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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.


Undergraduate Panel Notes

A couple weeks ago in Math 495 (Teaching College Mathematics) we had a panel of undergraduate students at UCLA talk about their experiences in the math department. We tried to get a good cross section of students, from those whose focus is math, to students who only took math classes as a departmental requirement. Given the diverse background of our panel, it was somewhat surprising to me how much they agreed about how a math discussion section should be run. Below I’ve summarized the panel’s advice to TAs as to what makes for good recitation class.

  1. Be organized. Write a brief lesson plan/list of topics on the board before class and follow it. The students rely on this structure to remind them of the big picture for the class. (Author’s note: Connie Chung, the undergraduate advisor in the math department said that disorganization is the most common issue that students cite in poor course evaluations. So it is unsurprising that this was the first piece of advice given by the undergraduate panel.)
  2. Do concrete examples, and do a lot of them. Don’t dwell too much on theory in discussion–a brief (~5 minute) review usually suffices. Students tend to have more difficulty in connecting the theory with applications than understanding the theory itself.
  3. Do examples you find “interesting” for upper division classes, but stick with more “standard” examples for lower division courses. Most students in lower division classes are in your discussion section to see how to solve the types of problems likely to appear on their homework/exams.
  4. Communicate with the instructor/lecturer for the course. Make sure that you are reviewing material that the students have seen in lecture. This is usually more helpful than giving a preview of material to come.
  5. Give detailed solutions or don’t bother giving a solution at all. The students want to see problems solved to completion, not just set up and cast aside. Often students struggle with the details in finishing a problem. Only when they see a full detailed solution are they confident they understand how to solve the problem.
  6. Gauge the level of your class. Don’t assume that because nobody asked questions about an example or topic that everyone understands it. Often, the students who don’t understand are afraid to speak up. So ask questions like, “What should I do next?” rather than “Are there any questions?” If you have to wait a while for suggestions on how to proceed, you need to spend more time on that topic.
  7. Use icebreakers. Get the students talking to each other and open avenues for them to interact more. This is best done at the beginning of the term to get everyone talking on day 1. (Author’s note: I always hated ice breakers as a student because I thought they were a waste of time. I was surprised that our panel unanimously supported them and found them valuable. I may be a convert yet…)
  8. Your discussion session should be informal. Students want to be comfortable asking and answering questions
  9. Always start the class with “easy” examples. This will make sure the students are not overwhelmed early on, and will help you diagnose problems if they students have difficulty. (See #6).
  10. Make your lessons interactive. If you want to solve a problem as an example, don’t jump right into the solution. Write the question on the board, and give students time to think about how to solve it. Solicit suggestions from the class on how to approach the problem, and help guide them to a solution. The students get the most out of the class when they are actively participating in the problem solving, rather than just copying down your solution. (This approach also helps you understand better what your students need the most help with. See #6.)
  11. Choose content/applications from student interest. For example, if you have a lot of physics majors in your class, they might want to see examples of how the material in your course is relevant to mechanics or electricity and magnetism. You might need to do a bit of homework yourself to find connections with biology, etc, but the students will appreciate it.
  12. You are never going too slow. (Not a single panelist had ever been in a discussion where they felt the TA was going too slowly. Not. One.)

Looking over all of the advice and suggestions our undergraduate panel gave the new TAs in the math department, there are very few things I found surprising. Nonetheless, I thought the advice was more impactful coming from students rather than professors or TAs. Overall, the students preferred discussion sections that employed active learning. Many studies show that active learning generally leads to higher student achievement, and the anecdotal evidence from the panel shows that not only is active learning effective, but the students prefer it (at least in discussion sections). A recent series of articles/blog posts from the AMS serve as a great primer for active learning strategies in math.