Friday, July 29, 2016

Rule #4: Respect Your Class Equipment

As the summer draws to a close, it's about time that I move away from spherical geometry and more towards my upcoming classroom and its rules.

Table of Contents
1. Better Together: California Teachers Summit 2016
2. History of the LAUSD Academic Calendar
3. The Academic Calendar Beyond the LAUSD
4. Definition of Summer
5. Managing the Academic Calendar and 80-Minute Block
6. The Willis Unit
7. The Tomb of the Unknown Soldier
8. The Loneliness of the Long Distance Runner
9. More Comments from Traditionalists
10. Rule #4: Respect Your Class Equipment

Better Together: California Teachers Summit 2016

One thing I've noticed is that during the summer, many members of the Math Twitter Blogosphere, or MTBoS, often write about the teacher conferences that they attend. As I consider myself a member of the MTBoS, I'll continue this tradition by discussing a conference that I attended.

Today was the second annual California Teachers Summit. No, I didn't attend the first summit last year because at the time, I was just a sub. But now I'm about to become a full-time teacher, and as a beginning teacher, I knew I could benefit from attending a teacher conference like this one. The theme for this year was Better Together, and this rings especially true for new teachers like me. I will be a better teacher when I work together with my more experienced colleagues.

The California Teachers Summit was held at several dozen locations throughout the state. I signed up for the location at California State University, Dominguez Hills, because there is actually where I earned my credential. The Summit was divided into three breakout sessions, but since I already had other plans for today, I only attended the first breakout session.

This session was titled "First Year Tips: How to Survive From Day 1" -- since I'm getting ready to start my first year of teaching, this was a no-brainer. The presenter of this session was Nicolette (sorry, I missed her last name). Although she has been teaching for well over a decade, she says that she has experience several "first years of teaching" as she made the transition from middle to elementary school, as well as from a charter to a public school. She is currently a fourth grade teacher in the nearby city of Eagle Rock.

Nicolette asked a group of us newbies what our biggest concern was as we get ready to begin our teaching careers. It goes without saying that a big concern of ours is Classroom Management. She recommended that we read The First Days of School by Harry and Rosemary Wong. This book is well-known among teachers, and indeed I myself own an old copy of their book (dated 1998). I once saw a newer version of the Wongs' book, but I never did purchase it. As the title implies, the Wongs believe that the first few days of school are critical in setting the tone for the new school year. And Chapter 3 of their book is called "How You Can Be a Happy First-Year Teacher" -- I'll definitely read this chapter over again before starting my first year of teaching.

One newbie teacher told Nicolette that her biggest fear was issues with parents -- in particular, the parents won't like it when they find out that their child's teacher is a first-year teacher. This is tricky -- after all, most patients probably wouldn't prefer a first-year doctor either, but if no one ever went to a first-year doctor, first-years could never become experienced doctors. Likewise, every experienced teacher must have once had a first classroom. Nicolette told us that we can reassure parents by introducing ourselves to them early in the year -- indeed the Wongs say the same in their Chapter 13.

Finally, Nicolette introduced us to the following website, where us newbie teachers can find resources created by other teachers:

As new teachers, other teachers are always a great resource, whether it's at the Teachers Pay Teachers website, the MTBoS, or best yet, a mentor on campus.

If you are a California teacher, I highly recommend attending the third annual California Teachers Summit next year. Maybe in 2017 I'll actually attend the entire day (especially since lunch is served)!

History of the LAUSD Academic Calendar

At the start of this post, I wrote "as the summer draws to a close," but today is only July 29th. As I grew up, I always attended schools on the Labor Day Start Calendar -- on that calendar, the last week of July was approximately the midpoint of summer break. I'm still getting used to the Early Start Calendar, with school starting in less than three weeks. Recall that the topic of today's post is school and time, and so I begin by discussing the history of the LAUSD school calendar. Again, remember that I'll be working at a charter middle school, not LAUSD proper. Still, my school calendar is nearly identical to that of LAUSD, so I'll write about the district calendar.

For years the LAUSD, like most Southern California schools, had a Labor Day Start Calendar. In fact, the day after Labor Day (a Tuesday) was a day of preparation for teachers, and all students would start school on Wednesday -- all students on the Traditional Calendar, that is. I wrote back in my Father's Day post that some district schools had a Year-Round Calendar instead. Let me explain more about the Year-Round Calendar.

The population of the district increased greatly as the children of the Baby Boomers (the generation we now know as the Millennials) reached school-age. District officials realized that, instead of building new schools, it would be more efficient if schools would operate year-round rather than close for the summer.

The Year-Round Calendar is based on there being 240 weekdays available in the school year. If you think about it, there are about 52 weeks in a Gregorian year. If we assume that winter break is two weeks and other holidays (Monday holidays, for example) are the equivalent of two more weeks, then this leaves 48 weeks, or 48 * 5 = 240 days, of school.

Now each student attends school for 180 of these 240 days. As the fraction 180/240 reduces to 3/4, this means that students need to attend school for three-quarters of the calendar year. Now on the Traditional Calendar, it's easy to have the students attend 3/4 of the year -- just have them go to school during three of the four seasons (fall, winter, and spring) and give them the fourth season off (summer, of course).

Here's how students can attend 3/4 of the year on the Year-Round Calendar -- we divide all students into four tracks, labeled by the first four letters A, B, C, and D. At any point three of the tracks attend school while the fourth track is off. The result is that all students attend school for 3/4 of the year, or 180 days, just like the Traditional Calendar students.

You might think that the A-Track student would take, say, the summer off, B-Track students would take the fall off, C-Trackers the winter, and D-Trackers the spring. But the actual Year-Round calendar in the LAUSD wasn't as simple. Students didn't have a single three-month break -- instead, the break was divided in half, so they had two breaks, each a month-and-a-half, during the year.

The school year was deemed to run from July to June -- so each student would move up a grade level as June turned into July. A-Track would take the first break, from the early July to mid-August. Then C-Track would be off from mid-August to late September, then B-Trackers are off from early October to mid-November, and then D-Trackers take their vacation from mid-November through winter break. Winter break marked the midpoint of the year -- it divided the semesters for all students. Then the pattern would repeat itself for second semester -- first A-Trackers would be off from winter break to mid-February, and so on.

The Year-Round Calendar no longer exists in the district, except at a single high school (located in the nearby city of Bell). Let's look at the 2016-17 calendar at that last high school to understand what a Year-Round Calendar looks like:

Dates School is in Session:

August 15th - December 20th
February 16th - June 30th

July 5th - September 26th
November 2nd - December 20th
January 3rd - March 30th
May 18th - June 30th

July 5th - August 12th
September 27th - December 20th
January 3rd - February 15th
March 31st - June 30th

July 5th - November 1st
January 3rd - May 17th

At some schools, having only 3/4 of the students attend at a time was enough to avoid overcrowding, but at others, this was not sufficient. These schools introduced what was known as a Concept 6 Calendar, which divided students into three tracks, with only two tracks on at a time.

Notice that two-thirds of 240 is only 160 days. The Concept 6 Calendar officially had the students on track for 163 days -- the extra three days being squeezed in the calendar somehow. For example, the Four-Track Calendar I linked to above gives the students a week off for Thanksgiving, but if the Concept 6 Calendar still existed (which it doesn't), mostly likely the Monday, Tuesday, and Wednesday of that week would be the three extra days.

With only three tracks, each break would be two full months, instead of a month and a half. A-Track would be off July and August, B-Trackers would be off September and October, C-Trackers would be off November and December, and then the pattern repeats with A-Trackers off January and February, and so on.

Notice that while every Year-Round school had a winter break, there is no spring break on any version of the Year-Round school. This can result in very long stretches without a holiday -- for example, we see that the A-Trackers at the last remaining Year-Round school start the second semester on a Thursday, then they attend Friday before taking President's Day off. After President's Day, they don't get another day off until Memorial Day, more than three months later! (It's also a bit strange that there would be a week off for Thanksgiving but not for Easter -- the difference of course is that Thanksgiving is on Thursday so there must be days off for Turkey Day, while Easter is on a Sunday and so it doesn't effect the school week at all.)

Five years ago, the LAUSD voted to abolish all but one Year-Round school. At the same time, the district was to adopt the Early Start Calendar. But the new calendar met union resistance, and so the only change that was made that year was the introduction of Thanksgiving holiday week. I noticed that the previous year, there had been furlough days due to budget cuts, and the furlough days were placed during Thanksgiving week. Even when the budget was restored, Thanksgiving week was so popular that it was made a part of the new calendar. The following year was when Early Start was ultimately implemented.

The district recently released the results of a survey regarding the school calendar. The first question was, "In your opinion, the school year should start..." with the choices Early August, Mid-August, Late August, and after Labor Day. Among teachers, the current Mid-August calendar was the most popular with high school teachers, while the Labor Day Start was favored by elementary teachers. Of course, this makes sense because the reason for starting school early is for first semester finals to be taken before winter break, so teachers who have finals -- high school teachers -- are the ones who want school to start early.

Many people also want to take the weather into account when considering the school calendar. In many years, the hottest day of the year is after the first day of school (on an Early Start Calendar), while the hottest part of the city is the San Fernando Valley (of "valley girl" fame). The results of the parent survey reveal that the Labor Day Start Calendar is popular in Local District 3 (a portion of the LAUSD), which is completely in the San Fernando Valley. Labor Day is also popular in Local District 4, which is only slightly in the Valley, as well as District 7, which is in the south, nowhere near the Valley. It could be that Districts 4 and 7 are adjacent to other districts (other than LAUSD) that have either Labor Day or Late August Starts.

There was also a question about a "Modified Early August" Calendar. This calendar has two semesters with the winter and summer breaks being of equal length. It is similar to a calendar with the first semester following the Early Start Calendar and the second semester following a Labor Day Start Calendar (i.e., it starts in February). It would also be similar to the A-Track calendar at the last remaining Year-Round high school. But this calendar wasn't popular with anyone.

The final question was, "The first semester should end before winter break." For this question, "I strongly agree" was the most common response, even among groups that chose Labor Day Start. I've mentioned this before -- both "the first semester should start after Labor Day" and "the first semester should end before winter break" are both desirable ideas. The only problem is that there just aren't enough weeks between Labor Day and Christmas to make up a full semester. I once pointed out that Labor Day to Christmas is approximately 2/5 of the year, so perhaps the school year ought to be divided into five quinters rather than four quarters. Then the first quinter can begin after Labor Day and the second quinter can end before Christmas.

Under the old Labor Day Start Calendar, each semester was about 19 weeks long (not counting winter or spring breaks). Recall that the school year contains 180 days, which is about 36 weeks, but there are about two weeks' worth of holidays, so the year is actually 38 weeks in length. An Early Start Calendar would have to begin around the first Wednesday in August in order for the two semesters to be equal -- this is about 16 weeks before Thanksgiving, and there are always three weeks between Thanksgiving and winter break, for a total of 19 weeks.

But the Early Start Calendar implemented four years ago started a week later, which resulted in the first semester having only 18 weeks and the second semester having 20 weeks. Even that calendar faced backlash, and so last year the calendar to start yet another week later. Now the calendar is set to begin on a Tuesday between the 14th and the 20th of the month (with Monday a preparation day), and so the split is now 17-21. It appears that 17 weeks is the shortest the first semester can be while still credibly calling it a "semester." Now 17 weeks is 17 * 5 = 85 days, but with various holidays it ends up being only 79 or 80 days, with at least 100 days in the second semester.

The Academic Calendar Beyond the LAUSD, and the Definition of Summer

I notice that there are a few other districts here in Southern California that still use what are referred to as "Year-Round Calendars." But on these calendars, all the students at a school are on a single track -- the breaks are just spread out more. In some ways, these schedules are vestiges of older Multi-Track Calendars that are being phased out (for example, the Modified Early August proposal of LAUSD being based on the old A-Track schedule).

In San Diego, for example, a few schools divide the year into three "trimesters." The first trimester begins in late August and ends in mid-December. The second trimester begins after MLK Day (so the winter break is four weeks long) and ends in late March. The third trimester begins in late April and ends on July 21st. I like to call this the "British Calendar" because it actually resembles the school calendar used in the UK, which also divides the year into three terms. (The UK also six half-terms though, which aren't included in the San Diego calendar.) Yes, school in the UK this year lasted all the way until July 21st (or at least the week before the 21st) -- this explains why the release of the final books in the Harry Potter series (including the "eighth book" published this year) are always in late July, right when British kids are off for the summer. But San Diego is planning on eliminating the Year-Round Calendar in the next few years, with all schools reverted to the traditional calendar.

Meanwhile, in Chula Vista (right next to San Diego), there's another Year-Round plan. A few years ago, I was surprised when I heard that the Chula Vista Little League players (who were on their way to the World Series) had trouble practicing because school had already started for them. So I looked it up and saw that in this city, the Year-Round Calendar is divided into four quarters. The first quarter began on July 20th and ends in mid-September. The second quarter begins in early October and ends in mid-December. After a three-week winter break, the third quarter begins in early January and ends in mid-March. The fourth quarter ends in early April and ends in early June. I like to call this the "Australian Calendar" because it actually resembles the calendar used in that country. In fact, if we add six months to the first day of school (in order to take the difference between Northern and Southern Hemisphere seasons into account) we obtain January 20th. Many schools in Australia did begin the first of their four terms around January 20th (or at least the week after the 20th).

Putting these together, notice it's possible for there to be two children, born the exact same day and living less than a mile apart (one in San Diego, the other in Chula Vista). Last Wednesday, one of the children attended the first day of school as a sixth grader -- and the very next day, the other child celebrated the last day of school as a fifth grader!

Wednesday, July 20th: First day of the 2016-17 school year in Chula Vista
Thursday, July 21st: Last day of the 2015-16 school year in San Diego

(Notice that in Chula Vista, the second day of school was Pi Approximation Day.) Unlike San Diego, Chula Vista currently has no plans to abolish the Year-Round Calendar.

So we see that there are various kinds of academic calendars -- some so different that even two adjacent districts can have "summer break" fail to overlap even by a single day. This mess raises the question, what exactly is the definition of "summer" anyway?

Definition of Summer

We think of summer as the time of year when the days are longest -- conversely, winter is the time of year when the days are shortest. But when are the days the longest, anyway? Naturally, the longest day of the year is the summer solstice, which is on June 20th or 21st. But if you think about it, by symmetry we expect the days leading up to the solstice to be as long as the days after the solstice -- for example, the day length for May 31st (three weeks before the solstice) should be the same as that for July 12th (three weeks after the solstice).

So if we want "summer" to be the quarter when the days are longest, then the season should begin halfway between the spring equinox and summer solstice, and end halfway between the solstice and the fall equinox. As it turns out, many pre-Christian cultures gave names to these days. The day on which summer begins, halfway between the spring equinox and the solstice, is called Beltane, and the day on which it ends, halfway between the solstice and the fall equinox, is called Lammas. Similarly, winter begins halfway between the fall equinox and the winter solstice, or Samhain, and ends halfway between the solstice and the spring equinox, or Imbolc.

Thus Beltane is in early May, Lammas in early August, Samhain in early November, and Imbolc in early February. These dates are used for a variety of modern holidays -- for example, Beltane is also known as May Day, Samhain is known as All Saint's Day, and Imbolc is Groundhog Day. (It's possible that the Assumption of Mary is tied to Lammas.) This explains why we need a groundhog to determine the start of spring, because Imbolc marks the end of winter -- the darkest quarter of the year. In China, Imbolc is known as Lichun -- which explains why Chinese New Year always occurs in late January or early February. It's intended to mark the new moon closest to Lichun -- which explains why it's also known as the Spring Festival.

If this is the case, then you may be wondering, why do we usually think of summer as beginning on the summer solstice and ending on the fall equinox -- meaning that summer is when the days start getting shorter during the entire season?

The reason for this is that the temperature always lags behind the light. The hottest day of the year is often months after the longest day of the year, and the coldest day of the year is often months after the shortest day of the year. This effect is most pronounced near the coasts due to the high specific heat of water -- it always takes extra time for the temperature of water to react to the sunlight.

For example, many people dream of a White Christmas, but on the East Coast, it's often said that a White Easter is more common than a White Christmas (or as the saying goes, "Green Christmas, White Easter"). This is because due to the largest temperature lag at the coasts, the coldest day of the year isn't merely after Christmas, but after Imbolc, and is actually closer to Easter than to Christmas, especially when Easter is early.

Here in Southern California, it doesn't snow, but coastal temperature lag is most noticeable here for the summer season. The weather is often cool and cloudy in June -- known as the June Gloom -- and often becomes hot and windy in the fall -- known as the Santa Ana winds. Because of this, the hottest day of the year is usually after Lammas (August), and sometimes even after Labor Day.

Suppose we wanted to take weather into consideration, and we wanted to come up with a different calendar for each state based on its climate. We look at the following:

-- Coastal states should have later school starts than landlocked states. This reflects the fact that warmer weather starts and ends later on the coasts.
-- Snowy states should have earlier school starts than rainy or dry states. This reflects the fact that we don't want make-up days due to snow to last past July 1st, and especially not July 4th. (This is also why I don't just blindly say "make summer vacation the hottest months" -- here in Southern California the hottest months are August and September, but we certainly don't want our school year to run from October to July.)

Given these two constraints, it turns out that one of the best states for a Labor Day Start Calendar may be my home state of California. Hot California Augusts fuel the desire to wait until Labor Day to start school, and there's no chance of snow make-up days pushing the end of the year into July. For years, California schools mostly began after Labor Day, but as we know, the current trend is for school to start in August here.

The last holdouts for the old Labor Day Calendar is in the Northeast -- mostly in New Jersey, New York, and New England. But this area is of course highly prone to snow (Nor'easters), and so the worry is always that there will be too many snow days threatening to push school into July. (In fact, in the Northeast the last day of school is often Tau Day, June 28th.)

The Midwest has always been the best region for Early Start Calendars. In these states, the last day of school is often in May. This explains why national AP tests must be given in May -- the test must be before the last day of school in the Midwest. This is much to the chagrin of coastal states where there's often a month of wasted time after the AP is taken.

We know that the main driver of the Early Start Calendar is high school finals. But still, I can't help but notice that while the old calendar ran from September to June (approx. the fall equinox to the summer solstice), the new calendar often runs from August to May (approx. Lammas to Beltane). It is almost as if the old summer break was designed to track the hottest days (from solstice to equinox) while the new summer break is designed to track the lightest days (from Beltane to Lammas).

But the goal of the Early Start Calendar is to have the first semester finals before Christmas -- that is, the winter solstice. (Most scholars don't believe that Christ was actually born on December 25th, but that Christmas is ultimately a winter solstice festival.) So it makes sense that if we want the winter solstice to be the midpoint of a school year that spans about 3/4 of the year, then it should begin 3/8 of a year before the winter solstice (which is Lammas) and end 3/8 of a year after the winter solstice (which is Beltane). If we want the school year to run from fall equinox to summer solstice, then the semester point needs to be Imbolc, not the winter solstice. (Recall I once posted a link to a joke proposal to move Christmas to February, which would solve the problem -- since again, Christ most likely wasn't born on December 25th anyway.)

So that explains why I'm getting ready for Back to School in August. Before I leave the topic of calendars, notice that the Year-Round Calendar still used at one LAUSD high school is naturally divided into eight parts, so it fits the eight seasonal markers. So A-Trackers are off from near the summer solstice to Lammas, and D-Trackers are off from just after Halloween (which is actually Samhain) through Christmas (the winter solstice).

Managing the Academic Calendar and 80-Minute Block

I've mentioned before (when discussing the Unexpected Hanging Paradox) that students believe that they are entitled to the last 10% of any stretch of academic time. Therefore, they feel that they shouldn't have to do work the last 10% the school year (which works out to be the last 18 days), the last 10% of a "term" (which here I'm using in the British sense as the period between major breaks, so here I mean the last week before winter break or spring break), the last 10% of the week (Friday afternoons), and the 10% of the period (the last eight minutes for me).

If we think about it, students believe that they are entitled to the first 10% of any stretch of academic time as well. So students will complain about working the first 18 days of school, the first week after winter or spring breaks, Monday mornings, and the first eight minutes of the period.

Of course, this is all nonsense. If there were really no work the first 10% and last 10% of the school year, then that means there would be only 144 days of school, not 180. And deducting the weeks before winter and spring breaks would mean that there would be only 124 days of school.

When I begin teaching, one thing that I will tell the students is that "We do math everyday in this class, from the first day of school until the last day of school." Seen another way, here are the only days when students don't have to do math:

-- Saturdays
-- Sundays
-- Wednesdays, for 7th graders (since 7th grade math block doesn't meet on Wednesday)
-- Holidays (by which I mean non-school days, others students won't do math on Halloween)
-- Any SBAC or other special schedule days when math block doesn't meet

Does this means that there will be no days of fun in my classroom? Well, we can certainly have fun in my class, as long as it's math. If you believe that math is never fun, then that means that we will never have fun in my class (but still I hope that at least that Pi Day will be fun, if even for just a day).

I want to write about the smallest and most common unit of time to manage -- the block. Most blocks will be 80 minutes in length, and I want the students to do something mathematical for the entire eighty minutes. This is what I want the typical 80-minute block to look like:

10 minutes: Warm-Up
10 minutes: Go over homework/previous day's lesson
20 minutes: New lesson (Foldable note taking)
10 minutes: Music break
20 minutes: Guided practice
10 minutes: Closure/Exit Pass

Notice that about halfway during the block is a "music break." This is a compromise between my goal that students work the entire 80 minutes and the reality that students won't want to concentrate on math the entire 80 minutes.

Back on Pi Approximation Day, I linked to a video where the digits of pi were used to make a song:

3.1415926... = E-C-F-C-G-high D-D-A...

(Over a year ago, I wrote about the controversy when someone tried to copyright pi!) The video also used some of the notes to make chords, as in:

3.1415926... = Em-(C)-Fmaj-(C)-Gmaj-(high D)-Dm-(A)...

Well, I wish to do the same in my own classroom. Every few days, I take a random number and convert the digits into notes:

.3842213648 = E-high C-F-D-D-C-E-A-F-high C...

Using chords, this becomes:

.3842213648 = Em-(high C)-Fmaj-(D)-Dm-(C)-Em-(A)-Fmaj-(high C)...

I then write these notes and chords into a simple song, which I then play on my guitar. Afterwards, I add some simple lyrics, which must be about math. Usually, I'll try to write three verses -- one each for sixth, seventh, and eighth grade math. Then I'll play the song and sing the corresponding verse during each block.

Now you can see that this accomplishes several goals. First, it provides some students with a break, yet allows other students to reinforce the math they just learned. Second, it provides me with a time incentive -- if we can cover the new lesson in less than 20 minutes, then there will be extra time for the music break (so that students might want to cooperate more). Third, if I can come up with some catchy or funny lyrics, students might remember the material better. I admit that I'm not a comedian, but perhaps the students might come up with funny lines to add to the song. Just imagine months later when students claim that they never learned something, only for me to prove them wrong when I play the song which taught them what they'd forgotten!

For certain lessons, a song from the old show Square One TV may be appropriate. For example, Module 8 of the eighth grade Illinois State text is "Tessellate a Structural Design," so at that point I can play the Square One TV Tessellations song. In any case, if the students ask me to play some famous or popular song, I'll tell them that I only play math songs.

I know that all of this will be much work. But I believe that it will be worth it in the long run -- to have students more engaged during the entire block.

The Willis Unit

How will I deal with the other 10% periods of time when students won't want to work? Well, as far as Monday mornings and Friday afternoons are concerned, my participation points system (mentioned in previous posts) is designed to encourage students to work during those times. In particular, I will allow students with the most points to earn a small reward (possibly candy, as I mentioned back in my Father's Day post).

My original idea was to give out rewards to each period once a week, using the following pattern:

Monday -- 1st Block
Tuesday -- 2nd Block
Wednesday -- 3rd Block
Thursday -- 4th Block
Friday -- 5th Block

Notice that first block would receive a reward on Monday -- when these students are probably tired and unmotivated to work. And fifth block would receive a reward on Friday -- when these students are probably excited and unmotivated to work.

Based on my school's schedule, fifth block is always P.E., so I wouldn't actually be giving out any rewards on Fridays. Monday's first block and Tuesday's second block are the same seventh graders, so I won't give out rewards both days -- instead, I'll give them out only on Mondays, when they'll need more motivation. Wednesday's third block is eighth grade and Thursday's fourth block is sixth grade, so this is when they'll receive their rewards.

Even though there is no reward on Fridays, I'll continue to use Fridays as activity days -- recall that the Illinois State text is full of activities and projects (and I may use some activities from texts other than Illinois State as well). This will continue the tradition I established on the blog of letting Friday be an activity day.

The weeks surrounding winter and spring breaks will be difficult, but then again, it's months before I have to worry about those weeks. But right away, I'll have to deal with the first 10% of the school year, when students will complain that they shouldn't have to do math during the first day, or week, or 18 days of school.

I refer to the first 18 days of the school year as the Willis Unit. I named this time of year after the last name of the first student I met, a girl, who wanted to talk me when I moved as a freshman. But if you prefer, you can take the name Willis to refer to Paul Willis, the British education theorist:

(It's also possible to call it the Wong Unit, since after all the Wongs wrote about the first days of school in their book.) According to the link, Willis wrote about an "anti-school subculture":

-- Kids who join an anti-school subculture go against the main culture (norms & values) of the school
-- These responses include not doing homework, disrupting lessons, messing around, breaking school rules
-- They tend to be placed in the bottom sets or streams [academic tracks -- dw] and invariably are working-class
-- They gain their status with their peers by doing the above as the school system has labeled them as 'failures'

And of course, this sounds just like many of the students who will be in my middle school classes. I gave this type of students a name in a previous post -- the U-kids. So the Willis Unit, the first 18 days of school, will be critical to my success as a teacher, to make sure that this anti-school subculture doesn't take over my classroom (especially since, as I suspect, many kids make the choice to join the anti-school subculture during their middle school years).

Here are my plans for managing the Willis Unit:

-- I must make sure that the students understand the rules and how I will enforce them. They must know that the class will be run for the benefit of the E-kids, not the U-kids.
-- Students are to know that they can and will be successful in my class. Notice that my sixth graders will be the last grade to take its first scheduled test, which will be on Day 17, and the tests will be returned to them graded on Day 18. This is where the name Willis Unit comes from -- it actually refers to the first unit of the lesson plan. By the end of the Willis Unit, every student will have received a test grade -- and if they see a low grade on that first test, they may choose to enter Willis's anti-school subculture. So during the Willis unit, I need to make sure that as many students are successful on that first test as possible.
-- I will also remind students of the importance of learning math, in the hopes that students will choose to reject the anti-school subculture.

The Tomb of the Unknown Soldier

I've mentioned before that I might take ideas from Danica McKellar's Girls Get Curves series to help motivate my girls to learn math. But this doesn't mean that I won't try to motivate the boys or get ideas from boy sources.

In particular, I was reading last September's issue of Boys Life magazine, which attracted me because it was the STEM issue. There were many articles about how STEM can shape the future -- for example, one article was about setting up colonies on the moon, and another was about a candy scientist who works in Hershey, Pennsylvania. In explaining how students can get on this career path, the candy engineer suggested that students take as many math and science classes as possible.

But ironically, the two articles which inspired me the most from this magazine have nothing to do with STEM at all. One was an article about role players in the NFL -- players like kickers, punters, and return specialists. Earlier this month, I wrote that every student should try to be the top student in the class, just as every player wants to win a championship.

Originally, I wanted to write "just as every player wants to be the MVP." The only problem is that there are football players -- such as kickers, punters, and returners -- who essentially have no chance of ever being MVP. The same is true in baseball -- we rarely give MVP awards to pitchers, especially not relievers, most especially not middle relievers. Perhaps in basketball, which is less positional, we can say that any player can theoretically become the MVP.

Still, kickers, returners, and relievers all contribute to the success of the team -- despite their different skill sets. Likewise, all students can contribute to the success of the class -- despite their different skill sets.

Now the other magazine article that inspired me was about the Tomb of the Unknown Soldier. I remember learning about it in elementary school -- in Arlington, Virginia, there lie three American soldiers who fought in World War I, World War II, and the Korean War. They died in battle and their identities are uncertain, hence the name "Unknown Soldier."

But one thing I didn't know about the tomb before reading the article is that there is a guard there. His duty is to march back and forth at the tomb, exactly 21 steps in each direction, over. He must do it no matter what the weather, and continue marching heavily armed until the Changing of the Guard, which is 24 hours later.

Of course, many people have heard of the British Changing of the Guard, but I never knew that there's an American Changing of the Guard as well. The British guards are famous for being highly disciplined, and the American guard is no different. I don't know how long a British guard's shift is, but the article tells us that the American guard's shift is 24 hours -- and that he must work three out of every nine days. Think about it -- the guard must keep on marching. He can't eat for 24 hours -- he can't sleep for 24 hours -- he can't even use the restroom for 24 hours. And all of this is just so he can guard, not a living person (like the Queen), but a tomb!

Why am I writing about this? Well, today's post is all about time in the classroom. Many of my students will eat and sleep in the classroom and ask for passes to the restroom. Apparently, they can't go 80 minutes without doing these things. And yet the tomb guard can go 24 hours -- that is, eighteen times the length of the class -- without doing any of these!

So we ask, why can't students go 80 minutes without going to the restroom, anyway? Well, let's think about it -- do teenagers need to go to the restroom as often on weekends or in the summer as they do on school days? Seriously, I doubt it. Students really ask for the restroom passes because they are bored in math class -- that is, they are much more likely to want to go to the restroom when they are bored than when they are entertained. If math class and school could be made more entertaining, then the students would be able to go hours without going to the restroom.

But again, we must compare this situation to that of the tomb guard. Don't you think that the tomb guard sometimes gets bored out there, doing nothing but math for 24 hours (albeit the simplest kind of math, namely counting to 21)? Again, the guard does shake off his duties merely because he happens to be bored, or even tired.

I'm proud to say that I completed three full years of school -- Grades 10-12 -- without asking for a restroom pass even once. How did I do it? Each day, I would use the restroom:

-- Before school
-- During nutrition
-- During lunch
-- After changing from street clothes to P.E. clothes (as I was on the track team)
-- After changing from P.E. clothes back to street clothes

That's five times per day that I went to the restroom. Now 5 * 180 = 900, so that's nine hundred times I used the restroom per year, and that was for three years (Grades 10-12) . So that's a grand total of 2,700 times that I used the restroom -- without missing even one second of class time! Again, I might have been bored and didn't enjoy certain classes (like history), yet I didn't ask for restroom passes. I would much, much rather preserve a reputation as one of the brightest and best-behaved students in the class and earn as many A's, B's, and E's as I could.

I could tell this story in class, but I don't think it's fair to compare myself as a 10th-12th grader to my students who are 6th-8th graders -- on average, four years less mature. And I don't claim perfection during my own years in Grades 6-8 -- I didn't ask for restroom passes very often, but I almost certainly did use the restroom at some point during class during these years.

Still, I do wish to minimize passes in the classroom. As a sub, I'd say that the most annoying request for restroom passes was when students would ask for one before class even begins -- and this was especially the case when it's a class right after nutrition or lunch, when they could have just used the restroom five minutes earlier without asking for a pass at all. Dealing with restroom passes right at the start of class often throws me off -- it makes it harder to take attendance, plus it means that I'm not getting the students on task, which then allows the U-kids to take over the class. (Actually, I should say the other U-kids, since as far as I'm concerned, anyone who asks for a pass right at the start of class is a U-kid.)

I was considering letting each student start each trimester with, say, two restroom passes, and then each student can earn an additional pass for each A that he or she earns. (This would include Dren Quizzes, on which every student should be earning an A.) But this means that I'd have two different things to keep track of -- restroom passes and participation points.

It's much easier on me just to make restroom passes part of the participation points system. Basically, each restroom pass costs the student a point. If the student doesn't return in, say, five minutes, then it costs another point. This fits into the existing point system -- so if the student doesn't return and has already used up his or her points, a detention is assigned.

This plan will become more severe at certain times:

-- On Wednesdays, when classes are only 50 minutes long, students should be able to go 50 minutes without going to the restroom even if they can't go 80 minutes. On Wednesdays, a point is deducted every three minutes, rather than five minutes.
-- Right after nutrition and lunch, again I will deduct points more often.
-- If an administrator tells me that I'm sending too many students out, or if there is a disturbance in the restroom, then again I will deduct points more often.
-- I'm considering having a rule that if a student asks for a pass before class even begins, then that student can't go at all during the 80-minute block. If the student then leaves the room anyway, then this would count as leaving the room without permission (or defiance).

There may be one time when this plan becomes more lenient -- during the music break. I still deduct points, but I will only deduct one point for the entire ten-minute music break. Also, this will be the only time that I let two students use the restroom rather than just one.

The Loneliness of the Long Distance Runner

As I alluded to above, I was on the track team in high school. Indeed, I was a long distance runner -- I ran Cross Country in the fall and the 800 and 1600 in the spring. Last year, I wrote about the movie McFarland, a Cross Country film. I recommend you watch that movie if you want to learn more about the One True Sport (as everything else is just a "game").

Throughout my middle school years I'd never thought of myself as a distance runner, but after performing well in the mile run as an eighth grader, I was recommended to join the Cross Country team as a freshman. To me, three miles seemed like an impossibly long distance to run, but still, I joined the team. In the fall, I finished my first race in under 25 minutes, and a few weeks later, I was running under 21 minutes at the Dana Hills Invitational. But unfortunately, I didn't run in the last race of the season, League Finals, because that was the week when I switched schools.

I did join the Track team at my new school, and ran in the distance races. The following summer, I worked hard and trained with the Cross Country team. As a sophomore, I broke 20 minutes at the Woodbridge Invitational. And the following year, I broke 19 minutes at the same race.

But I consider my greatest running accomplishment to be during my senior year. At a time in their careers when many runners stopped improving (except for the varsity runners, whose times were already under 17 minutes), I kept on getting faster. At the last race of the season, League Finals, I broke 18 minutes. I'll never forget crossing the finish line for the last time and realizing that I had run my fastest race ever. Meanwhile, my best time for the 1600 was just over five minutes.

One thing I've noticed is that distance runners tend to be above average academically. Indeed, in the movie, all seven McFarland runners end up going to college! I believe there's a reason for this -- distance running requires endurance -- the same type of endurance that can help a student survive a long two-hour final as well as a 15-to-20-minute race.

Again, I point out that many students believe that they are entitled to have the last 10% of a period, week, term, or school year free. Now let's apply this to a Cross Country or Track race -- we'll just stop running 0.3 miles away from the finish line in Cross Country, or 160 meters away from the finish line in the 1600. Yes, that would definitely look silly, and such a runner would not be able to win the race -- yet that is exactly what a student who believes in having the last 10% of academic time free.

By the way, the Olympics is coming up. In Track, much of the emphasis will naturally be on the sprint races, the 100, 200, and 400 (especially the sprinter Allyson Felix). But I recommend that you watch the distance races as well. I remember watching the distance races at the Olympic trials and saw some amazing results -- for example, Bernard Lagat won the 5000 meters (about the same distance as our cross country races), but what was amazing is that he did so as a 41-year-old! And another interesting middle distance runner is Brenda Martinez, who was tripped up during her 800 trial but recovered to qualify for Rio in the 800.

More Comments from Traditionalists

Yesterday, the traditionalist Bill responded to an article at the Joanne Jacobs website. As it turns out, the article is about academic time -- so it fits the topic of today's post perfectly. As usual, let me link to the article and then the Jacobs website where Bill commented:

According to the article, college students spend only 2.76 hours per day studying. Here is Bill's response to the article.

Bill says:

The average time to study per class in a STEM major (non-core coursework) was usually at least 90 minutes per class per day, or an average of 18-20 hours a week (at a minimum), and if you’re in lab science courses, you usually had a lab course meeting once a week for 3 or 4 hours at a time (esp. in Chem/Bio/Physics), outside of class, given a 12-14 credit hour per semester (which would also require 3-10 credits in Summer session for lower division coursework in the 1st two years).

Bill's point, of course, is that if these students really want to leave college actually knowing enough to be competent at a STEM career, they should be spending at least double the time on education as they actually are.

Here are a few other interesting comments, written by posters other than Bill. The following is an article about teacher pay:

Deirdre Murphy says (skipping to the part I wish to focus on):

If you compare them [teacher --dw] to doctors and nurses and engineers, they get paid less, but…those are all people who can do math…..
In a sense, the low pay may be a reflection of how little math the career entails. Pay seems to correlate with math, and as the researcher said, most teachers aren’t very math-y.

...except math teachers, that is. Still, the point I wish to make is that my students should work hard to learn math, because, as Murphy writes, "pay seems to correlate with math."

Meanwhile, earlier I wrote about how we, as teachers, will have to deal with cell phones. Well, here will be a major time-waster for our students -- Pokemon. Soon will be the first time since the release of the super-popular app that students will be in school:

At the link (and this was brought up at the California Teachers Summit), it is suggested that teachers try to take advantage of the popularity of Pokemon Go in their classrooms, but this may be tricky.


” if we were to turn Pokémon into a school subject, ‘certain children, many of them poor, would all of a sudden have trouble learning Pokémon’.”
If it becomes a school subject it will no longer be “cool”, the thrill of self discovery will disappear and play will become work, not to mention the possibility of it being poorly or boringly taught in school. Do we need to trick children into an education?

Anyway, here is the fourth rule that I wish to implement in my classroom.

Rule #4: Respect Your Class Equipment.

In my next post (in about a week), we will look at my fifth and final classroom rule. Enjoy your Lammas -- er, the last few days of your summer break.

Wednesday, July 27, 2016

Spherical Geometry (Legendre 502-506)

Today is our final spherical geometry post. In today's post we will cover the last five spherical geometry propositions in Legendre, which will be numbered 502 to 506. Of these five propositions, two are corollaries to the main theorem from our last spherical geometry post (namely that the area of a triangle is equal to its "excess") and two more theorems are called "scholia." The only main theorem for today is a generalization of that previous theorem -- how is the area of any spherical polygon (not just a triangle) related to its angle sum?

Let's begin with Proposition 502:

502. Corollary I. The proposed triangle will contain as many triangles of three right angles, or eighths of the sphere (494) [Legendre actually means 495 here -- dw], as there are right angles in the measure [the excess -- dw] of this triangle. If the angles, for example, are each equal to 4/3 of a right angle, then the three angles are equal to four right angles, and the proposed triangle will be represented by [an excess of] 4 - 2 or 2; therefore it will be equal to two triangles of three right angles, or a fourth of the surface of the sphere.

Recall that Legendre uses the right angle as a unit of angle measure, and the triangle with three right angles as a unit of area (that's what Proposition 495 is for). This is convenient because a triangle with three right angles has an excess of one right angle -- recall that the "excess" is the sum of the angles of a spherical triangle minus what the sum would have been had the triangle been Euclidean (two right angles or 180 degrees). So the area of the triangle (using the equilateral right triangle as a unit) is always equal to the excess of the triangle (using the right angle as a unit).

Now that Pi Approximation Day has come and gone, we'll return to using pi radians as our circle constant, and so the excess of a triangle is its angle sum minus pi. Using radians, we can rewrite the example given by Legendre as:

If the angles, for example, are each equal to 2pi/3, then the three angles are equal to 2pi, and the proposed triangle will be represented by an excess of 2pi - pi or pi; therefore it will be equal to two triangles of three right angles, or a fourth of the surface of the sphere.

Or we can rewrite the example using degrees:

If the angles, for example, are each equal to 120 degrees, then the three angles are equal to 360 degrees, and the proposed triangle will be represented by an excess of 360 - 180 or 180; therefore it will be equal to two triangles of three right angles, or a fourth of the surface of the sphere.

OK, we now move on to Proposition 503:

503. Corollary II. The spherical triangle ABC is equivalent to a lunary surface, the angle of which is (A + B + C)/2 - 1; likewise, the spherical pyramid, the base of which is ABC, is equal to the spherical wedge, the angle of which is (A + B + C)/2 - 1.

This follows from some of Legendre's earlier propositions. Recall that a lunary surface, or lune, has its area equal to twice its angle measure, while a triangle has as its area equal to its defect. Since the defect of a triangle is A + B + C - 2 (right angles), we take half of this to find the angle of a lune with the same area. And a spherical pyramid and a spherical wedge are obtained by starting with a triangle and a lune, respectively, and digging towards the center of the sphere. So if the triangle and lune have the same area, the pyramid and wedge must also have the same volume.

Legendre's next proposition, a "scholium," continues with spherical pyramids:

504. Scholium. At the same time that we compare the spherical triangle ABC with the triangle of three right angles, the spherical pyramid, which has for its base ABC, is compared with the pyramid which has a triangle of three right angles for its base, and we obtain the same proportion in each case. The solid angle at the vertex of the pyramid is compared in like manner with the solid angle at the vertex of the pyramid having three right angles for its base. Indeed, the comparison is established by the coincidence of the parts. Now if the bases of pyramids coincide, it is evident that the pyramids themselves will coincide, as also the solid angles at the vertex. Whence we derive several consequences; [deleted -- dw]

Let's skip down to the last thing that Legendre writes about solid angles:

"The angle at the vertex of the pyramid, whose base is a triangle of three right angles, is formed by three planes perpendicular to each other; this angle, which may be called a solid right angle, is very proper to be used as the unit of measure for other solid angles. This being supposed, the same number, which gives the area of a spherical polygon, will give the measure of the corresponding solid angle. If, for example, the area of a spherical polygon is 3/4, that is, if it is 3/4 of a triangle of three right angles, the corresponding solid angle will also be 3/4 of a solid right angle."

If we write Legendre's statement using radians, we obtain the following:

The angle at the vertex of the pyramid, whose base is a triangle of three angles of measure pi/2, is formed by three planes perpendicular to each other; this angle, which may be called a solid right angle, has a measure of pi/2 steradians. This being supposed, the same number, which gives the area of a spherical polygon, will give the measure of the corresponding solid angle. If, for example, the area of a spherical polygon is 3pi/8, that is, if it is 3/4 of a triangle of three right angles, the corresponding solid angle will also be 3pi/8 steradians.

And so we introduce a new unit of solid angle measure, the "steradian." Let's think about how ordinary radians are defined -- the radian measure of an angle equals the length of the arc of the unit circle subtended by the angle. Likewise, the steradian measure of a solid angle is equals the area of the polygon on the unit sphere subtended by the solid angle.

Notice that if we're considering circles other than the unit circle, we usually think of radian measure as the ratio of the arc length to the radius. But on the sphere, recall from the Fundamental Theorem of Similarity that the surface area of a sphere (or a spherical polygon) varies not as the radius, but as the square of the radius. Therefore, steradian measure is the ratio of the polygon area to the square of the radius -- hence another name for steradian, the "square radian." Fortunately, most of the time we will assume that the sphere is the unit sphere of radius 1.

A solid right angle measures pi/2 steradians. Therefore, each of the solid angles of a cube (of which there are eight) measures pi/2 steradians. The entire sphere measures 4pi steradians, since the surface area of the unit sphere is 4pi.

Officially, steradians are considered part of the SI system of measurement, which means that metric prefixes may be used with steradians (abbreviated as sr). As usual, we can take the earth to be our sphere for the purpose of visualizing various steradian units. For example, I found the following measures at another website:

4pi sr (= 1.2566 dasr) = entire earth's surface
1 dasr (decasteradian) = Americas plus liquid water on earth
1 sr (steradian) = Oceania plus Asia excluding Russia
1 dsr (decisteradian) = Algeria plus Libya
1 mcsr (microsteradian) = Costa Mesa, CA

Sorry, I just had to give the microsteradian example because Costa Mesa happens to be right here in Southern California. (Notice that the prefix "micro-" is often written as mc in ASCII, but the official symbol is the Greek letter mu.)

Is it possible to use degrees to convert Legendre's statement -- that is, can there possibly be such a thing as a "stedegree" or square degree? Yes, it's possible, but the conversion from steradians to square degrees is a bit awkward. Think about it -- there are three feet in a yard, yet there are nine square feet in a square yard (since 9 = 3^2). Likewise, just as there are (180/pi) degrees in a radian, there must be (180/pi)^2 square degrees in a square radian (steradian). So let's measure our solid right angle in square degrees:

solid right angle = pi/2 sr
                           = (pi/2 sr)(180 deg/pi rad)(180 deg/pi rad)
                           = 90 deg(180 deg/pi rad)
                           = (16200/pi) deg^2

This works out to be approximately 5156.62 square degrees. So we can rewrite Legendre as:

The angle at the vertex of the pyramid, whose base is a triangle of three angles of measure 90 degrees, is formed by three planes perpendicular to each other; this angle, which may be called a solid right angle, has a measure of 16200/pi square degrees. This being supposed, the same number, which gives the area of a spherical polygon, will give the measure of the corresponding solid angle. If, for example, the area of a spherical polygon is 3pi/8, that is, if it is 3/4 of a triangle of three right angles, the corresponding solid angle will be 12150/pi square degrees.

Obviously, it's much easier to use steradians than square degrees. And so in practice, square degrees are almost never used.

Let's move on to Proposition 505, which is the most important theorem of the day. Before we look at the proof, let's look at the statement of the theorem:

505. The surface of a spherical polygon has for its measure the sum of its angles minus the product of two right angles by the the number of sides in the polygon minus two.

We write this using symbols as follows -- let s = angle sum, n = number of sides, and with degrees:

A = s - 180(n - 2)

And now that term 180(n - 2) should look very familiar -- it is exactly the sum of the angles of a polygon in Euclidean geometry! And indeed, if we define the "excess" of any polygon to be its angle sum minus what that sum "ought to be" in Euclidean geometry, we can write the proposition as:

505. The surface of a spherical polygon has for its measure its excess.

We've already proved this theorem for triangles -- that is, when n = 3. And in a way, we've already proved this theorem for the n = 2 case as well. Think about it -- in Euclidean geometry there is no such thing as a 2-gon, but we can plug in n - 2 into the equation s = 180(n - 2) to obtain s = 0 -- that is, the sum of the angles of a 2-gon in Euclidean geometry is 0 (which is why it doesn't exist).

Now in spherical geometry a 2-gon does exist -- the lune. The two sides of a lune are semicircles that intersect at two antipodal points -- for example, if the two sides are meridians, then the vertices are the North and South Poles. Now the excess of a lune equals its angle sum minus that of a Euclidean "2-gon" (which is zero). So the excess of a lune is its entire angle sum. A lune has two congruent angles, one at each pole, and we've already shown its area to be twice the measure of its angle. Thus the area of a lune indeed equals its excess.

So now let's give Legendre's proof of Proposition 505. As it turns out, we prove it almost the same way that we prove the angle sum of a Euclidean polygon -- by dividing it into triangles. The only difference is that the excess of a Euclidean polygon is always zero, while in spherical geometry the excess equals the area.

Demonstration. From the same vertex A, let there be drawn to the other vertices the diagonals AC, AD; the polygon ABCDE will be divided into as many triangles minus two as it has sides. But the surface has for its measure the sum of its angles minus two right angles, and it is evident that the sum of all the angles of the triangles is equal to the sum of the angles of the polygon; therefore the surface of the polygon is equal to the sum of its angles diminished by as many times two right angles as there are sides minus two. QED

We may compare Legendre's proof to the proof of the Polygon-Sum Theorem (Euclidean geometry, of course) given in Lesson 5-7 of the U of Chicago text. If we wish, we may even combine the Euclidean and spherical proofs, as follows:

Sum(ABCDE...) = Sum(ABC) + Sum(ACD) + Sum(ADE) + ...
                           = pi + Excess(ABC) + pi + Excess(ACD) + pi + Excess(ADE) + ...
                           = pi(n - 2) + Excess(ABCDE...)

and then we replace all the "Excess" terms with 0 for Euclidean and "Area" for spherical geometry.

The final theorem to cover is a "scholium" -- basically the same theorem except given as multiples of the Legendre's favorite unit, the right angle:

506. Scholium. Let s be the sum of the angles of a spherical polygon, n the number of sides; the right angle being supposed unity, the surface of the polygon will have for its measure s - 2n(-2) = s - 2n +4.

We have finished our study (which took more than year!) of Legendre's spherical geometry. But there are a few more issues that I want to discuss before I leave spherical geometry altogether.

Last month, I mentioned that there's another way to look at spherical geometry. From the perspective of Legendre, the geometry of the sphere is basically a part of Euclidean geometry, as the sphere itself lies in Euclidean space. The other perspective is that of Riemann -- as in Bernhard Riemann, the German mathematician who lived about a generation after Legendre.

Riemann is probably best known for the Riemann hypothesis, an unsolved conjecture. In fact, mathematicians have sought a solution for so long that it is the first of the Millennium Problems, with a million-dollar prize to be awarded to the one who gives a solution.

But today we will discuss the Riemann sphere, a model of non-Euclidean spherical geometry. For Riemann, we don't consider a sphere as sitting in a Euclidean universe -- instead the Riemann sphere is the entire non-Euclidean universe. Last month, I wrote:

And so another way to approach spherical geometry -- without referring to the sphere itself or any of its Euclidean properties -- is to start with some postulates that describe spherical geometry and use these to prove the theorems of spherical geometry. We aren't allowed to use phrases like "great circle" which betray knowledge of the underlying sphere. Instead, we must refer to great circles by the role that they serve in spherical geometry -- as lines.

Here are some new postulates that I suggested might work to generate a spherical geometry:

-- Any two lines intersect.
-- Every line has exactly two poles. (Here "pole" is an undefined term.)
-- Every point is the pole of exactly one line.
-- The pole of a line never lies on the line.
-- If l and m are lines and a pole of l lies on m, then the other pole of l lies on m, and both poles of m lie on l. (Two lines l and m satisfying this property are defined to be perpendicular.)

Using these postulates, we can prove the following theorem:

Theorem: Any two lines intersect in at least two antipodal points.

Notice that here we need to define "antipodal points." Well, that's easy -- every line has two poles and every point is the pole of exactly one line. So the other pole of the line of which the point is a pole is its antipodal point. To make it easier, let's use the word "equator" to refer to the line of which a point is the pole. So the antipodes of a point is just its equator's other pole.

Let a and b be the two lines. Our postulates already give us one point at which they intersect, say P, so P lies on both a and b. Let p be the equator of P. Then p has another pole, P', the antipodes of P.

Now another postulate states that if l and m are lines and a pole of l lies on m, then the other pole of l lies on m. Thus since p and a are lines and a pole of p (namely P) lies on a, the other pole of p (namely P') lies on a. Likewise P' lies on b. So a and b intersect at both P and P'.

But as good as these postulates sound, I was unable to prove that the two lines must intersect in at most two antipodal points. Of course, I could just start adding postulates to guarantee that two lines intersect in exactly two points, but continuing to add axioms is a bit ad hoc.

Instead, it's more elegant if we just start with Euclid's five postulates, and keep only the postulates that hold in Riemann's spherical geometry. Here's a link to Euclid's postulates:

Postulate 1.
To draw a straight line from any point to any point.
Postulate 2.
To produce a finite straight line continuously in a straight line.
Postulate 3.
To describe a circle with any center and radius.
Postulate 4.
That all right angles equal one another.
Postulate 5.
That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Now I've mentioned before that in the other form of non-Euclidean geometry, hyperbolic geometry, the first four postulates of Euclid hold, but the famous Fifth Postulate fails -- and indeed, we usually think of Euclidean geometry as one that satisfies the Fifth Postulate and non-Euclidean geometry as one that violates it.

But we've seen before on the blog that spherical geometry doesn't fit this idea perfectly. Indeed, we see that there are propositions of Euclid that fail in spherical geometry, even though the proof avoids the fifth postulate. The first theorem such is Proposition 16:

This theorem is often known as the Triangle Exterior Angle Inequality or TEAI:

There are geometries besides Euclidean geometry. Two of the more important geometries are elliptic geometry and hyperbolic geometry, which were developed in the nineteenth century. The first 15 propositions in Book I hold in elliptic geometry, but not this one.

Geometries satisfying the first four postulates are called "neutral geometry." Both Euclidean and hyperbolic geometry satisfy Postulates 1-4, so they both count as neutral geometry. But Proposition 16 fails in spherical geometry despite requiring only Postulates 1-4, so spherical geometry is not a neutral geometry.

A tricky question is, which of the first four postulates does spherical geometry violate? According to the link above, we see:

Elliptic geometry satisfies some of the postulates of Euclidean geometry, but not all of them under all interpretations. Usually, I.Post.1, to draw a straight line from any point to any point, is interpreted to include the uniqueness of that line. But in elliptic geometry a completed “straight line” is topologically a circle so that any pair of points on it divide it into two arcs. Therefore, in elliptic geometry exactly two “straight lines” join any two given “points.”

Also, I.Post.2, to produce a finite straight line continuously in a straight line, is sometimes interpreted to include the condition that its ends don’t meet when extended. Under that interpretation, elliptic geometry fails Postulate 2.

In spherical geometry (which is slightly different from "elliptic geometry," by the way), there are infinitely many lines through two antipodal points. So under my own interpretation, spherical geometry absolutely violates the First Postulate.

It appears that spherical geometry must violate either the First or Second Postulates, since Proposition 16, which fails in spherical geometry, depends on both of them. But we see that Proposition 5, the Isosceles Triangle Theorem, also depends on Postulates 1-2 -- and yet ITT is valid in both Euclidean and spherical geometry! Of course, the reason is obvious -- the First Postulate is invoked in Proposition 5 only to show the existence of a line joining two points, not its uniqueness.

The best way to see why Proposition 16, the TEAI, fails in spherical geometry is to take a triangle for which it fails and see whether we end up trying to join two antipodal points, which isn't unique.

Let ABC be a triangle, and let one side of it BC be produced to D.
I say that the exterior angle ACD is greater than either of the interior and opposite angles CBA and BAC.

Let ABC be a triangle for which we know TEAI fails -- for example, let's take Legendre's favorite triangle of three right (interior) angles. All the exterior angles are also right angles, and so no exterior angle is greater than any interior angle. We'll place B at the North Pole, and so both A and C end up on the Equator.

Bisect AC at E. Join BE, and 
produce it in a straight line to F.

Since E is the midpoint of AC, E must also be on the Equator.

Make EF equal to BE

And that does it. Since BE is a quadrant, so is EF, which places F at the South Pole. So B and F are now antipodal points, which renders the use of the First Postulate with B and F invalid.

I will consider the Second Postulate to be valid in spherical geometry (so as far as we're concerned, any line segment can be extended into a full "line," or great circle). Meanwhile, as I've pointed out before, I consider the Fifth Postulate -- believe it or not -- to be valid in spherical geometry! This postulate is vacuously true in spherical geometry, especially if we rewrite it as Playfair's Postulate (given a point and a line, there is at most one line parallel to the line through the point).

So to me, Postulates 2-5 hold in both Euclidean and spherical geometry, just as Postulates 1-4 hold in both Euclidean and hyperbolic geometry. Just as we have the name "neutral geometry" to refer to one in which Postulates 1-4 hold, I propose a new name, "normal geometry," to refer to one in which Postulates 2-5 hold. The name "normal" sounds a bit like "neutral" -- and besides, there are only two geometries that people "normally" think about, namely Euclidean geometry on a plane and spherical geometry to describe the earth (for example, when flying on an airplane). On the other hand, we don't "normally" think about hyperbolic geometry at all.

One theorem of normal geometry is "the sum of the angles of a triangle is at least 180 degrees," just as one theorem of neutral geometry is "the sum of the angles of a triangle is at most 180 degrees."

I've fascinated by what some normal proofs might look like. I'm curious, for example, whether the Fifth Postulate can be used to prove something nontrivial in normal geometry (that is, other than something like "if two parallel lines are cut by a transversal, then..." which is only vacuously true in spherical geometry).

But of course, this blog focus is no longer on geometry. It's about time that I spend my exclusive focus on my new middle school classroom.

Friday, July 22, 2016

Rule #3: Respect Yourself and Others (Pi Approximation Day Post)

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

It also marks two 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.

Let me a little introspective here. Writing on this blog has taught me many things about myself that I have not realized.

For starters, I realize that I change my mind a lot. I say that I'm going to post something about a geometry topic, then I change my mind and post something else -- and then I say that I'll post a third thing when I revisit that topic next year. From the perspective of the blog readers -- you -- it must be infuriating to read something in the archives that I say that I'll post and then move forward in the archives only to find out that I never posted it.

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. What are the Common Core Standards?
3. Am I for or against the Common Core Standards?
4. What is the "Illinois State text"?
5. Who are Hung-Hsi Wu and Fawn Nguyen?
6. Who are "the traditionalists"?
7. What are E's, S's, and U's?
8. If I will be teaching all three Grades 6-8, why will this blog cover only my 8th grade classes?
9. What will I do about cell phones in the classroom?
10. How will I state my most important classroom rule?

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

I am David Walker. I have recently earned my clear credential in Single Subject Math here in my home state of California, and I have just been hired as a full-time teacher. I will be starting my first full year of teaching at a charter middle school in Los Angeles.

This already explains some of why I change my mind so frequently. For the first two years of this blog, I was a substitute teacher, and my posts were either about whatever I taught in that day (if it was math) or otherwise I would default to Geometry. This year, I plan on using the calendar for the LAUSD district, which my charter follows (with one slight exception). Day 1, the first day of school, will be on August 16th in this district.

Notice that even though I refer to the LAUSD several times in this post, my school is officially a charter school and not an LAUSD school.

2. What are the Common Core Standards?

Common Core refers to the new ELA and math standards taught in many states, including California. In particular, I refer to the Core's new focus on translations, rotations, and reflections in geometry -- these transformations were not emphasized before the Core. Transformations appear in the standards for eighth grade and above. Because of this, even though I will teach all three middle school grades (sixth, seventh, and eighth), I will mostly blog about my eighth grade class on the blog.

The Common Core is neutral in that it supports either a traditional pathway (Algebra I, Geometry, Algebra II) or an integrated pathway. On the traditionalist pathway, of course it's mainly the Geometry class where the new geometric transformations appear. On an integrated pathway, the transformations appear in both Math I and Math II classes.

Notice that there is actually very little difference between Common Core 8 and Integrated Math I. Because of this, my blog readers who are high school teachers can apply the assignments to their Math I classes as well.

3. Am I for or against the Common Core Standards?

There are some things that I don't mind about the Common Core and some things I'd change. I know that some people oppose the Core for political reasons. I try to avoid politics here on the blog, but unfortunately the Core is inherently political. In particular, opponents of the political party that was in power when the Core was first adopted tend to dislike the Core as well.

During the first two years of this blog, I devoted many posts to the Common Core debate. But now I am a full-time teacher at a Common Core school, so I will no longer write posts debating the Core, and indeed, I will be toning down the politics once the school year begins.

4. What is the "Illinois State text"?

I use the phrase "Illinois State text" to refer to the IMaST program created by the Center of Mathematics, Science, and Technology at Illinois State University. It is the text used at several middle schools in LAUSD proper, as well as my charter school.

The following comes directly from the preface of the teacher's edition of the Illinois State text:

"The Creative Core Curriculum in Mathematics with STEM, Literacy and Arts Teacher STEM Project Edition Grade Eight book is unique. It consists of a series of learning cycles and readings specifically designed to teach all of the Common Core Content standards for 6th grade mathematics. Each learning cycle guides the students through a series of activities leading to conceptual understanding, not just memorization of facts. Concepts are taught through the activities, not just practiced or applied through supplemental projects. Although conducting activities can take longer than traditional pedagogy, it is actually a more efficient use of time since students are better able to comprehend and remember mathematical concepts."

For example, Cycle 6 of the 8th grade text is "The Capacity of Water-Carrying Structures." In this project, students actually construct four pipes using card stock -- one a triangular prism, one a rectangular prism, one a cylinder, and one a shape of their choosing. Students then compare their lateral surface areas and volumes to determine which pipe has the most efficient shape.

The Illinois State text is not the only text that I mention here on the blog, not by a long shot. But now that I am getting ready to teach full-time, I will be writing more and more about the Illinois State text and less about other texts. In particular, the Illinois State text is not the same as the U of Chicago text that I mention in most Geometry texts on the blog.

5. Who are Hung-Hsi Wu and Fawn Nguyen?

These are California math teachers who have commented on how geometry should be taught. During the first two years of this blog, I discussed these and other mathematicians many times. But again, my focus will be turning away from geometry and more towards the middle school math that I will be teaching.

Dr. Hung-Hsi Wu is a Berkeley mathematician. He has written extensively on how to teach Geometry according to the new Common Core Standards:

Wu's document describes both eighth grade and high school geometry. Naturally, my focus this year will be on the eighth grade portion of this document.

It is Dr. Wu who defines pi as the area of the unit disk -- this definition favors pi above the numbers tau and lambda. (Yes, today is Pi Approximation Day, so here's a pi reference.) Meanwhile, Wu has also created an extensive document on how to teach fractions under the Common Core:

Wu describes how to teach fractions to students in Grades 3-7, but by eighth grade, the study of fractions is complete. So I really won't discuss Wu's fraction method on the blog, as my focus will be on the eighth grade class.

Here is one more link from Wu's website. Even though he wrote this in 2005 -- before the advent of the Common Core -- Wu writes of key mathematical ideas in middle school (which in this case he defines as Grades 5-8). As it turned out, some of Wu's ideas were ultimately incorporated into the Common Core Standards. (As you can see, Wu advocated for the inclusion of transformations in Geometry years before there was ever a Common Core!)

Nowadays, I want to focus more on my fellow middle school math teachers and their blogs. The best known middle school math blogger (by far) is Fawn Nguyen. In fact, she teaches right here in Southern California. Here's a link to her blog:

Nguyen's most important post regarding 8th grade geometry and transformations is the following:

Nguyen writes:

My 8th graders are learning about rigid transformations. I want to add a bit more complexity to what our book is asking the kids to do. For example, the book is having them reflect a shape mainly across the x-axis or y-axis, or on a rare occasion, reflect it across “the horizontal line that goes through y = 3.” Well, right before this chapter, we’ve been working with writing and graphing linear equations, so I want kids to reflect a shape across any line, including one that may cut through the shape itself.

And just as Nguyen supplements her text with this transformations activity, I can use this activity in my own classroom in addition to the Illinois State text. Learning Cycles 7-8 in the 8th grade text are only indirectly related to transformations, and it does fit the project-based learning pedagogy of my classroom to add a project such as this one.

But as Nguyen is a fellow middle school teacher, I will often go to her blog for guidance regarding classroom management and other issues. Here's a link to her page describing Days 1-2 in her classroom from two years ago. (Unfortunately, she didn't post at all last August.)

Nguyen writes:

Tomorrow is the day. I’m excited to meet my new Math 6 students — all 71 of them, two classes of 36 and 35. I should know about half of the 33 Math 8 kids because I taught half of them in 6th grade.

So as we can see here, Nguyen teaches both sixth and eighth graders. I, of course, will be teaching seventh grade as well.

Nguyen writes that she begins the first day of school with a warm-up and an activity. I plan on doing the same, but my warm-up and activities will be different.

And so I will continue to look to both Wu and Nguyen for guidance through my first year of teaching.

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 two years of posting on the blog. But the traditionalist on whom I plan to focus the most goes by the username Bill. Indeed, as I begin to teach, I'll be writing more about Bill and less about the other traditionalists.

Bill often responds to articles posted at the Joanne Jacobs website. Here is a link to that site:

Here's a link to a recent comment of Bill's regarding college readiness. As usual, I provide first the link to the article (which is just a report from the ACT test, which doesn't allow comments) and then the Jacobs website (which does allow comments, and where Bill has commented).

Bill says:
Colleges these days are about being a business model, not getting an actual education (IMO). They’ll take your money until you either flunk out and don’t come back, or run out of it…
A student with a Composite ACT score of 15.1 is un-prepared for college level work, given that the minimum for success in college is a composite of at least 21-22 (higher than that is better, of course).
Additionally, I’d like to see the grades vs the ACT scores, I suspect they don’t match, if you have a ACT composite of 15 and a GPA of 3.8, I’d say the student got some grade inflation to help them out…

Meanwhile, Bill was quite outspoken in a thread about algebra and statistics:

Bill says:

But, I wonder about college students who can’t figure out 2x + 4 = 14. It’s not rocket science..
college students who cannot figure this out have NO business being admitted to a college in the first place, this is basic algebra (or pre-algebra even) and should have been completely mastered by at least the 9th grade, if a student was planning on attending college.
Basic Stats and Probability requires a basic working knowledge of algebra, plain and simple…trying to skirt around this issue will lead to students who won’t be able to master stats or probability.
On a more sad note, in my home state of Nevada only 9% of 11th graders have the necessary skills in reading, writing, english, and math to succeed in college per exam results…

In these two comments, we see that a concern of Bill is that students can get good grades in math without actually learning anything. I actually address this in one of my classroom rules. (Rule #2: Respect Your Honesty, or Avoid trying to get an A without learning anything.)

Actually, regarding 2x + 4 = 14, I'd say that students really are learning how to solve it. It's just that students forget how to solve it over the summer, so by the time they take the ACT or a college placement exam, they've forgotten it. Yes, 2x + 4 = 14 isn't rocket science, but it isn't bike riding (that is, something that once learned is never forgotten) either. (Instead, it's more like "use it or lose it.") I'd like to address this in my classes, but this is more of a challenge -- how much control will I have over what my students still remember years after taking my class?

Speaking of rocket science, Bill mentions it in his next comment.

Bill says:

There was an illustration in Calculus for Dummies…a football QB throwing a pass to a player and missing (consequences aren’t usually tragic), but aiming a spacecraft so that it can reach the moon or another planet is way worse if you miss (and that’s what calculus can do, since it’s nothing more than very advanced algebra, and allows you to handle rate of change issues)…
Calculus – the agony and dx/dt (running really fast now)…:)

I agree with Bill here -- if we want students to learn something, we should tell them why, in order to answer the question "When is this (say, Calculus) used in real life?"

In Bill's final comment, he tells us why so many students are struggling in math:

Though this problem starts in grades K-5 (and in the home) as many parents are themselves ‘math challenged’ and when students haven’t mastered add, subtract, multiply, divide, percentages, and fractions by the time they have left elementary school, they’re going to struggle the rest of their lives in math, period, and that can have life altering consequences, when they find out how much math is involved in some careers (all of the STEM disciplines and things like data analytics, business intelligence and so on require a solid working knowledge of mathematics beyond Algebra II/Trig).
Though perhaps I’m biased, but when I watch people struggle to figure out basic math concepts or percentages, it’s a sure sign they didn’t get good math instruction in K-5 and/or at home, and many elementary school teachers don’t have the strongest math skills.

I will often use the word "dren" to describe these students struggling in basic math. A "dren" is a reverse-nerd, or someone who hasn't mastered elementary math. I usually say K-3 instead of K-5, in order to avoid calling anyone who doesn't know fractions "drens." Bill doesn't use the word "dren," but I will use it to summarize Bill's concerns in a single syllable.

Most traditionalists dislike the Illinois State math texts -- that is to say, they dislike project-based learning texts, not necessarily the Illinois State texts per se (as most traditionalists have probably never heard of IMaST). Even though a traditionalist like Bill writes about using Calculus to get to the moon, he most likely wants Calculus classes to focus only on the concepts, not projects.

One more traditionalist whose ideas I'll take into the classroom is Katharine Beals. Unlike Bill, she doesn't post at the Jacobs website, but instead has her own blog:

Beals writes a series, "Math problems of the week," where she often complains about Common Core questions -- and more recently, answers. The following is a link to a Common Core test question taken from her home state of Pennsylvania:

But there's more to it than that. This is a sixth grade statistics question, and usually she ends her weekly post with an "Extra Credit" that is usually sarcastic or rhetorical. Beals writes:

Is knowing the meaning of "quartile value" an indicator of college and career readiness [that is, the stated goals of the Common Core -- dw]?

Here is the answer to this question (the answer Beals hopes for, that is):

No, of course students don't need to know anything about quartiles -- or statistics -- for that matter, to be ready for college and career. The only things that students need to know are algebra and calculus, and so anything in the sixth grade curriculum that doesn't prepare the students for algebra, including all of stats, should be dropped from the curriculum. Indeed, more students would be ready for Algebra I in eighth grade if the sixth and seventh grade classes spent less time on stats and more time on Pre-Algebra.

Naturally, I disagree with Beals here. First of all, typing in "quartile" into Google gives the following non-Common Core, real-world links on the first page:

Furthermore, I assume that Beals has no problem with other topics, such as division of fractions, included in the sixth grade curriculum.

Here's my first Extra Credit Question:
Of fraction division and quartiles, which topic will cause more sixth graders to ask "When will we ever use this in real life?" and which topic will the students instead sit down and try to learn?

But there is one thing idea I will take from Beals and include in my own classroom. By the time we get to the eighth grade, some of the stats topics (such as trend lines) are more challenging.

Here's my second Extra Credit Question:
Of polynomials and trend lines, which topic will cause more eighth graders to ask "When will we ever use this in real life?" and which topic will the students instead sit down and try to learn?

Unlike Beals, I don't claim to know the answer to this question. And so I want to test it out in my own eighth grade class. When we reach the unit on stats, I'll simultaneously teach (some of) the students polynomials and the rest trend lines. On the test, students will be able to choose whether they wish to answer polynomial questions or trend line questions. Then based on the results, we'll find out whether eighth graders are better off learning Algebra I (as Beals and the traditionalists believe) or learning stats (which I believe is more useful in real life).

7. What are E's, S's, and U's?

So far, none of the classroom rules that I've mentioned so far address classroom behavior -- which should strike you as odd. After all, shouldn't classroom rules be all about behavior?

The letters E, S, and U are the marks given in the LAUSD for work habits and cooperation. As you may expect, E, S, and U stand for "excellent," "satisfactory," and "unsatisfactory" respectively. So just as my Rule #1 basically tells students to strive for an A in every class, today's rule tells students to strive for an E in every class.

I'd like to say that I always earned E's in all my own classes growing up, but of course this is nowhere near being true. I still remember receiving "check-marks" (somewhat like U's) in many of my elementary classes (often in "works well with others" or something similar). In fact, I remember finally receiving the "Student of the Month" reward for being the best-behaved student in my class -- during May of my sixth grade year. This was the last possible month that I could have earned the award at my K-6 elementary school, as there was no separate award for June.

In the seventh grade, I never received any U's, but I did get an N once -- here N stands for "needs improvement" and represents a mark between S and U. (N's are not used in the LAUSD.) From the seventh grade on, I always tried to behave whenever I was in front of a teacher, but the N was for something I did when I was away from a teacher -- failed to turn in enough homework. (So technically the N was for work habits, not cooperation.) After I got the N, I always made sure to meet the 75% homework threshold in order to avoid getting any more N's. (Believe it or not, I think the class in which I earned the N was Algebra I -- my academic grade was A, yet work habits was N!)

In high school I earned mostly E's -- though I did receive a few S's, but never any N's or U's. For the rest of this post, I will refer only to the LAUSD marks of E, S, and U, not N.

To me, the three LAUSD marks represent three types of students -- E-kids, S-kids, and U-kids:

-- E-kids consistently follow the rules, even when given the opportunity to break them.
-- U-kids consistently break the rules, even when given the opportunity to follow them.
-- S-kids sometimes follow and sometimes break the rules. If the teacher has a strong classroom management plan, the S-kids will act like E-kids and follow the rules. But if the teacher has a weak classroom management plane, the S-kids will act like U-kids and break the rules.
-- Most students are probably S-kids. And so this is why classroom management is important -- the ability of the teacher to manage determines the behavior for most of the students.

Notice that I've myself have been all three types of student. I was a U-kid in elementary school, mostly an S-kid in middle school, and an E-kid in high school.

Also, as a sub, I've been both types of teacher. On some days, my classroom management was strong enough to make the S-kids behave like E-kids. But unfortunately, on other days my classroom management was so weak, the S-kids behaved like U-kids. If you go back and read some of my posts from the first two years (using the "subbing" label), you'll recognize both my good days and my bad days (though I didn't use terminology like "S-kids" in those posts).

What did I do on my better days to make the S-kids act like E-kids? Well, first of all I made sure that the students always had something to do. And second, I made sure to hold the students accountable for doing that assignment.

In previous posts I've referred to a "Who Am I?" game and a group point system. As a teacher, I won't play the "Who Am I?" game everyday, but I will keep the point system. I will refer to these points as Participation Points and assign them to individuals.

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.

When I was a student teacher, I ended up covering three classes -- third period Algebra I, fifth period Algebra II, and sixth period Algebra II. (Notice that third period was right before nutrition, fifth period right before lunch, and sixth period at the end of the day -- yes, you can see how I got my detention idea!) I consider two of those classes to be less successful -- fifth period, because the class has 40 students, and third period, because the students acted like, well, freshmen. In those classes, the S-kids acted like U-kids.

But I consider the sixth period Algebra II class to be the most successful class I've ever taught. The students in that class were very motivated to outdo each other and answer questions so that they can earn the Participation Points. In that class, the S-kids acted like E-kids.

It's easy for me to say that sixth period was successful because the students were more mature than the freshmen, and there weren't 40 of them in the class. But looking back, I believe that the reason for the difference was psychological. Subconsciously, I was thinking, well, I've survived my freshmen and my class of 40, and so it's all downhill from here. And that thinking showed up in the way that I taught the class -- the students knew that my expectations were high and they lived up to them. I reckon that if that sixth period class had been the only class I had to cover for student teaching, it would not have been as successful, since it was that "downhill" feeling I had after covering third and fifth periods that caused me to teach sixth period effectively.

In my less successful subbing assignments, I didn't play any game or have a point system. I usually preferred playing "Who Am I?" only in math classes, and so it was often in those other classes when I had trouble managing the class. Sometimes, there's nothing I could have done to make the students behave -- for example, I mentioned before that my final subbing assignment was a science class that was learning about the reproductive system. I couldn't get the students to work because I had to spend all my effort checking permission slips and making sure that only students who had permission were given the lesson. I couldn't start playing my game, which would have required me to ask questions from the worksheet out loud enough for students without permission to hear. So the S-kids in the class inevitably acted like U-kids. On the other hand, in other classes I had no such excuse -- I could, and should, have set up a point system so that students would be motivated to work.

My goal is for every class to be like the sixth period Algebra II class that I student-taught. I need to think about what made that class successful and manage all my classes that way. It will be tough, as my middle school students will be several years younger than my Algebra II students. But I owe it to my E-kids to manage the class effectively.

8. If I will be teaching all three Grades 6-8, why will this blog only cover my 8th grade classes?

The name of this blog is "Common Core Geometry." Therefore, it's expected that I post something related to geometry.

When I first named the blog, it was before I received my first teaching assignment, so I had no idea what class I'd be teaching, or whether it would even be middle school or high school. All I knew was that the new Common Core Geometry standards would be a major stumbling block for teachers, and I wanted to help them out by creating this blog.

As I mentioned earlier, the new geometry standards based on transformations actually begin with the 8th grade standards. Therefore, I remain true to the original title and purpose of this blog by focusing on the 8th grade classes. But I will write posts about the entire 8th grade year though, not just the geometry units (and again, Common Core 8 is nearly the same as Integrated Math I).

I wrote that I will be trying to post three days per week after school starts. I've decided to attempt to post on a regular schedule based on the school calendar using the following pattern -- I will skip posting every third day. So my first post will be on Day 1 or Tuesday, August 16th, then my next post will be on Day 2 or Wednesday the 17th, and then I skip Day 3 or Thursday the 18th, before posting again on Day 4 or Friday the 19th. Other skipped posts in August will be Day 6 (Tuesday the 23rd), Day 9 (Friday the 26th), and Day 12 (Wednesday the 31st).

As you can see, whenever the day number is a multiple of three, I avoid posting that day. This will result in posting approximately thrice a week. During some weeks I might end up posting four times, while during certain holiday weeks there might be only two posts, but most of the time there will be three posts in a week.

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

Of course, as any teacher in this era will tell you, one major behavioral issue will be the students' using cell phones in the classroom. The traditionalist Bill makes his opinion of cell phones clear:

Bill says:

Smartphones are NOTHING but a distraction in the classroom, period…they should be prohibited except during breaks and lunch…

Of course cell phones are prohibited -- but just because something is prohibited, it doesn't mean that they won't appear in the classroom.

I am on the older edge of the Millennial generation -- if one can really call me a Millennial at all. (I define Millennials as those born in the old millennium but graduated high school in the new, but I graduated high school in 1999, the old millennium.) So my school years just barely missed the advent of the smartphone. The closest I can get is when I was working at the library just before I began working on my credential. Sometimes I would look up other websites on the library computers, until my boss told me to close the browser window. I noticed that as soon as I did so, I was much more attentive to patrons and was thus a much more effective worker. And so I have no choice but to incorporate a cell phone ban into my Participation Points system.

Notice that cell phones are a highly advanced form of technology -- indeed, I once read that there's more computing power in a tiny cell phone than in the A-bombs during World War II. In theory, a generation that enjoys cell phones and video games should love math and science, the subjects that make that technology possible. In theory, no one should ever type "I hate math" on the Internet, since without math there would be no Internet. But in practice, we know this is not the case.

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.

By the way, I've written that not only do I want to write more about young people going into STEM, but young females in particular. Here is an article about a girl from right here in Southern California who not used her math and science skills to create an actual app. She convinced her high school to start offering more STEM classes:

Still, I know that some members of our generation can't go seven waking minutes without using a cell phone, and yet teachers expect them to go seven hours without using one -- that is, 60 times as long as they naturally would. Indeed, I know that for some students, the only effective incentive that will motivate them to work is a few minutes' free time on their cell phone. Again, the idea is for me, after telling the students my library story, to remind them that the people who matter (employers) will criticize -- that is, make fun of -- those who can't go a few minutes without reaching for a phone, and those who do have self-control will get more A's and E's, and ultimately jobs and promotions.

10. How will I state my most important classroom rule?

This will be how I'll state the rule in my classroom:

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 -- in other words, by striving to earn E's in all their classes.

If this rule sounds familiar, it should -- it comes directly from Fawn Nguyen's classroom (see her first day of school link above). In Nguyen's class, though, this is Rule #1.

But today is Pi Approximation Day, and I like to eat apple pie since I associate apple pie with this time of year ("as American as apple pie," and this is the month that we celebrated Independence Day).

In honor of Pi Approximation Day, let me post some more videos about pi. The first few of these mention several ways to approximate pi.

Let's start with some scientific facts about the number pi (including ways to approximate it):

The narrator of this first video mispronounces the name Euler. I'm sorry, but I just can't help linking to a Fawn Nguyen post about the pronunciation of Euler:

The History of Pi. 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 her How to Bake Pi, say almost exactly the same thing.

The Infinite Life of Pi:

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.

Here is a longer video, but it contains some more series approximations of pi:

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:

Here are some formulas for the number pi. (One of them is the approximation 22/7.)

Let's wrap up with one more Pi song, a longer one this time:

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

Finally, this link is not a video, but it's one of my favorite links for Pi Approximation Day. We know that 22/7 is approximately equal to pi. As it turns out, 22/7 is actually more than pi -- and we can find out why 22/7 > pi using calculus. It's possible that an AP Calculus student -- a senior on the SteveH plan -- might be able to calculate this integral:

A few other Pi Approximation Day (aka "Casual Pi Day") links:

(The last post was written by Tanya Jeffries -- another female computer scientist!)

And so I wish everyone a Happy Pi Approximation Day.