A TOK Lecture on Mathematical Thinking

Students in our International Baccalaureate program here at RM are required to take a core class called Theory of Knowledge (TOK) which is kind of a philosophy class for high school students–or, at least the epistemology piece.

In some schools, this course is taught by math teachers. Here at RM, no math teachers currently teach TOK, which is too bad. So I volunteered to put together a guest lecture on Mathematical Thinking. I’ve tried it out once with a TOK class and I gave the lecture for some of my math teacher colleagues today after school. I plan to give the lecture to more TOK classes this spring.

I thought I’d share it with the MTBoS as well, so here it is. Feel free to read, comment on, or borrow my materials. I think other IB math teachers would especially benefit:

Geometry Is Beautiful

Header of Dimensions

Find nine 14-minute videos on geometry here. They’re breathtakingly beautiful!.

  • 1 Mapmaking: Stereographic projection of Earth
  • 9 Proving: the essence of mathematics
  • 2 M.C. Escher: Stereographic projection of the Platonic solids
  • 3 Four-dimensional polyhedra: Simplex, hypercube, 24-cell, 120-cell, 600-cell
  • 4 The three-sphere and 4D polyhedra viewed stereographically
  • 5 Complex numbers: Now a sphere is a “complex projective line”
  • 6 Transformations with complex numbers: Some Möbius (linear fractional) transformations, some fractals (quadratic)]
  • 7-8 Hopf fibration of three-sphere

Each video starts very simply and gets harder, but the computer graphics are so incredibly stunning that you will stay for the pictures even after you lose the narration. They are somewhat cumulative. Don’t jump into the middle. Number 9 can be watched at any time. The authors recommend last or second.

[Posted & edited by Dr. Gene Chase. Reviewed in American Mathematical Monthly, vol. 120, no. 3, March 2013, pp. 288-290.]

Fearless Symmetry

I come to you today with a recommendation for the book Fearless Symmetry by Avner Ash and Robert Gross. I started it this summer and finally had a chance to finish it over the Christmas break. I didn’t understand the last half-dozen chapters, but my dad did warn me that would happen. I wouldn’t even attempt reading it unless you’ve already been exposed to some undergraduate mathematics. But if you have, or if it’s been a while and you need a refresher, I highly recommend the book.

In the book, Ash and Gross attempt to explain some of the math underlying Wiles’ proof of Fermat’s Last Theorem. So you can understand why the math gets a bit hard at the end.

Along the way, you’ll get a very conversational, well-written, fun-loving introduction to the Absolute Galois Group of the Algebraic numbers. This is a group that is so complicated and messy and theoretical that we can only explicitly write down two elements of the group. In order to talk about it, we need representations, which the authors also introduce in a gentle way. In particular, we need linear representations.

Elliptic curves become very important too. I have studied elliptic curves in two of my classes before, but I really liked the way they introduced them here: We know everything about linear equations (highest exponent 1), and everything about conics (highest exponent 2 on x and y), but suddenly things become very interesting when we allow just ONE of the exponents (on x) to jump to 3. These are elliptic curves. Amazingly, you can define an arithmetic on the points of an elliptic curve that yield both a GROUP and an algebraic VARIETY. Incredible. Of course, the authors introduce what a variety is too.

After reading this, I also gained a much bigger view of abstract algebra–a course I’ve taken, but I found myself guilty of seeing the trees but not the forest. I loved the way Ash and Gross introduce the group SO3 and relate it to A4 with the rotations of a sphere inside a shell. Very nice visualization!

I could go on, but just know that there are lots of little mathematical gems scattered throughout this book. It’s a refreshing jaunt through higher-level mathematics that will demystify some of the smart-sounding words you’ve been afraid to ask about :-).

Go check it out!

87th Carnival of Mathematics

The 87th Carnival of Mathematics has arrived!! Here’s a simple computation for you:

What is the sum of the squares of the first four prime numbers?

That’s right, it’s

Good job. Now, onto the carnival. This is my first carnival, so hopefully I’ll do all these posts justice. We had lots of great submissions, so I encourage you to read through this with a fine-toothed comb. Enjoy!


Here’s a post (rant) from Andrew Taylor regarding the coverage from the BBC and the Guardian on the Supermoon that occurred in March 2011. NASA reports the moon as being 14% larger and 30% brighter, but Andrew disagrees. Go check out the post, and join the conversation.

Have you ever heard someone abuse the phrase “exponentially better”? I know I have. One incorrect usage occurs when someone makes the claim that something is “exponentially better” based on only two data points. Rebecka Peterson has some words for you here, if you’re the kind of person who says this!

Physics and Science-flavored

Frederick Koh submitted Problem 19: Mechanics of Two Separate Particles Projected Vertically From Different Heights to the carnival. It’s a fun projectile motion question which would be appropriate for a Precalculus classroom (or Calculus). I like the problem, and I think my students would like it too.

John D. Cook highlights a question you’ve probably heard before: Should you walk or run in the rain? An active discussion is going on in the comments section. It’s been discussed in many other places too, including twice on Mythbusters. (I feel like I read an article in an MAA or NCTM magazine on this topic once, as well. Anyone remember that?)

Murray Bourne submitted this awesome post about modeling fish stocks. Murray says his post is an “attempt to make mathematical modeling a bit less scary than in most textbooks.” I think he achieves his goal in this thorough development of a mathematical model for sustainable fisheries (see the graph above for one of his later examples of a stable solution under lots of interesting constraints). If I taught differential equations, I would  absolutely use his examples.

Last week I highlighted this new physics blog, but I wanted to point you there again: Go check out Five Minute Physics! A few more videos have been posted, and also a link to this great video about the physics of a dropping Slinky (see above).

Statistics, Probability, & Combinatorics

Mr. Gregg analyzes European football using the Poisson distribution in his post, The Table Never Lies. I liked how much real world data he brought to the discussion. And I also liked that he admitted when his model worked and when it didn’t–he lets you in on his own mathematical thought process. As you read this post, you too will find yourself thinking out loud with Mr. Gregg.

Card Colm has written this excellent post that will help you wrap your mind around the number of arrangements of cards in a deck. It’s a simple high school-level topic, but he really puts it into perspective:

the number of possible ways to order or permute just the hearts is 13!=6,227,020,800. That’s about what the world population was in 2002. So back then if somebody could have made a list of all possible ways to arrange those 13 cards in a row, there would have been enough people on the planet for everyone to get one such permutation.

I think it’s good to remind ourselves that whenever we shuffle the deck, we can be almost certain that our arrangement has never been created before (since  52!\approx 8\times 10^{67}  arrangements are possible). Wow!

Alex is looking for “random” numbers by simply asking people. Go contribute your own “random” number here. Can’t wait to see the results!

Quick! Think of an example of a real-world bimodal distribution! Maybe you have a ready example if you teach stat, but here’s a really nice example from Michael Lugo: Book prices. Before you read his post, you should make a guess as to why the book prices he looked at are bimodal (see histogram above).

Philosophy and History of Math

Mike Thayer just attended the NCTM conference in Philadelphia and brings us a thoughtful reaction in his post, The Learning of Mathematics in the 21st Century. Mike wrote this post because he had been left with “an ambivalent feeling” after the conference. He wants to “engage others in mathematics education in discussions about ways to improve what we do outside of the frameworks that are being imposed on us by those outside of our field.” As a secondary educator, I agree with Mike completely and really enjoyed his post. Mike isn’t satisfied with where education is going. In his post, he writes, “We are leaping ahead into the unknown with new educational models, and we never took the time to get the old ones right.”

Edmund Harriss asks Have we ever lost mathematics? He gives a nice recap of foundational crises throughout the history of mathematics, and wonders, ultimately, if we’ve actually lost any mathematics. There’s also a short discussion in the comments section which I recommend to you.

Peter Woit reflects on 25 Years of Topological Quantum Field Theory. Maybe if you have degree in math and physics you might appreciate this post. It went over my head a bit, I’m afraid!

Book Reviews

In this post, Matt reviews a 2012 book release, Who’s #1, by Amy N. Langville and Carl D. Meyer. The book discusses the ranking systems used by popular websites like Amazon or Netflix. His review is thorough and balanced–Matt has good things to say about the book, but also delivers a bit of criticism for their treatment of Arrow’s Impossibility Theorem. Thanks for this contribution, Matt! [edit: Thanks MATT!]

Shecky R reviews of David Berlinski’s 2011 book, One, Two Three…Absolutely Elementary mathematics in his Brief Berlinski Book Blurb. I’m not sure his review is an *endorsement*. It sounds like a book that only a small eclectic crowd will enjoy.


Peter Rowlett submitted this post about linear programming and provides a link to an interactive problems solving environment.

Peter Rowlett also weighs in on the recent news about a German high school boy who has (reportedly) solved an open problem. Many news sources have picked up on this, and I’ve only followed the news from a distance. So I was grateful for Peter’s comments–he questions the validity of the news in his recent post “Has schoolboy genius solved problems that baffled mathematicians for centuries?” His comments in another recent post are perhaps even more important though–Peter encourages us to think of ways we can remind our students that lots of open problems still exist, and “Mathematics is an evolving, alive subject to which you could contribute.”

Jess Hawke IS *Heptagrin Girl*

Here’s a fun-loving post about Heptagrins, and all the crazy craft projects you can do with them. Don’t know what a Heptagrin is? Neither did I. But go check out Jess Hawke’s post and she’ll tell you all about them!

Any Lewis Carroll lovers out there? Julia Collins submitted a post entitled “A Night in Wonderland” about a Lewis Carroll-themed night at the National Museum of Scotland. She writes, “Other people might be interested in the ideas we had and also hearing about what a snark is and why it’s still important.” When you check out this post, you’ll not only learn about snarks but also about creating projective planes with your sewing machine. Cool!

Mike Croucher over at Walking Randomly gives a shout out to the free software Octave, which is a MATLAB replacement. Check out his post, here. MATLAB is ridiculously expensive, and so the world needs an alternative like Octave. He provides links to the Kickstarter campaign–and Mike has backed the project himself. I too believe in Octave. I’ve used it a few times for my grad work and I’ve been very grateful for a free alternative to MATLAB.

The End 

Okay, that’s it for the 87th Carnival of Mathematics. Hope you enjoyed all the posts! Sorry it took me a couple days to post it–there was a lot to digest :-).

If you missed the previous carnival (#86), you can find it here. The next carnival (#88) will be hosted by Christian at checkmyworking.com. For a complete listing of all the carnivals, and more information & FAQ about the carnivals, follow this link.


Pi R Squared

[Another guest blog entry by Dr. Gene Chase.]

You’ve heard the old joke.

Teacher: Pi R Squared.
Student: No, teacher, pie are round. Cornbread are square.

The purpose of this Pi Day note two days early is to explain why \pi is indeed a square.

The customary definition of \pi is the ratio of a circle’s circumference to its diameter. But mathematicians are accustomed to defining things in two different ways, and then showing that the two ways are in fact equivalent. Here’s a first example appropriate for my story.

How do we define the function \exp(z) = e^z for complex numbers z? First we define a^b for integers a > 0 and b. Then we extend it to rationals, and finally, by requiring that the resulting function be continuous, to reals. As it happens, the resulting function is infinitely differentiable. In fact, if we choose a to be e, the \lim_{n\to\infty} (1 + \frac{1}{n})^n \, not only is e^x infinitely differentiable, but it is its own derivative. Can we extend the definition of \exp(z) \, to complex numbers z? Yes, in an infinite number of ways, but if we want the reasonable assumption that it too is infinitely differentiable, then there is only one way to extend \exp(z).

That’s amazing!

The resulting function \exp(z) obeys all the expected laws of exponents. And we can prove that the function when restricted to reals has an inverse for the entire real number line. So define a new function \ln(x) which is the inverse of \exp(x). Then we can prove that \ln(x) obeys all of the laws of logarithms.

Or we could proceed in the reverse order instead. Define \ln(x) = \int_1^x \frac{1}{t} dt . It has an inverse, which we can call \exp(x) , and then we can define a^b as \exp ( b \ln (a)). We can prove that \exp(1) is the above-mentioned limit, and when this new definition of a^b\, is restricted to the appropriate rationals or reals or integers, we have the same function of two variables a and b as above. \ln(x) can also be extended to the complex domain, except the result is no longer a function, or rather it is a function from complex numbers to sets of complex numbers. All the numbers in a given set differ by some integer multiple of

[1] 2 \pi i.

With either definition of \exp(z), Euler’s famous formula can be proven:

[2] \exp(\pi i) + 1 = 0.

But where’s the circle that gives rise to the \pi in [1] and [2]? The answer is easy to see if we establish another formula to which Euler’s name is also attached:

[3] \exp(i z) = \sin (z) + i \cos(z).

Thus complex numbers unify two of the most frequent natural phenomena: exponential growth and periodic motion. In the complex plane, the exponential is a circular function.

That’s amazing!

Here’s a second example appropriate for my story. Define the function on integers \text{factorial (n)} = n! in the usual way. Now ask whether there is a way to extend it to (some of) the complex plane, so that we can take the factorial of a complex number. There is, and as with \exp(z), there is only one way if we require that the resulting function be infinitely differentiable. The resulting function is (almost) called Gamma, written \Gamma. I say almost, because the function that we want has the following property:

[4] \Gamma (z - 1) = z!

Obviously, we’d like to stay away from negative values on the real line, where the meaning of (–5)! is not at all clear. In fact, if we stay in the half-plane where complex numbers have a positive real part, we can define \Gamma by an integral which agrees with the factorial function for positive integer values of z:

[5] \Gamma (z) = \int_0^\infty \exp(-t) t^{z - 1} dt .

If we evaluate \Gamma (\frac{1}{2}) we discover that the result is \sqrt{\pi} .

In other words,

[6] \pi = \Gamma(\frac{1}{2})^2 .

Pi are indeed square.

That’s amazing!

I suspect that the \pi arises because there is an exponential function in the definition of \Gamma, but in other problems involving \pi it’s harder to find where the \pi comes from. Euler’s Basel problem is a good case in point. There are many good proofs that

1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + ... = \frac{\pi^2}{6}

One proof uses trigonometric series, so you shouldn’t be surprised that \pi shows up there too.

\pi comes up in probability in Buffon’s needle problem because the needle is free to land with any angle from north.

Can you think of a place where \pi occurs, but you cannot find the circle?

George Lakoff and Rafael Núñez have written a controversial book that bolsters the argument that you won’t find any such examples: Where Mathematics Comes From. But Platonist that I am, I maintain that there might be such places.

7 billion

Looks like by all accounts, there are now 7 billion people in the world today. At least that’s what wikipedia says. Here are two blog posts on the subject from the math blogging community. I have to say, I was surprised to see that wolframalpha doesn’t know anything about this important event (I’m sure it won’t be the last time alpha disappoints me). Anyway, I hope everyone feels humbled to be just one of the crowd.

And, happy birthday to Karl Weierstrass!

Why I hate the definition of trapezoids (again)

Sorry, I thought I got it all out of my system in my first post about trapezoids last week :-). Allow me to rant a bit more about trapezoids. First let me remind you of the problem. Many Geometry books, our school district’s book included, state the definition of a trapezoid this way:

“A quadrilateral with one and only one pair of parallel sides.”

In case you didn’t catch the point of my first post: I think this is a poor definition and should be abolished from all Geometry curriculum everywhere. Here are some pictures I recently came across on the internet depicting the hierarchy of quadrilaterals. These picture agree with the above definition. Let me just say once more, I completely and totally disagree with these pictures, and I think you should too. That is to say, all of the following pictures are WRONG.



And I could go on and on. Now here are two good ones.


To be fair, the first set of pictures are only partially wrong. They have good intentions. Typically, the first breakdown of quadrilaterals in those pictures is by “number of parallel sides.” The first lines that come off of the word ‘quadrilateral’ divide quadrilaterals into three categories usually:

  • No parallel sides (i.e. the kite)
  • Exactly one set of parallel sides (i.e. the trapezoid)
  • Two sets of parallel sides (i.e. the parallelogram)

So the pictures aren’t wrong, per say. They just depict different information. The problem comes when teachers ask, “Look at this diagram and tell me: Is every rectangle a trapezoid? Is every rhombus a kite?” The answer to both questions is ‘yes.’ But students instinctively answer ‘no’ when using the first set pictures, and you can see why.

The problem is a historic one. If you go back to Euclid’s Elements, Definition 22 in Book 1, you can see the origin of this problem right away (a translation from the Greek):

Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

In the above definition from Euclid, here are the (not perfect) translations of each figure:

  • Euclid’s square –> Our square
  • Euclid’s oblong –> Our rectangle
  • Euclid’s rhombus –> Our rhombus
  • Euclid’s rhomboid –> Our parallelogram
  • Euclid’s trapezia –> Our…trapezia/trapezium?

The last definition is a bit confusing, since we don’t have a very well-agreed upon name for this figure. But notice that ALL of Euclid’s definitions are exclusive. A square is never an oblong. A square is never a rhombus. Each of the above quadrilaterals, according to Euclid, is mutually exclusive. These exclusive definitions persisted into the 19th century.

But sorry Euclid, no one likes your definitions anymore. I hate to say it, because everyone loves Euclid.

In his defense, he wasn’t using these names for the same purpose we do. Nothing about his language is very technical and he doesn’t say ANYTHING else substantial about these definitions. He doesn’t use them to make categorical statements about quadrilaterals or to give properties that might be inherited. The names he uses are of little consequence to the rest of his work.

Can we lay this issue to rest yet? A parallelogram is always a trapezoid. Say it with me,

A parallelogram is a trapezoid.

A parallelogram is a trapezoid.

A parallelogram is a trapezoid.

Anything you can say about a trapezoid will be true about a parallelogram (area formulas, cyclic properties, properties about the diagonals). A parallelogram is a trapezoid.

For more posts on this topic, visit here and here.

Give the Babylonian’s some credit

So says this CNN article from Friday.

Over 1,000 years before Pythagoras was calculating the length of a hypotenuse, sophisticated scribes in Mesopotamia were working with the same theory to calculate the area of their farmland.

Working on clay tablets, students would “write” out their math problems in cuneiform script, a method that involved making wedge-shaped impressions in the clay with a blunt reed.

These tablets bear evidence of practical as well as more advanced theoretical math and show just how sophisticated the ancient Babylonians were with numbers — more than a millennium before Pythagoras and Euclid were doing the same in ancient Greece.

“They are the most sophisticated mathematics from anywhere in the world at that time,” said Alexander Jones, a Professor of the History of the Exact Sciences in Antiquity at New York University.

He is co-curator of “Before Pythagoras: The Culture of Old Babylonian Mathematics,” an exhibition at the Institute for the Study of the Ancient World in New York.


[Hat tip: Mr. Gherman]