Newsletter #68     What's in a Game?

It came to me recently that there has been remarkably little evidence this quarter to support the hypothesis of my continued existence, much less that of on-going events of educational interest. Sometime in the near future I will address the latter (yes! plenty going on!) but this time around I am going to do the prequel routine with another AWM education column. I will give a lead-in, though, because the context for the article has a fair amount of local interest.

A number of years ago Jim King, who regularly teaches the course in geometry for future high school teachers, began lobbying for the addition to our course list of a class whose content would be of interest to someone already teaching and whose timing would be such that people in that category could actually take it. His efforts came to fruition a few years ago with the creation of Math 497, which meets from 4:30 to 7:00 once a week. It has been going along nicely, with a steady supply of in-service teachers, a mixed array of on-campus students and a spectacular variety of mathematical topics, ranging from hands-on calculus to spherical geometry to a historical perspective on some important theorems. One notable feature has been the amount of pleasure the people planning to teach and the people already teaching take in each other's presence in the course. Some day I would love to emulate New Mexico State University and set up the course in such a way as to formalize partnerships between those two populations, but for the moment the mechanics of that overwhelm me. And meanwhile, I have been having a grand time in the quarters that I have taught it alternating between terra cognita (variations on elementary probability) and foreign turf that I wished to learn more about in good company. One such topic was the uses of games in the mathematics classroom, and to that I have devoted the current quarter. Also the current AWM column, which follows:

What's in a Game?
A thought by any other name would feel as neat.

Over the past decade, I have been increasingly intrigued by the uses of games in the classroom. They seem to me to have a notable potential for engaging students and enticing them into discovering new concepts or applying old ones but also for giving an impression of mathematical activity which is purely illusory. My interest was first piqued when I ran into them in the French mathematics education research program of Didactique. An alternative name for Didactique is the Theory of Situations, and a lot of the situations used both for teaching and for experimentation are based around games. One that is particularly key and accessible is the Race to Twenty, a version of Nim in which two players take turns adding either one or two to an ongoing sum with the objective of being the one to make the sum twenty. This game and the meticulous use made of it became the centerpiece in my efforts to explain Didactique to various collections of American colleagues. What struck me most in their responses was the degree to which on our side of the Atlantic we value the spontaneity and energy a game can generate. This resulted in a low-key experiment over a number of years in which I used Nim in teaching my own class of future elementary school teachers. They learned a lot, and I learned far more, especially in the area of how great the distance is between the first sighting of the kind of strategy that Nim requires and the ability to use that strategy in other contexts not to mention recognizing contexts in which it can be used.

This issue continued bumping around at the back of my mind until this year, when I took advantage of one of the perks of academia: I opted to expand my thinking process by declaring a seminar on the subject. It wasn't quite the intimate conversation I had envisioned, because the notion of a course on games caught the fancy of an even wider spread of students than I had expected, but the result has been a lively exchange of ideas. The class contains people now teaching in schools from elementary through community college, undergraduates who plan to go into the teaching of elementary or secondary school, regular maathematics majors, a couple of engineers, and someone who tutors math as a hobby. It is also nicely mixed with respect to gender, which forced me to be instantly conscious of one of the hazards of using games, to wit, the statistically supported observations of the differences in how males and females respond to competition. It took a couple of weeks of adjustments to arrive at a comfortable functioning level, and to my shame and sorrow I did lose one teacher who said she felt overwhelmed and outclassed, but eventually a careful balance of randomized and de-randomized groupings and some tactics for funneling competitive urges into collaborative ones produced a highly positive learning environment. Given the background of the situation it seemed reasonable to start with Nim, so we did that. Predictably enough, we became instantly engrossed in strategies, and recognizing when to reapply a strategy, and the impact on the game of apparently minor changes in the rules. Eventually we tore ourselves away and spent some time on a pair of dice games and their attendant probabilities. Then we moved into Pico-Fermi, which is a paper-and-pencil version of Mastermind in which one player tries to guess a sequence of letters chosen by the other, aided by clues in the form of a "Pico!" for each correct letter in a given round and a "Fermi!" for each letter which is not only correct but in the correct location in the sequence. What made this game fascinating in our context was a pair of attributes: it is enormously adjustable, and clearly offering it at too simple a level produces boredom and offering it at too complex a level leads to wild guessing, while offering it at the right level is conducive to all manner of interesting deductive reasoning.

With all of these under our belts we were finally set to delve into the questions on which the course was built: do these belong in the classroom? If not, why not? If so, when? And how do we help students connect the games with the rest of their learning, especially their mathematical learning? Two class members provided me with the perfect ammunition by stating in their class follow-up reports: "Playing the [Pico-Fermi] game was a lot of fun, but I'm still not entirely convinced that there is much math to be learnt from the game" and "We could not relate it to math. What it reminded me of was one of the sections on the GRE. Which is a lot of problem solving and critical thinking, but not math." I couldn't possibly have invented a better pair of statements for launching a discussion, so I put them on the overhead and asked "So what is math and how do you think these games relate to it?" An intense half hour later, some were still trying to articulate an opinion, others had begun to adjust theirs, and everyone had realized the range and the pertinence of the replies to that question. For homework they had then to think some more and write up those thoughts. Some, reasonably enough, were still bewildered, but others had extremely cogent comments to contribute: "As students play the game they are developing a strategy that systematically brings them to the correct answer. This is exactly what I think we are teaching when we teach math." "Whether or not the game is math is arguable, but it's not as important as whether or not the game would improve a child's ability to understand math better and be able to learn math concepts more quickly." "Whether this kind of thinking is mathematical, I can only say, I think it is. I think it is because when I am getting somewhere in the game, it feels like the same kind of thinking that is involved in solving some other math problems."

As the quarter comes to an end, I have been reflecting on my choice to run a mathematics class about the choice to run a mathematics class using games. I am struck by some parallels: both offer easy access to student engagement and both offer exciting possibilities for using that engagement to deepen the students' understanding. This leads to an obvious concern: the third notable characteristic of teaching with games is that they can be so entertaining that the learning in the lesson gets swept gently under the carpet. I had easy ammunition to shoot down that concern for this particular class, though, thanks to the presence of class members who could write reports like: "I spent two class periods playing the game and increasing the level of difficulty. When one of my students commented that his head hurt from thinking, I felt Pico-Fermi was a worthy activity for the mathematics classroom."

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