After subbing the first day of a two-day absence at the middle school yesterday, today I'm covering the last day of a three-day absence. So both of these could have become true multi-day assignments for subs, yet neither is a multi-day.
Even though this is a US history class, today's video is all about Mexico -- The Storm that Swept Mexico (namely Pancho Villa). It's a two-hour video that the students have watched all week. On Tuesday, the regular teacher assigned 25 video notes. The sub assigned 25 more notes on Wednesday and then 10 notes yesterday.
Since the video is almost complete, it doesn't take the full period today to finish. Therefore I decide to follow the other subs and assign only five notes (half as many notes as yesterday for the half-period that it will take to finish). But there's a caveat -- since yesterday's sub noted too much talking and electronic use in the earlier periods, I tell the students that I will increase the required notes by one for each phone or Chromebook that they use during the video.
This works -- the two morning periods cited yesterday end up talking less today, and I don't need to raise the number of notes. They're not necessarily the best classes of the day, but they are certainly the most improved from yesterday.
In the last class before lunch, a true emergency occurs (as opposed to yesterday's so-called restroom emergencies) -- one girl vomits on the desk. (Just last week, I heard some teachers in the lounge tell horror stories about students who throw up in class, and now it happens to me.) Another student escorts her to the office, and she eventually is sent home early.
But the incident does distract this class from the video. This period was one the best classes with the earlier subs, but after today's incident, students begin to talk. I must raise the number of notes to six when one student takes out a phone -- and I nearly increase it to seven when the girl next to him takes out her phone just as the video is about to end. (There's no point in adding another note this close to the end, since there's nothing to write about.)
I'm not sure whether there's anything I could have done. Even the strongest classroom managers have problems when students suddenly become distracted.
Let's do another randomly chosen Stewart chapter today:
Chapter 6 of Ian Stewart's Calculating the Cosmos: How Mathematics Unveils the Universe is called "The Planet that Swallowed its Children." As usual, Stewart begins with a quote:
"The star of Saturn is not a single star, but is a composite of three, which almost touch each other, never change or move relative to each other, and are arranged in a row along the zodiac, the middle one being three times larger than the lateral ones, and they are situated in this form: oOo." -- Galileo Galilei, Letter to Cosimo de' Medici, 30 July 1610.
This chapter is all about the planet Saturn and its most noticeable feature -- its rings. Here's how Stewart begins:
"When Galileo first pointed his telescope at Saturn, and drew what he saw, it looked like this...You can see why he described it as oOo in his excited letter to his sponsor Cosimo de' Medici."
Of course Stewart includes a drawing here, but I don't show it -- oOo is sufficient for us too. He published his discovery in a secret code:
"If anyone later made the same discovery, Galileo could then claim priority by deciphering this as Altissimum planetam tergeminum observavi: 'I have observed the most distant planet to have a triple form.' Unfortunately Kepler deciphered it as Salve umbistineum geminatum Martia proles: 'Be greeted, double knob, children of Mars.'"
Oops, that's the wrong planet -- but luckily for Kepler, Mars does have two "children" (moons). As for Saturn, Galileo eventually realized that the "children" of Saturn weren't moons, but rings that can be seen only through a telescope:
"This fact alone tells us that the rings are very thin compared to the planet, but we now know that they're very thin indeed, a mere 20 metres."
It was Giovanni Cassini who studied the rings of Saturn in greater detail:
"The innermost ring is the B ring, the outermost is the A ring. Cassini also knew of a fainter C ring inside the A ring."
Kepler was able to calculate how fast the rings were moving around the planet:
"The speed of a ring particle in kilometres per second is 29.4 divided by the square root of its orbital radius, measured as a multiple of the radius of Saturn."
In modern times, NASA sent out the Voyager I probe to explore the ringed planet:
"Rich Terrile, one of the scientists working with the images, noticed something totally unexpected: dark shadows like fuzzy spokes of a wheel, which rotate."
It turns out that these shadows are previously undiscovered moons of Saturn. We now know that the ringed planet has 53 named moons and dozens of other satellites that have yet to be named.
At this point, Stewart describes a diagram which I can't show you:
"The diagram show the outer edge of the A ring, and the slanting white lines are regions where the particle density is greater than average."
It was long believed that Saturn is the only ringed planet:
"But in 1977 James Elliot, Edward Dunham, and Jessica Mink were making observations with the Kuiper Airborne Observatory, a transport aircraft equipped with a telescope and other apparatus."
The trio ultimately discover that Uranus has rings as well. The author describes what happens when a moon of Saturn or Uranus interact with the rings:
"It looks like the moon is repelling the ring, but actually the effect is the result of gravitational forces that slow the ring particles down."
The Voyager 2 probe viewed the moons of the two ringed planets in more detail:
"This makes the dynamics of the ring more complex, and it also means that the tidy explanation of a narrow ring held in place by shepherd moons is too simplistic."
We now know that all of the gas giants have rings, though some are very faint:
"Moreover, Douglas Hamilton and Michael Skrutskie discovered in 2009 that Saturn has an absolutely gigantic, but very faint, ring, far larger than the ones Galileo and the Voyagers saw."
It's even likely that some planets orbiting other other stars (exoplanets) have rings. So we ask, how do ring systems form?
"They may have formed when the original gas disc coalesced to create the planet; they could be relics of a moon that has been broken by a collision; they could be remains of a moon that got closer than the Roche limit, at which tidal forces exceed the strength of the rock, and broke up."
Stewart concludes the chapter by comparing ring formation to fossil formation:
"Each 'fossil' provides evidence for what happened in the past, but you need a hypothesis to interpret the evidence, and you need mathematical simulations or inferences or, better still, theorems to understand the consequences of that hypothesis."
Lesson 6-5 of the U of Chicago text is called "Congruent Figures." In the modern Third Edition, we must backtrack to Lesson 5-1 to learn about congruent figures.
This is what I wrote last have about today's lesson:
Lesson 6-5 of the U of Chicago text is on congruent figures. Congruence is one of the most important concepts in all of geometry, especially Common Core Geometry.
As I mentioned many times on this blog, the word congruent is defined very differently in Common Core Geometry than under previous standards. We all know what it means for two segments to be congruent -- that is, that they have equal length -- or for two angles to be congruent -- that is, that they have equal measure. The new definition of congruent appears to be original to the Common Core, and yet, it isn't. Years before the Core, the U of Chicago text used the following definition of congruent -- indeed, it is mainly because of this definition that I chose the U of Chicago as the textbook on which this blog is based:
Definition:
Two figures, F and G, are congruent figures [...] if and only if G is the image of F under a translation, a reflection, a rotation, or any composite of these.
And there we have it -- this definition of congruent predates Common Core. But many opponents of Common Core do not like this new definition. One such opponent is Ze'ev Wurman, a member of the commission here in California that reviewed the Common Core standards. Although his views are posted at several websites, one of the best Wurman articles I found is at this link:
http://www.libertylawsite.org/2014/03/27/the-common-cores-pedagogical-tomfoolery/
Skipping down to the discussion of the math standards -- since, as Wurman himself points out, math is his area of expertise -- the author begins with some elementary school standards. For example, Wurman gives this standard:
1.OA.6: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
Wurman states that this standard should have stopped after the first sentence. Instead, it goes on to prescribe some nonstandard algorithms for addition. I already discussed much of this in many previous posts. Since much of what Wurman writes about grades 1-3 echo what I wrote about the lower grades, I am in full agreement with Wurman for the lower grades.
But then Wurman moves on to Common Core Geometry. Here's what he writes about it:
A true content standard would simply say “Students prove triangle congruence” or, perhaps, “Students understand triangle congruence,” leaving the method of instruction to the teacher. Instead, Common Core not only dictates how to teach congruence, it insists on a specific experimental method of instruction that has an established a track record of failure where it was invented [...]
[emphasis Wurman's]
He then gives a link to a PDF file about the Russian mathematician and scientist A.N. Kolmogorov, whom the PDF credits as the creator of geometry based on transformations. Wurman implies that this geometry was tried out in Russia (i.e., the Soviet Union) and was a big failure.
As I mentioned many times on this blog, the word congruent is defined very differently in Common Core Geometry than under previous standards. We all know what it means for two segments to be congruent -- that is, that they have equal length -- or for two angles to be congruent -- that is, that they have equal measure. The new definition of congruent appears to be original to the Common Core, and yet, it isn't. Years before the Core, the U of Chicago text used the following definition of congruent -- indeed, it is mainly because of this definition that I chose the U of Chicago as the textbook on which this blog is based:
Definition:
Two figures, F and G, are congruent figures [...] if and only if G is the image of F under a translation, a reflection, a rotation, or any composite of these.
And there we have it -- this definition of congruent predates Common Core. But many opponents of Common Core do not like this new definition. One such opponent is Ze'ev Wurman, a member of the commission here in California that reviewed the Common Core standards. Although his views are posted at several websites, one of the best Wurman articles I found is at this link:
http://www.libertylawsite.org/2014/03/27/the-common-cores-pedagogical-tomfoolery/
Skipping down to the discussion of the math standards -- since, as Wurman himself points out, math is his area of expertise -- the author begins with some elementary school standards. For example, Wurman gives this standard:
1.OA.6: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
Wurman states that this standard should have stopped after the first sentence. Instead, it goes on to prescribe some nonstandard algorithms for addition. I already discussed much of this in many previous posts. Since much of what Wurman writes about grades 1-3 echo what I wrote about the lower grades, I am in full agreement with Wurman for the lower grades.
But then Wurman moves on to Common Core Geometry. Here's what he writes about it:
A true content standard would simply say “Students prove triangle congruence” or, perhaps, “Students understand triangle congruence,” leaving the method of instruction to the teacher. Instead, Common Core not only dictates how to teach congruence, it insists on a specific experimental method of instruction that has an established a track record of failure where it was invented [...]
[emphasis Wurman's]
He then gives a link to a PDF file about the Russian mathematician and scientist A.N. Kolmogorov, whom the PDF credits as the creator of geometry based on transformations. Wurman implies that this geometry was tried out in Russia (i.e., the Soviet Union) and was a big failure.
Why should we use the transformation approach? In pre-Core Geometry, we must define the word congruent three times -- first for segments, then for angles, and finally for figures. But in some ways, this is an ad hoc approach. In the Common Core, we only define congruent once, and it applies to segments, angles, and figures all at once.
In the Common Core, congruent means "identical up to isometry" (and later on, we see that similar is defined as "identical up to a similarity transformation"). There are many concepts in college-level mathematics that are defined similarly -- such as topologically equivalent ("identical up to homeomorphism"), equinumerous ("identical up to bijection"), and so on. Furthermore, the Lebesgue measure of a set is defined so that two sets such that there is an isometry mapping one to the other have the same measure.
So in some ways, the Common Core definition is more rigorous than the pre-Core definitions. Also, in some ways, the Common Core definition predates Kolmogorov by a wide margin -- Euclid himself used it as the Principle of Superposition in his proof of SAS (Proposition I.4).
In Hilbert's formulation of Euclid's axioms, congruence is a primitive notion -- that is, it is undefined just as point, line, and plane are. Actually, it's two undefined terms, since Hilbert considers segment and angle congruence separately. As I mentioned before, we can't define an undefined term, but instead we know what it means through the use of axioms or postulates. Hilbert provides six axioms of congruence -- these cover the Equivalence Properties and Segment and Angle Congruence Theorems as given in this section, some of the Point-Line-Plane and Angle Measure Postulates, and SAS. We notice that Hilbert's congruence is completely nonmetric -- there is no notion of distance or angle measure anywhere.
So which formulation should we use? This is a Common Core site and so I use the Common Core definition of congruence, but in the long run, which is best for the students? The usual guiding principles is that if a concept is easy for the students to understand and leads to a higher concept, then the students should learn how to prove it. But if the lower concept is difficult for the students, it should be made into a postulate and not proved in class.
So we can see a full continuum, from more proofs to more postulates:
- Common Core: SAS, ASA, SSS all proved (using transformations)
- Hilbert (supported by David Joyce): SAS postulated, ASA, SSS proved (using SAS)
- Status quo: SAS, ASA, SSS all postulated (most texts)
- Minimalist: SAS, ASA, SSS not mentioned (isosceles/parallelogram properties postulated)
The argument from Wurman and other Common Core opponents is that proving SAS, ASA, SSS from transformations only confused students (which would be the reason why this would have been a big failure in the Soviet Union) and that they should be assumed as postulates. But then, we wonder, why not go one step further and state that all proofs confuse students, so that all proofs involving SAS, ASA, SSS should be dropped, and the properties of triangles and parallelograms assumed? Why is the status quo, where SAS, ASA, SSS are assumed and used in proofs, exactly the right level of complexity for the students?
Well, this is what I hope to find out through this blog. It could be that these Core opponents are correct, and that the status quo level of complexity is exactly appropriate for high school students taking geometry. To me, this is not as clear-cut as elementary math, where the standard algorithms for addition and subtraction are clearly superior to the nonstandard algorithms. This is the reason that I agree with the traditionalists for K-3 math, but not high school math yet.
As for the other theorems proved in this chapter, the Equivalence Properties of Congruence is proved in a way that is standard for many types of transformations -- by using the identity, inverse, and composite functions. The Segment and Angle Congruence Theorems are proved using reflections only, since the text states (in the "Shorter Form" of the definition of congruence) that only reflections, or a composite thereof, are needed to establish congruence. But sometimes it's easier for students to visualize other transformations -- for example, in the Segment Congruence Theorem, one can simply translate X to Z, so that X' and Z coincide. Then one can rotate W' to Y, so that W" andY coincide. In the text, both of these are reflections instead.
Notice that this lesson, 6-5, is the first lesson in which the word congruent appears. The U of Chicago text is careful to use phrases such as "of equal length (measure)" instead of congruent.
I've mentioned before that many people -- both teachers and subjects -- use the words equal and congruent interchangeably. There are two distinctions to make -- one is that numbers (including lengths and angle measures) are equal, while segments and angles are congruent. The other is that we don't know that any figures are congruent until we know of an isometry mapping one to the other, which the Segment and Angle Congruence Theorems provide.
In this course, the latter distinction has priority. I admit that I myself have called angles "equal" (when it's their measures that are equal) on this blog -- because I don't want to call them "congruent" until reaching the Segment and Angle Congruence Theorems. I am especially guilty of this when I write phrases such as "Angle A = Angle B" because it's so much easier than trying to write an angle symbol in ASCII. Occasionally, I would underline a slash: m / A = m / B is the best I can do. Of course, I can't really draw a congruent sign at all, unless I write ~= and you just imagine that the tilde is directly above the equal sign.
I've mentioned before that many people -- both teachers and subjects -- use the words equal and congruent interchangeably. There are two distinctions to make -- one is that numbers (including lengths and angle measures) are equal, while segments and angles are congruent. The other is that we don't know that any figures are congruent until we know of an isometry mapping one to the other, which the Segment and Angle Congruence Theorems provide.
In this course, the latter distinction has priority. I admit that I myself have called angles "equal" (when it's their measures that are equal) on this blog -- because I don't want to call them "congruent" until reaching the Segment and Angle Congruence Theorems. I am especially guilty of this when I write phrases such as "Angle A = Angle B" because it's so much easier than trying to write an angle symbol in ASCII. Occasionally, I would underline a slash: m / A = m / B is the best I can do. Of course, I can't really draw a congruent sign at all, unless I write ~= and you just imagine that the tilde is directly above the equal sign.
Now all of this was five years ago. So what does Wurman have to say this year?
Today is an activity day, but I keep changing it slightly each year. Part of the problem is a calendar quirk -- this is the year that Thanksgiving moves from its earliest possible date (November 22nd) in the prior year to its latest possible date (the 28th) in the current year. But the first day of school in my district isn't similarly six days later than last year. Thus last year, Lesson 6-5 was on the last day before Thanksgiving break, and the activity was all about drawing turkeys -- something fun to do on the last day before nine straight days off. But this year, Lesson 6-5 is not the last day before the holiday.
This year, I'm posting the version of the activity that makes no reference to turkeys. (On the other hand, it's still correct to say that Day 65 is near the end of the third quaver this year.)
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