Phases of the Moon

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In my previous post, I made the error of assuming that most readers would know how the phases of the Moon work, which is by no means an understanding that can be taken for granted.  I wonder why this bit of information is presented to kids when they’re so young.  Understanding the geometry of the Sun, the Earth and the Moon, and the kinematics—which means the geometry in motion— is not easy for a young person who is still struggling with the thought that the Earth is not flat.

Anyone who understands the phases of the Moon perfectly well may feel free to skip this post; it is entirely educational, and doesn’t have all the opinions that people are accustomed to seeing here.

First of all, the phases of the moon are not things that really happen to the Moon in quite the same way as one might think.  Let’s try to visualize the phases of the Earth first.  We know perfectly well that half the Earth is lit up at any given time of day.  If we were flying around the Earth in a rocket or Space Shuttle, depending on where we’re looking from, we would see the Earth completely lit up by the Sun, partially lit up, or we might be looking at the night side, and the Earth would not be lit up at all.

To make things even clearer, suppose we’re in a room with a bright floodlight at the far end of the room, and someone is walking about the room holding a tennis ball.  At any time, exactly half the tennis ball will be illuminated.  Suppose we call the line that separates the lighted side from the dark side the Night/Day line, if we make the person holding the tennis ball to just stand still for a minute, and stand right opposite the Night/Day line (which is actually a circle that goes round the ball), we should see something that looks like a half-moon.  If we move by 45 degrees (centered at the tennis-ball, of course) away from the lamp, we would see a ‘quarter-moon;’ if we move 45 degrees towards the lamp, we would see a three-quarter moon.  If we look at the tennis ball from the side of the lamp (being careful not to let our shadow interfere with the scene), we would see a full moon, and so on.

Now, if we were to observe the Earth, the Moon and the Sun from outer space, ideally from up, way above the North Pole, what we would notice is that the relative positions of the Earth, the Moon and the Sun do not change very much.  In fact, you could watch for a couple of days, and not notice much motion.  But you will notice that the Earth is spinning.  That is by far the fastest thing that is happenning.  It will appear to go around one complete rotation in roughly 24 hours, not surprisingly.  This means that the point of view of the Moon from anyone on the Earth doesn’t change much for a day or two.  If we could keep running round the Earth to keep the Moon always in sight, it will appear to have the same phase for a day or two.  Actually, if we wait until the Moon is directly overhead, and then start running, it will be the same time of day wherever we are, because the Sun will also appear to be motionless.  (This happened to me on New Year’s Eve of 2000; I took a plane out from London, and when I arrived in New York, it was still New Year’s Eve.)

Let’s represent the Sun, the Earth and the Moon by points, and call them S, E, and M respectively.  Over a week, the angle SEM will change about 90 degrees, and in a little less than a month, the angle will be back where it was.  This period of time is called a Lunar Month for this very reason.  Whatever the phase of the moon was at any given moment, the phase is exactly the same one Lunar Month later.

If you’re comfortable with the idea that the phase of the Moon depends on only the angle SEM, which slowly goes from 0 to 360 degrees over a Lunar Month—which is about 28 days—we can now connect the dots.

If the Moon is exactly overhead at 6:00 PM, it has to be a half moon, because it is being illuminated squarely from one side.

If the Moon is exactly overhead at 12:00 Midnight, it has to be a full moon, because it will be illuminated from ‘behind our heads,’ the Sun being on the opposite side of the Earth from us (and the Moon), and the Sun is “looking” at the Moon in exactly the same direction as we are.  (This is why all eclipses of the Moon take place when there is a full moon.)

If the Moon is exactly overhead at, er, let’s see: at 9:00 PM, it will be a 3/4 -moon.

If the Moon is exactly overhead at Noon, it will be a new moon, which is a moon which is all black, or rather, in shadow.

If the Moon is exactly overhead at 3:00 AM, it is essentially the same as if it were overhead at 9:00 PM; if the Moon is exactly overhead at 3:00 PM, it is essentially the same as if it were exactly overhead at 9:00 AM: a quarter-moon.  The two phases are called First Quarter or Last Quarter, I’m not sure which is which.

The figures I’m providing are not as clear as I had planned, mostly because I’m squeezing in so much detail into each diagram, to save space.  (Ha ha; there is so much of it, it seems unnecessary...)

In the first diagram, the angle between the line to the Moon and the line to the Sun is about 90 degrees,

In the second diagram, the angle between the line to the Moon and the line to the Sun is about 135 degrees, which is about 3/4 of one hundred and eighty degrees.

In the last diagram, the angle is 180 degrees, which is when you get a Full Moon.  When the Sun is behind the Moon, you get a New Moon, which is a dark moon, illuminated from behind.  (We still see it, lit by Earthlight, light reflected out to the Moon from us, the Earth.  Astronauts say that Earthlight is very beautiful, and of a bluish cast.)

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A strange progression, from understanding to puzzlement!

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Here is an interesting phenomenon I have observed!  It takes place over several generations, so you have to take it for what it’s worth.

A teacher explains a certain phenomenon to her students.  She understands it perfectly, at least to whatever extent it is possible to understand a thing.  For the sake of clarity, suppose it is about the phases of the Moon.  “So, you see,” she says, “if the Moon happens to be exactly overhead at 6:00 PM, when you can assume the Sun is just setting, the line from the Moon to the Earth will be at 90 degrees to the line from the Sun to the Earth!”

The majority of the students, of course, will see where this is going, and while the Sun may not be just setting, in principle the Moon-Earth-Sun angle is going to be roughly a right angle, as close as makes no difference.  Now, since the line from the Sun to the Moon, and from the Sun to the Earth are nearly parallel (that’s probably a tough sell, too; some of the slower kids will get hung up on the fact that if the Sun lies on both lines, how can they be parallel?  They cannot understand that they’re never going to experience more parallel lines than these two, since the Sun is so far away.  (Oddly enough, kids will not have any problem regarding two sides of a street as being parallel, though in theory they’re actually little bits of enormous circles, and as such the term ‘parallel’ does not even apply to them.)

Anyway, the point is, that if it so happens that the Moon is dead overhead at 6:00 PM, it will have to be a half-moon.  Throughout the night, both the Sun and the Moon will drift Eastwards, but they will essentially maintain the right angle, since it is the Earth that is rotating, while the Sun and the Moon are moving far too slowly to make much of a difference to the angles.

You can imagine that the teacher explains the phases of the Moon in greater detail, and some of the students get it, but others will not.  However, if the teacher is to be considered a good teacher, by the standards that have come down to us, she will give the class some rules of thumb whereby even the students who have not understood the phenomenon can make predictions about the phases of the Moon.

Unfortunately for us, and for humanity at large, you can imagine that it is some of those kids who just didn’t get it, but who were able to use the rules of thumb effectively, who go on to take up the profession of teaching.  Many teachers teach for the love of it.  The rest do it because it is the most convenient thing to do.

Teachers, by and large, are a conscientious lot, and one can imagine that, in advance of having to teach about the phases of the Moon, a teacher will put in a fair attempt at trying to understand the whole business, and what eluded her as a student might easily be within her grasp as an adult, and her students may be so lucky as to have the opportunity to understand the geometry of the phenomenon.  But over several generations, it is quite possible that only the rule of thumb survives.

Years later, it is possible that the principles of a particular phenomenon still hold, but whatever condition allowed the creation of Rules of Thumb no longer applies.  For example, the rules of thumb governing health insurance are probably no longer valid, especially in light of the changed situation regarding pre-existing conditions.  Insurance analysts who relied on Rules of Thumb must make up a whole new set of RoT's to use when they’re not in a mood to really think, on a morning after a night of heavy drinking.  (Actuaries drink like fish.  I’ve seen them.  There’s a rule of thumb for you!)

So if the assumptions on which a Rule of Thumb is based change enough, people will be puzzled, and lose faith in the entire education they have received, which in most cases is just the entire body of Rules of Thumb on which they operate.  Kids these days actually think that the teacher’s entire responsibility is to pass along some rules of thumb (and some professors have bought into this idea), and are puzzled and frustrated when anyone tries to give them the Big Picture.  We have an entire population of kids saying: “Just the facts, Ma'am.”  They mean: Just the rules of thumb, please; I don’t have time for anything else.

The phenomenon of only the rule of thumb surviving is, of course, not by any means restricted to astronomy.  All around us we see examples of people relying on rules of thumb:

• Buy low, sell high.
• Charge whatever the Traffic Will Allow.
• Cheating the Government is OK, because it’s Our Money, Anyway.

Some of these are only Rules of Thumb if you’re not very moral, to begin with.

• Righty Tighty, Lefty Loosey.  (Unless you’re working with certain plumbing fixtures or bicycle pedals!)
• Red Sky In The Morning, Take Warning.
• Taxes are bad for Small Business.
• Government is bad for Health Care.

The rules of thumb for diet and nutrition are the most variable.  Some people avoid sugar, but eat tons of meat, unaware that protein is ultimately turned into either other proteins, or to sugars, especially if you eat a lot of protein.  And unless you eat only egg-white, protein carries along with it a lot of fat (no matter how hard you try to trim it away), and a teaspoon of fat has as much calories as a quarter cup of sugar.  (I’m not absolutely certain of the equivalence; I’ll have to check it and get back to you.  Another scary rule of thumb!!!)

Rules of Thumb for responding to Economic Conditions are pretty interesting, and suspicious, too.  Economists have a myriad ways of advising whether or not to raise taxes, whether or not to raise interest rates, whether or not the Fed should tighten the money supply, whether or not investors should buy stocks versus bonds.  The very fact that these superstitions are so widespread—and make no mistake: they’re superstitions—means that when something unexpected happens, all economists react the same way, just like panic-stricken people in a burning cinema, and of course all the nervous investors on Wall Street are watching each other, just so they can make the same lame-brain move as everyone else.

I used to have a rule of thumb, borrowed from a famous mathematician in Cambridge of the 1920's: give easy exams, but grade hard.  You quickly learn which students know what they’re doing, and which do not.

This really works, provided the kids do not do detailed postmortems of the final examinations for the benefit of their juniors.  Can you imagine the conversation?
“I got a B!  I sort of knew a little of what was going on, but I just didn’t have the time to put in much work all semester, but I must have nailed the final!  He gives easy finals, y’all!”
So of course the next class doesn’t do a stroke of work, anticipating an easy final.

So I give really difficult finals.  They certainly are harder to grade, but with a lot of work, it is possible to decide which students really know what they’re doing, and which ones are just applying Rules of Thumb.

[Added later:

I’ve seen much the same phenomenon even among my close friends, when we go to a seminar or an informational meeting.  It is almost the expected thing to do, to ignore almost everything except what is known as the Takeaway.  What is the Takeaway?  It is the whole event condensed into a Rule of Thumb.  Distilling everything into trivial niblets is a way of life, for those of us who just can’t bother to deal with, or can’t handle, the subtleties of modern life.  It is certainly true that people whose job is giving information have lost the knack of delivering the information tellingly and succinctly.  But even if the speaker was a genius, there is always that executive summary that everyone demands.  Executives are people who do not have a lot of brains, but have a lot of responsibility, and most of the time they work with rules of thumb (more’s the pity).  But once everybody wants to Takeaway only the Rule of Thumb, we are in deep trouble.]

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Rationing the Educational Edge

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Why do we educate everybody, in the USA, despite the fact that many of our citizens could not could afford a private education for their children?  (Why do we lavish any sort of education on the poor and downtrodden, if we don't even issue them a free automobile, that mark of citizenship?)  The long history of public education in the USA brings out the different view the Founding Fathers of the nation had of what would make an excellent republic, than the view we believe those who have grabbed the reins of power in modern America to have.  To those of us who are accused of being liberals, it seems clear that the Founding Fathers, even though they were by no means unanimous in their vision, wanted what was best for the general population, while at the same time protecting private property, which they saw as each person's right to keep what was his hard-earned wealth.  Modern oligarchs, in contrast, seem to take the position that no one deserves anything they can't afford.

As society evolved, and cynics were able to break down the philosophical bases of the American Republic into dollars and cents terms, what was written into the constitution and encoded in early laws was seen to be a very uneasy compromise between what was good for all, versus what could be afforded by each one.  Today, the few who own most of the wealth of the nation look with great skepticism at their financial obligations: Why, they ask, should those of us who have the greatest wealth, have to fund highways, hospitals, schools, the arts, and science and research?

In a recent Bill Moyer's interview, he discussed Diane Ravitch's book Reign of Error, in which she points out the dangers of permitting the continued migration of Education, which has been a part of the social contract, and a cooperative venture so far, into the private sector (and its exploitation for profit).

Let's look at the problem from the point of view of the Fiscal Conservative.  On the one hand, when we pay for public education with tax dollars, and hand over the monitoring of whether the dollars are spent effectively to government inspectors, the taxpayers are likely to see what goes on in schools as ongoing apparent ineptitude.  This is not how we do things in my sweatshop, says the Fiscal Conservative.  But lately, some of them have looked at education, and they see an opportunity to make money.

So far, thank goodness, not every entrepreneur sees education as a money-bin, but Diane Ravitch clearly sees an alarming trend, namely that consortiums of  hedge fund managers are taking over charter schools in certain cities, and removing public education entirely from the supervision of those who used to be responsible for educational quality control.

Public education is complex, and unfortunately the simple metrics that the private sector tends to advocate for the measurement of government "efficiency" cannot take into account all the parameters of what goes into the quality of what is called "Student Learning Outcomes."  But teachers, out of desperation, have taken it upon themselves to study the government data (and of course government is mostly helpless to resist the bottom-line that the most vociferous taxpayers put forward in the name of The Public,) and have noticed the very plausible trend that teachers who have students who live below the poverty line have classes that perform below average.

From the point of view of private enterprise and capitalist economics, it seems like a good idea to segregate students from below the poverty line, and students better able to afford quality education.  Though this seems the most "practical" solution to the problem, from a philosophical point of view, this sort of segregation goes against all the instincts of anyone who is even slightly liberal in his or her views.  But Diane Ravitch is noticing that there is not a strong stand against this tendency to privatize education within the Democrat Party.  Many Democrats have bought into the conservative line, and are eager to play along with the plan to privatize public education, and in some cities, completely replace public schools with Charter schools, which, Ms Ravitch asserts, do not have better results than public schools, as they claim in their marketing.

Unfortunately this question has many aspects: an educational aspect (what will work best for my kid's education?  Must I sacrifice my own children to my political philosophy?), a political significance (what is the right thing to do for the nation's children?  Can we save all the public schools, or focus on private schools, which seem to promise a bigger bang for the buck?), and an economic aspect (how is the nation's and the taxpayer's money best spent?  Is it worth putting good money after bad into the bottomless pit that public education has become?) and a social justice aspect --I'm struggling for a phrase here, and this is the closest I can get--(Is it right to make a child's education depend on the parent's ability to pay?)  So the privatizing forces have multiple ways of pushing forward their agenda.

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