission. Even colleges that don't formally require calculus are likely to send students rejection letters if they apply without having calculus on the schedule, especially if they apply to STEM majors. Therefore they want the classes leading up to senior year to prepare students to take AP Calculus -- including Algebra I in the eighth grade. They oppose Common Core because its eighth grade math standards do not comprise a full Algebra I class. I only partially agree because I don't want seniors who can't handle AP Calculus or eighth graders who can't handle Algebra I to be labeled failures. Just as AP Calculus can open the door to STEM, it can close the door to a non-STEM career if students are forced to spend too much effort trying to pass their high school math classes.

OK, so I support the traditional pedagogy with regards to elementary math and oppose it with regards to high school math. But what's my opinion regarding traditionalism in middle school math -- you know, the grades that I'll actually be teaching? In middle school, I actually prefer a blend between the traditional and progressive philosophies. And the part of traditionalism that I agree with in middle school is making sure that the students do have the basic skills in arithmetic -- since, after all, that is elementary school math, and with elementary math I prefer traditionalism.

I've referred to many specific traditionalists during my three years of posting on the blog. The traditionalist who is currently the most active is Barry Garelick. Here is a link to his most recent post:

Garelick teaches middle school math right here in California. In this post, he criticizes the famous businessman Elon Musk for encouraging Project-Based Learning instead of traditional math lessons.

He highlights two commenters in this post. One of them is SteveH, a traditionalist in his own right. I consider SteveH to be a co-author of Garelick's blog since he posts there so often.

But today I wish to focus on the other commenter, Richard Phelps. Here's what he writes:

My high school physics class — in the early seventies — comprised project after project. Teacher called it “Harvard Physics.” Each class period was filled with setting up equipment and building contraptions, in small groups. Essentially, it was a “lab only” course. I learned less in that course than in any other over 4 years of high school, and I’m including gym in the comparison. Typically, an entire class period was devoted to delivering just one fact or concept — “authentically” — that could have been simply told to us in less than a minute or, with some discussion, in just a few minutes — un-authentically. Moreover, so much of our attention was focused on building the contraptions and getting them to work that, in most cases, the one factoid we were supposed to learn was lost in the morass of mostly irrelevant information.

Some traditionalists are satisfied by returning to traditional math while conceding that projects are at least useful in science. But apparently, to others like Phelps, even science should be project-free. To Phelps and other science traditionalists, students should spend most of science class either taking notes based on a teacher's lecture (which could include a lab demonstration, but only the teacher is performing the lab, not the students), or answering questions out of the text. Anything else, to the science traditionalists, is not learning.

Here's my problem with what Phelps is saying -- I don't know what state he is in, but I know that Garelick is a fellow Californian. Here in the Golden State, the two major university systems (UC and CSU) have what's known as the a-g requirements for college admission. Let's look at the a-g requirement for science:

Laboratory Science ("d")
Two units (equivalent to two years or four semesters) of laboratory science are required (three years are strongly recommended), providing fundamental knowledge in two of the following disciplines:
[emphasis mine]
-- Biology
-- Chemistry
-- Physics

So we see that the UC and CSU systems expect high school students to take laboratory science. A student who takes the lecture science courses that Phelps prefers would not be admitted to a UC, since the UC explicitly states that it expects incoming students to have had laboratory science.

Last year, I was hired as a math teacher at my middle school. But I suddenly found out that there was no science teacher, and I unexpectedly had to teach science as well as math.

Meanwhile, the Illinois State text that I mentioned above is strongly project-based, and I was required to have students complete a project once every two weeks. It is undoubtedly a curriculum of which the traditionalists would disapprove.

Now here's the thing -- one day, I tried to pass out a worksheet for science, and my eighth graders complained about it. I wrote about this in my January 27th post:

-- On [January] 13th, I decide to print the students a Study Island worksheet on Human Interactions. The idea is that this lesson would bridge the gap to the Green Team next month. But then partway through the lesson, the students complain that it was too boring and refused to work on it. One girl who had transferred in from another school told me that her old school had a real science teacher who gave interesting projects, such as an Edible Cell Model. I agree to try something similar in my class.

The girl told me that my lessons weren't "fun" enough. When I tried to explain that lessons should not be "fun," but instead should allow them to learn a lot (as Garelick, SteveH, and Phelps would argue), she countered that she'd learned much from her previous class, yet she nonetheless had fun then.

I suspect if I were to tell this story to a traditionalist, he'd tell me that this was insufficient evidence that science classes should be fun. "So," our hypothetical traditionalist would ask, "when you finally gave the project, the girl was engaged, right?"

Well, let's find out:

-- On [January] 17th (after the three-day weekend) I introduce an Edible Molecule Project to be completed at home, just like the Edible Cell Model from last year. But the students reject this as well, telling me that they can't do it since they haven't learned anything about molecules yet. This includes the new girl, who probably never wanted to do the project in the first place. She only brought it up in order to tell me how her old teacher was a much better science teacher than I am (which is true, since she was a genuine science teacher and I'm not).

"Oh," our hypothetical traditionalist responds, "there were other issues in the class, and so this doesn't show that science teachers should give more projects."

And part of the problem was that as a natural math teacher, I rarely covered science, and so it's understandable that the students would reject something that was unexpected. I suspect that if I had covered science the entire year (as I should have), my science lessons would have been accepted -- regardless of whether they were project-based or traditional.

I plan on writing about my eighth grade class -- indeed about one particular student. Earlier I wrote that I'd post it soon, and in fact I'd had nearly the entire post already prepared for posting. But that was the day I found out that I wasn't going to be hired at the new school. The major disappointment weakened my will to post, especially about teaching -- and besides, I would need to spend extra time applying for teaching jobs. Instead, I wrote about topics from books written by both Theoni Pappas and Eugenia Cheng.

Notice that the Illinois State curriculum covers both math and science. As I think about my science failure more and more, I now realize that I should have dropped the Illinois State math projects and covered only the science projects. Indeed, this is essentially what my counterpart did at our sister charter school. I could have met the two-week project requirement with only science projects and prepared the students for lab science "d" in high school to boot, and this would have provided me with a loophole to keep the math class more or less traditional.

By the way, I wonder when Phelps believes that students should have their first lab science class. I'm certain that he'd admit that Ph.D candidates need to do original research in labs. So it remains to be seen whether Phelps would accept lab science for lower division, upper division, and MS candidates.

7. What are number bases?

I write about number bases today because they are related to both the Pappas topic (on computers) and the traditionalists (and you'll find out why soon).

Pappas explains:

"The binary system (base two), which uses only 0s and 1s to write its numbers, held the key to communicating with electronic computers -- since 0s and 1s could indicate the 'off' and 'on' position of electricity."

Earlier in the book, Pappas counts from zero to twelve in binary:

0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, ...

As she explains, the first mathematician to work in binary was the famous Scottish mathematician John Napier. This was before computers, so he used a chessboard to display his binary numbers.

Pappas writes:

"For example, to add 74 + 99 + 46, each number is written out in a row of the chess board by placing markers in the appropriate squares of the row so that the sum of the markers' the number they represent. 74 has markers at 64, 8, and 2 since 64 + 8 + 2 = 74."

In modern binary notation, we would write 74 as 1001010. Pappas proceeds:

"After each number is expressed on the chess board, the numbers are added by gathering the markers vertically down on the bottom row. Two markers sharing the same square equal one marker to their immediate left. So two '2' markers produce one '4' marker. Working from right to left, any two markers sharing the same square are removed and replaced by one marker in the adjacent square at the left. At the end of this process, no square will have more than one marker. The sum of the values of the remaining markers represents the sum of the numbers."

Here's the connection to traditionalism -- notice that this is actually the standard algorithm for adding in base two. The line about "any two markers sharing the same square are removed and replaced by one marker in the adjacent square" corresponds to "carry the one" in the standard algorithm.

And so we see that by learning about other number bases, we can shed more light on how young students learn arithmetic in base ten.

There are other number bases besides base two, of course. There is, in fact, an entire message board devoted to base 12, or dozenal:

This website is the de facto home of number bases in general. Indeed, the members of this site come up with their own words to describe the properties of number bases. This actually deserves a post of its own, and indeed I'll be making such a post soon. I admit that lately on the blog, "soon" essentially means "whenever." Well, let's just say I'll post about number bases before we reach the last page of the Pappas chapter on computers (page 221, which corresponds to August 9th).

8. Why wasn't my first year of teaching as successful as I'd like?

I've been reflecting on this for the past four months. And I've realized something -- whenever I have a bad day in the classroom, whether it was this year or back when I was a sub, the reason for that bad day can be summarized in a single sentence:

The students were led to believe that they could do whatever they wanted, then they resisted when they found out that they couldn't.

I remember my first day of substitute teaching, years ago in an eighth grade math class. Several of the students refused to do any work, and I kept nagging at them to do it. The students eventually rebelled, and during one period, they pulled a classic sub trick -- they all jumped up at the same time. I asked myself, what was I thinking making the students do work when I'm just a sub? And so I thought it was better not to expect too much from students -- at least the students wouldn't rebel against me the way they did that first day. I was wrong to believe this -- yet it permeated my entire line of thinking when I was in the classroom, all the way until this year.

The real reason the students jumped up that was that I didn't make it clear from the start of the period that I expected them to be quiet and do work. When I was taking roll, the students continued to talk, yet I didn't interrupt roll to tell them to stop. From this, they concluded that they could do whatever they wanted, and so they were shocked when I tried to punish them for not doing their work. If I had made it clear that there was to be no talking even during the roll, they probably would not have jumped up when I told them to work.

I don't write about classroom management often on the blog. This year, however, I wrote extensively about management in several posts labeled MTBoS (Math Twitter Blogosphere).

In last year's FAQ, I wrote about E's, S's, and U's -- conduct grades used by the LAUSD. My charter school didn't use these conduct marks, yet I still mentally divide my students in into three groups corresponding to these conduct grades.

Also in that FAQ, I wrote about a participation points system that I would use this year. I was hoping to use this as my primary management system, but it failed. This is what I wrote:

At the start of each unit (that is, right after the test), each student has two Participation Points. I will award a point each time a student gives a correct answer, to a Warm-Up or any question that I ask during the main part of the lesson.But if a student fails to participate or otherwise misbehaves, I'll deduct a point. If the student's Participation Point total drops below zero, I will begin assigning consequences, beginning with a minute of detention for each additional point below zero. I expect these detentions to be the most effective in my eighth grade classes, because based on my school's block schedule, Math 8 always meets right before nutrition or lunch, so students will want to avoid those detentions. (Math 7 also meets before nutrition on certain days of the week.) Failure to show up to detention results in a doubling of the detention time, and further consequences occur when the total detention time exceeds a certain amount. 

The problem was that in all three classes (but especially sixth grade), there were many students who were smart, yet very talkative. They would run up their Participation Point total by answering questions, and so when they lost points, it was never enough to earn them a detention.

Since the students weren't punished, they thought that it was okay to talk. Soon it was similar to the worst days as a sub -- since they believed talking was OK, they were shocked when it was time for them to work, and I ended up yelling at them. My biggest fear from last year was realized -- all of the S-students became U-students.

When I came up with the point system, I wanted to make it as easy as possible. So I gave them a single score to represent both academics and behavior. This is what Eugenia Cheng warned about in her Beyond Infinity, when she wrote about using one variable to represent two dimensional data.

And so I should have kept track of academics and behavior separately. I could still have awarded points for correct answers. But I should never have deducted points for bad behavior. Behavior needed to be addressed with a discipline hierarchy separate from Participation Points.

9. What did I do about cell phones in the classroom?

I addressed this as part of last year's FAQ. I wrote:

And so, at the end of any detention earned due to cell phone use, I will require the student to say "Without math, there wouldn't be any cell phones" before releasing them. And if by chance I must confiscate a phone, I will require the student to say the same before returning the phone.

And yes, I really did make them say this last year. But I had a completely separate problem with cell phone use this year.

I would see a student apparently typing something on a cell phone. I would tell her to put the phone away, and then she'd claim that she didn't have a phone out. Then I'd say that I see her with a phone out -- only for her to reveal that she'd been pressing on her cell phone case, not the phone itself.

The idea here is that I can't reliably tell whether a student has a phone case or actual phone out, and so the only fair thing to do is let students use phones whenever they wanted. The argument was that it's far better to let ten students use phones with no punishment whatsoever than to punish even one innocent student for merely having a phone case out.

Of course, students aren't entertained by playing on phone cases -- they are entertained by playing on actual phones. The sole purpose of playing with the case was to "neuter" the no cell phone rule -- by exploiting a loophole to prove that the rule is "unfair."

In reality, a student playing on a phone case isn't innocent. Such a student is trying to create a classroom where she can use her phone with no punishment. Her goal obviously isn't to make herself smarter, learn a lot of math, or become a valedictorian.

Thus a student who plays with a cell phone case deserves punishment -- yet I was powerless to do so, since nowhere in my rules was it stated, "no cell phone cases."

This leads to our last FAQ item:

10. How should have I stated my most important classroom rule?

This is what I wrote in last year's FAQ:

Rule #3: Respect yourself and others.

Students respect themselves and each other as well as me, the teacher, by following all rules and allowing others to learn the material.

This rule worked for Fawn Nguyen, the teacher from whom I got this rule. But it didn't work in my classroom at all. Of course, a student playing with a phone case isn't respecting the teacher, but I needed a rule that would require the student to put the case away immediately.

Here's a much better rule:

Rule #1: Follow all adult directions.

And so if I say, "Put the phone case away," the student couldn't counter that phone cases were against the rules, because my direction was to put the case away.

If I ever find myself in the classroom again, this will definitely be my first and most important rule.

OK, here are some Pi Approximation Day video links:

1. Draw Curiosity

Notice that this video, from last year, actually acknowledges Pi Approximation Day.

2. Reed Reels

This video mentions that even Legendre -- the author of the geometry text we've been discussing in some of my recent posts -- contributed to the history of pi. In particular, Legendre proved that not only is pi irrational, but so is pi^2 (unlike sqrt(2), which is irrational yet has a rational square, namely 2).

This video also mentions a quote from the mathematician John von Neumann: “If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” We've seen Eugenia Cheng, in both of her books, say almost exactly the same thing.

3. Fact Retriever

This video gives ten amazing facts about pi.

4. Numberphile

No Pi (Approximation) Day is complete without a Numberphile video. In this video, we learn that pi is not just irrational, but is also transcendental -- that is, there doesn't even exist a polynomial f, with integer or rational coefficients, such that f (pi) = 0 (unlike sqrt(2), which is algebraic). This is even stronger than Legendre's result that pi^2 is irrational.

5. Sharon Serano

Well, I already gave ten facts about pi, and so this video is twice as good.

6. Vi Hart

No, I can't avoid another one of my favorite math YouTube posters, Vi Hart. Of course, in this video she argues that there's nothing special about pi at all

7. Danica McKellar

I've mentioned McKellar's video in my past Pi (Approximation) Day posts, and it's a favorite, so I'm posting it again this year.

8. A Song Scout

This is another pi song based on its digits. Unlike Michael Blake's song (listed below), it is in the key of A minor rather than C major.

9. Michael Blake

I like the idea of using music to teach math, so here's a favorite, Michael Blake's "What Pi Sounds Like." I tried to find a way to incorporate songs such as this one into the classroom.

10. The Ancient Melodies

Going back to number bases, this song uses the digits of pi in dozenal, or base 12. Dozenal was chosen because we commonly use the 12EDO scale in Western music.

Bonus. Radomir Nowotarksi

This song is actually based on the Fibonacci sequence, not pi. I mention it here because in a previous post, I wrote about a Numberphile video which featured a song where the notes (1 = C, 2 = D, 3 = E, up to 7 = B) were based on reducing the Fibonacci numbers mod 7. I never found that video, but here Nowotarski is doing the same thing.

Nowotarksi uses the Lydian mode in this song rather than the major scale. This means that note 4 corresponds to F# rather than F. Like Michael Blake, Nowotarksi places a chord on each note, but these are different chords due to the use of F# over F. Where Blake uses a D minor triad (D-F-A), Nowotarksi uses a D major triad (D-F#-A).

I've mentioned that many songs based on digits are manipulated so that they end on Note 1, which is the tonic of the scale. But Blake's song ends on Note 5, since five is the last digit before the first appearance of zero (which he was tried to avoid). Song Scout uses G# for both zero and seven -- and his song actually ends on G#. The dozenal song changes the last digit from Note 5 to Note 4 so it could end on Eb (which serves as a tonic in that section). The Fibonacci song repeats, and each section ends on Note 7.

And so I wish everyone a Happy Pi Approximation Day.

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