Introduction
Today is New Year's Eve, so this marks my final post of 2022. It's only my eighth Yule Blog post though -- but with eight posts in two weeks, I remain on pace to finish the twelfth post in my third week of break.
When 2022 began, I made the following prediction (on the Stats blog):
But the pandemic didn't end in 2021, and now many consider 2021 to be a Murphy's year too. Case in point, when legendary actress Betty White died yesterday, many people tweeted that it was just one last bad thing that had to happen in 2021.
That being said, why should we believe 2022, the third year (n + 3) of the pandemic, to be any better than the first two years? After all, the year begins with omicron raging more strongly than ever. As far as I know, some celebrity will die on December 31st, 2022, and the response will be that it's one last bad thing to happen in 2022 -- a year to be just as horrible as its two predecessors.
Well, I suppose that a celebrity did die today -- Pope Emeritus Benedict XVI. And yesterday a celebrity died -- journalist Barbara Walters. And the day before that a celebrity died -- soccer legend Pele. Here's an article about all three passings:
https://www.who2.com/last-minute-deaths-of-2022-pele-a-pope-and-barbara-walters/
So it appears that 2022 was a bit like what I predicted -- a Murphy's year (though perhaps not quite as Murphy-ish as 2020 or 2021) that ends with significant figures passing away.
Anyway, speaking of looking back to 2022, the year we're about to complete...
Yule Blog Prompt #12: What I Learned in 2022...
There is one lesson this year where I can truly say that I learned as much as my students. It was October 24th, the day of the "slope walk" in my Math I class. Because it was on a Monday (a non-posting day), I only briefly mentioned it in my post that Tuesday:
By the way, yesterday's Math I lesson was interesting. The TOSA set up a calculator-based laboratory in the library, and invited the Math I classes to go on a "slope walk" -- it was my turn yesterday. The students must walk according to a given linear equation -- so for y = 2x + 1, for example, they must begin at the 1-foot mark and walk 2 feet per second. A motion sensor captures the movement and sends the data to the calculator, and the kids get to evaluate each other on their accuracy.
I'm a bit surprised that with so many MTBoS posts about the use of technology in the classroom, I don't see much mention of calculator-based laboratories on the MTBoS.
Anyway, the activity worked as described -- the calculator and motion detector were placed on a desk, and several yardsticks were placed on the floor (with the labels changed so that 1, 2, 3, 4, 5, and so on, feet were marked). As the student walked along the yardsticks, the TI-84 would keep track of the position (in data list L2), velocity (L3), and acceleration (L4) of the student at each unit of time (stored in list L1).
The teacher on special assignment (TOSA) completely set up the calculator and activity for my second Math I class that day, helped me during fourth, then left me on my own for sixth. So as the day went on, I learned more about how to set up the lesson. In second period, he began by calling on one student volunteer to perform the slope walk, and then called other students to guess the equation that the first volunteer had walked.
And so in fourth period, I tried to begin my class the same way. I used my own TI calculator (not the one set up for the slope walk, but the one with my random name generator) to choose the student to make the first slope walk, and then to choose the kids to guess the equation. But unfortunately, the first few kids refused to take a guess, which got me upset that they weren't participating -- likely either talking to each other or playing on phones, not paying attention. And so I began to argue with them.
I admit that the argument was a continuation of a dispute that had happened earlier in that class, but I don't quite remember what it was. It was either a group of students who left their Warm-Ups blank, or who had been caught talking on the previous Friday's quiz and refused to write down their standards as a consequence of cheating.
At any rate the TOSA called me out on arguing here. He pointed out that once a guess has been made, the other students are more likely to participate and correct the first student's guess once they see the guess and the data graphed on the TI calculator. So the best idea is just to get a first guess up there -- perhaps calling on a stronger (not random) student to take the first guess, or even have the students discuss it with each other and graph the first equation I overhear. Indeed, it's better to have a weaker student be the one to make the first walk, not the first guess -- a weaker student might not know how to come up with linear equations, but definitely knows how to walk at a more or less constant speed. This is a situation where choosing students at random isn't the best idea.
And this was the more valuable lesson for me to learn that day -- how to avoid arguments in the class and get students to participate, not just how to set up the TI calculator. Indeed, from that moment on, I avoided most arguments in fourth period, which quickly became my best behaved Math I class. I'm glad that the TOSA was the one who criticized me for arguing, rather than the principal -- who would go on to observe that period a month and a half later for my evaluation.
Instead, most arguments involving Math I occur in either second or sixth periods. I still need to find a good way to get those students to follow my instructions without resorting to arguments.
Quinters on the 10-Day Calendar
Let's get back to Calendar Reform and quinters. We will move on to a calendar with ten-day weeks and figure out how to divide the 180-day school year into five 36-day quinters.
First, we need a ten-day calendar. A simple one I found is the 6 * 6 * 10 calendar:
https://calendars.fandom.com/wiki/6*6*10_regular_calendar
It's just like the Sexagesimal Calendar, except with six ten-day weeks rather than ten six-day weeks. As the author doesn't state the names of the days of the week, we might call them Oneday to Tenday, or perhaps Zeroday to Nineday (since the counting in this calendar starts with zero).
Suppose we let Zeroday to Fiveday be the school week, and Sixday to Nineday be the weekend. Then each month would contain six weeks of six days to give us 36-day quinters -- but then there's no week left over at the end of the year for a vacation week. That's the cost of having four-day weekends.
Also, there's no relation stated here between months and seasons. We might assume that the first month (Zeromonth?) starts on the winter solstice (like the Sexagesimal Calendar). Then Threemonth starts on the summer solstice and counts as summer break -- the lone long break of the year. The other five months are the school quinters.
Here's a link to another ten-day week -- the Metric Week:
https://zapatopi.net/metrictime/week.html
This site proposes dividing two straight years into 73 weeks, with one week split between the years -- even years start on Zeroday and odd years on Fiveday. The only blank day needed is Leap Day. The site also proposes having either seven weekdays (Zeroday-Sixday) or five weekdays, but neither five nor seven divides the 36-day quinter evenly. So I might prefer having six weekdays (like 6 * 6 * 10 above).
But the site says nothing about months (or even how to label the weeks if there are no months). As far as we know, the author intends to use Metric Weeks with the Gregorian year.
There are a few ways to expand this into a full calendar. Since the ten-day week is called a "Metric Week," the natural idea is to have ten weeks per month, or 100 days each. Then there are three long months and one short month of 65 (or 66) days. If we align the short month with the summer vacation, then the resulting school year is functionally equivalent to the 6 * 6 * 10 calendar (except that the long months correspond to trimesters, not quinters).
It's also possible to use 10 as the number of months per year, not the weeks per month. But each month would need to have 36 or 37 days, which doesn't align with the 10-day week easily.
What sorts of months align with the weeks then? We could have either nine 40-day months or twelve 30-day months. Nine 40-day months align with the Modern Calendar and appear in the 10-10 Calendar:
https://calendars.fandom.com/wiki/10%E2%80%9310_calendars
But as we've seen, quinters don't work very well with nine 40-day months.
Twelve 30-day months appear in the Tenstrong Calendar -- and it's noted that this idea is an old one, dating back to the French Revolution:
https://calendars.fandom.com/wiki/Tenstrong_Calendar
But each 30-day month would essentially be half of a 60-day month in the 6 * 6 * 10 calendar, and so the resulting school calendar once again would be similar.
Quinters on the 11-Day Calendar
Of course, the ten-day week isn't our target week length. My ultimate goal is to divide my original calendar, the Eleven Calendar, into quinters.
So far, we've seen that as we moved from nine- to ten-day weeks, the fourth weekend day forces us to eliminate all breaks except summer break. An eleventh day -- making six school days and five weekend days per week -- would cut into summer break even more.
Indeed, we see a very easy way to divide the year into quinters. Each quinter would contain two months with six weeks each. That fills ten of the eleven months, so the eleventh month is summer break. So this would be just like 6 * 6 * 10, except the summer break is now an even smaller fraction of the year -- one-eleventh, rather than one-sixth.
In past posts, I divided the three-week summer break into three one-week breaks, and then spaced those weeks equally throughout the year. This would give us three short breaks instead of a long one -- and trimesters rather than quinters.
What we'd really like to do is keep the quinters, but have shorter weekends so that the 36 days within each quinter take less time to complete, leaving us with a break at the end of each quinter. We can still combine this with the eleventh month to provide us with a summer break.
Hmm -- the Eleven Calendar contains 11 months of 33 days each. Hey, that sounds familiar -- yes, it reminds of the 352/384 Calendar, which has 11 months of 32 days each. So perhaps we should just modify that calendar -- add a 33rd day of to each month, and voila! So the resulting calendar would have eight-day weeks instead of eleven (with the 33rd as a blank day).
And there's a way to keep 11 days in this calendar as well. Instead of having holidays ten days apart (the so-called "new," "half," and "full moon" days that have nothing to do with the actual lunar phases), we have holidays that are eleven days apart. Indeed, this was one idea I had with the original Eleven Calendar -- the first three days of the week in the calendar are "Muslim Sabbath," "Jewish Sabbath," and "Christian Sabbath."
Suppose we had a calendar with holidays on the Christian Sabbaths -- the 3rd, 14th, and 25th. We can then divide the 33 days of each month into weeks -- say Threeday to Eightday, with Oneday and Twoday as the weekend. In this case, the school week would be six days, except that if one of those days is a Sabbath, then the week is reduced to five days.
So the first school week would go from the 3rd to the 8th. The 3rd is a Sabbath, so the first school week contains five days, from the 4th to the 8th. The second week, from the 11th to the 16th, has a midweek Sabbath on the 14th. The third week goes from the 19th to the 24th. While the 25th is the Sabbath, we can observe this Sabbath on the 24th, so we again have five days. The only week with a full six-day week is the last week, from the 25th to the 32nd. It's the longest week, but we make up for it by adding the blank day on the 33rd to the weekend.
The the next three weeks are the 4th-8th, 11th-16th (sans the 14th), and the 19th-23rd days in the second month. This gives us seven weeks with five days each except for one six-day week, so it's a grand total of 36 days in the quinter. The holiday at the end of the quinter runs from Sabbath to Sabbath, from the 25th to the 3rd of the next month.
We do end up with a working Quinter Calendar, though it's a bit complex. The school days are based on an eight-day week, while the Sabbaths follow an eleven-day week. It's been pointed out that any calendar that breaks the seven-day week is troublesome because followers of the Abrahamic faiths will continue to follow the seven-day Sabbath cycle. Thus if we're going to overlay two different week lengths anyway, one of them should be the seven-day cycle. In fact, as we'll soon see, it's the 11-day week that's dead weight here -- overlapping the seven- and eight-day weeks makes a workable calendar.
Suppose that we have overlapping seven- and eight-day cycle. The seven-day cycle contains the standard day names, and suppose that you're a Christian who want to take Sundays off. The eight-day cycle contains eight days, with six consecutive days as school days.
Each quinter, as we've seen above, contains seven school weeks. It then follows that exactly one of these seven school weeks starts on Monday, exactly one on Tuesday, one on Wednesday, and so on though we don't know which day, say, the first week begins (and it will vary from quinter to quinter).
The six-day school week that begins on Monday goes Monday-Saturday -- it contains no Sabbath, and so all six days are school days. All the other weeks contain a Sabbath, and so each of those weeks contain five school days. So once again we have seven school weeks -- one with six days, the rest with five -- again giving us 36 school days in the quinter.
You might ask, wouldn't that blank day on the 33rd mess things up? Yes it will -- so instead of having every month be 33 days, let's alternate between 32 and 34 days. The first month of 34 days is the summer month, and then each quinter consists of a 32- and a 34-day month, with the only blank (with respect to the eight-day week) days on the 33rd and 34th of the long months.
And by going 34-32-34-...-32-34 days, the entire year contains not 363 days, but 364 days -- which is a multiple of seven, so it lines up with the seven-day weeks. And since 364 days is too short for the shorter year, we must eventually add -- you guessed it -- a Leap Week. which can be made to line up with the Leap Week in any other type of Calendar (Usher, Pax, etc.).
Of course, this calendar isn't without its problems. One week in each quinter begins on a Saturday, so we'd have a lone school day before the Sabbath (and the temptation to take that day off too). And a week that starts on Wednesday would end with a lone day on Monday -- another day that many students will try to take off (so you better hope that's not finals week).
And by this point we've long since diverged from the Eleven Calendar as I originally planned it. The only thing eleven-ish about this calendar is that there are eleven months. The number of days per week is no longer eleven, and the number of days per month is no longer a multiple of eleven.
Finally, the one thing about the Fixed Festivity Week Calendar (and any "Fixed Festivity" calendar listed on the Calendar Wiki) is just that -- the "festivities" (holidays) are fixed. The original purpose of the calendar was to avoid having fixed holidays -- days when airlines, amusement parks, and other businesses raise their prices.
This is why I originally wanted a hybrid week -- weeks when different shifts of students are attending school at different times. But during the pandemic, hybrid schedules became a reality -- and as we found out, the hybrid schedule turned into chaos. So I no longer wish to have hybrid on my calendar.
But still, we might have different school schedules implemented at the district level. Then it will be harder for businesses to jack up their prices since many kids are off at different times. The only requirement will be that there should be one month off for summer, the other ten months divided into five quinters, and 36 days of school in each quinter.
We'll keep the 3rd, 14th, and 25th as off days -- but let's just call them Holidays instead of Sabbaths, in order to avoid the temptation of overriding these with a seven-day Sabbath cycle. Some of these Holidays might correspond to Christian (or other religious) holidays, so many people might wish to go to church on this Holidays anyway.
On my Eleven Calendar, the months are labeled March-January. My March starts at the same time as on the Gregorian, so the last 2-3 days of Gregorian February (including Leap Day) are blank days. March is mostly aligned in the two calendars (except the last two days of my March), but each of my months start progressively later in the corresponding Gregorian month. My January only lines up with the last week of Gregorian January, plus the first 26 days of February (before the blank days).
So now we need 33 holidays, three in each of the eleven months. I was considering using many of the holidays from the Fixed Festivity Day (not Week) Calendar:
https://calendars.fandom.com/wiki/Fixed_Festivity_Calendar
There are already 29 holidays here, so we need just four more holidays. One reason for taking the Day version rather than the Week is that the FFW version combines several holidays into one week. For example, the spring holiday week goes Shrove Monday, Fat Tuesday, Ash Wednesday, Maundy Thursday, Good Friday, Holy Saturday, Easter Sunday (so holidays from the start and end of Lent are combined into a single week). The FF version combines the early Lent days into a single February holiday and the late Lent days into a single March holiday -- in other words, it takes into account that the holidays occur in different weeks, not a single week like the FFW version.
The FF holidays come from differing traditions -- for example, "Independence" is in July to celebrate the US Fourth of July, but "Thanksgiving" is in October to match the Canadian holiday, and "Labor" is in May to match International Labor Day. Instead of US Labor Day, "Children/School" is at the start of September to mark the start of the school year (like Liber on the Sexagesimal Calendar).
So let's create our holiday schedule. One thing notable about many of the US federal holidays is that they fall slightly more than a month part -- consider Labor-Columbus-Veterans Day in the fall, as well as MLK-Presidents in the winter, and Memorial-Independence in the summer. In fact, these holidays are close to be 1/11 of a year apart (instead of 1/12), so they fit on an Eleven Calendar.
So let's place Memorial Day on May 25th in our calendar. The Independence Day works out to be June 25th (which corresponds to July 2nd, aka Adams Day, in our calendar). And the other holidays also fall on the 25th of their months, although since my months start later, the holidays are in the named months preceding their Gregorian months, so Labor Day is August 25th, Columbus Day is September 25th, Vets Day is October 25th, MLK Day is December 25th, and Prez Day is January 25th.
I've already named a date for Christmas -- December 3rd, which matches Gregorian December 25th and is hence the right day for Christmas. July is the summer month, and so August become the first month of the school year. Yes, this is before Labor Day, but it's later than the start of Gregorian August. And recall that marker of the school start is Children/School Day, not Labor Day, so we can place that holiday on August 3rd.
Of course, some people might object that Independence Day ought to be the Fourth of July -- and perhaps rearrange the calendar so that holidays occur on the 4th, 15th, and 26th of their respective months, and even align July (rather than March) to the Gregorian year, so that the Fourth of July really is the Fourth of July (thus making June the skipped month rather than February). This plan has the advantage of placing all the above-named federal holidays in their correct months (from Labor Day on September 4th, up to Memorial Day on May 4th), but throws off Children/School Day and Christmas.
I will keep my original plan with March aligned and holidays on the 3rd, 14th, and 25th. Then I'll use the holidays from Fixed Festivity to fill the rest of the calendar, starting from our new Christmas holiday on December 3rd.
Here is the calendar that I came up with. Three holidays per month (on the 3rd, 14th, 25th) are listed, most coming from the Fixed Festivity (FF) Calendar, with commentary:
December: Christmas; Epiphany/Baptism; MLK (New Year's has been moved, since my New Year is in March. Christmas to Epiphany form the Twelve Days of Christmas using inclusive counting, where both Sabbath holidays are included.)
January: Candlemas/Equality; Valentine/Groundhog; Presidents (FF is illogical here -- "Candlemas" and "Groundhog" ought to be the same, as both are Gregorian February 2nd. The middle holiday should be Candlemas/Groundhog -- this is 40 days after Christmas using exclusive counting, where all Sabbath holidays are excluded. Shrove/Lent goes with Presidents' Day, just like Usher Calendar.)
March: New Year; Spring/Lady/Annunciation; Good/Holy/Maundy (Actually I prefer Palm here, since it's a week before Easter. Lady/Annunciation ought to be March 25th in both calendars, but FF places it early in order to avoid Holy Week. Cesar Chavez Day fits here as well.)
April: Easter; Father/Men; Remembrance (No quinter break lines up with Easter. FF lists Labor Day here, but we use US Labor Day instead. And some nations use "Remembrance" to refer to US Veterans Day here instead. The newest LAUSD holiday fits here --- Armenian Genocide Remembrance Day.)
May: Mother/Women; Ascension/Parents; Memorial (Ascension is 40 days after our Easter, excluding Sabbath holidays. This is also Month 3, Day 14, so it can serve as a new Pi Day, although it's now a Sabbath holiday rather than a school day. Pentecost goes with Memorial Day, just like Usher Calendar.)
June: Corpus Christi/Family; Midsummer; Independence. (Juneteenth fits with Midsummer, just like Gregorian Calendar, and places it on a June fourteenth.)
July: World/Unity; Assumption/Transfiguration; Freedom/Liberty. (This is summer break.)
August: Children/School; Ruler/Region; Labor. (This is the first month of school.)
September: Peace/Rest; Thanksgiving; Columbus. (Since "Thanksgiving" here means Canadian, it ought to be the same as Columbus.)
October: Community/Civic; Saints/Hallows/Reformation; Veterans. (The Reformation was on October 31st, which is why it's placed with Hallows here.)
November: Carnival/Joy; Dead/Souls; Advent/Nicholas. (FF actually places Mexican Day of the Dead with US Veterans Day -- also a day of the dead, as in war dead.)
Links to Other Yule Bloggers and Juneteenth
Shelli, the leader of the Yule Blog Challenge, made her twelfth Yule Blog post yesterday. It's part of her series "My Favorite Friday":
http://statteacher.blogspot.com/2022/12/myfavfriday-favorites-of-2022.html
So Shelli is the clear winner of the Yule Blog Challenge as the first to a dozen posts. Moreover, she wasn't satisfied with twelve, and so she makes it a baker's dozen today, with a post about preparing for the new year:
http://statteacher.blogspot.com/2022/12/preparing-for-2023.html
In this post, Shelli writes:
I report back to school on Monday for meetings and collaboration time, then kids come on Tuesday. Like most teachers over the break, I've been thinking about what to do on the first day of the semester. I like this time as a reset and definitely plan to do some non-curricular tasks and to revisit our classroom norms.
In other words, Shelli must work on Monday, even though it's the legal federal holiday (namely New Year's Observed). But at least the students don't have to attend on the federal holiday.
But this makes it seem OK to have a Quinter Calendar (for the usual Gregorian Calendar) with the last teacher day on June 18th, even if this is Juneteenth Observed. Having teachers works on observed federal holidays (January 2nd, June 18th) isn't equivalent to having them work on the actual holidays (January 1st, June 19th). I doubt Shelli's school will make her report on Monday, January 1st, 2024, since that is the actual New Year's Day.
This reminds me of the UCLA calendar -- I'm surprised I haven't checked out to see how my alma mater is observing the new federal holiday:
- In 2021, the new holiday was just barely announced a few days in advance. So UCLA observed it a week late -- on Monday, June 28th.
- In 2022, the holiday was observed on Monday, June 20th -- the observed federal holiday. It did mean that the first day of summer school was a Tuesday rather than the usual Monday.
- In 2023, a Leap Week was added to the calendar. So Juneteenth, June 19th, falls during the week off between spring quarter and summer school, rather than the first week of summer school.
From 2024-2027 this pattern continues -- Chavez Day falls during spring break (the week between winter and spring quarters), and Juneteenth during summer break (between spring quarter and summer).
But then in the posted calendar for 2028, three strange things happen:
- Winter Quarter begins on Tuesday, January 4th, instead of its usual Monday. (I'm not sure why January 3rd is to be avoided, especially considering that the quarter began January 3rd, 2022.)
- Spring break is the week before Chavez Day, rather than the week of Chavez Day.
- Summer break is the week before Juneteenth, rather than the week of Juneteenth.
The resulting calendar still isn't perfect though. In 2033, Chavez Day will fall on its earliest possible date, March 25th and Juneteenth on its latest possible date, June 20th. This is the opposite problem from the K-12 Quinter Calendar concerning early Juneteenths. But we have until 2033 to figure that one out (for example, Chavez Day could be observed on April 1st -- the day that it will be observed in LAUSD that year -- instead of March 25th). The students who will attend UCLA in 2033 are mostly in elementary school now.
Rapoport Question of the Day
Today on her Mathematics Calendar 2022, Rebecca Rapoport writes:
How many papers did Ron Graham and Paul Erdos write together (additional co-authors allowed)?
This doesn't even count as a research problem -- a Google search tells me about these two famous mathematicians (both of whom have been mentioned on this blog before), but no website anywhere tells me how many papers they co-authored. Both are prolific authors so it doesn't surprise me that they co-authored many papers, but no source reveals that they've written exactly 31 papers together -- which must be the answer, since today's date is the 31st.
Here's an easier question -- what is Ron Graham's Erdos number? Well, since they've co-authored at least one paper, Graham's Erdos number must be 1.
Conclusion
In my next post, I'll return to my COVID What If? stories. But here's one thing I do want to say about my past as a student, and an unexpected connection to the present day.
A month ago, I got in touch with one of my old friends from the district we attended for K-8. I first met him in the first grade, and the last class we had in common was eighth grade History.
The reason we got in touch is, as it turns out, he has a son who's now a freshman at the very school where I currently work! That's how he found me -- he discovered my LinkedIn page, saw where I'm now working, and contacted me to tell me about him.
My friend's son isn't in any of my Math I classes, though. I actually spoke to his Math I teacher recently since we have the same conference period -- she informed me that my friend's son is one of the top students in her class. (I reckon she taught the slope walk lesson to him correctly.) And I saw the young son myself a couple of times since this discussion -- he plans on joining the Track team in the spring.
Anyway, I wish everyone a happy new year, and for a less Murphy-ish 2023 than we had in 2022.
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