Saturday, December 26, 2015

Calendar Reform

I hope you had a wonderful Christmas. But now it's the day after Christmas, often called Boxing Day in many English-speaking countries. Beginning today, my thoughts start turning towards New Year's Day, calendars -- and Calendar Reform.

As I wrote last year, calendars and Calendar Reform have always fascinated me. Many people may have noticed that there was a full moon on Christmas Day this year. I've also noticed that many holiday specials depict Santa Claus and his reindeer-pulled sleigh flying through a sky lit by a full moon. An interesting question is to ask, therefore, is when was the last time there was a Christmas full moon, and when will it happen again?

This is a question that Calendar Reformers seek to answer, and possibly solve. Is it possible to create a brand-new calendar, one on which Christmas always falls on a full moon date. Or, perhaps an easier task is one on which Christmas always falls on the same day of the week. This year, it fell on a Friday, which is nice as it leads to a three-day weekend. Last year it fell on a Thursday, which was inconvenient (unless you lived in one of those countries celebrating Boxing Day as well, in which case you got a four-day weekend).

Today I will repeat the post I wrote last year, when I mentioned several calendar proposals. Then on New Year's Day, I will post what I hope is my own original Calendar Reform proposal. But first, let's start with a video clip from my favorite TV show, The Simpsons:

"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 -- Boxing Day -- since after all, it's the week 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.

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 in years when Veteran's Day falls on a Tuesday, many students and teachers unofficially take Monday off -- and many schools and districts officially close that day as well. This problem also occurs at offices, where, as I said at the start of this post, workers want to take off two extra Fridays because Christmas and New Year's Day were on Thursdays last 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 abissextile, 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 December 13th) and Chinese holidays such as Lunar New Year, which occurred very late in 2015.

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.)

Incidentally, it's the Metonic Cycle that answers the question I gave at the start of this post -- when was the last time there was a Christmas full moon? There was a full moon on Christmas 19 years, or one Metonic Cycle, ago -- so it was back in 1996. The next Christmas full moon will be one Metonic Cycle from now, or 2034.

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 toproposed 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 contained 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 SeptemberOctoberNovember, andDecember. 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 --HumanusSanctus, 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 two years ago 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 2016. Look out for my original proposal for calendar reform on Friday, New Year's Day -- and let's find out how many boxes my proposal will end up checking.

Thursday, December 24, 2015

Yes West Virginia, There Is an Alternative to Common Core

Last year, I established a tradition of posting my first winter break post on Christmas Eve, and I'm doing so again this year. Last year, I posted an alternative to Common Core which I called "my gift to you," and I'm doing so again this year.

Last year, I proposed that we replace Common Core Math with Singapore Math -- a curriculum that is popular with many traditionalists. Singapore Math pushes more advanced math down into lower grades so that students can take Calculus as seniors.

I also wrote about why so many traditionalists are so obsessed with senior year Calculus. I linked to the websites of certain STEM colleges in this area. such as the California Institute of Technology (Caltech) and Harvey Mudd College. Let's link to those websites once again:

I even pointed out the irony in that Harvey Mudd calls its first-year math curriculum "Common Core" yet the Common Core Standards, as written, don't adequately prepare students for the Harvey Mudd Common Core.

So we see our dilemma here. Unless we want schools like Caltech and Harvey Mudd to have zero applicants, we need to allow high schools to teach senior year Calculus and middle schools to teach eighth grade Algebra I, not Common Core 8. But we don't really want to force students who have no intention of becoming Mudders or Techers into eighth grade Algebra I and senior year Calculus.

Last year on Christmas Eve I gave everyone the gift of Singapore Math.  But since then, I've had to take that gift right back. The Singapore Math course for the equivalent of eighth grade is a super advanced course -- students there learn the equivalent of a full Algebra I course, yet there's as much geometry in the course as Common Core 8. My home state of California nearly recommended a course similar to Singapore's eighth grade course, with enough geometry to pass the SBAC test based on Common Core 8, yet enough algebra to make it to Calculus by senior year. The class was deemed too difficult for eighth graders to pass, and so the proposal was dropped. Maybe Singaporeans can pass such a class, but it's not realistic to expect American eighth graders to pass the course.

It's apparent that we'd want some sort of tracking system so that those headed to Harvey Mudd or Caltech can take higher math than those headed for non-STEM careers. There are two ways to distinguish higher-track math from lower-track math -- depth and speed. The Presidential Birthday and Consistency plan that I proposed recently -- based on the Sidwell Friends curriculum -- emphasizes depth much more than speed. In particular, we see that freshmen on both the middle and high tracks take some form of Geometry (as the Integrated Math I course on the high track is mostly geometry), while seniors on both tracks take Calculus, of either the AP or non-AP variety. We see how speed distinguishes the low track from the higher two tracks -- students on the low track don't take Geometry until sophomore year and therefore don't take Calculus at all.

But many traditionalists want to see the highest track distinguished from the others by speed. They feel that students can get into Algebra and Calculus much faster than the Common Core and other curricula allow, if only teachers and administrators would let them.

Using speed-based tracking, everyone learns the same material to the same depth, but they learn them at much different speeds. I've hinted at speed-based tracking here on the blog before, but I didn't include it in the Presidential Birthday plan because speed-based tracking deviates from Sidwell's curriculum, and as soon as we deviate from Sidwell, charges of hypocrisy may appear (as the president would have his own children learn from a different curriculum from the one that he is imposing on the rest of the country).

So let me propose a new speed-based tracking plan. What should I call this new plan? Instead of Presidential Birthday plan, let's change "Birthday" to "Christmas" since this is actually my Christmas gift to you this year. I also want to change "Presidential" to "Gubernatorial." With the passage of the Every Student Succeeds Act, the states will have more power in selecting a curriculum. So we can think of this as what I'd propose if I were elected governor, rather than president.

Speaking of states, West Virginia is the latest to abolish Common Core. The proposed standards to replace Common Core in the Mountain State include many things traditionalists like -- memorizing the times tables by third grade is emphasized more strongly in West Virginia than in Common Core, and, for a non-math example, West Virginia has new standards for cursive writing.

And there are explicit standards in West Virginia for Calculus. Indeed, here is a link to the standards for high school math:

The West Virginia standards do contain some form of tracking. In particular, we see that both West Virginia and Sidwell Friends have an Algebra III course for students who barely pass Algebra II and so aren't ready for Pre-calculus. By contrast, Algebra III classes aren't common here in California.

Of course, since I'm here in California, I shouldn't discuss what's happening in other states. The Gubernatorial Christmas plan that I'm about to describe only applies to my home state of California.

Oh, and here's one last thing to discuss before I can describe the plan. I keep mentioning Presidential Consistency to ensure that students get as good an education as the president's own daughters. But there's no need for me to mention Gubernatorial Consistency -- our current governor, Jerry Brown, never had any children. Our lieutenant governor, Gavin Newsom, does have three children -- his oldest child was born in September 2009. So she just barely missed California's new kindergarten cutoff by a few weeks last year -- instead, she may have been in our state's Transitional Kindergarten, to begin true kindergarten this year. And our State Superintendent is Tom Torlakson -- he has two daughters, but considering his age, he's likely has school-age grandchildren. These are the state officials I'd consider if I wanted to incorporate Consistency Core into possible state standards.

And so let's begin the Gubernatorial Christmas standards. Our goal is to divide the students into tracks and then teach the same content at different speeds depending on the track. But before we can determine who's "above grade level" or "below grade level," we must define "grade level."

Now even though these Gubernatorial Christmas standards are not compatible with the Presidential Birthday standards, I'm tempted to define the middle track of the Presidential Birthday (i.e., Sidwell) standards as on grade level. Or -- since this is California after all -- we can define the old pre-Core California standards to be grade level. Notice that both middle-track Sidwell and pre-Core California consider Algebra I to be on grade level for eighth grade.

But -- no matter what the traditionalists may want -- I still don't like the idea of declaring any senior who can't do Calculus to be "below grade level" -- not even if we take it to be "non-AP Calculus" (which I still don't know exactly what that is -- my guess that it focuses on polynomials and e^x only is just that, a guess). For these standards, I prefer declaring Algebra I to be on grade level for freshmen, so seniors only have to make it to Precalculus to be considered on grade level. This isn't terrible, since there would be many opportunities for students to accelerate and thus make it to Calculus, Caltech, and Harvey Mudd.

Now the heart of these new standards is the testing. I have mentioned a scoring system similar to this proposal a few times here on the blog, but now I'm posting it again. The following is based on computer testing just like the PARCC or SBAC, but students don't simply receive scores on a scale of 1-4 as the SBAC provides. Instead, students receive a three- or four-digit score that's much more descriptive as to whether a student is below, on, or above grade level:

400: Ready to begin 4th grade math
500: Ready to begin 5th grade math
600: Ready to begin 6th grade math
700: Ready to begin 7th grade math
800: Ready to begin 8th grade math
900: Ready to begin Algebra I
1000: Ready to begin Geometry
1100: Ready to begin Algebra II
1200: Ready to begin Pre-calculus
1300: Ready to begin Calculus

Many people complain about computer-adaptive testing, but this scoring system actually justifies the use of computers in testing -- the order of the test questions can be tailor-made to each student based on their responses. Suddenly, the need to test on the computer is justified.

Now this isn't compatible with the Presidential Birthday plan, but we ask ourselves, is this compatible with the Every Student Succeeds Act? The new version of NCLB still requires states to test their students every year from third to eighth grades, as well as once in high school (typically during the junior year). In previous posts, I often proposed that eighth graders (and if there are any seventh graders) in Algebra I shouldn't have to take the Common Core test -- that immediately solves the problem of trying to shoehorn the geometry of Common Core 8 into the Algebra I class. Under this plan, eighth graders can still take a computerized test which would allow them to test either eighth grade math, Algebra I, or whatever level math they have learned.

The computer test should max out at 1300 or Calculus, since the students already have a Calculus test they can take -- the AP Calculus test.

Now the results of the Gubernatorial Christmas tests should be used for placement into math classes for the following year -- which implies that the scores should be given promptly. What I expect is that tracks will ultimately develop based on patterns in the various student scores. For example, the middle track would consist of students who gain about 100 points per year, as per the above chart.

There's likely to be a group of students who can't gain 100 points per year -- the lower track. Let's say that they end eighth grade with 900 points, as they're supposed to, but once they reach Algebra I, they struggle and can gain only 80 points per year rather than 100. So let's look at their scores:

Freshman year: 900-980
Sophomore year: 980-1060
Junior year: 1060-1140
Senior year: 1140-1220

So we see that these low-track students cover most of Algebra I as freshmen. Then as sophomores, they finish Algebra I and start Geometry. In junior year, the first semester is devoted to Geometry and the second semester to Algebra II. Finally, as seniors they finish Algebra II and begin just a little bit of Pre-calculus -- a course that's approximately equivalent to the Algebra III courses that we can find on the East Coast. So in a way, the low-track students are following a sort of Integrated Math plan, since they see both Algebra and Geometry during their sophomore and junior years. This justifies Integrated Math in a way that the Common Core doesn't.

It may be tempting to come up with an accelerated schedule for high-track students which would allow them to reach Calculus and beyond. But let's keep in mind that since high school math is harder than elementary math, we expect most acceleration to occur in the early years (but neither Common Core nor my Presidential Birthday plan make such acceleration easy).

This means that a student is more likely to reach Calculus by reaching Algebra I in eighth -- maybe seventh, perhaps even sixth -- grade and proceed one year at a time until they reach Calculus, than by starting Algebra I as a freshman and accelerating to Calculus. This plan allows students to accomplish this by scoring high enough in elementary school to reach Algebra I in middle school.

So there is less need for Integrated Math on the high tracks, but it's still possible. The eighth grader I tutored last year covered Algebra I the first quarter and then Geometry the rest, so this would be like a student who began the year near 980 and worked his way up to around 1060 by the end of the year.

In fact, I like to divide each course into ten units and have each unit correspond to 10 points on the scoring scale. So a score of 1060 means that a student has completed eight units of Geometry and is now ready to begin Unit 7. If we base this on the middle path Geometry course that I posted last week, notice that there were eleven units listed there. Well, the Introduction to Geometry unit is probably hard to test, so there are only ten testable units:

1. Triangle Congruence
2. Using Tools of Geometry and Triangle Properties
3. Parallel Lines
4. Polygon Properties and Circles
5. Area Formulas
6. Transformations, Tessellations, and Area
7. Volume Formulas
8. Pythagorean Theorem and Volume
9. Triangle Similarity
10. Similarity and Trigonometry

Under this plan, it's possible for a student to start Algebra I as early as sixth grade, which would allow a student to reach Calculus as a sophomore. That way, the student can take the state test as a freshman (thereby meeting the "once in high school" test that the Every Student Succeeds Act needs) and then the AP Calculus AB test the following year. Calculus as a ninth grader would be awkward since then there would be no ESSA test in high school -- but Calculus as a freshman works backward to Algebra I in fifth grade, which is itself awkward since it would be difficult for elementary schools to offer Algebra I.

Indeed, as long as students begin taking ESSA tests in third grade, this allows students to accelerate as early as fourth grade. It may be difficult to offer accelerated classes in elementary school unless the school adopts something like the path plan that I discussed earlier. And yes, this testing and tracking plan works for both ELA and math, but this blog always focuses on math. (Below third grade, tracking may be difficult as long as students aren't required to test.)

Now I've posted two mutually incompatible plans to replace Common Core -- Presidential Birthday and Gubernatorial Christmas. You may be wondering what's next -- the Mayoral New Year's plan, or should it be the King's St. Patrick's Day plan? I can keep writing plans I'd implement if I were the president, governor, pope, or emperor of the universe, but none of those will ever happen. But there is, in fact, a way for me to implement the Gubernatorial Christmas plan as a classroom teacher.

I remember last year subbing in a sixth grade class. The students were working on a worksheet on one-step equations. One of the equations was awkward: 5 - x = 2. The correct answer is x = 3, but many students were convinced that the answer was x = 7. The problem is that obtaining the correct answer x = 3 requires manipulation with signed integers that sixth graders haven't mastered yet -- they'd just barely seen negative integers if at all. and they certainly hadn't performed much arithmetic with negative numbers yet. In short, the equation 5 - x = 2 is inappropriate for sixth graders.

And so here's what I did -- when explaining the question, I just called it a "seventh grade equation," and then I explained how to solve it. This allows me to teach the students who might understand the signed integers how to solve it (so that they are accelerating somewhat towards seventh grade math, if only for that one question) without alienating those who don't get it (since the question was labeled as 7th grade, so it's OK if they didn't understand it for another year).

So far I haven't given such problems on the blog, but I may in the future. Notice that this is actually easier to implement in an Integrated Math class (since if students don't understand, say, a geometry question in Math I, they'll see geometry again in Math II), yet I based the Gubernatoral Christmas on the traditionalist Algebra I-Geometry-Algebra II sequence.

With any tracking plan, I always worry about those who would using tracking to enforce segregation, but this test -- just like the special Geometry test I mentioned last week -- allows students to challenge their placement simply by scoring higher on the test, since the computer-adaptive test automatically gives more difficult questions if the easier ones are answers correctly.

But still, tracking leads to controversy. In the news last week, there were some comments made by Justice Antonin Scalia on the topic of affirmative action. Just as it's impossible to discuss Common Core without getting into politics, it's impossible to discuss tracking without into demographics, especially race. I try to avoid politics on the blog unless the post is labeled "traditionalists," and I want to discuss racial demographics even less, unless necessary when talking about tracking. This is why I'm burying this discussion deep in a post in the middle of the holidays rather than dirtying up a Geometry-labeled post with this sort of discussion.

Here is a link to an article referring to Justice Scalia's comments:

The fundamental argument for mismatch theory is that non-academic preferences in admissions to a higher education institution does not properly provide beneficial service to its intended receivers.

Since I want to discuss race as little as possible on the blog, let's go back to the old analogy that I posted back in October, about monkeys, fish, and trees. Let's say that there are two schools in our world -- one teaches how to climb trees, and the other teaches how to swim in a pond. Obviously, the monkey will do well at the tree-climbing school, while the fish will obviously fare much better at the pond-swimming school. To admit the fish to the tree-climbing school would clearly be a mismatch, hence the name of the theory, "mismatch theory."

So the fish will struggle at the tree-climbing school. In the pond-swimming school, of course, the fish will swim much better. It will be much happier when it is actually passing the swimming tests as opposed to failing the climbing tests, and it will be much happier when it is graduating from the swimming school and finding a job where its swimming skills will come in handy.

That is, the fish will be happy 29 out of 30 days of the month. The problem is that the one day of the month when the fish isn't happy is payday, when -- if you remember from October -- it finds out that the monkey is making ten times as much money as it is.

And that's the problem I have with mismatch theory. To me, all sorts of large demographic gaps are acceptable (tracking, college admissions, etc.) as long as they disappear by as soon as we reach the dollar sign on the paycheck. We admitted the fish to the climbing school because we wanted to give the fish a chance for a high-paying job. If we don't admit the fish, then it will complain that we never gave it a chance to make ten times the money.

But this problem has no easy solution. A school system can't change the fact that the job that the monkey is best suited for pays only one-tenth as much as the job the fish is best suited for.

Yet this is the problem with Justice Scalia's comments. He prefers that students who would do better in the college be the ones admitted, even if this leads to huge demographic gaps. Notice that this is ultimately related to tracking in the K-12 system, if members of certain demographics are placed on tracks that don't prepare them well for elite colleges and high-paying jobs. (Notice that Scalia refers to "classes that are too fast for them" -- the speed-based tracking that I'm discussing in today's post.)

To me, since the race gap has no easy solution, I'd focus on fixing the gender gap first. So let's discuss this gap in a little more detail.

When the Disney movie Beauty and the Beast first came out, I remember thinking how Gaston, the villain, tells the female lead Belle that it's not proper for a woman to read. The story after all takes place in the 18th or 19th century, and so Gaston's attitude is typical of his era.

Let's contrast this with my favorite TV show, The Simpsons, and recall how Lisa is the smart one while Bart is, let's say more street smart than book smart. Bart's attitude is the direct opposite of Gaston's -- to Bart, it's not proper for him, a male, to read. And indeed, Bart's attitude is more typical of his era -- in the 20th and 21st century, we are in a situation where in high schools, there are more females than males on tracks that lead to college.

Sure, another trend is that males tend to be better in math and science than in reading. So we see that in the STEM subjects, these two opposing forces cancel each other out -- the California SBAC results bear this out, as there is no significant gap between boys and girls in math. ELA, of course, is a different matter, the two trends reinforce each other, and so girls scored significantly better than boys on the ELA SBAC tests.

So how did we get from the Gaston-Belle trend in the 19th century to Bart-Lisa in the 21st? My theory is that males have always preferred action -- doing things. In the 19th century when most women were housewives, reading was considered doing things. But by the 21st century, as technology and sports have advanced, reading is not considered doing something, and so boys decide that they don't like to read. Notice in both the Gaston-Belle and Bart-Lisa cases, the males are the ones who decide to be the gender that reads or not.

So women outnumber men in colleges, but males may prefer vocational training and working with the hands -- and some of these jobs may pay just as well as those that require college. This is something to consider when considering the only gaps that matters to me -- the wage gaps.

Most of this post criticizes a comment made by Justice Scalia, a Republican-nominated judge. I will keep this post as balanced as possible and criticize a comment made by a Democrat.

Former First Lady and presidential candidate Hillary Clinton recently said that all schools that are below average should be closed down. Of course this is silly, since we expect about half of schools to be below average simply by chance. This is a great time to me to remind the readers that for either the Presidential Birthday or Gubernatorial Christmas plans, teachers are only judged on test scores to the extent that students are -- for example, if test scores count as 50% of a teacher's evaluation, then they should count as 50% of a student's grade.

So it goes without saying that if test scores are to play any role at all in whether a school remains open or shut down, then they should play a proportionate role in a student's grade. Closing down a whole school affects all the teachers at the school, whether effective or not. And so I must strongly disagree with Clinton's statement here.

This concludes my post. I wish the readers of this blog a very Merry Christmas!