I taught a class tonight that didn't work. It was somewhat fun, and somewhat well developed in presenting applications of game theory. But overall: :P I'm writing this post to note down thoughts before I forget, and I'm making it public because I generally think that various try-fail-learn processes should be more open/public. Also, I've been feeling awkward lately that I present myself so unilaterally as an 'experienced teacher' now, and not a student/new and learning teacher or in any other mind-set... more on that in the future perhaps.

I've decided to try for 2 focus goals of self-improvement in teaching each season. This season, my goals are (1) Time management via consciously prioritizing academic and non-academic goals, and (2) creating situations where my students feel ownership for what they're making.

Game Theory part III of IV

This activity was part of a 2 hour session for 45 students in 6th-8th grade.
1) The students cut out paper candy bars and then ranked each type (30 types) of candy bar 0-3 so that the sum of all of the types was 50.
2) In groups of four, I asked them to design an agreed 'fair' method for distributing the candy
3) They implemented the method, and the person to get the most total value of candy (using their own value sheet to scale) in each group, wins a real prize.

Time management went OK. I cut the activity a bit short (~1:15 total out of the 2 horus for the session) and extended some other parts of the class that were working better.

Review: Freedom to Experiment is one of the four foundational freedoms that I strive for when I teach (the 4 are: freedom to fail, experiment, of effort, and to try on different identity roles). But in order for an activity to feel like a valid experiment, there needs to be meaningful choices that the students understand when they are making those choices, and there needs to be feedback, ideally interesting, immediate feedback in response to their choices, and then a chance to go back and make those choices again, more informed. Here's where I think I went wrong:

1) Part 1 mechanics: there were too many types of candy and the max rank wasn't high enough. It should have been something closer to 6 types of candy, 15 points total, rank each from 0-10 I think.
2) Part 2 mechanics: I should have gone over more examples before setting students out to make their own. They had a lot of choice here, and I wanted them to have 'ownership' of their methods, but only a few groups really seemed to construct - most just fished around for something reasonable-seeming and then used that.
3) Part 3 mechanics: OK, but I should have had a sheet for describing more data about the results of using the algorithm -- showing which kinds of preferences or strategies would have the advantage and how and why.

Next time:
I think I should have had a first part of class dedicated to working with 3 or 4 very different division algorithms, letting students get a feel for which are best in different circumstances. Then part 2 could become combining these methods to make something more fair for everyone. Part 3 needed more organization, some follow-up questions with exact answers in addition to the . And I should have left time to play a 2nd time, without letting people change their ranks in-between, to illustrate the impact of making preferences public.

Unexpected:
I spent 10-15 minutes in a few random blocks telling stories about how game theory has come up in scenarios I've seen:

Real estate needing to buy all of some set of houses in order to make any of those houses valuable, a sport conference leader needing to choose a mechanism for coaches drafting their teams, the Splash class lottery algorithm that I worked on as a college student, etc.

These stories hushed the room, and I think I could do more with them. I could ask students how they would 'game' each system, and then how to patch those gaps to make that kind of 'cheat' impossible.

Next time:
1) Real-life game theory examples
2) Trying out 3 or 4 simplified algorithms in groups of 4, with a smaller candy-set
3) Give groups time to build one algorithm from the existing ones
4) Have students implement it
5) Discuss what students chose to do and who it advantaged
6) Play again
7) Discuss the result of more-public value information

Big picture -- If I can make this class into a really excellent one, I think it would make a great introduction to the game theory sequence I've been building. I'm working on 8-12 class sequences right now, of sessions that can stand alone if they need to, but that tie together under a theme as well.

A) Designing Something Fair -- Real Life Game Theory, Types of Fairness
B) Designing Something Fair -- Pet (instead of candy) Intro
C) Designing Something Fair -- Design your own fair division algorithm
D) Designing Something Fair -- Fair Voting Lab

E) Comparing Strategies -- Monty Hall Intro
F) Comparing Strategies -- Prisoner's Dilemma (Iterative Play AI Design)
G) Comparing Strategies -- RPS and Rock Scissors (Mixed strategies)
H) Comparing Strategies -- Real life Game Theory, Nash Equilibria

I) Perfect Play -- Say 16
J) Perfect Play -- 2-pile Nim (poison points)
K) Perfect Play -- Dots and Boxes
L) Perfect Play -- Dots and Boxes AI + Tournament

Higher level Topics:
* Advanced Combinatorial Game Theory
- perfectly logical pirates
- proofs about combinatorial games
- gomoku and variants

For context, I'm making an activity for high school students in an afterschool math program. The activity is aimed at estimating, understating, and working with really big numbers, and this part of the program (4 sessions) is focused on bijections and infinties, with a theme of how mathematicians think about topics that are too large or abstract to handle with intuition, and for how mathematicians make choices that shape what the mathematics we study looks like.
( More Details )

The list so far:

I HAVE NO IDEA, BUT I'D LIKE TO USE THEM IF I CAN FIGURE OUT AN APPROXIMATION:
* The approximate number of primes between 0 and 2^57,885,161 − 1 (???)
* If two 'opponents' at Go are actually collaborating to make the game as long as possible, approximately how many turns might the game last (on a 19x19 board)? (???)
* The approximate number of different chess games for which the board never returns to a previously experienced state. (???)
* The approximate number of atoms in everything living on the earth (???)

I THINK THESE ARE RIGHT:
* The approximate circumference of the earth in miles (~10^4)
* The approximate number of sheets of paper in a stack as tall as the height of the Transamerica Pyramid in SF (~10^6)
* The approximate number of seconds in a human life (~10^9)
* The approximate number of people in the world right now (~10^10)
* The approximate number of neurons in a human brain (~10^11)
* The approximate number of seconds between now and when a dinosaur last lived (~10^15)
* The approximate number of ants in the world right now (~10^16)
* The approximate number of calculations that can be done by the world's current fastest computer (The Tianhe-2 supercomputer) in 1 minute (~10^18)
* The approximate number of stars in the universe (~10^30)
* The approximate number of atoms in the earth and everything on it (~10^50)
* The approximate number of different ways a deck of 52 cards might be shuffled (~10^68)
* The approximate predicted number of atoms in the universe (~10^80)
* The approximate number of rabbits that would exist in the universe after a year if, starting with 2 rabbits, ever day, every pair of rabbits alive mate and produce 6 offspring, and no rabbits ever die. (~10^219)
* The approximate value of the largest known prime number (~10^17425169)

See more at my ask MetaFilter page.

Weighing "Failing Gracefully" highly is one of the best strategies I picked up in college.  It means that among many options for how to do things, giving a strong priority (a "weight") to those that, even if they don't reach fruition, still make a significant positive impact.  Projects that have zero utility until they are completely finished are dangerous - because in reality, few things have the chance to finish before the world changes around them making a 'perfect finish' impossible.  And with a limited amount of time to work on projects, it's also a huge deal when the partial project, put on hold, still has utility in that state.

Last night I ran into a project that has the highest potential to succeed /and/ to fail gracefully that I'ves seen in a long time.  It's not a new idea.  It's simply a local school that wants me to come in once a week for 4 hours to do 4 sequential classes, one with part of each grade.  And the concurrent math classes would be split so that students who need more support with the current school unit have a much higher student to teacher ratio, while students who are ready to move on don't get board/distract the class - instead they come to my session and do something likely related to their teachers' curriculum - but open ended research and problem-solving centric. :)

This role has about 60% of what appealed to me about working at MoMath - outreach to students who might not yet know that math is awesome, non-traditional curricula, visibilty to other teachers and scalability.  And then it also has some remarkable positive differences - that I'd be working with the same group of students, building a community and relationships with them over an entire year for instance.  Also, more freedom in what and how to teach.  And it's /so/ tempting that the scalability is still there... just doing this for 4-5 schools myself is a lifelong career that I think I'd be pretty happy with, in part because of how much free time it would leave me with to do any projects I saw fit to do.  :)  But, the large scale dream would be collecting people to do this with me - maybe from the math-circle teaching community.  The MC lesson plans are a good fit, and what I'm doing now -- developing MC-style lesson plans that can parellel core 'school-content' could be useful to other people.  I can see life that I would really like in this -- teaching 2-3 days/week, and coordinating the administration of a dozen people doing similarly, observing their classes 2-3 days/week.  Constantly building and rebuilding lessons to fit what new schools and teachers are doing. Working with teachers to expand their regular curriculum with the support of these weekly sessions.  :D  Wow -- it'a a position and a lifestyle that meets almost every metric I've got.

So yeah, I'm excited.  And... I think there's only a 5, maybe 10 percent chance that what I'm describing above actually happens.  But the failure is doing this myself at one school and then it doesn't go further than that.  And that 'failure' is also OK - very OK.  A lot of what I'm doing right now is like this -- I'm not sure what part will 'take off' if any, but the partial steps -- teaching for a lot of Math Circles and trying out other types of teaching/tutoring as well - the partial steps are also good things.  :)

I'm working with a friend, Reena, on a teacher-training video series for ESP and the first video which I am mainly responsible for producing is "Things that should increase with class difficulty" We were talking about this at 4am today and, since one of the sub-topics is writing a good class description, we were talking about the effect of adding "Hardcore" to a class title.  Earlier... yesterday?... one of my students had suggested that I should have called my class "Cannibalism" instead of Theoretical Computer Science, since that would have attracted more students and since the class features a proof of the incomputability of the halting problem via a carefully constructed turring machine running on its own encoding. <om nom nom> In any case, in the last hour, these two ideas have solidified into an awesome class that I'm going to teach for Splash this year:

Hardcore Cannibalism: The Mathematics of Things That Eat Themselves
Description:  <om nom nom.>

Topics:
  Fractals (in particular escape time fractals with recurrence relations, like the Mandelbrot set)
  A VERY brief sketch of that proof of the halting problem
  Recursive definitions (definitions which contain parts of themselves, like the Fibonacci numbers) and recursive algorithms and, of coruse
  The paradox of self-containing sets

Thus far, these topics are fairly arbitrarily chosen... although reena and I both found it a bit odd that most of the math I teach would fit into this class in some sense.  In any event, if you have ideas or math-topics that I should ad to the above list, comment to this post.  Thanks!
 
Delta: from Robert Assalay no less - Newton's Method of Approximation, to the extent that it can be done without calculus. :)

Notes I think I have learned about teaching (what I think I should do better)
(suggestions /very/ welcome)

1) Devote some time in the first few days to /really/ trying to figure out how the students think about the subject, then work to make ends meet
2) Have handouts with important lecture notes - including the formulas in assignments
3) Assignments should be a piece of paper physically given to students and projects should be a sheet of instructions and a sheet of grading keys
4) Review the last class in the first quarter of the next.
5) Give the students large blocks of time to work on things during class
6) Set up organization early on - ways for students to contact you, keeping track of the questions you can't answer without research

they'll be more later, this is just what I can think of in my insomnia now.