Today is the last day of school in my old district. It isn't the last day of school in my new district, where it is only Day 174. But the blog is following the old calendar. Usually, today is when I post a preview of the upcoming school year.
As of today, my employment situation hasn't changed. I will finish my year of subbing this week (with subbing likely both tomorrow and Friday, which are both non-posting days). And right now, as much as I'm hoping to have a full-time teaching position in the fall, things are still not looking good. I might not like it, but I must assume that next year will be yet another year of subbing.
Assuming I'm still a sub next year, my plans are to continue following calendar for my old district, even though most of my subbing calls are in my new district. Next year's school calendar in this district will be mostly the same as this year's. One main difference is that Thanksgiving will be later next year, so that Thanksgiving break will fall between Chapters 6 and 7 next year, rather than midway through Chapter 6. The second difference is that Easter will be earlier next year, so that the four-day holiday weekend will fall midway through Chapter 14 next year, rather than between Chapters 14 and 15.
There is one last difference between this year's calendar and next year's. In 2018, PSAT Wednesday fell between Chapters 3 and 4. This was convenient because Chapter 3 is shorter (with only six sections), and its material on equations of lines (Lessons 3-4 and 3-5) is helpful for the PSAT. But the second Wednesday of October 2019 is exactly Yom Kippur -- a day when some schools are closed (not my old district, but both my new district and the LAUSD). The College Board never gives the test on Yom Kippur. Therefore the PSAT will be later next year, so that the test will fall midway through Chapter 4 next year, rather than between Chapters 3 and 4.
This change means that there is an extra day during Chapter 3 with nothing to do, while one of the Chapter 4 lessons will land on PSAT day. Of course, I could just simply move the start of Chapter 4 a day early, but then the digit pattern would be lost. The new PSAT day will be Day 44, which with our digit pattern is Lesson 4-4, "The First Theorem in Euclid's Elements." In the district whose calendar the blog is following, PSAT day is a minimum day with no regular classes meeting after the test.
The most logical thing to do is to replace Lesson 4-4 with an activity on Euclid's Proposition 1 and give it on Day 40, the day that would have been PSAT day. I could also have moved this activity even earlier and give the Chapter 3 Test on Day 40 (just as Days 20 and 30 are test days) -- but then that would put the Chapter 3 Test on Yom Kippur. What if there are Jewish students in our classes? If the reason for the change in the PSAT date is to avoid Yom Kippur, then we really shouldn't give the Chapter 3 Test on that day either. So the Chapter 3 Test remains on Day 39, while Yom Kippur there is a less important activity that observant Jews can easily skip. (Recall that there is a three-day weekend at this time that is considered to divide the quarters -- so the PSAT will move from the last week of the first quarter to the first week of the second quarter.)
By the way, this is one reason that I wish the Islamic Calendar were lunisolar. The College Board currently accommodates the lunisolar Jewish holiday of Yom Kippur, and I reckon that they would try to avoid Muslim holidays as well if they fell during predictable seasons. By the way, I hope any Muslim readers of this blog are enjoying their Eid al-Fitr feast. But notice that finals week according to the blog calendar is around Eid -- in other words, any Muslim students in this district are forced to take finals during their big celebration this year.
One chapter whose dates won't change next year is Chapter 8. First semester finals will once again fall on Days 83-85. This means that Chapter 8 will once again straddle the semesters -- Lessons 8-1 and 8-2 being covered just before finals, and the new semester starting with Lesson 8-6, the first day after winter break (a Tuesday).
But Lessons 8-7 ("The Pythagorean Theorem") through 8-9 ("The Area of a Circle") are my big activity days. These were my most popular posts of the school year (by hit count). In theory, I should have only one activity day per week, but screw it -- these are my favorite activities, too. There is also a problem with the Chapter 8 Test. Day 90, the day of the Chapter 8, lands on a Monday, a week after the return from winter break. At least one school in the district whose calendar we're following has a minimum day on the first school Monday of January, so the Chapter 8 Test lands on a minimum day.
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Here's my solution for next year -- the first week after winter break will now be considered a week full of activities. This resembles the "Winterm" concept that some schools have at this time. Even Lesson 8-6, "Areas of Trapezoids," can be replaced with an activity. This also might be a good way to squeeze in Lesson 8-5, "Areas of Triangles," which is skipped by finals and the digit pattern -- a single activity day can be used to introduce both formulas. This means that we no longer have a tough area lesson on the first day after winter break.
The following Monday, the Chapter 8 Test will be reduced to a Chapter 8 Quiz instead. This quiz will cover the formulas taught during the previous week of activities, and is short enough to be given on a minimum day. Notice that the Chapter 15 Test was given on Day 160 -- and not only was that a Monday, but it was the first Monday of May, and hence another minimum day in my district! So I will reduce the Chapter 15 Test to a quiz as well. This also means that the students have a shorter quiz instead of a longer test going into the AP exams, which some of our students might be taking.
Therefore there will no longer be any chapter tests on Mondays -- only two quizzes (and the final, for those periods with a Monday final). I warned everyone last year that there would be Monday tests, but it just so happened that both such test Mondays (Days 90 and 160) were minimum days.
Some readers might wonder why I don't simply switch to my new district's calendar. In that district, school starts a week before Labor Day, and winter break doesn't divide the semesters -- instead, there are a full 90 days in the first semester. But I like the idea of the tough area formulas being pushed back to second semester, rather than the end of the first semester. Using my new district's calendar would also mean that my favorite area activities land the week before finals, with the lessons on pi (8-8 and 8-9) being completely skipped for the final (Days 88-90). Thus I believe that my old district's calendar fits the digit pattern and the lessons that I wish to teach.
Once again, I have no plans to teach probability on the blog, even though a true California Common Core Geometry course would contain lessons on probability (as we saw last week). If I really wanted to, I could switch to the modern Third Edition of the U of Chicago text. This text has only 14 chapters, which frees up Days 151-159 for probability. In the new text Chapter 3 has more sections, which makes it harder to give the Chapter 3 Test before Day 40 -- but then again, the PSAT won't be until Day 45, so this might work.
Come to think of it, maybe this might be a good year for me to switch to the Third Edition because of the change in the PSAT date. But do I really want to change nearly all of my previously posted lessons to fit the new text?
OK, that's enough about next year's U of Chicago Geometry course on the blog. I want to use the rest of today's post to revisit some of Hofstadter's Godel, Escher, Bach that we glossed over. After all, I did promise you that I'd go back to those chapters. My plan is to cover one Chapter today and another in my next post -- my first summer post. That will probably be all -- I don't want to drag our spring side-along reading book deep into the summer.
(By the way, I write the mathematician's name as Godel here. But technically, there should be an auslaut -- sorry, I'm meant an umlaut -- over the "o" in Godel. Some people spell the name in ASCII as "Goedel," where "oe" is considered equivalent to o-umlaut.)
For today I choose to cover Chapter 12, since I wrote very little about it in my April 25th post. We don't need to revisit Dialogue 12, since that's just a translation of Lewis Carroll's "Jabberwocky" into three languages, English, French, and German. (I do point out that some of the German words contain umlauts -- for example, the English nonsense word "raths" is translated as Rath' with a-umlaut.) So instead, we'll launch directly into the Chapter.
Chapter 12 of Douglas Hofstadter's Godel, Escher, Bach is called "Minds and Thoughts." Here's how it begins:
"Now that we have hypothesized the existence of very high-level active subsystems of the brain (symbols), we may return to the matter of a possible isomorphism, or partial isomorphism, between two brains."
So this Chapter is all about comparing different people's brains. The author is asking, can minds be mapped onto each other? He points out that even twins' brains aren't identical or isomorphic, but a partial isomorphism might be possible:
"It would seem an obvious conclusion that there is some sort of partial software isomorphism connecting the brains of people whose style of thinking is similar -- in particular, a correspondence of (1) the repertoire of symbols, and (2) the triggering patterns of symbols."
The author shows us a picture of a tiny portion of his "semantic network." Just like the networks of Lesson 1-4 of the U of Chicago text, it contains nodes ("symbols" or "vertices") and arcs (the connections between thoughts). Our goal is to compare different semantic networks, including local and global properties:
"Local properties require only a very nearsighted observer -- for example an observer who can only see one vertex at a time; and global properties require only a sweeping vision, without attention to detail."
Hofstadter now writes about how to translate Carroll's "Jabberwocky." He mentions that the other two languages don't always preserve the past tense of the English poem:
"Who can say whether remaining faithful to the English tense would have been better? In the German version, the droll phrase 'er an-zu-denken-fing' occurs; it does not correspond to any English original."
At this point, the author introduces something he calls "ASU's," or "Alternative Structures of the Union," which involve each person drawing his or her version of the entire USA:
"This arduous task will take months. To make things easier, you have a cartographer on hand to print everything in neatly."
Each person's own ASU corresponds to the thoughts in his or her brain. For example, my personal ASU will have Southern California look the same as that part of the real USA, since this is where I live and am familiar with. Small parts of areas that I've traveled to (including driving trips to Northern California, Kansas City, and Florida, and a flight to Maryland) will also look the same. The major cities that I've heard of will exist in my ASU. But rural areas that I've never heard of will be completely different in my ASU, as are the roads leading to those places.
"So the local-global distinction is not relevant here. What is relevant is the centrality of the city, in terms of economics, communication, transportation, etc."
Once again, Hofstadter emphasizes that the major, well-known features of ASU's are the same:
"The fact that all ASU's have some things in common, such as the East Coast, the West Coast, the Mississippi River, the Great Lakes, the Rockies, and many major cities and roads is analogous to the fact that we are all forced, by external realities, to construct certain class symbols and triggering paths in the same way."
The author asks, how much do language and culture channel thought? Many of use think the same way because we speak the same language. (He doesn't mention this, but the relationship between language and thought is a major theme of George Orwell's 1984.)
"A non-native speaker will have picked up words from dictionaries, novels, or classes -- words which at some time may have been prevalent or preferable, but which are now far down in frequency -- for example, 'fetch' instead of 'get,' 'quite' instead of 'very,' etc."
Hofstadter now considers trips and itineraries in ASU's. The roads in ASU's correspond to the various connections between thoughts:
"If one is to continue to use the ASU-metaphor, then, it is important to remember that the cities represent not only the elementary symbols, such as those for 'grass,' 'house,' and 'car,' but also symbols which get created as a result of the chunking ability of the brain -- symbols for such sophisticated concepts as 'crab canon,' 'palindrome,' or 'ASU.'"
The author tells us that some pathways are possible, some are merely potential, and others are completely preposterous. But even preposterous pathways -- indirect routes -- can be divided into direct stretches:
"On reflection, this is hardly surprising, since it is quite reasonable that we should only be able to imagine fictitious things that are somehow grounded in the realities we have experienced, no matter how wildly they deviate from them."
Hofstadter now discusses different styles of translating novels. His main example is the translation of Dostoevsky's novel Crime and Punishment into English, where he considers how to translate a certain street name:
"Now we could be frank with the reader (who, it may be assumed, probably won't have the slightest idea whether the street is real or fictitious anyway!) and give him the advantage of our modern scholarship, writing 'Stoliarny Lane' (or 'Place')."
Now the author moves on to coding and high-level comparisons between programs. We want to know whether two different programs carry out the same task, but:
"Perhaps one programmer wrote in a machine language, the other in a compiler language. Are two such programs comparable?"
Here Hofstadter's solution would be to look at the semantic network of the computer's "brain" -- a memory dump:
"In the end, the programmer would understand the goals of the program and could describe it in high-level terms -- for example, 'This program translates novels from Russian to English,' or 'This program composes an eight-voice fugue based on any theme which is fed in.' Now our question must be investigated in the case of brains."
Hey, it could also be a four-voice fugue, just like the Bach Google Doodle from back in March. So anyway, let's follow the author's return to brains, potential beliefs, and potential symbols. He suggests that we might try counting the number of beliefs in a brain:
"If that is the kind of goal we will be striving for in a chunked description, then it is easy to see what kinds of troubles we will run up against."
Hofstadter now returns to his own character, the Crab, from his Prelude (the opening Dialogue of Part 2 of the book). The Crab has different reactions to playing different pieces of music.
"Other times, he will be quite excited by it, but this reaction requires the right kind of triggering from the outside -- for instance, the presence of an enthusiastic listener, to whom the work is new."
But the author's book is all about the self and self-reference. So he asks, where is the sense of self?
"For we would then be compelled to look for an explanation of the mechanism which does the perceiving of all the active symbols, if it is not covered by what we have described so far."
Hofstadter explains that the self is also described as a symbol in the brain. But it is so complex that he refers to "I" or "the self" as a subsystem, rather than a simple symbol, and describes what happens when two complex subsystems interact with each other:
"Then they both attempt to communicate with a third subsystem of my brain -- my self-symbol -- and it is at that point that the 'I' inside my brain gets wind of what's going on; in other words, it starts picking up a chunked description of the activities of those two subsystems. Typical subsystems might be those that represent the people we know intimately."
To the author, subsystems in different people consist of shared code:
"There would be tricky repercussions connected with representations in him of representations in me of representations in him of one thing or another."
Hofstadter tells us that the self-symbol plays the role of consciousness, or the "soul":
"This means that it has to have symbols for mental activity -- in other words, symbols for symbols, and symbols for the actions of symbols. Of course, this does not elevate consciousness or awareness to any 'magical,' nonphysical level."
At this point the author quotes J.R. Lucas, an Oxford philosopher. Hofstadter points out that the views of Lucas are opposite his own. Here is part of the Lucas article "Minds, Machines, and Godel":
Lucas:
"At one's first and simplest attempts to philosophize, one becomes entangled in questions of whether when one knows something one knows that one knows it, and what, when one is thinking of oneself, is being thought about, and what is doing the thinking."
...
"In saying that a conscious being knows something, we are saying not only that he knows it, but that he knows that he knows it, and that he knows that he knows that he knows it, and so on, as long as we care to pose the question: there is, we recognize, an infinity here, but it is not an infinite regress in the bad sense, for it is the questions that peter out, as being pointless, rather than the answers."
...
"If the mechanist produces a machine which is so complicated that this ceases to hold good of it, then it is no longer a machine for the purposes of our discussion, no matter how it was constructed."
...
"In fact we should say briefly that any system which was not floored by the Godel question was eo ipso not a Turing machine, i.e., not a machine within the meaning of the act."
Hofstadter concludes the chapter as follows:
"In the following Chapters, we shall come back to many of the topics touched on so tantalizingly and fleetingly in this odd passage."
But of course, we've already seen much from those Chapters.
Thus concludes this last school year post. The first summer blog entry will be coming up soon, and I'll expand upon one last Hofstadter chapter in that post.
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