If you recall from my November 5th post, this special ed teacher has only one class of his own (a US History class), with the other four classes of co-teaching. The history class watches Kung Fu Panda II, while the English classes are reading Crispus (a coming of age story set in medieval times) in seventh grade and The Diary of Anne Frank in eighth grade.
The other two classes are seventh grade math classes. These classes are beginning to learn about prisms and their surface areas. This corresponds roughly to Lessons 9-2 and 10-1 of the U of Chicago text, except that it's prisms only -- cylinders are saved for eighth grade under the Common Core.
Of course, I can't resist singing "All About That Base and Height" during this class. This time, I sing just the second verse since the first verse is mainly on volume. If I'd had more time, I would have sung the 2D version of this song (the one from YouTube) first, since it s about the areas of rectangles and triangles (the bases of the prisms). Then afterward I could jump to the second verse of the 3D version of the song for surface area. I end up singing the song mostly in second period -- first period (rotating after lunch today) was so tired that the resident teacher must repeat the lesson several times, leaving me less time to sing songs.
The regular teacher tells me that she's spreading out this tricky lesson over several days. This is in contrast with how I tried to teach it to eighth graders back at -- oops, ixnay on the arterchay athmay!
As far as classroom management is concerned, there's not much to say since most of the classes have a resident teacher. Sixth period (rotating before lunch today) is very noisy when the resident teacher tries to teach the online interactive English lesson. In fifth period history -- the only class that is without a co-teacher (but still with an aide) -- one student tries to switch seats during the movie. The aide and I ask him to move to the correct seat. But when he's still sitting one seat away from the correct seat, I say, "Your correct seat is over there," and he replies sarcastically, "I know."
There are some unnecessary phrases that I say at times, and "Your correct seat is over there" is one such phrase. The student knows where he belongs -- he just doesn't want to be there. I know that I haven't mentioned the focus resolutions in a while, but actually, I want to focus more on avoiding extra phrases, using "teacher look" or "Because I said so" instead.
This post has been labeled "traditionalists," because the traditionalists have been active yesterday as well as today. This means that even with spring break over in both districts (and hence I'm done with my spring break extracurricular activities), today's post will fall victim to Hofstadter's Law -- I won't have time to write everything I want to say about today's Hofstadter chapter.
Dialogue 10 of Douglas Hofstadter's Godel, Escher, Bach is called "Prelude..." It is the Prelude to Part II of the book. By the way, Part I is titled "GEB" and Part II is titled "EGB."
Dialogue 10 begins as follows:
"Achilles and the Tortoise have come to the residence of their friend the Crab, to make the acquaintance of one of this friends, the Anteater. The introductions having been made, the four of them settle down to tea."
Tortoise: We have brought along a little something for you, Mr. Crab.
Crab: That's most kind of you. But you shouldn't have.
This Dialogue introduces a new character, the Anteater:
Anteater: I wonder what it could be.
Crab: We'll soon find out. (Completes the unwrapping, and pulls out the gift.) Two records! How exciting! But there's no label. Uh-oh -- is this another of your "specials," Mr. T?
The Tortoise tells them that Bach first played the music on the records. Then Achilles tells him that the records have something to do with a remarkable new result in mathematics:
Achilles: It's a very simple idea. Pierre de Fermat, a lawyer by vocation but mathematician by avocation, had been reading in his copy of the classic text Arithmetica by Diophantus, and came across a page containing the equation
a^2 + b^2 = c^2
He immediately realized that this equation has infinitely many solutions a, b, c, and then wrote in the margin the following notorious comment: ...
OK, I don't need to repeat what that marginal comment is here on the blog. I've mentioned Fermat's Last Theorem many times on the blog -- the fact that the equation:
a^n + b^n = c^n
has no natural-number solution for a, b, c at least 1, n at least 2.
Anyway, in the Dialogue, Achilles claims that the Tortoise has both proved and disproved Fermat's Last Theorem! (Notice that Hofstadter wrote this Dialogue well over a decade before Wiles and his proof of FLT, which makes this Dialogue a bit dated.)
Crab: Oh, what a shame that you don't have them here. But there's no reason to doubt what you have told us.
Achilles now claims that he can make Bach music appear on the record, by some strange process he calls "acoustico-retrieval":
Crab: I see. But I don't see where number theory enters the picture or what this all has to do with my new records.
Achilles tells him that apparently, the solution to FLT can be used as a secret code to make Bach's music play on the record. So the four friends play the record:
Crab: Don't tell me it's a recording of Bach playing his own works for harpsichord!
Indeed, it's Bach's Well-Tempered Clavier, which consists of 24 preludes and fugues -- one in each major and minor key.
Crab: Ah, yes, well do I remember those long-gone days of my youth, the days when I thrilled to each new prelude and fugue, filled with the excitement of their novelty and beauty and the many unexpected surprises which they conceal.
At this point Achilles compares Bach's fugues to an Escher print -- Cube with Magic Ribbons:
Tortoise: In which there are circular bands having bubble-like distortions which, as soon as you've decided that they are bumps, seem to turn into dents -- and vice versa?
Achilles: Exactly.
The athlete tells the others that he can interpret Bach's fugue in two ways as well:
Anteater: Just as when you look at the magic bands, eh?
Anyway, the discussion continues:
Tortoise: Oh my, look at this! I just turned the page while following the music, and came across this magnificent illustration facing the first page of the fugue.
Crab: I have never seen that illustration before. Why don't you pass it 'round?
Notice that Hofstadter doesn't reveals what this mysterious illustration is yet. Instead, he just concludes the Dialogue as follows:
Achilles: Well, I guess the prelude is just about over. I wonder if, as I listen to this fugue, I will gain any more insight into the question, "What is the right way to listen to a fugue: as a whole, or as the sum of its parts?"
Tortoise: Listen carefully, and you will!
"The prelude ends. There is a moment of silence; and..."
Chapter 10 of Douglas Hofstadter's Godel, Escher, Bach is called "Levels of Description, and Computer Systems." Here's how it begins:
"Godel's string G, and a Bach fugue: they both have the property that they can be understood on different levels. We are all familiar with this kind of thing: and yet in some cases it confuses us, while in others we handle it without any difficulty at all."
This Chapter -- and indeed Part II -- is all about computers and artificial intelligence. The author's first example involves computers that can play chess:
"It used to be thought -- in the 1950's and on into the 1960's -- that the trick to making a machine play well was to make the machine look further ahead into the branching network of possible sequences of play than any chess master can."
The reference is a bit dated, by the way. Hofstadter goes on to add:
"Computer chess programs which rely on looking ahead have not been taught to think on a higher level: the strategy has just been to use brute force look-ahead, hoping to crush all types of opposition. But it has not worked."
But in the years since this book was written, it has worked. Indeed, it was just a few years after Wiles proved FLT that Deep Blue defeated Grandmaster Kasparov. Since then, computers have crushed the best human players. Just as we've seen with Watson on Jeopardy, when it comes to increasing technology, the computer always wins.
The idea is for the computer to perceive information in "chunks":
"At this highest level, the description is greatly chunked, and takes on a completely different feel, despite the fact that many of the same concepts appear on the lowest and highest levels."
At this point, Hofstadter details all the levels at which information can be chunked. At the lowest level, information consists of "bits":
"Physically, a bit is just a magnetic switch that can be in either of two positions. You could call the two positions 'up' and 'down,' or 'x' and 'o,' or '1' and '0' ... The third is the usual convention."
At the next level, bits are chunked into words, called "machine language." The authors gives examples of various instructions:
ADD the word pointed to in the instruction, to a register.
PRINT the word pointed to in the instruction, as letters.
JUMP to the word pointed to in the instruction.
Here ADD, PRINT, and JUMP are words consisting of bits. The next level up is called "assembly language," where the computer coder actually uses words such as ADD, PRINT, and JUMP:
"Therefore, a program in assembly language is very much like a machine language program made legible to humans."
As an analogy, Hofstadter gives an example from music:
"A pianist who plays the notes G-E-B E-G-B is also playing an arpeggio in the chord of E minor. There is no reason to be reluctant about describing this from a higher-level point of view. We have two modes of describing what the CPU is doing."
Notice that GEB and EGB are the names of the two parts of this book -- and they're just the initials of the three title characters. It just a coincidence that they spell out an E minor chord (but of course, Bach would have recognized them as such.)
"The next level of the hierarchy carries much further the extremely powerful idea of using the computer itself to translate programs from a high level into lower levels."
Most of the computer languages that you've heard of are high-level languages. The author mentions "Algol" as one of the first such languages. He compares the conversion from Algol to machine language to writing an equation for a word problem:
"Actually, getting from a word problem to equation is far more complex, but it gives some inkling of the types of 'unscrambling' that have to be carried out in translating from a high-level language to a lower-level language."
Another example of a high-level language mentioned by Hofstadter is LISP. Notice that Logo is actually a dialect of LISP. The author adds that LISP is one of the best languages of artificial intelligence -- and indeed, Brian Harvey mentions this on his Berkeley Logo website.
Some languages, such as Algol (and C++, and Java) are compiled into machine language:
"The first compilers were written in assembly language, rather than machine language, thus taking full advantage of the already accomplished first step up from machine language."
Hofstadter tells us that as long as a program works, it doesn't matter at which level of abstraction we consider it to be running:
"It is when something goes wrong that it is important to be able to think on different levels. If, for instance, the machine is instructed to divide by zero at some stage, it will come to a halt and let the user know of this problem, by telling where in the program the questionable event occurred."
Machine Language Level:
"Execution of the program stopped in location 1110010101110111"
Assembly Language Level:
"Execution of the program stopped when the DIV (divide) instruction was hit"
Compiler Language Level:
"Execution of the program stopped during evaluation of the algebraic expression '(A + B)/Z'"
The author mentions other levels of abstraction:
"Then there is the level of the operating system, which fits between the machine language program and whatever higher level the user is programming in."
Windows is probably the best known operating system. Mac OS is another one. The author compares the user of an operating system to a passenger on a flight:
"Here again, it is when something goes wrong -- such as his baggage not arriving -- that the passenger is made aware of the confusing system of levels underneath him."
He continues:
"If there is, in the cushion underneath the programmer, a program whose purpose is to 'guess' what the programmer wants or means, then it is quite conceivable that the programmer could try to communicate his task and be totally misunderstood."
And this is part of Artificial Intelligence:
"...generalize from examples, correct some misprints or grammatical errors, try to make sense of ambiguous descriptions, try to second-guess the user..."
Once again, Hofstadter's Law forces us to skip to the end of the chapter:
"Is consciousness an epiphenomeon? To understand the mind, must one go all the way down to the level of nerve cells?"
Let's look briefly at the traditionalists. Barry Garelick has posted twice so far this week:
https://traditionalmath.wordpress.com/2019/04/22/from-the-annals-of-ed-school-dept/
https://traditionalmath.wordpress.com/2019/04/23/more-from-the-annals-of-ed-school-dept/
The first post has drawn no comments, but here he introduces his latest blog series:
A new series that comprises a collection of things heard and overheard in ed school, uttered by students and professors alike. Ed school is the place where discarded and discredited psychological theories go to thrive.
So let's go directly to the second post:
In my Educational Psychology class, I gave a presentation on constructivism, showing the difference between minimal guidance, and guided instruction, and evidence that inquiry-based approaches are ineffective. The professor lauded me with praise afterward and said it really got her thinking, plus she really was intrigued with Singapore Math (which I used as examples of explicit and guided instruction).
One major traditionalist, Ze'ev Wurman, comments, but doesn't say much:
Ze'ev Wurman:
NAP publications can be freely downloaded in PDF. This one too..
But the main commenter in the thread is Chester Draws. He used to comment more regularly on the Garelick blog but not recently -- his last comment was a year ago, almost to the day.Anyway, here's what he writes in today's comment:
Chester Draws:
For example, “If students had simply been given problems to solve on their own (an instructional practice used in all the sciences), it is highly unlikely that they would have spent time efficiently.” is not a call for inquiry. Nor is “People must achieve a threshold of initial learning that is sufficient to support transfer. This obvious point is often overlooked and can lead to erroneous conclusions about the effectiveness of various instructional approaches. It takes time to learn complex subject matter, and assessments of transfer must take into account the degree to which original learning with understanding was accomplished.”
I suspect the professor is confusing the book’s call for “student centred learning” with the other aspects of progressive education. But How People Learn is quite consistent that knowledge is fundamental, not skills, and that teachers need to actively teach.
Like most traditionalists, Draws assumes that if "teachers actively teach," then students will listen -- but I start from the opposite assumption, namely that if teachers give direct instruction, then students who find it boring will just tune out.
Case in point is today's seventh grade math class, especially the first period. The lesson is more or less traditional -- the teacher tells the students how to find surface area and the students do it. At the end of class, the teacher asks the students how to find surface area (after she's shown them). Most of the students don't answer, as if they're waiting for one or two "nerds" to answer her questions. Once again, no one learns anything from a worksheet that's left blank.
This might be a spot for an "inquiry-based" lesson. Two questions that I've seen posted often on the web is "What do you notice?" and "What do you wonder?" The idea is that students are more inclined to answer these questions ("I notice that the prisms are 3D figures." "I wonder whether there's an easier way to find surface area.") than the traditionalists' questions. "How people learn" is by doing -- not by leaving things blank.
Draws mentions "transfer" in his comment. This is the idea that traditional lessons teach students knowledge that they can transfer to more difficult problems, as opposed to, say, project-based lessons that teach students only to answer only the specific questions asked in the project. But once again, leaving a worksheet blank doesn't teach anything that transfers (unless by "transfer" we mean "teaches students how to leave other worksheets blank").
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
The complement of a 67-degree angle is _____.
The measures of complementary angles add up to 90 degrees. Thus the complement of a 67-degree angle measures 90-67, or 23 degrees. Therefore the desired angle is 23 degrees -- and of course, today's date is the 23rd.
This is what I wrote last year about today's lesson:
We now reach the final chapter of the U of Chicago text, "Further Work with Circles." This year, I follow the digit pattern and so we cover Chapter 15 on Days 151-159. But for those who start PARCC next week and need to complete Lesson 15-3, we'll reach it on Thursday.
Lesson 15-1 of the U of Chicago text is on Chord Length and Arc Measure. The key theorem of this lesson is:
Chord-Center Theorem:
a. The line containing the center of the circle perpendicular to a chord bisects the chord.
b. The line containing the center of the circle and the midpoint of a chord bisects the central angle determined by the chord.
c. The bisector of the central angle of a chord is perpendicular to the chord and bisects the chord.
d. The perpendicular bisector of a chord of a circle contains the center of the circle.
Proof:
Each part is only a restatement of a property of isosceles triangles.
a. This says the altitude to the base is also a median.
b. This says the median to the base is also an angle bisector.
c. This says the angle bisector is also an altitude and a median.
d. This says the median to the base is also an altitude. QED
Why didn't the text just say "diameter" instead of "the line containing the center of a circle"? I assume it's because a diameter is a segment, but bisectors are lines. Here is the other key theorem:
Arc-Chord Congruence Theorem:
In a circle or in congruent circles:
a. If two arcs have the same measure, they are congruent and their chords are congruent.
b. If two chords have the same length, their minor arcs have the same measure.
The U of Chicago text points out that we can't use the terms "are congruent" and "have the same measure" interchangeably. Two angles are congruent if and only if they have the same measure, but two arcs with the same measure aren't necessary congruent. A 50-degree arc of a tiny circle is nowhere near congruent to a 50-degree arc of a large circle. The theorem tells us that an additional condition is needed -- the circles must be congruent also.
But what does it mean for two circles to be congruent? The U of Chicago text proves that two circles are congruent if and only if their radii are equal. Recall that in Common Core Geometry, we can only show two figures congruent by showing that some isometry maps one to the other:
Lemma:
Two circles are congruent if and only if they have equal radii.
Proof of Lemma:
If two circles X and Y have equal radii, then one can be mapped onto the other by the translation mapping X to Y. So they are congruent. Of course, if they do not have equal radii, since isometries preserve distance, no isometry will map one to the other. QED
Proof of Part a of Arc-Chord Congruence:
In circle O, you can rotate Arc AB about O by the measure of Angle AOC to the position of CD. Then the chordAB rotates to CD also, and AB is congruent to CD, Thus in a circle, arcs of the same measure are congruent and have congruent chords. QED
Part b is left in the text as an exercise. A hint is given -- the measure of an arc equals the measure of its central angle. This suggests that we could use a traditional two-column proof via SSS:
Given: AB = CD in Circle O
Prove: measure Arc AB = measure Arc CD
Proof:
1. AB = CD 1. Given
2. AO = CO, BO = DO 2. All radii of a circle are congruent.
3. Triangle AOB = COD 3. SSS Congruence Theorem
4. Angle AOB = COD 4. CPCTC
5. Arc AB = CD 5. Definition of arc measure
The text warns us that in circles with different radii, arcs of the same measure are not congruent -- they are similar. This isn't proved in the text, but notice that one of the Common Core Standards directs students to "prove that all circles are similar." So let's do so right here:
Theorem:
All circles are similar.
Proof:
If the two circles are concentric, then their common center is also the center of a dilation, with the scale factor obviously R/r, with r the smaller radius and R the larger radius. If the two circles have different centers, then we can translate one of the circles so that its center matches the other, then perform the dilation. QED
Actually, a single dilation will work if the centers aren't the same, but it's difficult to locate the center of this dilation, so it's easier just to translate first.
Here is my first worksheet for Chapter 15:
Lesson 15-1 of the U of Chicago text is on Chord Length and Arc Measure. The key theorem of this lesson is:
Chord-Center Theorem:
a. The line containing the center of the circle perpendicular to a chord bisects the chord.
b. The line containing the center of the circle and the midpoint of a chord bisects the central angle determined by the chord.
c. The bisector of the central angle of a chord is perpendicular to the chord and bisects the chord.
d. The perpendicular bisector of a chord of a circle contains the center of the circle.
Proof:
Each part is only a restatement of a property of isosceles triangles.
a. This says the altitude to the base is also a median.
b. This says the median to the base is also an angle bisector.
c. This says the angle bisector is also an altitude and a median.
d. This says the median to the base is also an altitude. QED
Why didn't the text just say "diameter" instead of "the line containing the center of a circle"? I assume it's because a diameter is a segment, but bisectors are lines. Here is the other key theorem:
Arc-Chord Congruence Theorem:
In a circle or in congruent circles:
a. If two arcs have the same measure, they are congruent and their chords are congruent.
b. If two chords have the same length, their minor arcs have the same measure.
The U of Chicago text points out that we can't use the terms "are congruent" and "have the same measure" interchangeably. Two angles are congruent if and only if they have the same measure, but two arcs with the same measure aren't necessary congruent. A 50-degree arc of a tiny circle is nowhere near congruent to a 50-degree arc of a large circle. The theorem tells us that an additional condition is needed -- the circles must be congruent also.
But what does it mean for two circles to be congruent? The U of Chicago text proves that two circles are congruent if and only if their radii are equal. Recall that in Common Core Geometry, we can only show two figures congruent by showing that some isometry maps one to the other:
Lemma:
Two circles are congruent if and only if they have equal radii.
Proof of Lemma:
If two circles X and Y have equal radii, then one can be mapped onto the other by the translation mapping X to Y. So they are congruent. Of course, if they do not have equal radii, since isometries preserve distance, no isometry will map one to the other. QED
Proof of Part a of Arc-Chord Congruence:
In circle O, you can rotate Arc AB about O by the measure of Angle AOC to the position of CD. Then the chord
Part b is left in the text as an exercise. A hint is given -- the measure of an arc equals the measure of its central angle. This suggests that we could use a traditional two-column proof via SSS:
Given: AB = CD in Circle O
Prove: measure Arc AB = measure Arc CD
Proof:
1. AB = CD 1. Given
2. AO = CO, BO = DO 2. All radii of a circle are congruent.
3. Triangle AOB = COD 3. SSS Congruence Theorem
4. Angle AOB = COD 4. CPCTC
5. Arc AB = CD 5. Definition of arc measure
The text warns us that in circles with different radii, arcs of the same measure are not congruent -- they are similar. This isn't proved in the text, but notice that one of the Common Core Standards directs students to "prove that all circles are similar." So let's do so right here:
Theorem:
All circles are similar.
Proof:
If the two circles are concentric, then their common center is also the center of a dilation, with the scale factor obviously R/r, with r the smaller radius and R the larger radius. If the two circles have different centers, then we can translate one of the circles so that its center matches the other, then perform the dilation. QED
Actually, a single dilation will work if the centers aren't the same, but it's difficult to locate the center of this dilation, so it's easier just to translate first.
Here is my first worksheet for Chapter 15:
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