I wrote earlier that there would be a test given on the Thursday before a four-day weekend. Thus this is that test day. It's one of only two tests that comes just before a weekend -- the other was the Chapter 6 Test given on the last Friday in November. Unfortunately, there's been more tests given just after the weekend on Monday than just before the weekend.
Meanwhile, in Godel, Escher, Bach, there are two Chapters left in Part I. It's so convenient to let the four-day Easter weekend separate Parts I and II, so I need to post two Dialogues and two Chapters today in order to do so.
But there's that dang Hofstadter's Law again! I have hardly enough time to post one chapter in a day, and now I must post two chapters. Then again, by doing so I'll be able to take four full days off from posting starting tomorrow. So I must sacrifice today to gain an stress-free holiday weekend.
Dialogue 8 of Douglas Hofstadter's Godel, Escher, Bach is called "Crab Canon." It begins as follows:
Achilles and the Tortoise happen upon each other in the park one day while strolling.
In the interest of time, let me quote only their friend the Crab in today's post, since it's the first time that we're meeting him:
Crab: Hallo! Hulloo! What's up? What's new? You see this bump, this lump? Given to me by a grump. Ho! And on such a fine day. You see, I was just idly loafing about when up lumbers this giant fellow from Warsaw -- a colossal bear of a man -- playing a lute. He was three meters tall, if I'm a day. I mosey on up to the chap, reach skyward and manage to tap him on the knee, saying "Pardon me, sir, but you are Pole-luting our park with your mazurkas." But WOW! he had no sense of humor -- not a bit, not a wit -- and POW! -- he lets loose and belts me one, smack in the eye! Were it in my nature, I would crab up a storm, but in the time-honored tradition of my species, I backed off. After all, when we walk forwards, we move backwards. It's in our genes, you know, turning around and round. That reminds me -- I've always wondered, "Which came first -- the Crab, or the Gene?" That is to say -- "Which came last -- the Gene, or the Crab?" I'm always turning things round and round, you know. It's in our genes, that I must lope along on my merry way -- so off I go on such a fine day. Sing "ho!" for the life of a Crab! TATA! Ole!"
In this Dialogue, Achilles and the Tortoise keep discussing things that are reversible -- including the order of the four bases (T, C, G, A) in the Crab's DNA. In fact, the entire Dialogue is reversible -- for example, the first few lines are:
Tortoise: Good day, Mr. A.
Achilles: Why, same to you.
Tortoise: So nice to run into you.
Achilles: That echoes my thoughts.
And the last few lines are:
Tortoise: That echoes my thoughts.
Achilles: So nice to run into you.
Tortoise: Why, same to you.
Achilles: Good day, Mr. T.
The Crab's line is in the exact middle of the Dialogue.
There's also Crab Canon -- the name of both an Escher drawing and a Bach piece. (In the reversible Dialogue, the Tortoise mentions the Escher drawing and Achilles mentions the Bach piece.) The artwork is reflection-symmetric, and the piece can also be played forwards and backwards.
This is the sort of song that I'd like to try coding in Mocha. Unfortunately, BASIC doesn't allow us to read notes from the DATA lines backwards. Instead, we place the notes and their lengths in DIM arrays and then use FOR loops to play it forwards and backwards (STEP -1 is backwards):
10 DIM D(20),L(20)
20 FOR X=1 TO 20
30 READ D(X),L(X)
40 NEXT X
50 DATA 18,8,15,8,12,8,11,8,19,12
60 DATA 12,8,13,8,14,8,15,8,16,8
70 DATA 17,8,18,4,19,4,24,4,19,4
80 DATA 16,4,15,8,16,8,18,8,16,8
90 N=9
100 FOR X=1 TO 20
110 SOUND 261-N*D(X),L(X)
120 NEXT X
130 FOR X=20 TO 1 STEP -1
140 SOUND 261-N*D(X),L(X)
150 NEXT X
Bach wrote his song in the key of C minor. I wrote my version in 18EDL (white D minor) and then used N=9 to bring it down to C minor to match Bach's version.
I ad-lib a little when Bach plays a few notes not available in 18EDL. Actually, there is also a second line in eighth notes. Due to Hofstadter's Law, I don't have time to code it all out -- plus it includes a few more notes unavailable in 18EDL.
Chapter 8 of Douglas Hofstadter's Godel, Escher, Bach is called "Typographical Number Theory." It begins as follows:
"Three examples of indirect self-reference are found in the Crab Canon. Achilles and the Tortoise both describe artistic creations they know -- and, quite accidentally, those creations happen to have the same structure as the Dialogue they're in."
This Chapter is all about another formal system -- Typographical Number Theory (TNT). It's intended to represent N, or nonformal number theory.
For example, the author writes the following in N:
(1) 5 is prime.
(2) 2 is not a square.
(3) 1729 is a sum of two cubes.
(4) No sum of two positive cubes is itself a cube.
(5) There are infinitely many prime numbers.
(6) 6 is even.
He begins by converting these to more "elementary notions":
(1') There do not exist numbers a and b, both greater than 1, such that 5 equals a times b.
(2') There does not exist a number b, such that b times b equals 2.
(3') There exist numbers b and c such that b times b times b, plus c times c times c, equals 1729.
(6') There exists a number e such that 2 times e equals 6.
Now Hofstadter introduces the symbols of TNT. First, he begins with the natural numbers themselves which he calls numerals:
zero: 0
one: S0
two: SS0
three: SSS0
etc.
All of these are made up of 0 (zero) and S (the successor of). Next, we need some variables, which consist of letters a through e plus possibly some prime symbols:
e
d'
c"
b'"
a""
Addition and multiplication are noted as follows:
(b + c)
(b * c)
Actually, Hofstadter uses a raised dot for multiplication, but in ASCII I'll write it as *. And of course, we'll just use = for equals.
The author also uses symbols from yesterday's Propositional Calculus, except that instead of atoms P, Q, R, there are now equations using =:
2 plus 3 equals 4: (SS0 + SSS0) = SSSS0
2 plus 2 is not equal to 3: ~(SS0 + SS0)=SSS0
If 1 equals 0, then 0 equals 1: <S0=0 -> 0=S0>
Hofstadter also introduces quantifiers. For example, the statement:
(b + S0) = SS0
is open (that is, neither true nor false). To make it closed, we use quantifiers:
Eb: (b + S0) = SS0 (There exists b such that b + 1 = 2.)
Ab: (b + S0) = SS0 (For all b, b + 1 = 2.)
Hofstadter actually uses upside-down E and A for "there exists" and "for all," but it's traditional in ASCII to use regular E and A. Recall that Eugenia Cheng also uses these quantifiers in her third book on logic.
It's possible to use these symbols to translate (1)-(6) above. For example, "6 is even" becomes:
Ee: (SS0 * e) = SSSSSS0
and "2 is not a square" becomes:
~Eb: (b * b) = SS0
Using these quantifiers, we now introduce the five axioms of TNT:
Axiom 1: Aa: ~Sa=0
Axiom 2: Aa: (a + 0) = a
Axiom 3: Aa: Ab: (a + Sb) = S(a + b)
Axiom 4: Aa: (a * 0) = 0
Axiom 5: Aa: Ab: (a * Sb) = (a * b) + a
The author writes that the first axiom was first devised by Giuseppe Peano in the 19th century. He first came up with the Peano postulates, which Hofstadter writes in terms of Genies and djinns:
(1) Genie is a djinn.
(2) Every djinn has a meta (which is also a djinn)
(3) Genie is not the meta of any djinn.
(4) Different djinns have different metas.
(5) If Genie has X, and each djinn relays X to its meta, then all djinns get X.
The rest of this Chapter is quite complex. Let me link to some of the proofs that Hofstadter is able to prove in this TNT by using Metamath:
Proof of 1 + 1 = 2
Hofstadter's Version:
Statements Reasons
1. Aa: Ab: (a + Sb) = S(a + b) 1. axiom 3
2. Ab: (S0 + Sb) = S(S0 + b) 2. specification (S0 for a)
3. (S0 + S0) = S(S0 + 0) 3. specification (0 for b)
4. Aa: (a + 0) = a 4. axiom 2
5. (S0 + 0) = S0 5. specification (S0 for a)
6. S(S0 + 0) = SS0 6. add S
7. (S0 + S0) = SS0 7. transitvity (lines 3, 6)
This proof uses a few rules that are stated in the book but I haven't introduced yet (namely the transitivity and add S rules).
Metamath Version:
http://us.metamath.org/mpegif/o1p1e2.html
This proof is a little different, only because 1 has been previously defined to be S0 and 2 has been previously defined to be SS0. (Here "suc" means S.)
Here the author provides a long derivation -- addition is commutative. His proof has 56 steps, but here's a similar one from Metamath with 64 steps:
http://us.metamath.org/mpegif/nnacom.html
Hofstadter writes about tension and resolution in TNT:
"Could one devise a much fancier typographical system which is aware of the tensions and goals inside derivations? I would have preferred to show how to derive Euclid's Theorem (the infinitude of the primes) in TNT, but it would probably have doubled the length of the book."
Actually, number theorists should be hoping that TNT isn't complete:
"Thus, in principle, if TNT were complete, number theorists would be put out of business: any question in their field could be resolved, with sufficient time, in a purely mechanical way."
The author writes about Hilbert's program -- the desire to prove all true statements:
"The visions of a thin rope are based on the presumption that facts about number theory won't be needed in proving that such a typographical theory holds."
We'll skip to the conclusion (which is ultimately related to Godel's Theorem) so we can move on:
"Less metaphorically, we can say: Any system that is strong enough to prove TNT's consistency is at least as strong as TNT itself. And so circularity is inevitable."
Dialogue 9 of Douglas Hofstadter's Godel, Escher, Bach is called "A Mu Offering." It begins:
The Tortoise and Achilles have just been to hear a lecture on the origins of the Genetic Code, and are now drinking some tea at Achilles' home.
Achilles: I have something terrible to confess, Mr. T.
Tortoise: What is it, Achilles?
Achilles tells his half-shelled friend about a dream he had. It was about DNA, except that instead of A and T, the DNA was made up of Achilles and the Tortoise. The C might stand for the Crab, but he doesn't know who G is. Then he starts talking about Zen Buddhism:
Achilles: I have always had a yen for the yin and yang, you know.
Tortoise: Oh, splendid. Then perhaps I can finally become enlightened.
Achilles explains that Zen is all about enlightenment of the SATORI -- the state of "no-mind":
Tortoise: Hmm...Now THERE'S something I culd like about Zen.
Achilles: I had a hunch you'd get attached to it.
The athlete tells the Tortoise about a "koan" (or Zen parable) about a 119-year-old monk, Joshu:
Tortoise: A mere youngster!
Achilles: By your standards, yes. Now one day while Joshu and another monk where standing together in the monastery, a dog wandered by. The monk asked Joshu, "Does a dog have Buddha-nature, or not?"
Tortoise: Whatever that is. So tell me -- what did Joshu reply?
Achilles: 'MU.'
And that's the end of the parable. He also tells the Tortoise about two more Zen zoans:
Zoan #1:
A monk asked Baso: "What is Buddha?"
Baso said: "The mind is Buddha."
Zoan #2:
A monk asked Baso: "What is Buddha?"
Baso said: "The mind is not Buddha."
Some of the scenes clearly aren't related directly to Zen, but to logic (or both). The goal is to convert koans into strings via the "Geometric Code" in order to determine which one is true -- in other words, which one "has Buddha-nature":
Achilles: Yes -- there is a venerated book which lists the "Geometric Code." If you don't have a copy of this book, of course, you can't translate a koan into a string.
Tortoise: Evidently not. What is the origin of the Geometric Code?
Achilles: It came from an ancient master known as "Great Tutor" who my master says is the only one ever to attain the Enlightenment "Yond Enlightenment."
...
Tortoise: I was just wondering about those two koans -- I mean the koan and its un-koan -- the ones which say "This mind is Buddha" and "This mind is not Buddha" -- what do they look like, when turned into strings via the Geometric Code?
Achilles: I'll be glad to show you.
...
Tortoise: Now would I have to perform the operations in just the same ORDER as you did?
Achilles: No, any order will do.
...
Achilles: What happened?
Tortoise: I wanted to get rid of that knot.
Achilles: But instead of untying it, you tied another one, and then BOTH disappeared! Where did they go?
Tortoise: Tumbolia, of course. That's the Law of Double Nodulation.
This is an obvious pun of "Law of Double Negation." Meanwhile, Hofstadter ends this Dialogue with the last koan, from Zen master Kyogen:
Achilles: Yes -- there is a venerated book which lists the "Geometric Code." If you don't have a copy of this book, of course, you can't translate a koan into a string.
Tortoise: Evidently not. What is the origin of the Geometric Code?
Achilles: It came from an ancient master known as "Great Tutor" who my master says is the only one ever to attain the Enlightenment "Yond Enlightenment."
...
Tortoise: I was just wondering about those two koans -- I mean the koan and its un-koan -- the ones which say "This mind is Buddha" and "This mind is not Buddha" -- what do they look like, when turned into strings via the Geometric Code?
Achilles: I'll be glad to show you.
...
Tortoise: Now would I have to perform the operations in just the same ORDER as you did?
Achilles: No, any order will do.
...
Achilles: What happened?
Tortoise: I wanted to get rid of that knot.
Achilles: But instead of untying it, you tied another one, and then BOTH disappeared! Where did they go?
Tortoise: Tumbolia, of course. That's the Law of Double Nodulation.
This is an obvious pun of "Law of Double Negation." Meanwhile, Hofstadter ends this Dialogue with the last koan, from Zen master Kyogen:
Zen is like a man hanging in a tree by his teeth over a precipice. His hands grasp no branch, his feet rest on no limb, and under the tree another person asks him: "Why did Bodhidharma come to China from India?" If the man in the tree does not answer, he fails, and if he does answer, he falls and loses his life. Now what shall he do?
Tortoise: That's clear; he should give up Zen, and take up molecular biology.
Chapter 9 of Douglas Hofstadter's Godel, Escher, Bach is called "Mumon and Godel." It begins:
"I'm not sure I know what Zen is. In a way, I think I understand it very well; but in a way, I also think I can never understand it at all."
The Mumon mentioned in the Chapter title was a thirteenth century Zen master. Hofstadter points out that Mumon was a contemporary of Fibonacci. His name in Chinese means "No-gate." He wrote the Mumonokan in order to "explain" the Zen koans, but his "explanations" are even more opaque than the original koans!
This is an interesting chapter, but once again I don't have much time to write about it. I think I'll just skip all of the Mumon non-explanations, since that just wastes time.
But on this Thursday before Christianity's holiest day, it's nice to read about Zen Buddhism. About three months ago I mentioned Louis Sachar's There's a Boy in the Girl's Bathroom. Well,it's been a while since I've read the end of the book, but I remember after all these years that there was a mention of Zen Buddhism. (I forget what the discussion was all about, though.)
Regarding Escher and Zen, Hofstadter writes:
"In questioning perception and poising absurd answerless riddles, Zen has company, in the person of M.C. Escher."
Here he includes the following Escher pictures: Three Worlds, Dewdrop, Another World, Day and Night, Rind, Puddle, Rippled Surface, and Three Spheres II.
I will point out that this Chapter resolves last week's MU-puzzle.
But on this Thursday before Christianity's holiest day, it's nice to read about Zen Buddhism. About three months ago I mentioned Louis Sachar's There's a Boy in the Girl's Bathroom. Well,it's been a while since I've read the end of the book, but I remember after all these years that there was a mention of Zen Buddhism. (I forget what the discussion was all about, though.)
Regarding Escher and Zen, Hofstadter writes:
"In questioning perception and poising absurd answerless riddles, Zen has company, in the person of M.C. Escher."
Here he includes the following Escher pictures: Three Worlds, Dewdrop, Another World, Day and Night, Rind, Puddle, Rippled Surface, and Three Spheres II.
I will point out that this Chapter resolves last week's MU-puzzle.
Hofstadter begins by quoting Mumon: "If any of you has one eye, he will see the failure on the teacher's part." But why should it matter to have one eye?
Actually, it's not "one eye" that matter, but "one I." Here the I-count is the number of I's in a given word, beginning with 1 (the original axiom I). Then he points out:
(1) The I-count begins at 1 (not a multiple of 3);
(2) Two of the rules do not affect the I-count at all;
(3) The two remaining rules which do affect the I-count do so in such a way as never to create a multiple of 3 unless given one initially.
The conclusion -- and a typically hereditary one it is, too -- is that the I-count can never become any multiple of 3. In particular, 0 is a forbidden value of the I-count. Hence, MU is not a theorem of the MIU-system. The puzzle from Chapter 1 has no solution!
The rest of the chapter is all about Godel-Numbering. It's all about assigning numbers to statements to show that their proofs are impossible.
Hofstadter begins by showing how to assign a Godel-Numbering to the MIU-system. He starts by assigning a digit to each of its three symbols:
M becomes 3, I becomes 1, U becomes 0
MU becomes 30, MIIU becomes 3110, etc.
We then include these numbers in proofs:
Statements Reasons
1. MI 1. axiom 31
2. MII 2. rule 2 311
3. MIIII 3. rule 2 31111
4. MUI 4. rule 3 301
5. MUIU 5. rule 1 3010
6. MUIUUIU 6. rule 2 3010010
7. MUIIU 7. rule 4 30110
Then rules can be converted into numbers:
Rule 1: If a string ends in I, we can add U to the end.
Rule 1: If we have made 10m + 1, then we can make 10(10m + 1).
These statements involve numbers (which he calls MIU-numbers), and so these are actually complicated statements of TNT. It's possible to write the statement "b is a MIU-number" as a very, very complicated statement of TNT, which he calls MUMON. (The name means "MU monster" since these numbers are monstrously large, but of course it's just another one of Hofstadter's typical puns with Zen master Mumon.)
The goal now is to assign a Godel-Numbering to TNT itself. Hofstadter does this by assigning digits to each of the symbols of TNT:
0 becomes 666, S becomes 123, = becomes 111, and so on.
In his notation, Axiom 1 (0 is not a successor) becomes:
Aa: ~Sa = 0
626,262,636,223,123,262,111,666
and the Rule of Detachment becomes:
If x and <x->y> are both theorems, then y is a theorem.
If x and 212x633y213 are both theorems, then y is a theorem.
Now we have TNT-Numbers, which are a recursively enumerable set of numbers:
"Arithmetized TNT is actually extremely similar to the arithmetized MIU-system, only there are more rules and axioms, and to write out arithmetical equivalents would be a big bother -- and quite unenlightening, incidentally."
He writes that TNT is trying to swallow itself:
"And it is in the nature of any formalization of number theory that it's metalanguage is embedded within it. We can dignify that observation by calling it the Central Dogma of Mathematical Logic, and depicting it in a two-step diagram."
TNT => N => meta-TNT
The Godel statement becomes:
Hofstadter begins by showing how to assign a Godel-Numbering to the MIU-system. He starts by assigning a digit to each of its three symbols:
M becomes 3, I becomes 1, U becomes 0
MU becomes 30, MIIU becomes 3110, etc.
We then include these numbers in proofs:
Statements Reasons
1. MI 1. axiom 31
2. MII 2. rule 2 311
3. MIIII 3. rule 2 31111
4. MUI 4. rule 3 301
5. MUIU 5. rule 1 3010
6. MUIUUIU 6. rule 2 3010010
7. MUIIU 7. rule 4 30110
Then rules can be converted into numbers:
Rule 1: If a string ends in I, we can add U to the end.
Rule 1: If we have made 10m + 1, then we can make 10(10m + 1).
These statements involve numbers (which he calls MIU-numbers), and so these are actually complicated statements of TNT. It's possible to write the statement "b is a MIU-number" as a very, very complicated statement of TNT, which he calls MUMON. (The name means "MU monster" since these numbers are monstrously large, but of course it's just another one of Hofstadter's typical puns with Zen master Mumon.)
The goal now is to assign a Godel-Numbering to TNT itself. Hofstadter does this by assigning digits to each of the symbols of TNT:
0 becomes 666, S becomes 123, = becomes 111, and so on.
In his notation, Axiom 1 (0 is not a successor) becomes:
Aa: ~Sa = 0
626,262,636,223,123,262,111,666
and the Rule of Detachment becomes:
If x and <x->y> are both theorems, then y is a theorem.
If x and 212x633y213 are both theorems, then y is a theorem.
Now we have TNT-Numbers, which are a recursively enumerable set of numbers:
"Arithmetized TNT is actually extremely similar to the arithmetized MIU-system, only there are more rules and axioms, and to write out arithmetical equivalents would be a big bother -- and quite unenlightening, incidentally."
He writes that TNT is trying to swallow itself:
"And it is in the nature of any formalization of number theory that it's metalanguage is embedded within it. We can dignify that observation by calling it the Central Dogma of Mathematical Logic, and depicting it in a two-step diagram."
TNT => N => meta-TNT
The Godel statement becomes:
"G is not a theorem of TNT." (that is, G is a not a TNT-number)
He asks, is G a theorem of TNT, or not?
"We will make our usual assumption: that TNT incorporates valid methods of reasoning, and therefore that TNT never has falsities for theorems.
Hofstadter concludes the chapter by asserting that Mumon has the last word:
"We will make our usual assumption: that TNT incorporates valid methods of reasoning, and therefore that TNT never has falsities for theorems.
Hofstadter concludes the chapter by asserting that Mumon has the last word:
Has a dog Buddha-nature?
This is the most serious question of all.
If you say yes or no,
You lose your own Buddha-nature.
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
Two identical regular 40-sided polygons are placed side by side. The gap between them is an angle of __ degrees.
The exterior angle of a regular 40-gon is 360/40 = 9 degrees. Pappas likes to ask about the gap between two such polygons, which is double the exterior angle or 18 degrees. Therefore the desired angle is 18 degrees -- and of course, today's date is the eighteenth.
Because today is test day, it's also a traditionalists' post. Due to Hofstadter's Law, I don't have time to time to say much.
First of all, I do want to mention that there were two big comebacks in sports this week -- the only local team currently in the playoffs is the L.A. Clippers, who overcame a 31-point deficit to beat the highly favored Warriors. The other is in golf, where Tiger Woods won his first major championship in over a decade at the Masters.
I compare athletes who work hard to come back to students who work hard in math classes. I like to see more math students avoid giving up and be just like Tiger and the Clippers.
Meanwhile, there's a post about math at the Joanne Jacobs website, where traditionalists often post:
https://www.joannejacobs.com/2019/04/add-a-4th-year-of-high-school-math/
Requiring four years of high school math will prepare more students for college, some California State University trustees believe. But civil rights and college-access advocates say adding math requirements will make it harder for disadvantaged students to qualify, reports Larry Gordon on EdSource.
Here are some key comments:
GoogleMaster:
Bring back Business Math! More people need to know how to do basic bookkeeping.
wahoofive:
Social promotion is the real problem.
It’s not about high school. When kids are coming out of middle school not really knowing how to add fractions or understand percents, no amount of HS curriculum tweaking will help. The problem social promotion happens much earlier. I’m not pretending there’s any easy fix, but adding a fourth year of HS math definitely won’t help.
Momof4:
While I am fine with requiring 4 years of math, through calculus, for those heading to engineering, math and the hard sciences, I think demonstrated success through algebra II is fine for most – providing college readiness SATs. I stopped at algebra II, in the 60s, and had no trouble handling inorganic, organic,and biochem; and I regularly used algebra I-level math at work. Another year of HS math was unnecessary.
By the way, Momof4 is also the poster who regularly promotes sentence diagramming, as I alluded to earlier this week.
Ann in L.A.:
Calculus isn’t as important as a better grasp of statistics, which is the most important poorly-taught subject in school. Though you can’t really do full-on stats without calculus, having a 4th year math class which focused on getting the basics down could be very useful.
Bill:
Notice that Bill is usually the poster who writes that you can't do Stats without Algebra I. Here Ann goes a step further and says that you can't do Stats without Calculus!
Also, notice that at the Edsource link above, there's another major traditionalist, Zeev Wurman:
Zeev Wurman:
Another poster, SD Parent, also appears to be traditionalist-leaning:
SD Parent:
One could get whiplash following the roller coaster ride of math curriculum and course progression in California. The real problem is that students are not being taught math well in K-12. It would be better to focus on effective K-12 math instruction than expect students to suddenly understand math after a 4th year of mediocre instruction in high school.
It wasn’t that long ago that the state adopted Common Core Math and then forced a slow-down, essentially “dumbing down” the progression of students through the Algebra I-Geometry-Algebra II (or in the case of San Diego Unified, Integrated Math 1-3) by starting Algebra I in 9th grade. Later, the state decried remediation of math at the CSUs because it slowed down CSU graduation rates and made remedial students feel bad because they weren’t earning credits. Now the CSUs are effectively saying that high schools need to teach students math before they arrive at college by offering a 4th year of math. And so it goes…
Before we leave the traditionalists, I wish to provide one more link:
https://newmathdoneright.wordpress.com/2018/02/19/peano-axioms-help-learn-recursion-in-programming/
The author of this website apparently believes that children should be taught Peano axioms -- that's right, the same axioms that Hofstadter discusses in his book:
Here at New Math Done Right, we support languages like Lisp and Scheme which are based on fundamental mathematical logic like recursion. If you want to learn these languages and ways of thinking, learning the Peano Axioms and recursive logic will greatly help.
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
Two identical regular 40-sided polygons are placed side by side. The gap between them is an angle of __ degrees.
The exterior angle of a regular 40-gon is 360/40 = 9 degrees. Pappas likes to ask about the gap between two such polygons, which is double the exterior angle or 18 degrees. Therefore the desired angle is 18 degrees -- and of course, today's date is the eighteenth.
Because today is test day, it's also a traditionalists' post. Due to Hofstadter's Law, I don't have time to time to say much.
First of all, I do want to mention that there were two big comebacks in sports this week -- the only local team currently in the playoffs is the L.A. Clippers, who overcame a 31-point deficit to beat the highly favored Warriors. The other is in golf, where Tiger Woods won his first major championship in over a decade at the Masters.
I compare athletes who work hard to come back to students who work hard in math classes. I like to see more math students avoid giving up and be just like Tiger and the Clippers.
Meanwhile, there's a post about math at the Joanne Jacobs website, where traditionalists often post:
https://www.joannejacobs.com/2019/04/add-a-4th-year-of-high-school-math/
Requiring four years of high school math will prepare more students for college, some California State University trustees believe. But civil rights and college-access advocates say adding math requirements will make it harder for disadvantaged students to qualify, reports Larry Gordon on EdSource.
Here are some key comments:
GoogleMaster:
Bring back Business Math! More people need to know how to do basic bookkeeping.
wahoofive:
Social promotion is the real problem.
It’s not about high school. When kids are coming out of middle school not really knowing how to add fractions or understand percents, no amount of HS curriculum tweaking will help. The problem social promotion happens much earlier. I’m not pretending there’s any easy fix, but adding a fourth year of HS math definitely won’t help.
Momof4:
While I am fine with requiring 4 years of math, through calculus, for those heading to engineering, math and the hard sciences, I think demonstrated success through algebra II is fine for most – providing college readiness SATs. I stopped at algebra II, in the 60s, and had no trouble handling inorganic, organic,and biochem; and I regularly used algebra I-level math at work. Another year of HS math was unnecessary.
By the way, Momof4 is also the poster who regularly promotes sentence diagramming, as I alluded to earlier this week.
Ann in L.A.:
Calculus isn’t as important as a better grasp of statistics, which is the most important poorly-taught subject in school. Though you can’t really do full-on stats without calculus, having a 4th year math class which focused on getting the basics down could be very useful.
Bill:
Calculus is nothing more than advanced algebra and trig, but if you have mastered algebra I or algebra II, you should be able to solve this without too much difficulty:
(x->5) (x^2 – 25) / (x – 5)
lim
lim
Notice that Bill is usually the poster who writes that you can't do Stats without Algebra I. Here Ann goes a step further and says that you can't do Stats without Calculus!
Also, notice that at the Edsource link above, there's another major traditionalist, Zeev Wurman:
Zeev Wurman:
I wonderfully-sounding foolish idea.
CSU doesn’t need more than today’s 3 years that already include Algebra 1&2 and Geometry. Certainly not for majors in movie-making, history, or communications. And the STEM-intending students can–and do–take pre-calc or calc as the fourth year already. The problem with three years of math is not that the expected level of knowledge is too low, but that too many students do not meet existing expectations despite formally passing the required courses. They don’t meet the expectations because of lack of preparedness, not because they were not pushed to take another year of math to which they are unprepared.
Another poster, SD Parent, also appears to be traditionalist-leaning:
SD Parent:
One could get whiplash following the roller coaster ride of math curriculum and course progression in California. The real problem is that students are not being taught math well in K-12. It would be better to focus on effective K-12 math instruction than expect students to suddenly understand math after a 4th year of mediocre instruction in high school.
It wasn’t that long ago that the state adopted Common Core Math and then forced a slow-down, essentially “dumbing down” the progression of students through the Algebra I-Geometry-Algebra II (or in the case of San Diego Unified, Integrated Math 1-3) by starting Algebra I in 9th grade. Later, the state decried remediation of math at the CSUs because it slowed down CSU graduation rates and made remedial students feel bad because they weren’t earning credits. Now the CSUs are effectively saying that high schools need to teach students math before they arrive at college by offering a 4th year of math. And so it goes…
Before we leave the traditionalists, I wish to provide one more link:
https://newmathdoneright.wordpress.com/2018/02/19/peano-axioms-help-learn-recursion-in-programming/
The author of this website apparently believes that children should be taught Peano axioms -- that's right, the same axioms that Hofstadter discusses in his book:
Here at New Math Done Right, we support languages like Lisp and Scheme which are based on fundamental mathematical logic like recursion. If you want to learn these languages and ways of thinking, learning the Peano Axioms and recursive logic will greatly help.
Notice that Logo is technically a dialect of Lisp, which also explains why I linked to Logo programs on the day we read the recursion chapter in Hofstadter.
To me, teaching Peano to children is a bit extreme. Still, this is an interesting blog. For example, there are some old posts from over five years ago where the blogger states some of the TNT rules.
(Notice that most of the posts are from the year before I started this blog. There is one post dated New Year's Eve 2015 and another dated last year.)
To me, teaching Peano to children is a bit extreme. Still, this is an interesting blog. For example, there are some old posts from over five years ago where the blogger states some of the TNT rules.
(Notice that most of the posts are from the year before I started this blog. There is one post dated New Year's Eve 2015 and another dated last year.)
This is what I wrote last year about today's test:
Today is approximately the end of the fifth hexter -- the midpoint of the third trimester -- so it's a good test day.
Today is the Chapter 14 Test. Here are the answers to my posted test:
1. DE = 32, EF = 16sqrt(3).
2. TU = 16, US = 8sqrt(3), SK = 8, TK = 8sqrt(2).
3. 3/4
4. 3/5
5. 0.309
6. 0.625
7. 1/2
8. sqrt(3) (Some people may consider this question unfair, since the above question and both corresponding questions on the practice had rational answers, leading students to believe that they can just use a calculator to find the exact value rather than use 30-60-90 triangles.)
9-10. These are vectors that I can't reproduce easily here.
11. BC/AC (or a/b, if the students learned it that way).
12. AB and AD
13. ACD and CBD
14. This is a vector that I can't reproduce easily here.
15. (9, 6)
16. 115 feet, to the nearest foot.
17. (1, 4)
18. (3, -3)
19. (3, 2). (I hope students don't get confused here and solve these three backwards!)
20. This is a vector that I can't reproduce easily here.
Thus concludes Chapter 14. Stay tuned -- we're starting Chapter 15 on Tuesday, after the long Easter weekend.
Today is approximately the end of the fifth hexter -- the midpoint of the third trimester -- so it's a good test day.
Today is the Chapter 14 Test. Here are the answers to my posted test:
1. DE = 32, EF = 16sqrt(3).
2. TU = 16, US = 8sqrt(3), SK = 8, TK = 8sqrt(2).
3. 3/4
4. 3/5
5. 0.309
6. 0.625
7. 1/2
8. sqrt(3) (Some people may consider this question unfair, since the above question and both corresponding questions on the practice had rational answers, leading students to believe that they can just use a calculator to find the exact value rather than use 30-60-90 triangles.)
9-10. These are vectors that I can't reproduce easily here.
11. BC/AC (or a/b, if the students learned it that way).
12. AB and AD
13. ACD and CBD
14. This is a vector that I can't reproduce easily here.
15. (9, 6)
16. 115 feet, to the nearest foot.
17. (1, 4)
18. (3, -3)
19. (3, 2). (I hope students don't get confused here and solve these three backwards!)
20. This is a vector that I can't reproduce easily here.
Thus concludes Chapter 14. Stay tuned -- we're starting Chapter 15 on Tuesday, after the long Easter weekend.
No comments:
Post a Comment