So, I’m teaching a new course, Math 2033, Mathematical Thought, and it’s going great! I’d like to take a moment to write about it!

(This is one reason the MF has been kinda slow lately; another is that I’m chair) When it’s fully up and running, we’ll have about 150 students in one large section each semester (we’re starting with about 100). In a nutshell, it’s the Math Factor, as a course.

So, the list of topics is pretty familiar; from the podcast you are pretty well acquainted with the kinds of things I like to share: game theory, encryption, a little number theory, theory of computation & godel’s theorem, cardinality/infinity; plus more visual hands on things like topology, graph theory, symmetry, four-dimensional geometry, and so forth, and some baby programming in a playground IDE (scratch.mit.edu)

The real thing though is that my co-teacher and I have taken a kind of radical approach to the structure of the course—and our crazy ideas are working out great! In a nutshell, the students are guaranteed a C just for showing up and doing what they’re told (more on that in a sec), but to get an A, they have to become active collaborators in the building of the course, adding to the long-term infrastructure. I didn’t expect to give out very many A’s at all, but a surprising number of students seem up to the challenge. Part of the point is that this reward structure aligns the interests of the course directly with the interest of the students. Another nice thing is that it is much more like the reward structure of Real Life, far more so than most academic experiences: you can coast and do ok, but to really succeed, initiative and imagination are required. Interestingly, 20% of the class can’t even rise to the minimal standard of showing up, and will fail. 

Another nice thing is that students can bring to bear any of their own interests and abilities; we need such a wide range of things done—photography, writing, editing, leaders on our discussion board, organizers, all kinds of stuff. It is in fact possible to get an A by dragooning other students into harvesting, trimming and delivering a huge load of bamboo for some math sculptures the class will be making soon. The fact is, I have large ambitions for this, and no way to do more than a fraction of the work; students that help bring this off will be the ones that get an A.

SO, how does it actually work? The basic daily rhythm is that we give a lecture, usually with some sort of hands-on fun and games component. We then post a prompt or two on the class discussion board (hidden to the outside world). The students have 24 hours to post, and then another 24 hours to comment on each others ideas. This is the real heart of the course and the activity has steadily grown, reaching 1700 posts a couple of weeks ago. Wow!  (I shouldn’t exaggerate this though: some are really into it, many are trying to get by with as little as possible. I am aiming for a culture where slacking is gently disgraced, and we’re on track to get there)

As you can guess, this has completely lifted out of my ability to monitor; we have a number of ways this is digested and managed. For example, about half a dozen of the more thoughtful students are responsible for reading all of posts and trying to raise the level of discourse, and for creating useful summaries of the best ideas.

BUT that’s just the “internal” part of the course. Externally, open to the world, is a wiki, math2033.uark.edu which is going pretty well. I view this as a multi-year project, so this is a pretty good start. Most of what you see there is the product of about twenty students, and a few really do A LOT of work, including having developed the basic organizational framework. (So they get A’s for sure)

I’ve been spending a lot of time developing solid materials for use in the course, such as this sample handout, on the Halting Problem.

The students seem pretty pumped. It’s working!!! 

(We passed an important milestone last week; several students told me they ended up fooling around way too much with one of the optional assignments, messing them up in other classes! Perfect!)

The sum of the digits of 351 is 9 and the sum of the digits of 2 x 351 = 702 is also 9.

1) If a number has this property, can we always rearrange its digits and obtain another number with this property (513, 135, etc all have it)

2) Which powers of 2 have this property?

3) And most of all, can you give a simple characterization of the numbers with this property, in terms of just the digits themselves?

Check out these great gifts!

• Zome is an incredibly powerful construction system!
• the great puzzles of Puzzellation (available at Barnes and Nobles)
• The terrific puzzle computer game DROD
• The Magic Mirror Image Coloring Book
• The Riddles of the Sphinx by David J Bodycombe, an amazing compendium of puzzles, of hundreds of kinds, at all levels of difficulty, with historical essays to boot!
• Which leads us to Nikoli, the great Japanese puzzle co! (Rules can be found here)
• The Princeton Companion to Mathematics is a landmark classic. A must-have for every serious student, researcher or amateur.
• How Round is Your Circle just one of the many fantastic titles out on Princeton University Press
• AK Peters is another fantastic press, with a wide range of interesting math and CS titles, including, ahem, the Symmetries of Things.
• Binary Arts/ThinkFun is another source of great puzzles!
• And the authors Martin Gardner and Ivan Moscovitch are always fantastic!

Hope this helps and have fun!! Let us know how it works out!

Happy Holidays from the Math Factor!

Now, really, tell me, what good is a podcast if you can’t promote your beautiful new book?

We are very very pleased to announce the publication of The Symmetries of Things, a comprehensive, modern account of the mathematics of symmetry, complete with over 1000 illustrations!

This model is a three-dimensional shadow of a four-dimensional polyhedron. Though it looks quite complicated, it is merely divided into lots and lots of cells that are pretty simple: tetrahedra (blue in the fig below), dodecahedra (brown), and prisms (yellow and green).

Towards the boundary of the ball, the cells are increasingly foreshortened; this is exactly analogous to the way a two-dimensional picture of a three-dimensional polyhedron is a division of a disk into polygons that are increasingly foreshortened towards the edge of the picture.

Just as the archimedean polyhedra are formed of polygons, the archimedean polytopes are formed of archimedean and regular polyhedra:

At some point we hope to get around to explaining this better.

In the meantime, we enthusiastically recommend

Here are some sites for intermediate to advanced adventurers; if someone knows of a beginner’s guide to the fourth dimension or has another link to recommend, please let us know!