Category Archives: edutainment

computer science edutainment math teaching

Numberphile video on the Josephus Problem

Recently, the following Numberphile video on the Josephus Problem has been making the rounds on math-related social media. I watched the video, and I thought Daniel Erman did a remarkably good job at explaining how to solve a mathematical problem. Daniel’s approach is similar to the techniques described in Polya‘s “How to Solve It.” Yet the particular story that Daniel tells also has an appealing narrative arc.

Daniel’s video adheres to the following principles, which I think are fairly universal in mathematical problem solving.

  1. Start with a concrete problem. If the problem has a nice story to go along with it, all the better. The Josephus Problem is a great example of a concrete mathematical question. Given a method by which the soldiers kill one another and the number of soldiers, where should Josephus stand to be the last living soldier?
  2. Formalize and generalize the problem. What is special about the number 41? The mechanism by which the soldiers kill one another works just as well for any number of soldiers, so consider the problem for \(n\) soldiers.
  3. Consider simpler versions of the general problem. Now that we have the general \(n\)-soldier Josephus problem, we can easily work out a few examples when \(n\) is small. To quote Polya, “If you can’t solve a problem, then there is an easier problem you can’t solve: find it.” This process of finding simpler and simpler related problems until you find one you can solve is to me the most important general problem solving method.
  4. Solve enough of the “simple” problems until you see a pattern. Solving the simpler problems gives one both data and intuition that will allow you to conjecture about a general solution.
  5. Generalize the pattern as much as you can so that it fits the examples you’ve solved. Even if the pattern doesn’t give a complete answer (for example, Daniel’s observation that if \(n\) is a power of \(2\), soldier \(1\) is the last living soldier), even a partial solution is likely valuable to understanding a complete solution.
  6. Prove your generalization of the pattern to obtain a solution to the general problem. Often, this doesn’t happen all at once. The Numberphile video happens to give a particularly elegant solution in a very short period of time. Don’t get discouraged when not everything falls into place the first time you try to solve the problem!
  7. Apply your general solution to the original concrete problem.

In my own research, I follow the strategies above. In particular, Polya’s advice regarding finding and solving simpler problems (steps 3 and 4) is maybe the most valuable single piece of problem solving advice I know of. I think math could be characterized as the art of generalizing simple observations. Often, the simple observations arise by wasting a lot of paper trying to solve simple problems.

The narrative outlined in the steps above is also valuable from a pedagogic standpoint. By starting with a tangible (if slightly morbid) problem, the student/participant immediately has some intuition about the problem before beginning formal analysis. In my experience, one of the biggest challenges students face is connecting abstract statement and theorems to concrete problems. By introducing the concrete problem first and using this problem to motivate the formal development needed to solve the problem, students can begin using their imagination earlier in the problem solving process. This makes learning more interactive, memorable, and effective.

edutainment

The Geometry of Music

I saw this charming video explaining the geometry of two note chords in music. Interestingly, it turns out that two note chords naturally live on the Möbius strip.

I would love to see a visualization of Bach’s Inventions on a Möbius band… Apparently the video above uses Flash (ew) so it won’t play on certain mobile devices. If it doesn’t load, you can see the original post/video here.

computer science edutainment

Computer Musings by Donald Knuth

Donald Knuth’s diverse achievements lend him demigod status in many circles. His series The Art of Computer Programming is admired by theoretical computer scientists, programmers, and hackers alike for its panoramic yet detailed treatment of algorithms. His academic research record is impressive. His TeX typesetting system forms the core of what is likely the most widely used technical typesetting software in the world. Given Knuth’s penchant for the aesthetics of typesetting, his website is perhaps not what one would expect–maybe he has strong sense of irony.

Every year Donald Knuth gives a special seminar on whatever topic he finds interesting. You can view these computer musings and other lecture series here.

computer science edutainment expository math

Information Theory

I have just uploaded the beginnings of an essay on information theory to my website. You can see the essay (in its currently incomplete state) here. Information theory was first described by Claude Shannon in his groundbreaking 1948 paper, A Mathematical Theory of Communication. What is particularly surprising about Shannon’s truly remarkable paper is its completeness. Not only does Shannon suggest a mathematical model for (digital) communication and information, but he produces a huge array of fundamental results for his model. It is exceptionally rare that such a complete theory comes into being so fully formed, like the birth of Athena. A decade ago, UCSD produced documentary about Shannon’s life and work, which I have posted below.

edutainment

The Storytelling of Science

Some of the biggest names in popular science came together for a panel as part of the Origins Project at ASU. Bill Nye, Neil deGrasse Tyson, Richard Dawkins, Brian Greene, Ira Flatow, Neil Stephenson, Tracy Day and Lawrence Krauss shared some of their favorite stories about science. I’ve always enjoyed how the folklore of science can grant humanity and pathos to what might otherwise be viewed as an austere subject. The stories shared by the participants are at once entertaining and inspiring.

Following the presentations by the panelists, the floor was opened up for questions. The subsequent discussion among the panelists ran the gamut of topics from the politics of science to philosophy. They hit upon some of my favorite topics such as the role of governmental funding in basic research, the unreasonable effectiveness of mathematics in the sciences, and the perils of watering down science for popular consumption.

Part I of the video shows the prepared presentations, while Part II includes the lively and informal responses to audience questions.

 

edutainment math

Dimensions Video Series

In this month’s issue of the American Mathematical Monthly, there is a review of a series of animated videos about geometry, called Dimensions. The aim of the videos is to introduce the geometry of 4-dimensional space, and in particular the 3-sphere which we can view as living in 4-dimensional space. I haven’t finished watching the entire series, but what I’ve seen so far is pretty impressive. The entire series runs about two hours. It is freely available on youtube — see below for the first installment.

edutainment math

Möbius Transformations


Möbius transformations (also called linear fractional transformations) are maps from the complex plane to itself of the form
\[
f(z) = \frac{a z + b}{c z + d}
\]
where \(a, b, c, d \in \mathbf{C}\) and \(a d – b c \neq 0\). This definition extends to functions on the Riemann sphere, where the geometry of such transformations becomes more apparent. The following short video shows off a remarkable correspondence between symmetries of the sphere and Möbius transformations.

edutainment teaching

Richard Feynman: The Messenger Series

While I’m posting educational videos this morning, I might as well post one of my favorite lectures series of all time: Richard Feynman’s Messenger Series, given at Cornell University in 1964. I consider Feynman to be one of the greatest expositors of science of all time. He had an uncanny ability to distill concepts of physics to their essence and highlight the conceptual simplicity of highly technical ideas. Additionally, he is a very charismatic speaker and his lectures are a pleasure to watch.

edutainment math

Ramanujan: Letters from an Indian Clerk

There are few more compelling, mysterious, and ultimately tragic stories in mathematics than that of Srinivasa Ramanujan. He was mathematical prodigy from Madras, India. With very little formal training, he came up with results that continue to baffle the mathematical community. 25 years ago (in honor for the 100th anniversary of his birth) the BBC produced a documentary about his life and mathematical legacy. Now that video is on youtube: