Which is not a Platonic solid? Figure 1, 2, 3, or 4?
1) Dodecahedron
2) Square Pyramid
3) Cube
4) Octahedron
The U of Chicago text only discusses this in a Exploration question as part of Lesson 9-7 -- it's not part of the main curriculum:
"A regular polyhedron [or Platonic solid] is a convex polyhedron in which all faces are congruent polygons and the same number of edges intersect at each of its vertices."
The dodecahedron, cube, and octahedron are all Platonic solids. But the square pyramid isn't, because its faces are congruent. Indeed, four of its faces are triangles and one is a square. Therefore the right answer is 2) -- and of course, today's date is the second. Notice that on the Pappas calendar, multiple choice questions usually appear during the first week of each month -- otherwise the number of answer choices would overwhelm us, since we'd have to provide 31 answer choices on Halloween.
Chapter 12 of Eugenia Cheng's The Art of Logic in an Illogical World, "Fine Lines and Gray Areas," begins as follows:
"One night during my first term at university another fresher was found in the kitchen just before midnight eating a bowl of cereal. He explained that, according to the best-before date, his milk was about to go off at midnight."
The point of this story, of choice, is that there's no fine line or exact moment when the milk changes from unspoiled to spoiled. Likewise, there's no single moment when one person falls in love. Cheng quotes from Jane Austen's Pride and Prejudice as Mr. Darcy says to Elizabeth:
"I cannot fix on the hour, or the spot, or the look, or the words, which laid the foundation. It is too long ago. I was in the middle before I knew that I had begun."
Cheng reminds us that although logic may be black and white, the world rarely is. There are often gray areas:
"But I worry about the world turning into certainties that are almost certainly flawed. We should understand different ways of dealing with gray areas and become better at working with their nuance, instead of longing for the false promise of black and white clarity."
As with many of her examples, the author likes to return to cake. She writes:
- It won't hurt to eat one small piece of cake.
- And however much cake I've already eaten, it can't hurt to eat one more mouthful.
As Cheng points out, we can then prove that it's okay to eat arbitrarily much cake. This is an example where the logic of the situation pushes us to one of two extreme positions, either:
- it is not okay to eat any cake at all, or
- it is okay to eat infinite amounts of cake.
The trouble is the gray area. She tells us that this is what happens when we try to eat one more bite of cake, or when a child tries to stay up an extra two minutes:
"But really, the bedtime itself is spurious -- it is an arbitrary line that has been set in a gray area between 'sensible bed time' and 'much too late.' One way to get round this logic is just to shrug and say just because something is logically implied, that doesn't mean I'm going to believe it."
At this point Cheng refers to the "deductive closure" -- the deductive closure of a set of axioms is the set of all statements that can be proved from those axioms. She tells us that having a deductively closed set of beliefs is part of being a logical human being -- but gray areas are problematic.
The author asks, so where do we draw the line? When it comes to eating cake, she writes that she should draw the line on the safe side:
"This is especially true because I am liable to stray over my line a but, so I should put a little buffer zone in to be on the safe side."
Now Cheng proceeds to write about grade boundaries -- something that's important to us teachers:
"In the UK system, students graduate from university with a degree that is classed as first class, 'upper second' class, 'lower second' class, or third class, otherwise known as first, 2:1, 2:2, or third. But where should the boundaries be drawn?"
Try changing this to "US Common Core exams" (like SBAC or PARCC) and suddenly we wonder where to draw lines -- the cut scores between 1, 2, 3, and 4 (and 5 for PARCC). She points out:
"No matter where you place it, someone will argue that it's unfair to the person just below it, and as a result the line tends to shift further and further down. There is no logical place to put that line. I think the only logical thing to do is get rid of the lines and publish either averages on a fully sliding scale or percentiles instead."
She also writes about Obama. How could he be the first black president if his mother is white? Now according to Cheng, "black" here is being used to mean "non-white""
"At least talking about white people and non-white people is a genuine dichotomy, where black and white is a false one."
Again, I remind you that Cheng writes about race and politics throughout her book. If you prefer not to read this, then I suggest that you avoid this blog for the next week and skip all posts that have the "Eugenia Cheng" label.
This is a great time for me to put in that disclaimer, since Cheng now goes into sexual harassment:
"It can be especially hard to draw lines when dealing with people who do not respect boundaries. This can happen if you are the kind of person who likes being generous and being able to help people."
And of course, all the hearings and FBI investigations are all about whether a certain man crossed such a line with a certain woman 36 years ago. Again, the author reminds us that it's difficult to know exactly when a line has been crossed:
"Mr. Darcy's inability to draw the line for love also applies to hurt -- we don't necessarily know where is the exact moment that something is going to start hurting us. We only know it will definitely hurt us a lot, like if inappropriate physical contact becomes rape."
And so Cheng draws her line closer to the safe side, thus leaving a buffer zone where's it's not necessarily dangerous yet, but for extra protection. (Note that this refers to when the contact is actually happening and not necessarily in a hearing 36 years after the fact.)
The author now proceeds to discuss Body Mass Index, defined as mass over height squared:
"The first thing that people object to about this is that it doesn't take into account how muscular you are, and so very strong athletes are liable to have a high BMI as muscle is very dense."
For women like Cheng, the cutoff to remain healthy is about 25. But to remain on the safe side, Cheng tries to remain around a BMI of 24:
"One interpretation of what I'm doing is treating the line itself as something hazy, so I try to stay far enough on the good side of the line that I'm out of its hazy range."
The author's next topic is mathematical induction. I've mentioned induction on the blog several times before, including last year during another side-along reading book (Stanley Ogilvy). Cheng returns to her favorite example, baked goods:
"No matter where you place it, someone will argue that it's unfair to the person just below it, and as a result the line tends to shift further and further down. There is no logical place to put that line. I think the only logical thing to do is get rid of the lines and publish either averages on a fully sliding scale or percentiles instead."
She also writes about Obama. How could he be the first black president if his mother is white? Now according to Cheng, "black" here is being used to mean "non-white""
"At least talking about white people and non-white people is a genuine dichotomy, where black and white is a false one."
Again, I remind you that Cheng writes about race and politics throughout her book. If you prefer not to read this, then I suggest that you avoid this blog for the next week and skip all posts that have the "Eugenia Cheng" label.
This is a great time for me to put in that disclaimer, since Cheng now goes into sexual harassment:
"It can be especially hard to draw lines when dealing with people who do not respect boundaries. This can happen if you are the kind of person who likes being generous and being able to help people."
And of course, all the hearings and FBI investigations are all about whether a certain man crossed such a line with a certain woman 36 years ago. Again, the author reminds us that it's difficult to know exactly when a line has been crossed:
"Mr. Darcy's inability to draw the line for love also applies to hurt -- we don't necessarily know where is the exact moment that something is going to start hurting us. We only know it will definitely hurt us a lot, like if inappropriate physical contact becomes rape."
And so Cheng draws her line closer to the safe side, thus leaving a buffer zone where's it's not necessarily dangerous yet, but for extra protection. (Note that this refers to when the contact is actually happening and not necessarily in a hearing 36 years after the fact.)
The author now proceeds to discuss Body Mass Index, defined as mass over height squared:
"The first thing that people object to about this is that it doesn't take into account how muscular you are, and so very strong athletes are liable to have a high BMI as muscle is very dense."
For women like Cheng, the cutoff to remain healthy is about 25. But to remain on the safe side, Cheng tries to remain around a BMI of 24:
"One interpretation of what I'm doing is treating the line itself as something hazy, so I try to stay far enough on the good side of the line that I'm out of its hazy range."
The author's next topic is mathematical induction. I've mentioned induction on the blog several times before, including last year during another side-along reading book (Stanley Ogilvy). Cheng returns to her favorite example, baked goods:
- It's fine to eat 1 cookie.
- If it was fine to eat some number of cookies, it's okay to eat 1 more.
Therefore it's fine to eat any number of cookies.
Cheng also writes this by letting P(n) be the statement "It is fine to eat n cookies":
- P(n) is true.
- P(n) => P(n + 1).
Then by the principle of mathematical induction, P(n) is true for all whole numbers n. She proceeds:
"This is fine for whole numbers, but it gets tricky if you're trying to deal with a sliding scale that includes all the numbers in between, or even just all possible fractions."
And indeed, Cheng once passed out cookies to 20 students. The cookies she handed to adjacent students are almost the same size, yet the last cookie is twice the size of the first cookie!
The author now discusses fuzzy logic, where there are truth values between "true" and "false":
"It also might be true in the case of probability, where we can't be certain what the truth is, we can only be a certain percentage sure, with the rest being in some doubt."
This applies to weather forecasts (chance of rain), as well as percentages given on a test (as opposed to just pass/fail):
"Fuzzy logic is currently used more in applied engineering than in math, to deal with gray areas in control of digital devices."
Examples include rice cookers and heating/air conditioning units.
Cheng now returns to cookie sizes. She shows us a picture of her batch of cookies -- the one where each cookie has a slightly different size:
"This way I can meet my own needs and other people's at the same time, without actually having to know what anyone else's perfect size of cookie is -- I can be sure it's in there somewhere. This is an application of the intermediate value theorem, a theorem in rigorous calculus that math students usually study as undergraduates. It says that if you have a continuous function that starts at 0 and goes up to some number a, it must take every value in between."
Last year, Cheng talked to an art student in Chicago who created visual illusions and tried to see if viewers would believe if they were real or digitally manipulated:
"The question was to find the sweet spot where people would be really unsure which it was. I realized that she could invoke the intermediate value theorem: make a series of pieces starting with one that was obviously a physical construction, gradually becoming less obvious until she ended with one that was obviously physically impossible so must be a digital manipulation."
And of course her "sweet spot" was somewhere in between.
Cheng returns to races. What should we call someone who is mixed Asian and white, depending on whether that person is more Asian or more white?
"As described throughout this chapter there are various possibilities, each of which has advantages and disadvantages."
The goal, Cheng tells us, is to bridge the gap between the two sides, since in a way, we are all living on a gray bridge between black and white:
"If we all acknowledge that, and build even more bridges, I think we will achieve better understanding."
Her final example of such bridge-building is on social services:
"The person who believes that everyone should take responsibility for themselves might be able to acknowledge that some particularly 'worthy' people need help, perhaps members of the military who have been injured during active service."
Cheng concludes by telling us how we might draw such a person onto our bridge:
"The first step is to think about understanding a difficult argument by comparing it with a more understandable one that has something in common, that is, it is analogous in some way. This is the subject of the next chapter."
Lesson 3-4 of the U of Chicago text is called "Parallel Lines." (It appears as Lesson 3-6 in the modern edition of the text.)
In the past, I was always unsure how I wanted to teach parallel lines. Over the years, I kept changing the way I presented this topic, for various reasons. The U of Chicago text teaches parallel lines the way it's done in most Geometry books, with two postulates:
Corresponding Angles Postulate:
If two coplanar lines are cut by a transversal so that two corresponding angles have the same measure, then the lines are parallel.
Parallel Lines Postulate:
If two lines are parallel and cut by a transversal, corresponding angles have the same measure.
But Dr. David Joyce, an education critic, wrote that there should only be one parallel postulate. In fact, we know that not only does Euclid have only one Parallel Postulate (his famous fifth postulate), but he doesn't even need the postulate it in his proof of the "Corresponding Angles Postulate." Instead he uses something called the Triangle Exterior Angle Inequality, or TEAI. On the other hand, his proof of the "Parallel Lines Postulate" is based on his fifth postulate.
Another education writer, Dr. Hung-Hsi Wu of Berkeley, also proves the Corresponding Angles Postulate without need of any parallel postulate. Instead, Wu uses Common Core transformations -- specifically 180-degree rotations -- to attain this result.
Four years ago, I used Wu's approach and based my parallel lessons on these half-turns. But after that, I felt that this was unsatisfactory for two reasons:
In the past, I was always unsure how I wanted to teach parallel lines. Over the years, I kept changing the way I presented this topic, for various reasons. The U of Chicago text teaches parallel lines the way it's done in most Geometry books, with two postulates:
Corresponding Angles Postulate:
If two coplanar lines are cut by a transversal so that two corresponding angles have the same measure, then the lines are parallel.
Parallel Lines Postulate:
If two lines are parallel and cut by a transversal, corresponding angles have the same measure.
But Dr. David Joyce, an education critic, wrote that there should only be one parallel postulate. In fact, we know that not only does Euclid have only one Parallel Postulate (his famous fifth postulate), but he doesn't even need the postulate it in his proof of the "Corresponding Angles Postulate." Instead he uses something called the Triangle Exterior Angle Inequality, or TEAI. On the other hand, his proof of the "Parallel Lines Postulate" is based on his fifth postulate.
Another education writer, Dr. Hung-Hsi Wu of Berkeley, also proves the Corresponding Angles Postulate without need of any parallel postulate. Instead, Wu uses Common Core transformations -- specifically 180-degree rotations -- to attain this result.
Four years ago, I used Wu's approach and based my parallel lessons on these half-turns. But after that, I felt that this was unsatisfactory for two reasons:
- The Corresponding Angles results are more apparently related to translations than rotations -- after all, if two parallel lines are cut by a transversal, then a translation clearly maps one corresponding angle to another. Indeed, I often wonder why we use the same word "corresponding" to refer to angles formed by a transversal and to refer to parts of congruent (or similar) triangles. In the case of congruent (or similar) triangles, there is by definition an isometry (or similarity transformation) mapping one to the other. Well, in the case of angles formed by a transversal, there is a translation mapping one to the other. This justifies use of the term "corresponding angles."
- The idea of delaying the Parallel Postulate until it is needed comes from the idea of "neutral geometry," based on theorems proved using only the other four postulates of Euclid. But we found out that there are two types of neutral geometry -- Euclidean geometry (where the fifth postulate is true) and hyperbolic geometry (where it is false). If we're going to mention non-Euclidean geometry at all, I'd prefer spherical geometry -- as in the spherical earth -- over hyperbolic geometry. But unfortunately, hyperbolic geometry isn't neutral. At least one of the first four postulates fails in spherical geometry -- whereas the fifth postulate, ironically, does hold.
[2018 update: And of course over the summer I wrote about spherical geometry.]
Should I attempt to solve these two problems? In the end, these issues don't matter unless they actually affect teaching in the classroom. The second problem, with neutral geometry, matters only if we want, say, to introduce our students to non-Euclidean geometry at the end of the year -- and if were to do that, we'd want to show them spherical, not hyperbolic, geometry. (I've mentioned before how some high schools actually do this.)
Should I attempt to solve these two problems? In the end, these issues don't matter unless they actually affect teaching in the classroom. The second problem, with neutral geometry, matters only if we want, say, to introduce our students to non-Euclidean geometry at the end of the year -- and if were to do that, we'd want to show them spherical, not hyperbolic, geometry. (I've mentioned before how some high schools actually do this.)
[2018 update: Over the summer, I came up with the conclusion that the answer is "no," especially for the second question.]
It would be nice if there was a clean break between "natural geometry" (that is, Euclidean+spherical geometry) and Euclidean geometry so we can say, "these Euclidean results still hold in spherical geometry while those don't" -- just as there's a clean break between "neutral geometry" (that is, Euclidean+hyperbolic geometry) and Euclidean geometry (simply by introducing the parallel postulate).
[2018 update: What was Cheng just saying about drawing "lines" and "clean breaks"? Actually, this is a discussion about axioms, which was yesterday's Cheng chapter. The problem is that an axiomatization for spherical geometry isn't as straightforward as it is for the other two types.]
But does the first issue really matter? I wanted to use translations to demonstrate the Corresponding Angles Postulate in my class...
[2018 update: What was Cheng just saying about drawing "lines" and "clean breaks"? Actually, this is a discussion about axioms, which was yesterday's Cheng chapter. The problem is that an axiomatization for spherical geometry isn't as straightforward as it is for the other two types.]
But does the first issue really matter? I wanted to use translations to demonstrate the Corresponding Angles Postulate in my class...
...oops, ixnay on the arterchay athmay! So let's just skip this discussion from last year and proceed directly to the worksheets:
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