Wednesday, December 31, 2014

Calendar Reform

"It all started on the thirteenth hour, of the thirteenth day, of the thirteenth month.  We were there to discuss the misprinted calendars the school had purchased."

-- Marge Simpson, The Simpsons, "Treehouse of Horror VI"

One of my more esoteric hobbies is Calendar Reform. Sometimes I wonder, what would our lives be like if we had a different calendar -- one with, say, a thirteenth month, like Homer's "Smarch." I often like to think about Calendar Reform around this time of year -- New Year's Eve -- since after all, it's the day when we take our old calendars down and put up new ones.

In particular, since this is a school blog, I would like to point out how certain problems in the school year could be solved via Calendar Reform:

-- The Early Start Calendar. Many people, students and teachers alike, don't like the idea of starting school in August, and would rather wait until after Labor Day in September. But the reason for the Early Start Calendar is so that the semester can end before Christmas. That way, students don't have to try to remember what they learned before winter break in order to pass their January finals.

Even schools that start in August seldom start early enough so that exactly 90 days, or one-half of the school year, are completed before winter break. Originally, I counted backwards 90 days from winter break and started Day 1 so that Day 90 would be the last day before the holiday. But then I switched so that my dates matched the actual calendar for a local school district. As you can see, the first semester was only 84 days, so there are 96 days during the second semester.

(As an aside, notice that since the second semester is 96 days, this would make each quarter exactly 48 days, each quaver exactly 24 days, and each hemidemisemiquaver exactly three days.)

There are only two ways to get a full semester in before Christmas -- either make the first day of school earlier, or make Christmas later. Schools do the former and start school in August, since Christmas can't be changed -- at least, not without Calendar Reform.

-- Block Schedules. Many high schools have an A/B Block Schedule, where students have half of their classes on A Days and the other half on B Days. The problem is, with a Monday-Friday school week, if two days are A Days and the other two are B Days, what about the fifth day? Different schools do different things, including all classes meeting on the fifth day, the fifth day alternating between A and B, or even a pure block schedule alternating between A and B with no relationship between A/B and day of the week,

All of this could be avoided if there were an even number of days in the week. But the only way to accomplish this is Calendar Reform.

-- Veteran's Day Weekend. I mentioned earlier how, with Veteran's Day on a Tuesday this year, many students and teachers unofficially took Monday off -- and many schools and districts officially closed that day as well. This problem also occurs at offices, where workers want to take off last Friday and this Friday because Christmas and New Year's Day are on Thursdays this year. Independence Day, on the Fourth of July, also suffers when the holiday falls on an inconvenient day of the week.

When a holiday is tied to a particular day of the week, such as Thanksgiving on Thursday, its date must necessarily change. In 2012 Thanksgiving was on its earliest possible date, November 22nd, while in 2013 it was on its latest possible date, November 28th. I believe that much of Christmas creep -- where stores seem to be advertising for the holidays earlier every year -- is caused by this problem with the calendar. In the year when Thanksgiving moves from the 22nd back to the 28th, many stores who might have waited until after Turkey Day for a sale start it before the holiday instead, and it remains before the holiday even the next time Thanksgiving is on the 22nd again. A local radio station that switched to Christmas music the week before Thanksgiving in 2012 made the switch two weeks before Turkey Day last year, and so on. The accumulation of such changes over the years and decades means that Christmas ads are now earlier than ever before. ("Franksgiving" refers to President FDR, who changed the date of Thanksgiving in 1939, a year when Thanksgiving would have moved from November 24th the previous year to November 30th, for this very reason.)

(This happens with other holidays as well. Some stores felt the need to advertise for Easter early in 2008 when the holiday fell on March 23rd -- and kept on doing so ever since, even in 2011 when it fell on April 24th.)

Of course, even if Thanksgiving or Easter were fixed to a certain date, many retailers might decide to start the sales earlier one year anyway -- but they wouldn't feel as pressured to do so as they are when the holidays are early one year and late the next.

-- Paydays. Many districts in this area follow one of two payday schedules. One is quadriweekly, where payday comes every four weeks, on Friday. Notice that since there are 52 weeks in a year, there are actually 13 quadriweekly paychecks per year. Therefore, there is no relationship between the date of the month and payday -- some checks occur early in the month, while others occur late in the month. The problem is that rent and mortgage payments are typically due on the first of the month, so the checks don't line up exactly with the bills.

The other payday schedule is monthly -- many schools pay on the fifth of the month, since the first four days are ostensibly for processing the checks. But this still often means that bills are due four days before teachers get the money to pay them. Unfortunately, no Calendar Reform can make that four-day processing window disappear.

Still, we see some problems that teachers have that can be solved with Calendar Reform. There are many proposals for Calendar Reform floating around. The best way for me to state them is to observe the following link:

Notice that the author of this site is actually criticizing Calendar Reform -- he is saying that all calendar reform is bad and that we should just leave the calendar the way it is. The link contains a list of reasons why a particular proposal won't work. Every proposal will cause at least one of the boxes to be checked, so that every proposal would be rejected.

In the comments, someone joked that one should try to make a calendar proposal for which every box is checked. This is likely impossible, since many of the boxes are actually opposite solutions to the same problem. For example, the box:

no, we don't know what year the Big Bang happened

refers to a proposal where someone lets Year 0 be the year of the Big Bang. But then we have:

BC and AD aren't
that is, BC and AD aren't actually before Christ and anno Domini, in the year of the Lord, since according to scholars, Christ was most likely born around 5 BC. But a calendar that sets Year 0 to the year of the Big Bang probably wouldn't have BC or AD to begin with. Similarly:

nobody cares what year you were born

refers to a calendar whose author sets Year 0 to the year of his or her own birth. Such a calendar would reference neither the Big Bang nor the birth of Christ. So it would be difficult to create a calendar for which all three boxes are checked.

Still, an excellent way for me to introduce you to the various Calendar Reform proposals is to look at each item on the list and for me to give a proposal which would cause that item to be checked -- in other words, the calendar that inspired the author of the list to include that item.

So let's begin with the third item on the list, since it's an important one:

the solar year cannot be evenly divided into solar days

The length of the solar (or tropical) year is 365.2421897 days, so this is true. What this means is a calendar with only 365 days with fail to sync with solar years (which is, after all, the first item on the list) after several years have passed.

This problem has been known since the days of Julius Caesar. And so the ancient Roman emperor created a new calendar, the Julian Calendar, in which an extra day is added to the calendar once every four years. This leap day occurred as a doubled February 24th -- six days before the first (or kalends) of March, and so such a year became known as a bissextile, or "double sixth," year. In 1970, the Roman Catholic Church changed it so that February 29th is now Leap Day.

Because of Leap Day, someone may be either 365 or 366 days old on one's first birthday, either 730 or 731 days on one's second birthday, and either 1095 or 1096 days old on one's third birthday. But on one's fourth birthday in the Julian calendar, one must be exactly 1461 days old. Because of this, I sometimes refer to one's fourth birthday as one's "first Julian birthday," so that one's eighth birthday becomes the "second Julian birthday," and so on. Notice that someone born on Leap Day itself can be described as celebrating only the Julian birthdays.

So any calendar with Leap Days, such as the Julian calendar, would check this box. The reason for leap days is that the solar year cannot be evenly divided into solar days.

the solar day cannot be evenly divided into SI seconds
the length of the solar day is not constant

These two go together. Even though we think of days as being divided into 24 hours, hours into 60 minutes, and minutes into 60 seconds, the length of the day is not constant. It's always changing -- indeed, it's slowing down. After all, if you spin a globe, it eventually slows down and stops. And so it is with the earth.

Even though it takes a very long time for the earth to slow down appreciably -- for example, the dinosaurs experienced 22-hour days -- it has been slowing down noticeably. In particular, every few years, the clock is one second off. To make up for this, a Leap Second is added. The most recent Leap Second was added on June 30, 2012. No one knows when the next Leap Second will be, since it's based on actually observing the rotation of the earth.

So any calendar with Leap Seconds would check this box. The reason for leap seconds is that the solar day cannot be evenly divided into SI seconds.

the lunar month cannot be evenly divided into solar days
the solar year cannot be evenly divided into lunar months
These two go together. They refer to calendars such as the Chinese and Hebrew calendars, in which new months and years begin at the new moon. This is why we have Jewish holidays such as Hanukkah, which occurred very early in 2013 ("Thanksgivukkah") but later this year (not ending until Christmas Eve) and Chinese holidays such as Lunar New Year, which will occur very late in 2015 (February 19th, making the current Year of the Horse a "double spring, double rain" year).

A lunar calendar normally has twelve months, but once in a blue moon there is a thirteenth month, a Leap Month. It occurs about seven times every 19 years (the Metonic cycle, named for the Greek astronomer Meton, who calculated it). In particular, we know that a Leap Month in the Hebrew calendar occurred between Hanukkah 2013 and 2014, and a Leap Month in the Chinese Calendar occurred during the Year of the Horse.

(One personal note about the Metonic Cycle. Back when I was 18, I was curious as how to find the date of Easter -- which is influenced by Passover and thus the lunisolar Hebrew calendar. I was also curious as to what the phase of the moon was on the day I was born. As it turned out, it was easy. I only had to look ahead on the calendar for my next birthday, which happened to be my 19th -- that is, my first Metonic birthday -- and saw that it was a new moon. Therefore, I was born at new moon.)

So any calendar with Leap Months would check this box. The reason for leap months is that the solar year cannot be evenly divided into lunar months.

(Notice that the Islamic calendar would not check this box, as there are no leap months. But its twelve months are short of a full year. So the Islamic calendar would check the first box instead, since its lunar years don't sync up with the solar year.)

having months of different lengths is irritating
having months which vary in length from year to year is maddening

These two go together. Notice that these refer to our current calendar -- the three links in the second item link to "February," "Common year," and "Leap year." So our current calendar is one which would check these boxes!

But our current calendar is not the Julian calendar. February has a Leap Day every four years in the Julian Calendar. But why is it every four years? It's because the length of the tropical year is nearly 365.25 days, and .25 is equal to one-fourth. Yet, as I mentioned earlier, the length of the tropical year is not 365.25 but 365.2421897 days. That difference was noticeable enough that by the 16th century, the first day of spring was around March 10th or 11th, and not March 20th or 21st as it was when the Julian Calendar was created.

So the Catholic Church decided to reform the calendar. Pope Gregory XIII dropped ten days so that the seasons would begin when they were supposed to, and then added more rules to determine which years would have leap days. In particular, years that are divisible of 100 do not have Leap Days (even though 100 is divisible by 4) unless they are also divisible by 400. So 1600 had a Leap Day, but not 1700, 1800, or 1900. The year 2000 had a Leap Day, but 2100 won't.

Most of the readers of this blog weren't born yet in 1900, and most of us will be long dead by the time 2100 comes around. And so these Gregorian Exceptional Years don't affect any of us -- for our lifetimes, the Julian leap rule suffices. But still, the calendar we actually use is the Gregorian, not the Julian -- because of the skipped days, the date is different in both calendars. In the Julian Calendar, today is December 18th, not 31st as in the Gregorian. Many churches still use the Julian Calendar to determine Christmas and Easter -- most noticeably the Orthodox churches. These churches have no loyalty to the pope, so why would they follow Pope Gregory's calendar?

So far, the only calendars I've mentioned are existing calendars. By doing so, I've described the history of Calendar Reform thus far. The next boxes definitely refer to proposed calendars -- indeed, some of the most common proposals will check these boxes.

the solar year cannot be evenly divided into seven-day weeks
having one or two days per year which are part of no month is stupid
having one or two days per year with no day of the week is asinine
This is the big one, and the cause of many of the problems that I listed above. The solar year cannot be evenly divided into seven-day weeks -- seven divides neither 365 nor 366 evenly. The closest number that seven does divide is 364. It's because of that extra day or two that Christmas must change its day of the week every year, and why Thanksgiving must change its day of the month every year, and so on.

The first attempt to solve this problem is known as the World Calendar:

A full description of the calendar comes from this link:

In this improved calendar every year is the same. 
• The quarters are equal: each has exactly 
91 days, 13 weeks or 3 months. 
• The four quarters are identical in form with an 
ordered variation within the three months. 
• The three months have 31,30, 30 days respectively. 
• Each month has 26 weekdays, plus Sundays. 
• Each year begins on Sunday, 1 January; 
each working year begins on Monday, 2 January. 
• Each quarter begins on Sunday, ends on Saturday. 
• The calendar is stabilized and made perpetual 
by ending the year with a 365th day following 
30 December each year.  This additional day is  
   dated ‘W’, which equals 31 December, and 
called Worldsday, a year-end world holiday. 
• Leapyear Day is similarly added at the end of 
the second quarter.  It is likewise dated ‘W’, 
which equals 31 June, and called Leapyear Day, 
another world holiday in leap years.

In this case, it's these blank days -- the days dated 'W,' that make every year the same. Christmas in the World Calendar is always on Monday, and Thanksgiving is always on the 23rd. But it's these blank days that lead to checks on our checklist. They definitely have no day of the week -- the 'W' dates occur between Saturday and Sunday. Whether they are part of no month is debatable, since the link above does refer to them as the 31st days of June and December. So the World Calendar gets at least one and possibly two checks here.

Notice that the World Calendar does not get a check for "the solar year cannot be evenly divided into seven-day weeks," since the calendar doesn't attempt to divide them evenly (that's what the blank days are for). What sort of calendar gets a check here? Let's think about it. Consider what we've discussed in this blog entry so far:

The solar year can't be evenly divided into solar days -- that's why some years have Leap Days.
The solar day can't be evenly divided into SI seconds -- that's why some days have Leap Seconds.
The solar year can't be evenly divided into lunar months -- that's why some years have Leap Months.

Following this pattern, we should have:

The solar year can't be evenly divided into 7-day weeks -- that's why some years have Leap Weeks.

And that's the calendar that gets a check here -- a Leap Week calendar. There are several Leap Week calendars around. One that gained a little publicity about three years ago is the Hanke-Henry Permanent Calendar, posted at the following site:

Like the World Calendar, the Hanke-Henry Calendar begins on a Sunday, but the latter's months follow a 30-30-31 pattern while the former's go 31-30-30. This means that Christmas will always be on Sunday, and Thanksgiving will always be on the 24th. But the main difference is, instead of blank days, it has a Leap Week, called Xtr Week, at the end of December.

How often does Xtr Week occur? It occurs every six or five years. Henry writes:

I am indebted to Irv Bromberg for pointing out that a simple way exists to test whether a year contains a Xtr (or Extra) month: if the corresponding Gregorian year either starts on a Thursday, or ends on a Thursday, that year contains a Xtr (or Extra).

According to this rule, 2015 will contain a Xtr, since the Gregorian year 2015 both starts and ends on
a Thursday. Henry writes a program in Fortran that calculates whether a year has a Xtr. Let me rewrite it in TI-BASIC:

:Input Y
:If (R=4 and I=Y
) or (R=3 and I=

Entering 2015 outputs 1, indicating that there will be an Xtr Week. Entering 2016, 2017, 2018, and 2019 all output 0. The next Xtr Week will be in 2020.

When Henry posted his calendar three years ago, many news sites picked up the story. The following website, IO9, is typical. It's interesting to read the comments:

Many of the comments involve other proposed changes that will be covered in other items on our calendar checklist. Of those relevant to the calendar itself, many people weren't enamored with the idea of Christmas always being on Sunday. When given a choice, posters tend to choose days like Wednesday or Friday for Christmas, rather than Sunday. I tend to agree -- when Christmas is on Sunday, some schools don't break until December 23rd, so that kids actually have to go to school two days before Christmas! Other schools break on the 16th, but then have to take an extra day off on Monday, January 2nd. So Christmas on Sunday wreaks havoc on the school schedule as well.

So any calendar with Leap Weeks would check this box. The reason for leap weeks is that the solar year cannot be evenly divided into seven-day weeks. But most calendars won't check both this box and the box for days without a month or day of the week (blank days), since these are opposite solutions to the same problem.

your name for the thirteenth month is questionable

Aha, so there's our "Smarch"! Actually, there's a reason for having 13 months in a year. Recall how some schools pay teachers quadriweekly, so there are 13 paychecks in a year. So a 13-month calendar would have months exactly four weeks each.

One 13-month plan is the International Fixed Calendar:

The new month here is called "Sol" and occurs between June and July. Like the World Calendar, every month begins on Sunday and ends on Saturday. (Notice that Homer Simpson's "Smarch" also begins on Sunday and ends on Saturday. Presumably "Smarch" occurs after December, since Homer notes how lousy the weather is.)

I assume that any 13-month plan will result in a check for this box. Supposedly, this box is checked only if the name of the 13th month is questionable, but since the author of the list is opposed to all Calendar Reform, this box would be checked for any 13-month plan. (It might have been more honest for this item to read, "thirteen is an unlucky number of months" or something like that.)

The link on this item is to the name "Undecimber." I agree that it's a questionable name, since its name actually means eleventh month. But if we're going to criticize "Undecimber" as questionable, then we should similarly attack September, October, November, and December. An octagon in geometry has eight sides, so why is October the tenth month? Before Julius Caesar, the first month of the year was March, not January.

One day when I was in a local library, there was a display where someone posted a 13-month Calendar Reform plan. This plan drops July and August so that September through December are no longer misnomers. So now there are three new months to name -- Humanus, Sanctus, and Spiritus.

All of the calendars mentioned so far have blank days, just like the World Calendar. It's possible to have a 13-month plan with a Leap Week instead. One such calendar is the Pax Calendar:

Here the new month is called Columbus and occurs between November and December. The Leap Week is called Pax and occurs after Columbus.

With the new month of Columbus, notice that there's an extra month between Labor Day in September and Christmas in December. This means that we can fit the whole first semester between these two holidays and solve the Early Start problem! The five months September, October, November, Columbus, December make up 20 weeks, and let's declare the last week in December to be winter break. (Notice that December 25th occurs, much to the delight of the posters in the Henry thread, on Wednesday.)

This gives us 19 weeks, or 95 school days. So we can afford five more days off. Of course, Labor Day is one of these days. Many schools reserve the day after Labor Day for teacher preparation, so there's our second day. And of course we take off Thursday and Friday for Thanksgiving. These are November 26th and 27th in this calendar. The last day off could be another day for teacher preparation (as many schools start on Thursday after Labor Day). Or the extra day could be Wednesday, November 11th for Veteran's Day. This would actually divide the quarters evenly as there are 45 days from Labor Day to Veteran's Day and 45 more from Veteran's Day to Christmas.

Another calendar similar to the Pax Calendar is the New Earth Calendar:

The extra month occurs between June and July, except that it's called Luna, not Sol. Also, its months begin on Monday, not Sunday. Notice that this would put the 5th of the month on Friday -- meaning that the both payday schedules (quadriweekly and monthly) coincide in this calendar.

the lunar month cannot be evenly divided into seven-day weeks
Despite the New Earth Calendar having a month called Luna, it's not a lunar calendar. I don't know of any Calendar Reform proposal where there are months with 28 or 35 days, but designed to fit into a lunar calendar.

every civilisation in the world is settled on a seven-day week
This is a big one. Some Calendar Reformers propose weeks longer or shorter than seven days. One of the most common week-lengths, instead of seven, is six days. Here is a typical six-day calendar:

Six-day calendars tend to have twelve months with five weeks each. This gives us 360 days, so there must be five blank days. Different calendars distribute the five blank days differently.

The calendar I chose to link above drops Saturday. It declares Friday and Sunday to be the weekend, so that both Muslims and Christians can have their respective Sabbaths off. For schools, notice that this provides a four-day school week, from Monday to Thursday. This is convenient for A/B block schedules, where there can be two A days and two B days every week.

"daylight saving" doesn't

that is, daylight saving doesn't save daylight. I've already discussed DST back in November, in my first post after the time change.

Many of the items in this section refer to the biannual clock change, to which this author of this list is apparently opposed. But notice that my preferred alternatives -- year-round DST and the Sheila Danzig plan -- don't necessarily avoid checks either. Year-round DST results in the following box being checked:

local "midnight" should be the middle of the local night

because year-round DST puts the middle of the local night at 1 AM, not midnight. Also, the Danzig plan, which puts some time zones at year-round DST and others at year-round standard time, would result in a check here:

nobody would agree to pick your time zone over theirs

since those whom Danzig places in year-round standard time might prefer year-round DST instead, and vice versa.

Notice that the Hanke-Henry calendar places the entire world in a single time zone -- the Greenwich time zone. Naturally, this would place checks in both of the above boxes.

no, we don't know what year the Big Bang happened

Someone actually mentioned this on one of the comment threads! But I assume that it's actually a parody of the Holocene Calendar, where Year 0 is set to the Ice Age:

This simply places a 1 in front of all the dates. So today is New Year's Eve 12014, to be followed by New Year's Day 12015.

a leading zero on the year number only delays the inevitable

This refers to the Long Now, which is similar to the Holocene Calendar except that a 0 is placed in front of all dates instead of a 1. So today is New Year's Eve 02014, to be followed by New Year's Day 02015.

planetary-scale engineering is impractical

This one may sound weird, but it actually appeared in the comments at the International Fixed Calendar link above -- emcourtney posted:

Why don't we just boost the Earth into a slightly lower 336 day orbit around the sun, That way we can have 12 * 4 * 7 calendar with no sloppy leftovers! Why tinker with the calendar when you can tinker with orbital mechanics instead.

On the TV show Futurama (created by the Simpsons creator), a group of robots pushed the earth slightly farther from the sun in order to prevent global warming. This made the year a week longer, and this extra week was declared "Robot Party Week."

not every part of the world has four recognisable seasons

This refers to a calendar which seeks to put New Year's Day at a solstice or equinox. For example, the six-day-a-week calendar above begins at the spring equinox, and other months begin at the fall equinox and the solstices.

"sunrise" and "sunset" are meaningless terms at the poles

This refers to a calendar where days begin at sunrise or sunset. Notice that the Jewish and Islamic calendars have days beginning at sunset. As it turns out, these calendars do cause problems near the poles, where observers of fasts such as Yom Kippur and Ramadan don't know when to break the fast when the sun doesn't set.

Greenwich is not unambiguously inferior to any other possible prime meridian

This refers to a calendar where another prime meridian is chosen. For example, the Florence, Italy, meridian is chosen because it would put its opposite meridian, the International Date Line, out in the in the Bering Strait so that it no longer intersects Russia.

everybody in the world is already used to sexagesimal time divisions

This refers to changing the time to metric time rather than time based on 60 -- which goes back to the ancient Babylonians. Principal Skinner on the Simpsons once tried to introduce metric time. The following calendar proposes metric time:

This calendar also proposes the Florence-Bering prime meridian as mentioned above.

they tried that in France once and it didn't take

This refers to the French Revolutionary Calendar of the late 18th century. It contains ten-day weeks, with five blank days, and starts at the fall equinox -- so several other boxes are checked as well.

A Slate article from last year mentions this calendar.

I could go on forever about calendars, but this post is already bloated enough.

Happy New Year, and have a wonderful Gregorian year 2015.

Wednesday, December 24, 2014

Common Core Debate (Grades 8-12)

On this Christmas Eve, I post my gift to you -- my vision of how mathematics should be taught in the highest grades, 8-12 -- the grade span that I want to teach, and the focus of this blog. In this grade span there are two major issues that are commonly debated:

Integrated vs. Traditional Pathway -- The traditional sequence of courses goes Algebra I, Geometry, Algebra II, Pre-calculus, and then Calculus. But many states, districts, and schools have been turning towards an integrated approach more common outside the United States. Even though Common Core supports both pathways, some states (such as Utah) and districts (such as my own) use the new standards as a reason to move to an integrated pathway. I've discussed this here on this blog before.

Eighth Grade Algebra and Twelfth Grade Calculus -- Some people criticize the Common Core Standards because they feel that it doesn't adequately prepare students for STEM -- that is, a college major and ultimately a career in Science, Technology, Engineering, and Mathematics. Notice that AP Calculus is not required for admission at most schools -- but one exception right here in Southern California is Harvey Mudd College:

"A year of calculus at the high-school level is an entrance requirement for HMC, so familiarity with limits, differentiation and integration is assumed in all mathematics courses."

Harvey Mudd is, technically speaking, a liberal arts college -- its one of the five undergraduate Claremont Colleges -- yet it focuses on STEM. All of its majors are STEM majors, and therefore, all of its students are required to take three semesters of math. Ironically, Harvey Mudd calls these classes the Common Core math classes, but they have nothing to do with the Common Core State Standards that we discuss on this blog. Indeed, the question we ask is, are students who take the high school Common Core math classes well-prepared to take the Mudd Common Core math classes?

Oh, and it goes without saying that many Mudders take the Putnam exam that I mentioned on the blog earlier this month:

I have the pleasure of having known a high school classmate of mine who attended Harvey Mudd, and he is now a math teacher at our high school.

Of course, most colleges are not Harvey Mudd. But the argument is that many selective colleges, while not literally requiring AP Calculus, have the class as a de facto requirement in that the students who are admitted and are successful in STEM majors have taken calculus in high school. And so this is why many are concerned that Common Core doesn't adequately prepare students for calculus.

Unlike the old California State Standards, where seventh grade math is the last pre-algebra class (so that students can take Algebra I in eighth grade), the Common Core Standards have eighth grade math as the last pre-algebra class, so that Algebra I is taken as a freshman. This means that students would only reach pre-calculus as seniors -- and so they couldn't be admitted into Harvey Mudd or any competitive STEM program.

Some schools since the adoption of Common Core have discouraged their students from taking any class higher than Common Core 8 in eighth grade. An argument is that the Common Core 8 class contains some topics previously taught in Algebra I, such as systems of equations:

Analyze and solve pairs of simultaneous linear equations.

while Algebra I contains some topics previously taught in Algebra II, such as exponential functions.

But recall that it's not really eighth grade Algebra I that matters to the STEM-minded traditionalists, but twelfth-grade calculus. Algebra I in 8th grade is significant only in that a student needs to be in Algebra I by eighth grade in order to reach calculus by senior year -- unless that student is willing to take two math classes in one year, or a math class in the summer. Pushing down a couple of Algebra I and II topics to Pre-Algebra and Algebra I, respectively, and then using that as a reason to delay Algebra I to ninth grade doesn't allow a student to reach calculus in high school. And so, to the traditionalists, this is a poor excuse to water down the math standards. To them, anything less than calculus in senior year constitutes watered-down standards.

And so here are the standards that I propose for secondary school. I decided to compromise between the traditionalist and progressive ways of thinking. For these standards, I took the traditionalist approach of reaching calculus by senior year, and combined it with the integrated pathway often favored by progressives. These are not mutually exclusive -- for example, a local magnet program, the California Academy of Math and Science, used to do this. They required incoming freshmen to have taken Algebra I in eighth grade, then offered three years of integrated math, and finally as seniors, the students took calculus -- not AP Calculus, but an actual college calculus course offered on the adjacent college campus, California State University, Dominguez Hills. (But I've heard that this integrated program, ironically, no longer exists.)

As I mentioned before, the integrated program is more common internationally -- and of course, other countries are known for offering algebra early. And so I look toward other countries for standards that may be acceptable. In particular, I've already discussed the Singapore standards, which are often held by traditionalists in high regard:

Four textbooks have been written for the Singapore standards -- New Elementary Mathematics.

For some reason, the third and fourth textbooks are no longer in print. The link above is for the second textbook, and there's also a link for the first textbook there. The first textbook, intended for seventh graders, reads like a traditional pre-algebra text. But the second textbook, for eighth graders, is an integrated course. The above link gives the 14 chapters as follows:

1. Indices (i.e., exponents)
2. Algebraic Manipulations (i.e., factoring)
3, Literal and Quadratic Equations
4. Word Problems
5. Graphs
6. Simultaneous Linear Equations
7. Inequalities
8. Congruent and Similar Triangles
9. Mensuration (i.e., volumes of pyramids, cones, and spheres)
10. Pythagoras' Theorem and Trigonometry
11. Motion Geometry (i.e., reflections, rotations, and translations)
12. Statistics I
13. Statistics II
14. More Algebraic Manipulations

Just looking at the beginning of this course is daunting. Not only are eighth graders expected to know these Algebra I topics, but they are some of the more difficult topics in the book. In the Glencoe Algebra I text, with which I'm familiar from my student teaching days, exponents appear in Chapter 7, and factoring appears in Chapter 8. But by "algebraic manipulations," Singapore includes algebraic fractions, which don't appear in Glencoe until Chapter 11, and are often skipped by teachers who run out of time at the end of the year.

But believe it or not, I know of someone who teaches her Algebra I class exponents, polynomials, and factoring right at the beginning of the year. I've mentioned her before on the blog -- Sarah Hagan:

I mentioned earlier that Hagan had a problem with her students overgeneralizing -- thinking that two negatives make a positive even when adding, so -3 plus -5 is +8 -- and ignoring "the sage on the stage" when she told them that they were wrong. But, as it turns out, Hagan's solution to this problem is to move on to exponents and polynomials early in the school year! As she writes:

The same students who have been struggling with all of the above have been rocking our last few lessons on naming polynomials and multiplying polynomials.  Why?  My current theory is that multiplying polynomials is something they've never been exposed to before.  So, they actually found it necessary to listen to my explanation...

Somehow, those same students who added -3 to -5 to get +8 were able to multiply (x - 3)(x - 5) to obtain x^2 - 8x + 15 -- with the correct sign on both the -8x and +15 terms!

Apparently, Hagan begins the year with the equivalent of Glencoe's Chapter 1 -- basically a review chapter (and that's when her students added -3 to -5 to get +8), then jumps right into the equivalent of Glencoe Chapters 7 and 8. Unwittingly, Hagan is following the Singapore text order in her class, at least up to Singapore Chapter 7 (after which the text moves to geometry). Also, I'm not quite sure about algebraic fractions -- rational expressions. Many teachers skip this lesson, and I see no evidence that Hagan teaches this to her Algebra I students at all. Then again, the Singapore chapters listed above don't distinguish between Chapters 2 and 14. It could be that the algebraic fractions don't actually appear until Chapter 14. There's no way for me to know for sure, since I don't have a copy of the Singapore text.

Hagan's class proves that students can be successful following the Singapore order. The key difference between Hagan and Singapore is that Hagan's class consisted of ninth graders, but the Singapore text listed above is for eighth graders. I'm not necessarily convinced that American eighth graders can be as successful as Hagan's freshmen following the Singapore order.

Sometimes I wonder whether pushing the Singapore texts back a year would be more realistic for American students. Not only would this put Singapore's Secondary Two class in our freshman year -- more in line with Hagan's Algebra I class -- but it would push Singapore's Secondary Four class back to our junior year. I found out from other sources the second half of Secondary Four consists mainly of review. Putting this review section in our Grade 11 would prepare our students for the SBAC -- which, as you recall, is for Grades 3-8 and 11 -- as well as for the SAT exam that many students take during the second semester of junior year.

But recall our goal -- to appease the traditionalists, we need calculus in senior year. As it turns out, Singaporeans who want calculus take the "Additional Mathematics" topics, with an extra class for pre-calculus in Secondary Three and an extra class for calculus in Secondary Four. But we're trying to avoid doubling up on classes -- otherwise, we'd just stick to Common Core, with its Algebra I freshman year and doubling up in math one year in order to reach calculus senior year.

Also, let's recall why Singapore needs its students to reach calculus by the end of Secondary Four -- the equivalent of our sophomore year. In Singapore, just as in Great Britain, students take many national exams at the end of the equivalent of our tenth grade. J.K. Rowling, in her famous Harry Potter series, has her wizard sit the O.W.L.s, or "Ordinary Wizarding Levels," when he is 15 years old (i.e., the same age as an American sophomore). Here Rowling was parodying the O Level exams that she had to take back when she herself was 15. We see the phrase "O Level Mathematics Syllabus" appear right at the top of the page at the first Singapore link above.

Here in the United States we don't have the equivalent of the O Level exams. Instead, many of our major tests, such as the SBAC and SAT, are taken junior year, not sophomore year. And so we don't need to fit the Singapore grade level sequence to the extent of preparing for national tests that we don't have at the end of sophomore year.

Then again, we notice that if we keep the Singapore texts for the years that they were intended for -- Grades 7-10 -- then the Additional Mathematics courses for pre-calculus and calculus fit right into the junior and senior years. There's no doubling up on math, and calculus is reached senior year, just as I promised the traditionalists.

This means that the Grade 11 SBAC (and PARCC) exams would focus on pre-calculus. Notice that there would be no reason to write Common Core (or "David Walker Core," or whatever I plan on calling my standards based on Singapore's) Standards for calculus, since these would already be covered (and therefore determined) by AP. There would be no Core exam for calculus -- but only the AP exam.

Recall that I recommended the Saxon texts for Grades 4-7. In many ways, the Saxon and Singapore curricula -- both highly recommended by traditionalists -- are interchangeable. Saxon, like Singapore, has an integrated pathway for high school. But notice that Saxon's texts are called Algebra 1/2 (i.e., one-half), Algebra 1, Algebra 2, and Advanced Mathematics. Geometry is actually integrated into the so-called Algebra texts. But Algebra 1/2 is pre-algebra, and Advanced Mathematics is essentially a pre-calculus course. So Saxon integrates four years into three. Therefore one could actually take Algebra 1/2 in eighth grade and still make it to Saxon's Calculus text by senior year. But the Saxon texts must move quickly in order to cover four years' worth of material in three.

But my concern is that so much Algebra I in eighth grade and pre-calculus for juniors is too advanced for many students. I wouldn't mind pushing back the classes so that so many students wouldn't have to fail such classes. But this is where the traditionalists argue that this is a zero-sum game -- if we drop pre-calculus and calculus, the slowest students may pass, but the brightest students are being held back. But if we stimulate the brightest students, the slowest students will fail.

Many traditionalists argue that it's the brightest students who have priority. Indeed, especially at the high school level, they sometimes say that students who can't keep up with a rigorous curriculum should be expelled as they would only disrupt the bright students who can keep up. Of course, simply expelling the disruptive students then opens up a can of worms, because then politics, class, and ultimately race suddenly become factors -- especially if the expelled students end up being disproportionately members of some demographic group.

One argument that the traditionalists come up with is the cure cancer argument -- or maybe I should say Ebola as this is more newsworthy. It goes like this -- let's say we have two students, a bright student who can keep up with a rigorous student, and one who can't. The latter student can only disrupt the gifted student, and so must be expelled. The gifted student, free from distractions and armed with a rigorous curriculum, is able to learn much advanced material and ultimately becomes a scientist who discovers a cure for cancer or Ebola.

But what about the student who was kicked out of school? Let's say this student, with an incomplete education, is lucky enough to avoid prison and find a minimum-wage job at -- well, you can probably figure out what companies come to mind here. But then this person ends up getting fired, let's say for taking too many sick days. For as it turns out, this person is diagnosed with Ebola -- that is, exactly the disease for which our scientist has found a cure!

And now the traditionalists will say "Aha!" The student who was expelled is actually better off, since after that expulsion, the gifted student was free to learn how to cure the disease with which the dropout is now infected. But I say otherwise. For this person now has no job, no money, and no access to the cure. That person now ends up dying of Ebola.

In other words, I have no problem with providing opportunities to the gifted students, provided that some opportunity is given to the slower students. Notice that I didn't say college -- I agree that trying to send everyone to college is a foolish idea. All I want is for all students to have the opportunity to get a job that would allow them to live a comfortable, middle-class life. If the gifted students discover cures for cancer, I want the slower students to be able have enough income and enough health care to access the cure should they end up catching that disease. If everyone has that opportunity, then things like politics and race would never even come up.

This leads, of course, to the tracking debate. I would support tracking only if it allows students on all tracks to attain, eventually, a comfortable, middle-class life. Simply expelling students, to me, doesn't constitute an acceptable form of tracking. But the tracking debate is very complex, and deserves a blog entry of its own.

Friday, December 19, 2014

Semester 1 Review and Semester 2 Preview (Day 84)

Last night I tutored my geometry student in a session delayed from Tuesday night. He has finally reached Chapter 3 of the Glencoe text, which is on Parallel and Perpendicular Lines. In some ways this is similar to the same numbered chapter of the U of Chicago text, but latter text has overall poor coverage of parallel lines in general.

The student has just finished Section 3-2 of the Glencoe text, "Angles and Parallel Lines." Even though he's finished this lesson, he asked for further clarification on a certain proof, which turned out to be the Same-Side Interior Angles Consequence -- actually, the Glencoe text uses the term "consecutive interior angles" instead of "same-side interior angles."

Unfortunately, this proof is not included in my worksheets for this blog. This is because it is not emphasized in the U of Chicago text on which my worksheets are based. The only reference to the same-side interior angles is in Section 5-5, in the context of trapezoids:

Trapezoid Angle Theorem:
In a trapezoid, consecutive angles between a pair of parallel sides are supplementary.

Notice that technically speaking, this text also uses the phrase "consecutive angles," and that Dr. Franklin Mason also uses the term "consecutive interior angles," so maybe this term is becoming more popular than the term to which I'm accustomed, "same-side interior angles."

Here is the two column proof that the U of Chicago provides for the Trapezoid Angle Theorem -- as usual, with the headings changed to "Statements" and "Reasons" and an extra step for "Given."

Given: AB | | CD
Prove: Angles 1 and D are supplementary
(Note: In the diagram, angle 1 refers to BAD, and angle 2 refers to BAE, where E is a point on the transversal line AD, chosen so that angles 2 and D are corresponding angles.)

Statements                                           Reasons
1. AB | | CD                                          1. Given
2. angle 1 + angle 2 = 180                   2. Linear Pair Theorem
3. angle 2 = angle D                            3. Corresponding Angles Consequence
4. angle 1 + angle D = 360                  4. Substitution (step 2 into step 1)
5. Angles 1 and D are supplementary 5. Definition of supplementary

I told my student about this proof, or something similar. Of course, if we follow the Dr. Hung-Hsi Wu plan, our proof would use the Alternate Interior Angles Consequence instead of Corresponding Angles, since AIA is the basic consequence that is used to prove the others.

Recall that much of my planning includes how I would revise my course for next year, based on how much my lessons were able to help an actual student. So far, I see one glaring omission -- namely a proof of this Same-Side/Consecutive Interior Angle Consequence. I will include it next year, and maybe I will use the name that is currently en vogue, "consecutive interior angles."

Here is how my course will begin next year, based on my tutoring sessions so far this year:

Days 1-3: Opening Activity (This is now three days because my first day of school is Wednesday.)
Day 4: Section 1-6
Day 5: Section 1-7
Day 6: Section 1-8 ("One-Dimensional Figures")
Day 7: Two-Dimensional Figures and Review (new)
Day 8: Quiz
Day 9: Section 2-1
Day 10: Section 2-2
Day 11: Section 2-3
Day 12: Review
Day 13: Chapter 1 Test
Day 14: Activity: Identifying Patterns (new)
Day 15: Section 2-4
Day 16: Section 13-1 ("The Logic of Making Conclusions," moved up)
Day 17: Activity
Day 18: Section 2-5
Day 19: Section 2-6
Day 20: Section 2-7
Day 21: Section 3-1
Day 22: Activity (possibly another quiz instead)
Day 23: Section 3-2
Day 24: Section 3-3
Day 25: Section 4-4 ("The First Theorem in Euclid's Elements," moved up)
Day 26: Review
Day 27: Chapter 2 Test

The new sections that I mentioned throughout the semester as being helpful to my student have been added in. A new test for Chapter 1 has been added in -- otherwise one will go the entire first quaver without a test. Of course, making Chapters 1-2 longer means making something else shorter.

My student moved on to Section 3-3 of the Glencoe text -- "Slopes of Lines." This is covered in Sections 3-4 and 3-5 of the U of Chicago text -- but I am saving it for 2nd semester.

And this brings me to my 2nd semester preview. So far, I have stuck mostly to the U of Chicago order -- almost to a fault. This explains why I need to make so many changes to my 1st semester plan next year. But I will jumble things up for 2nd semester, in order to give lessons in an order that makes more sense for Common Core:

Chapter 12. Similarity
Chapter 11. Coordinate Geometry
Chapter 13. Logic and Indirect Reasoning
Chapter 14. Trigonometry and Vectors
Chapter 8. Measurement Formulas
Chapter 15. Further Work with Circles
Chapter 9. Three-Dimensional Figures
Chapter 10. Surface Area and Volumes

This also reflects the order in many other texts -- similarity first, volume last. In Common Core, similarity must come before coordinates, so we begin with Section 12-2, "Size Changes (Dilations) Without Coordinates." Much of the similarity lessons will be based on Wu.

Merry Christmas! Stay tuned for a winter break post on Common Core Debate, Grades 8-12.

Thursday, December 18, 2014

The King of the MTBoS (Day 83)

Today and tomorrow, of course, are the second and third finals days-- since generally there are three finals days, each day a minimum day with two periods of two hours or so each. Since I already posted the final, today's a good day to take a step aside and discuss this blog's place in the MTBoS.

The MTBoS is the Math Teacher Blogosphere -- the set of all blogs by and for math teachers. Some of the links that I've posted so far, such as Math Hombre and Sarah Hagan's Math Equals Love, are full-fledged members of the MTBoS. As I've mentioned before, most members of the MTBoS blog about the special math lessons that they've had with their students, and this is why I strive to do the same on this blog, despite not yet having my own classroom.

But I have yet to post to the most popular math blog of all. Indeed, nearly every member of the MTBoS has linked to this website at least once on his or her own blog. And so I don't consider myself a full member of the MTBoS until I link to this site as well.

Who is this most popular math blogger, the King of the MTBoS? Of course, it's Dan Meyer -- more commonly known as dy/dan:

So who is Dan Meyer, and what makes his site so popular? Well, he is a fellow California high school math teacher who often gives presentations about how best to teach math. The best way to get to know Meyer is to look at his most recent post, where he discusses his most recent talk at the annual California Math Council (which occurred over my birthday weekend):

I suspect that upon seeing that title, "Video Games & Making Math More Like Things Students Like," many traditionalists will give pause. They feel that so much time is being wasted trying to entertain the students and not enough trying to educate them -- and that if we spent time only on the latter, math achievement will be as high as it was in the good old days of their own youth. So the last thing they'd want to see is trying to make math class more like a video game.

But I disagree with the traditionalists here -- and here's why. If one wants to teach a traditional class with nothing but direct, "sage on the stage" instruction, one doesn't need a blog or a presentation on how to teach that way -- just teach that way. The whole purpose of blogs and presentations is to give alternative, new ways of teaching. And if students are fascinated by video games, then maybe by making math class more like video games, students will be almost as fascinated by math class. The real choice often is not between a fun class and traditionalist direct instruction, but between a fun class and nothing at all. For the students whom Meyer is trying to reach, anything that's not as fun as a video game is simply ignored.

Of course, notice that programming video games is extremely math intensive, especially graphing -- and I should know, since about two years ago I met a former high school classmate of mine who told me that he worked for a video game company for a few years. So anyone who says "I hate math!" or even "I hate graphing!" should avoid playing video games, period, since if graphing in math classes were to disappear, so would video games.

But that's not what Meyer's lecture was about. It's not about how math can be applied to video games, but on how math class can be made as interesting as a video game -- or, for another example of modern entertainment, a movie.

The idea that math lessons should be more like movies is the cornerstone of Meyer's specialty -- the three-act lesson. The three-act structure goes back much further than movies -- it goes all the way to Aristotle and the ancient Greeks. Most plays and stories have three parts -- a beginning, a middle, and an end. More specifically, they have a Setup, a Confrontation, and a Resolution. Even video games have their underlying story.

And so Meyer's lessons are also structured this way. About four months ago, Meyer posted his most recent three-act lesson, "Dandy Candies":

This lesson begins with a Setup --  Dandy Candies tells students they’re going to package up 24 cubical candy boxes. It asks them, “Which of four packages uses the least amount of packaging? Which uses the least amount of ribbon?” The Confrontation is where students make guesses as to what the dimensions of the box need to be, and calculate the surface area. The Resolution is where the class compares and discusses their results in order to answer the question in the Setup.

This sounds like an interesting lesson. I'm considering incorporating it into my own course for this blog, since we cover surface area and volume second semester.

Today I subbed in a seventh-grade classroom. In most of the classes, the teacher's lesson plan was for the students to plot points on a coordinate plane to draw a Christmas snowman -- a very common assignment often given the day or two before a long school break. But many of the students were not on task -- and yet I bet that many of them will go home to play the same video games whose graphics depend on graphing on a coordinate plane.

But one class was a intervention class, a second math class that struggling students often take. For this class, there was no lesson plan except to continue the graphs from the main class. So I decided to play the game that I described back in October. Since yesterday the students took a test on integer operations, I decided to ask questions on this topic. But some of the students wanted to earn more points for their group, so they wanted me to ask a fraction question, even though I was only giving questions on integers. That's right -- students in a middle school math class were actually asking for fraction problems! Such is the power of making math class more like a game.

I once read someone brag on Facebook how they can remember funny movie lines for years yet can't remember what they learned in math class last week. And so I disagree with the traditionalists here. I am now a loyal subject of the MTBoS and its king, Dan Meyer, and its his philosophy that I strive to follow here on this blog.

Wednesday, December 17, 2014

Semester 1 Final Exam (Day 82)

I had problems writing and posting this final, because of my printer. Now my printer has been showing a "Clear carriage jam" error message that simply will not disappear. This is the first time that I have ever seen a paper jam message that doesn't go away simply by removing all of the paper from the printer.

Lately I've been handwriting many of my posts due to printer problems. This time, I really wanted to have a typed final, only to have it not print. I was able to print some of it, but I couldn't quite make it all the way to #50 before the printer failed. So I had to handwrite the last few questions, as well as correct a few errors by hand, including my TI-BASIC program that didn't print correctly.

This is not what I wanted. I hope that I'll be able to get the printer fixed in time for the start of the second semester right after winter break.

Here are the answers to the questions on the final:


Tuesday, December 16, 2014

Review for Final Exam, Continued (Day 81)

Today we continue our review for the final exam.

Most of the questions on this half, which cover Chapters 5 through 7 of the U of Chicago text, are mostly self-explanatory. Notice that in Question 47, we are given that ABCD is a trapezoid with one pair of opposite angles congruent and we are to prove that it is a parallelogram. In other words, we are using the inclusive definition where a parallelogram is a trapezoid. If teachers prefer the exclusive definition, they can change the Given section to: AB and CD are parallel and angles A and C are congruent, to prove that ABCD is a parallelogram.

For the bonus section today, let me include those last two proofs that I've been trying to squeeze in for a while, but couldn't until now. These are the proofs that the angle bisectors of a triangle are concurrent -- meeting at the incenter -- and that the medians are concurrent -- meeting at the centroid.

Just as with the circumcenter and orthocenter, we get these proofs from Dr. Hung-Hsi Wu. Let's do the incenter proof first. But before we can prove this, we must prove what Wu calls "Lemma 12," but what I will call the Angle Bisector Theorem:

Angle Bisector Theorem:
If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.

Notice that this requires the concept of distance between a point and a line. This is usually defined to be the perpendicular distance from the point to the line. So we are going to begin with an angle O and a point P on the bisector of angle O. Since we want to consider the perpendicular distance from P to the sides of angle O, we choose points A and B on the sides of O such that OA and OB are perpendicular to PA and PB, respectively.

Here is Wu's proof, converted to two-column format:

Given: Ray OP bisects angle AOB, Ray OA perpendicular PA, Ray OB perpendicular PB
Prove: PA = PB

Statements                                                      Reasons
1. Ray OP bisects angle AOB, Ray OA ...      1. Given
2. Ray OB reflected over line OP is Ray OA 2. Side-Switching Theorem
3. PB' perpendicular to Ray OA                     3. Reflections preserve angle measure.
4. PB' is exactly PA                                        4. Uniqueness of Perpendiculars Theorem
5. PA = PB                                                      5. Reflections preserve distance.

I'm actually surprised that the U of Chicago text doesn't prove this theorem. It fits very nicely with the Perpendicular Bisector Theorem in Section 4-5.

We will also need the proof of the converse of this theorem:

Converse of the Angle Bisector Theorem:
If a point is equidistant from the sides of an angle, then it is on the bisector of the angle.

Given: PA = PB, Ray OA perpendicular PA, Ray OB perpendicular PB
Prove: Ray OP bisects angle AOB

Statements                                                      Reasons
1. PA = PB, Ray OA perpendicular PA, ...     1. Given
2. OP = OP                                                     2. Reflexive Property of Equality
3. Triangle POA congruent triangle POB       3. HL Congruence Theorem
4. Angle POA congruent angle POB              4. CPCTC
5. Ray OP bisects angle AOB                         5. Definition of angle bisector

Notice that this follows a common pattern that we've seen so far in the text -- a theorem is proved using reflections and symmetry, while its converse is proved using a Congruence Theorem (and in this case, HL). We could have proved the forward theorem in Section 4-5, but the converse had to wait until Section 7-5 when we proved HL. (The Perpendicular Bisector Theorem was proved in Section 4-5, while its converse could be proved using the Isosceles Triangle Theorem of Section 5-1, but the converse of that theorem in turn appears in Section 7-3, and its proof used AAS.)

Now we can prove the incenter theorem:

Angle Bisector Concurrency Theorem:
The three angle bisectors of a triangle meet at a point, called the incenter of the triangle. The incenter is the unique point equidistant from the three sides.

Given: Ray AE bisects angle BAC, Ray BD bisects angle ABC, Rays AE and BD intersect at I
Prove: Ray CI bisects angle ACB, I equidistant from all three sides

Statements                                                      Reasons
1. Ray AE bisects angle BAC, ...                    1. Given
2. I equidistant from AC and AB                    2. Angle Bisector Theorem
3. I equidistant from BA and BC                    3. Angle Bisector Theorem
4. I equidistant from CA and CB                    4. Transitive Property of Equality
5. Ray CI bisects angle ACB                          5. Converse of Angle Bisector Theorem

Wu also proves uniqueness -- that is, that I is the only point equidistant from all three sides. He does this by using the Converse of the Angle Bisector Theorem to show that another point I' that is equidistant from any two of the sides must lie on the bisector of the angle those sides form. (That is, Wu proves the Converse of this Angle Bisector Concurrency Theorem!)

Now finally, we get to the Median Concurrency Theorem -- the only concurrency that is specifically mentioned in the Common Core Standards. This one is even more complicated -- it depends on the properties and sufficient conditions for a parallelogram in Sections 7-6 and 7-7. It also depends on what Wu calls Theorem 18 and the U of Chicago calls the Midpoint Connector Theorem:

Midpoint Connector Theorem:
The segment connecting the midpoints of two sides of a triangle is parallel to and half the length of the third side.

The U of Chicago proves this in Section 11-5, using coordinates. But Wu proves this without using coordinates at all, as follows:

Given: In triangle ABC, D and E are midpoints of AB and AC, respectively
Prove: DE | | BC, BC = 2DE

Statements                                                      Reasons
1. bla, bla, bla                                                1. Given
2. Let F be on ray DE such that DF = 2DE   2. Ruler Postulate (Point-Line)
3. E midpoint of DF                                       3. Definition of midpoint
4. ADCF is a parallelogram                           4. Sufficient Conditions (pgram test), part (c)
5. CF = AD                                                    5. Properties (pgram consequence), part (b)
6.  CF = BD                                                    6. Transitive Property of Equality
7. CF | | AD (same as CF | | BD)                    7. Definition of parallelogram
8. DBCF is a parallelogram                           8. Sufficient Conditions (pgram test), part (d)
9. DF = BC                                                    9. Properties (pgram consequence), part (b)
10. BC = 2DE                                                10. Transitive Property of Equality
11. DF | | BC (same as DE | | BC)                 11. Definition of parallelogram

In analyzing this proof, notice how similar steps 4-7 are to steps 8-11.

Now we can finally prove the Median Concurrency Theorem, Wu's Theorem 19:

Median Concurrency Theorem:
The three medians of a triangle meet at a point G, called the centroid of the triangle. On each median, the distance of G to the vertex is twice the distance of G to the midpoint of the opposite side.

Given: In triangle ABC, C', B', and A' are midpoints of AB, AC, and BC respectively.
(This way, the medians are AA', BB', and CC' as per Wu's notation.)
Prove: CC' meets BB' at a point G such that BG = 2GB'.
(Without loss of generality, the same proof will work if we change B or C to A.)

Statements                                                      Reasons
1. bla, bla, bla                                                1. Given
2. C'B' | | BC, C'B' = 1/2 BC                          2. Midpoint Connector Theorem (applied to tri. ABC)
3. Let M, N be midpoints of BG, CG            3. Ruler Postulate (Point-Line)
4. MN | | BC, MN = 1/2 BC                          4. Midpoint Connector Theorem (applied to tri. GBC)
5. C'B' | | MN                                                 5. Transitivity of Parallelism Theorem
6. C'B' = MN                                                 6. Transitive Property of Equality
7. MNB'C' is a parallelogram                        7. Sufficient Conditions (pgram test), part (d)
8. MB' and NC' bisect each other                  8. Properties (pgram consequence), part (c)
9. BM = MG, MG = GB'                               9. Definition of midpoint
10. BM = GB'                                               10. Transitive Property of Equality
11. BM + MG = GB' + GB'                          11. Addition Property of Equality
12. BG = 2GB                                               12. Segment Addition (Betweenness Theorem)

As I mentioned before, many texts use coordinates to prove theorems such as these. But we don't know that coordinates work unless we use similarity, and we don't know that similarity works, according to Wu, unless we first use theorems such as this one. It is at this point where Dr. David Joyce and Common Core are in the most agreement. In order to avoid circularity and ensure that no result is used before it is proved, we must avoid coordinates here.