-- Why is the shape of a manhole circular?
-- Why not a square, rectangle, hexagonal, or elliptical shape?
-- Is it because a circle's shape is more pleasing?
-- There is a mathematical reason.
-- What is your explanation?
This is the only page of the section "Why Are Manholes Round?" Indeed, this is the final section of the first chapter of the Pappas book -- next week we'll start Chapter 2.
And I gave the entire text on this page -- the only pictures are of the four manhole covers, one round and the other three as described in the second question. In particular, Pappas never answers the questions she asks in the title.
Think about it for a while, then click on the following link for the answer:
Apparently, this is a common job interview question!
Chapter 12 of George Szpiro's Poincare's Prize is called "The Cigar Surgeon." Here's how it begins:
"Hamilton was stuck. He had made good progress during the past two decades, but after much heroic effort he could advance no further."
Recall that the title, "The Cigar Surgeon," refers to the "cigar manifold," the one 3D manifold for which Hamilton is unable to prove the Poincare Conjecture. We left off at the point where we finally learn who has completed the proof:
"Who is this mysterious man who -- willy-nilly -- is about to become one of the icons of twenty-first century mathematics? Already as a child 'Grisha,' as his family and friends call him, seemed destined for greatness in science."
Szpiro is writing about "Grisha" -- Grigori Perelman, the Russian who proves Poincare. As we learn here, Grisha, unlike some of the other mathematicians we've seen, is not a "dren," but excels in math from his earliest years. Let's get on to his post-doc years:
"After a couple of years at Steklov, Perelman went abroad, and for mathematicians, as for many scientists, this means America."
Eventually Perelman obtains a position at Berkeley. But unlike most Californians, he doesn't like to drive, as he explains to his colleague Zlil Sela:
"The subject: Cars are unnecessary, should be avoided, and Sela should under no circumstances buy one."
Szpiro tells us that Perelman has a habit of refusing mathematical prizes, as Anatoly Vershik, a Steklov mathematician who tries to recommend him for such a prize, finds out:
"Vershik tried to reassure him. Thereupon, Perelman asked who the members of the jury were who had decided on the worthiness of the prize."
And Perelman tells the judges that the prize is just "razzmatazz," because the work for which he is awarded the prize is incomplete. The mathematician returns to his homeland, but not before he learns about the Poincare Conjecture:
"He had received job offers from first-class universities such as Stanford and Princeton but had refused them all."
Back in Russia, Perelman works on finishing Hamilton's proof of Poincare. Again, there are certain shapes that are problematic, called "singularities," that don't fit the conjecture. Some of these sre "tubes," like the hypothetical cigar manifold:
"They are either connectors between two parts of the manifold, or they are appendages, attached to the manifold at one end and capped off at the other."
Perelman's work is to show that the cigar singularity is in fact impossible:
"The story of how he did that is a historic gem. First of all, Perelman needed some special tools. One cool trick that he employed to inspect the manifolds on their way to going 'pop' had actually been around for decades and had already been used by Hamilton."
The mathematician borrows a concept from physics, "entropy." As a manifold undergoes Ricci flow, its entropy increases:
"Using his concept of entropy and some intricate mathematics, Perelman showed that manifolds cannot become too tightly rolled up."
Moreover:
"Hence only finitely many singularities can be removed before the manifold's volume would be reduced to zero."
Szpiro compares this to the story of Hercules and the Hydra:
"That is why Heracles and the Hydra belong to a mythological tale, whereas Perelman and the manifolds are real...well, at least Perelman is. The areas of the chopped necks can get smaller and smaller, but the areas of the necks between two bodies must have a definite size."
And so in finite time, the "Hydra," if you will, becomes a sphere -- and that ultimately proves the Poincare Conjecture. Perelman has done it -- he is our hero, our Hercules who finally completes the proof of the conjecture proposed a century earlier.
The rest of the chapter is about Perelman posting his proof in 2002. By 2003 he is famous, and he travels around the world to lecture about his proof. But of course, he doesn't like the limelight at all, and we recall from Chapter 1 that when he is offered a Fields Medal, he refuses it.
Szpiro describes the arXiv website where Perelman first posts his proof:
"There is no refereeing, though lately an endorsement system has been instituted. After a deluge of rubbish papers, the arXiv's advisory board decided that at least one person who has previously posted to the arXiv must give his or her blessing to a paper before it can be put into the arXiv."
And this includes "cranks" who claim to have proved the Poincare conjecture. But Perelman is aware of this:
"If you think Perelman's mode of disseminating his findings to the mathematics community a wee bit arrogant, consider this: Never -- not once -- is Poincare's name mentioned in the three arXiv postings."
And here is the conclusion of the paper:
"The paper's last, thirteenth, section is entitled 'The global picture of the Ricci flow in dimension three.'"
Regarding the Russian's fluency with the English language:
"Maybe not quite intelligible, and we'll take Perelman's word for the accuracy of the math, but there is no argument with the syntax."
Here's an excerpt of Perelman's paper as the mathematician describes it to his colleage Gang Tian:
(2) In a region, where singularity is forming in finite time, the injectivity radius is controlled by the curvature;
And indeed, regarding the paper's reception among mathematicians:
"In discussions among themselves, colleagues became more and more enthusiastic. Another few days after the first posting, a colleague sent an e-mail to Perelman asking whether it was true that his paper, and the ones that were to come would constitute a proof of the Geometrization Conjecture."
The mathematician's response was short and sweet: "That's correct." His first lectures on his paper were in April 2003 at MIT. Rob Kusner and two students were in the audience:
"The three described the event in their math department's newsletter: 'The MIT lecture theater is packed.'"
And as his fame grew:
"At a private dinner after one of the lectures, several people tried to convince Perelman to stay and work in the United States."
But of course, he doesn't take anyone up on that offer. Meanwhile, even though he just proved a famous problem in topology, former colleague Christina Sormani notices that Perelman doesn't think of himself as a topologist:
"He felt more comfortable with geometers and was attentive to questions of experts who already knew about Ricci curvature and Ricci flow."
Meanwhile, what happened to Richard Hamilton, who has been working on the problem?
"He had known for a long time that he was stuck and had openly solicited help. But after having spent the better part of two decades trying to solve the Poincare and Geometrization Conjectures, just to see a long-haired Russian wander in from obscurity to snatch away the ultimate prize, must have hurt, nonetheless."
Szpiro concludes the chapter by asking Christina Sormani where the Perelman is now:
"Sormani ventures, 'I suspect Perelman may be working much, much more than fifty hours a week, even if he pretends he is no longer doing mathematics at all.'"
This is what I wrote two years ago about today's test:
"Hamilton was stuck. He had made good progress during the past two decades, but after much heroic effort he could advance no further."
Recall that the title, "The Cigar Surgeon," refers to the "cigar manifold," the one 3D manifold for which Hamilton is unable to prove the Poincare Conjecture. We left off at the point where we finally learn who has completed the proof:
"Who is this mysterious man who -- willy-nilly -- is about to become one of the icons of twenty-first century mathematics? Already as a child 'Grisha,' as his family and friends call him, seemed destined for greatness in science."
Szpiro is writing about "Grisha" -- Grigori Perelman, the Russian who proves Poincare. As we learn here, Grisha, unlike some of the other mathematicians we've seen, is not a "dren," but excels in math from his earliest years. Let's get on to his post-doc years:
"After a couple of years at Steklov, Perelman went abroad, and for mathematicians, as for many scientists, this means America."
Eventually Perelman obtains a position at Berkeley. But unlike most Californians, he doesn't like to drive, as he explains to his colleague Zlil Sela:
"The subject: Cars are unnecessary, should be avoided, and Sela should under no circumstances buy one."
Szpiro tells us that Perelman has a habit of refusing mathematical prizes, as Anatoly Vershik, a Steklov mathematician who tries to recommend him for such a prize, finds out:
"Vershik tried to reassure him. Thereupon, Perelman asked who the members of the jury were who had decided on the worthiness of the prize."
And Perelman tells the judges that the prize is just "razzmatazz," because the work for which he is awarded the prize is incomplete. The mathematician returns to his homeland, but not before he learns about the Poincare Conjecture:
"He had received job offers from first-class universities such as Stanford and Princeton but had refused them all."
Back in Russia, Perelman works on finishing Hamilton's proof of Poincare. Again, there are certain shapes that are problematic, called "singularities," that don't fit the conjecture. Some of these sre "tubes," like the hypothetical cigar manifold:
"They are either connectors between two parts of the manifold, or they are appendages, attached to the manifold at one end and capped off at the other."
Perelman's work is to show that the cigar singularity is in fact impossible:
"The story of how he did that is a historic gem. First of all, Perelman needed some special tools. One cool trick that he employed to inspect the manifolds on their way to going 'pop' had actually been around for decades and had already been used by Hamilton."
The mathematician borrows a concept from physics, "entropy." As a manifold undergoes Ricci flow, its entropy increases:
"Using his concept of entropy and some intricate mathematics, Perelman showed that manifolds cannot become too tightly rolled up."
Moreover:
"Hence only finitely many singularities can be removed before the manifold's volume would be reduced to zero."
Szpiro compares this to the story of Hercules and the Hydra:
"That is why Heracles and the Hydra belong to a mythological tale, whereas Perelman and the manifolds are real...well, at least Perelman is. The areas of the chopped necks can get smaller and smaller, but the areas of the necks between two bodies must have a definite size."
And so in finite time, the "Hydra," if you will, becomes a sphere -- and that ultimately proves the Poincare Conjecture. Perelman has done it -- he is our hero, our Hercules who finally completes the proof of the conjecture proposed a century earlier.
The rest of the chapter is about Perelman posting his proof in 2002. By 2003 he is famous, and he travels around the world to lecture about his proof. But of course, he doesn't like the limelight at all, and we recall from Chapter 1 that when he is offered a Fields Medal, he refuses it.
Szpiro describes the arXiv website where Perelman first posts his proof:
"There is no refereeing, though lately an endorsement system has been instituted. After a deluge of rubbish papers, the arXiv's advisory board decided that at least one person who has previously posted to the arXiv must give his or her blessing to a paper before it can be put into the arXiv."
And this includes "cranks" who claim to have proved the Poincare conjecture. But Perelman is aware of this:
"If you think Perelman's mode of disseminating his findings to the mathematics community a wee bit arrogant, consider this: Never -- not once -- is Poincare's name mentioned in the three arXiv postings."
And here is the conclusion of the paper:
"The paper's last, thirteenth, section is entitled 'The global picture of the Ricci flow in dimension three.'"
Regarding the Russian's fluency with the English language:
"Maybe not quite intelligible, and we'll take Perelman's word for the accuracy of the math, but there is no argument with the syntax."
Here's an excerpt of Perelman's paper as the mathematician describes it to his colleage Gang Tian:
(2) In a region, where singularity is forming in finite time, the injectivity radius is controlled by the curvature;
And indeed, regarding the paper's reception among mathematicians:
"In discussions among themselves, colleagues became more and more enthusiastic. Another few days after the first posting, a colleague sent an e-mail to Perelman asking whether it was true that his paper, and the ones that were to come would constitute a proof of the Geometrization Conjecture."
The mathematician's response was short and sweet: "That's correct." His first lectures on his paper were in April 2003 at MIT. Rob Kusner and two students were in the audience:
"The three described the event in their math department's newsletter: 'The MIT lecture theater is packed.'"
And as his fame grew:
"At a private dinner after one of the lectures, several people tried to convince Perelman to stay and work in the United States."
But of course, he doesn't take anyone up on that offer. Meanwhile, even though he just proved a famous problem in topology, former colleague Christina Sormani notices that Perelman doesn't think of himself as a topologist:
"He felt more comfortable with geometers and was attentive to questions of experts who already knew about Ricci curvature and Ricci flow."
Meanwhile, what happened to Richard Hamilton, who has been working on the problem?
"He had known for a long time that he was stuck and had openly solicited help. But after having spent the better part of two decades trying to solve the Poincare and Geometrization Conjectures, just to see a long-haired Russian wander in from obscurity to snatch away the ultimate prize, must have hurt, nonetheless."
Szpiro concludes the chapter by asking Christina Sormani where the Perelman is now:
"Sormani ventures, 'I suspect Perelman may be working much, much more than fifty hours a week, even if he pretends he is no longer doing mathematics at all.'"
This is what I wrote two years ago about today's test:
Let's look at these final four questions in more detail. Questions 17 and 18 are graphing questions, except that one is transforming triangles, not snowmen. One of them is a glide reflection, while the other is a translation. I just hope that students won't be thrown off by seeing rules for each of these transformations, such as T(x, y) = (x + 5, -y).
Question 19 is about the cardinality of a set, N(S), which is mentioned briefly in Lesson 6-1 of the U of Chicago text, but I only discussed it briefly this year. Here's what I wrote about N(S) last year:
I decided that the only real reason that the U of Chicago introduces the N(S) notation for cardinality (number of elements in a set, previous question) is to prepare the students for function notation, so I might as well use it here. There's only one other place where I see n(A) used for number of elements in set A -- the Singapore Secondary Two standards!
Last year, I also wrote: (A Thanksgiving reference! These are the seven dates in November which could be turkey day!) Again, I originally wrote this test last year at Thanksgiving. Okay, this year we can pretend that the seven elements of S correspond to Christmas Day and the three days before and after Christmas.
The final question shows one more transformation -- which happens to be a dilation. Neither last year nor this year did I formally cover dilations. This is supposed to be a think-outside-the-box question where students should try to reason out what's going on. But think about it for a moment -- the graph makes it appear that (3, 3) is the image of (1, 1). So all students have to do is plug in x = y = 1 into each of the four choices and see that only choice (d) gives (3, 3) as the answer. (Of course, this year, the students have seen dilations because of Tom Turkey and Thanksgiving again.)
Students might consider the last four questions to be unfair. But even if they get all four wrong, it's still possible to get 80% -- the lowest possible B. So strong students who completed the review worksheet yesterday should still earn at least a C on this test.
Here are the answers to today's test -- the same answers I posted last year to the invisible test:
1. a translation 2 inches to the left
2. a translation 2 inches to the right
3. a rotation with center O and magnitude 180 degrees
4. a translation 8 centimeters to the right
5. true
6. angles D and G
7. triangle DEF, triangle GHI
8. Reflexive Property of Congruence
9. definition of congruence
10. Isometries preserve distance.
11. translation
12. translation
13. glide reflection
14. glide reflection
15.-16. The trick is to reflect the hole H twice, over the walls in reverse order, and then aim the golf ball G towards the image point H". In #15, notice that y and w are parallel, so reflecting in both of them is equivalent to a translation twice the length of the course. In #16, notice that x and y are perpendicular, so reflecting in both of them is equivalent to a 180-degree rotation.
17. glide reflection (changing the sign of y is the reflection part, adding to x is the translation part)
18. translation
19. 7
20. d (for dilation, of course!)
Now today's a test day, and it's been a while since I posted a topic about traditionalists, so let's make this our traditionalist post.
On Black Friday, Barry Garelick made the following post:
https://traditionalmath.wordpress.com/2017/11/24/how-to-write-a-pro-common-core-21st-century-skills-based-polemic/
You’ve probably seen this before. Someone, probably in their twenties, thinking they are holding their own among a group of highly educated people, prattling on about the uselessness of most college courses and disciplines and the value of a PhD. The group listens politely and after the outburst continues their conversation as if the young orator hadn’t said a word. The young person thinks that they are pretending they didn’t hear the polemic because they didn’t want to hear the truth.
I recently read a piece published by Achieve.org which unequivocally and uncritically supports the Common Core standards. One such piece caught my attention and since the writing of poorly informed and unscientific polemics seems to be the new standard, I thought I would provide a guideline on how to construct such papers, using this particular atrocity.
This post has drawn nine comments, and do I even need to tell you who made six of those comments?
SteveH:
Lane Walker redefines STEM (which was originally defined by educators and NOT STEM people – it’s a stupid term) and raises many strawmen, like being successful in math with just rote understandings. Can we once and for all lose that self-serving stupidity? Her goal is to redefine Algebra II to match the pseudo Algebra II of CCSS, but she never deals with needs of various career paths or how CCSS defines a no remediation in college math slope that starts in Kindergarten. I was amazed at the audacity of my son’s Kindergarten and first grade teachers when they lectured me on understanding in math. I never got that from my son’s traditional (generally from industry) teachers in his high school AP Calculus track.
[Well, it's been a while since SteveH has referred to Common Core as "pseudo Algebra II"!]
SteveH:
And finally, why do we never hear about individual style of learning applied to school choice? What seems to matter is NOT critical thinking, but rationalization and enforcement of their philosophy.
[Of course, traditionalists only care about "choice" because they want students and parents to choose the traditionalist pedagogy. If traditionalism were the dominant pedagogy in schools, then they wouldn't care about choice at all.]
SteveH:
Then she refers to Andrew Hacker’s “The Math Myth” and says: “I wholeheartedly agree with Hacker and others he quotes that Algebra 2, touted as the gateway to STEM opportunities, has done more to lower graduation rates than any other course.” Pseudo-Algebra II as a CCSS requirement and lower graduation rates have nothing to do with STEM career preparation. If you fail at Algebra II or are allowed to skip it, then many more than STEM careers will be eliminated.
[But if you require students to take true Algebra II and deny a diploma to those who fail it, then many non-STEM careers will be eliminated.]
Next, SteveH criticizes the proposed "pre-AP" curriculum:
Ninth grade?!? Way too late. Hello? Is anyone home? They don’t get it. One of the five classes (9th grade) is called Pre-AP Algebra 1. Really? Really?!? What is their curriculum sequence to AP Calculus- geometry in 10th grade, trig/pre-calc in 11th? Um, they don’t say.
[OK, SteveH does have a point here -- how can freshman Algebra I be "pre-AP" if it doesn't lead to the actual AP class, Calculus?]
SteveH:
How would that work for sports – have low expectation, fun soccer camps where there is no separation by level or ability? That’s generally the rule for early grades, but it very quickly changes. Are academics different even though all AP classes focus on mastery of skills and content. Pre-AP values skills and content knowledge, but they are allowed to be trashed in K-6 and then they talk about “social justice.” They try to claim the high ground of understanding and social justice, but they fail both. In high school they offer AP classes and never ask us parents how we had to support and push our kids to get them there. It must be magic fairy dust PBL engagement.
[SteveH seems to be advocating tracking again, and gives "social justice" as the reason that full-blown tracking isn't done in schools. Of course, if parents "track" at home as SteveH suggests, then the school's hands are clean as far as discriminatory tracking is concerned.]
I quote from only five of six SteveH's posts since the remaining post is short and repeats points he makes in the other five.
Question 19 is about the cardinality of a set, N(S), which is mentioned briefly in Lesson 6-1 of the U of Chicago text, but I only discussed it briefly this year. Here's what I wrote about N(S) last year:
I decided that the only real reason that the U of Chicago introduces the N(S) notation for cardinality (number of elements in a set, previous question) is to prepare the students for function notation, so I might as well use it here. There's only one other place where I see n(A) used for number of elements in set A -- the Singapore Secondary Two standards!
Last year, I also wrote: (A Thanksgiving reference! These are the seven dates in November which could be turkey day!) Again, I originally wrote this test last year at Thanksgiving. Okay, this year we can pretend that the seven elements of S correspond to Christmas Day and the three days before and after Christmas.
The final question shows one more transformation -- which happens to be a dilation. Neither last year nor this year did I formally cover dilations. This is supposed to be a think-outside-the-box question where students should try to reason out what's going on. But think about it for a moment -- the graph makes it appear that (3, 3) is the image of (1, 1). So all students have to do is plug in x = y = 1 into each of the four choices and see that only choice (d) gives (3, 3) as the answer. (Of course, this year, the students have seen dilations because of Tom Turkey and Thanksgiving again.)
Students might consider the last four questions to be unfair. But even if they get all four wrong, it's still possible to get 80% -- the lowest possible B. So strong students who completed the review worksheet yesterday should still earn at least a C on this test.
Here are the answers to today's test -- the same answers I posted last year to the invisible test:
1. a translation 2 inches to the left
2. a translation 2 inches to the right
3. a rotation with center O and magnitude 180 degrees
4. a translation 8 centimeters to the right
5. true
6. angles D and G
7. triangle DEF, triangle GHI
8. Reflexive Property of Congruence
9. definition of congruence
10. Isometries preserve distance.
11. translation
12. translation
13. glide reflection
14. glide reflection
15.-16. The trick is to reflect the hole H twice, over the walls in reverse order, and then aim the golf ball G towards the image point H". In #15, notice that y and w are parallel, so reflecting in both of them is equivalent to a translation twice the length of the course. In #16, notice that x and y are perpendicular, so reflecting in both of them is equivalent to a 180-degree rotation.
17. glide reflection (changing the sign of y is the reflection part, adding to x is the translation part)
18. translation
19. 7
20. d (for dilation, of course!)
Now today's a test day, and it's been a while since I posted a topic about traditionalists, so let's make this our traditionalist post.
On Black Friday, Barry Garelick made the following post:
https://traditionalmath.wordpress.com/2017/11/24/how-to-write-a-pro-common-core-21st-century-skills-based-polemic/
You’ve probably seen this before. Someone, probably in their twenties, thinking they are holding their own among a group of highly educated people, prattling on about the uselessness of most college courses and disciplines and the value of a PhD. The group listens politely and after the outburst continues their conversation as if the young orator hadn’t said a word. The young person thinks that they are pretending they didn’t hear the polemic because they didn’t want to hear the truth.
I recently read a piece published by Achieve.org which unequivocally and uncritically supports the Common Core standards. One such piece caught my attention and since the writing of poorly informed and unscientific polemics seems to be the new standard, I thought I would provide a guideline on how to construct such papers, using this particular atrocity.
This post has drawn nine comments, and do I even need to tell you who made six of those comments?
SteveH:
Lane Walker redefines STEM (which was originally defined by educators and NOT STEM people – it’s a stupid term) and raises many strawmen, like being successful in math with just rote understandings. Can we once and for all lose that self-serving stupidity? Her goal is to redefine Algebra II to match the pseudo Algebra II of CCSS, but she never deals with needs of various career paths or how CCSS defines a no remediation in college math slope that starts in Kindergarten. I was amazed at the audacity of my son’s Kindergarten and first grade teachers when they lectured me on understanding in math. I never got that from my son’s traditional (generally from industry) teachers in his high school AP Calculus track.
[Well, it's been a while since SteveH has referred to Common Core as "pseudo Algebra II"!]
SteveH:
And finally, why do we never hear about individual style of learning applied to school choice? What seems to matter is NOT critical thinking, but rationalization and enforcement of their philosophy.
[Of course, traditionalists only care about "choice" because they want students and parents to choose the traditionalist pedagogy. If traditionalism were the dominant pedagogy in schools, then they wouldn't care about choice at all.]
SteveH:
Then she refers to Andrew Hacker’s “The Math Myth” and says: “I wholeheartedly agree with Hacker and others he quotes that Algebra 2, touted as the gateway to STEM opportunities, has done more to lower graduation rates than any other course.” Pseudo-Algebra II as a CCSS requirement and lower graduation rates have nothing to do with STEM career preparation. If you fail at Algebra II or are allowed to skip it, then many more than STEM careers will be eliminated.
[But if you require students to take true Algebra II and deny a diploma to those who fail it, then many non-STEM careers will be eliminated.]
Next, SteveH criticizes the proposed "pre-AP" curriculum:
Ninth grade?!? Way too late. Hello? Is anyone home? They don’t get it. One of the five classes (9th grade) is called Pre-AP Algebra 1. Really? Really?!? What is their curriculum sequence to AP Calculus- geometry in 10th grade, trig/pre-calc in 11th? Um, they don’t say.
[OK, SteveH does have a point here -- how can freshman Algebra I be "pre-AP" if it doesn't lead to the actual AP class, Calculus?]
SteveH:
How would that work for sports – have low expectation, fun soccer camps where there is no separation by level or ability? That’s generally the rule for early grades, but it very quickly changes. Are academics different even though all AP classes focus on mastery of skills and content. Pre-AP values skills and content knowledge, but they are allowed to be trashed in K-6 and then they talk about “social justice.” They try to claim the high ground of understanding and social justice, but they fail both. In high school they offer AP classes and never ask us parents how we had to support and push our kids to get them there. It must be magic fairy dust PBL engagement.
[SteveH seems to be advocating tracking again, and gives "social justice" as the reason that full-blown tracking isn't done in schools. Of course, if parents "track" at home as SteveH suggests, then the school's hands are clean as far as discriminatory tracking is concerned.]
I quote from only five of six SteveH's posts since the remaining post is short and repeats points he makes in the other five.
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