Having a conversation about math is tough. Most people don’t tend to sit around and talk about their favorite algebra problems the way we might discuss A Tale of Two Cities or Crime and Punishment. It’s harder to debate the math issues of the day than it is to have a good argument about the causes of the Civil War. So naturally, when it comes to math class, dialogue has traditionally been much more limited than it has been in English or history classes: too often the plan has been “I, the teacher, do problems on the board and explain; you, the student, copy them down and listen.”
And yet we are recognizing as never before just how important it is for our students to talk about math. Scan through recent issues of any publication targeted to math teachers, and you’ll find numerous articles devoted to fostering classroom discourse. The reasons are numerous: we know that students who are actively involved in their own learning do better than passive learners; that students who converse about classroom topics process the material at a deeper level than those who don’t; and that classroom discourse can be an irreplaceable setting for self-reflection and self-evaluation. A student who can communicate freely about mathematical ideas is at a decided advantage when compared with a student who cannot.
So how do we get our kids talking? It’s often easier to advocate for mathematical discourse than to achieve it. The first and most obvious tactic is to get our students to come up to the board and explain the problems themselves, and that has a great deal of merit. As math teachers, though, we ought to be looking for more: We need to get our students to have conversations and debates with each other in math class.
Fortunately, there are great topics available to math teachers to get the conversation going. One such subject is game theory, a relatively new (and occasionally controversial) branch of mathematics that is concerned primarily with strategic decision making in competitive and cooperative situations.
Game theory got its start in the 1940s and ’50s and was pioneered by some rather interesting characters: Among its founding fathers were John von Neumann, one of the principal scientists of the Manhattan Project (and an infamous partier), and John Nash, whose battle with schizophrenia provided the plot for the Russell Crowe movie A Beautiful Mind. Since it got its start, game theory has found wide-ranging applications in areas such as economics, biology, political science, international relations, and military strategy.
While game theory may not seem, at first glance, like an obvious choice for a middle school classroom, I have found it to be an invaluable ally in getting my students to talk about math. Game theory is able to spark conversation in math class because it engages students in a very different type of problem solving than they are used to, even though many of the skills used are already familiar. Because of this, students are encouraged to engage in metacognitive discourse as they examine what it means to “solve” a math problem. Moreover, many problems from game theory are both interesting and fun to talk about; students who might not have shown much interest in math previously are frequently captivated by the competition and strategic thinking involved.
Take a classic game theory example called “Two-Thirds of the Average,” an exercise I typically do with seventh-grade honors pre-algebra students. The rules of the game are simple: Everyone in the class picks a number from 0 to 100. The numbers are collected, and the class average is calculated. The person who picked the number that is closest to two-thirds of the class average wins. We agree in advance to either two or three rounds of play. I usually ask for a couple of volunteers to calculate the averages, as well as to display the class data on a line plot on the board. After a brief piece of advice from me — “This game may seem like it comes down to luck, but think carefully about your strategy” — we’re quickly engaged in the competition.
After round 1 is played, we always end up with an average somewhere in the neighborhood of 50, and a winning guess somewhere around 33. Before moving on to the next round, we pause for some class discussion: asking “What were your strategies?” elicits a wide variety of responses and even a little debate. And then we focus in a bit: “For those who either won or came close to winning, what was your strategy?” With an eager group of students, it’s rare to have more than a few lucky guessers: At least one student will tell you that he or she believed most guesses in the class would be random and that consequently the average would fall right in the middle. This student found two-thirds of the middle and ended up winning or coming in a very close second.
Now, the results from round 1 are a pretty profound topic for conversation. We’ve just had a middle school student provide us with an intuitive grasp of normal distribution, one of the most important topics in statistics and a key tool in all areas of scientific study. Time permitting, a great discussion can ensue on that result alone. From a game theory standpoint, things get really interesting once round 2 is played: The class average usually moves to approximately 33 (keeping in mind the average of 50 from round 1), and the winning answer is in the neighborhood of 22.
What our new winner will tell the class is that he or she guessed (correctly) that a lot of people in the class would jump on the bandwagon and select something close to the winning number from round 1. Our new winner thought one step ahead, and took two-thirds of that number.
“Two-Thirds of the Average” is an example of what a game theorist would call a p-beauty contest, and it’s an exercise that models behavior that many economists have compared to the way investors behave in the stock market. It’s also not a bad introduction to evolutionary game theory, which biologists are increasingly using to explain how certain behaviors evolve. In nature, as in the markets, survival frequently comes down to thinking one step ahead of the competition. These and many other areas of scientific interest are available for class dialogue as we seek to apply our newfound knowledge of game theory to the real world.
More importantly, though, we can lead the class dialogue in a metacognitive direction. The skills we’ve used in the “Two-Thirds” game — data collection and graphing, measures of central tendency, etc. — are pretty standard fare for middle grades math curricula. The type of problem solving we’ve engaged in, however, is not. Students are eager to draw out the differences in class discussions: they are used to having a set of concrete steps to get from point A to point B, but in this exercise we’ve asked them to consider not only their own mathematical behavior but that of their classmates as well. We’ve also promised them that the “solution” isn’t set in stone.
There are epistemological considerations available to us as discussion leaders as well: “How do we know when we’ve arrived at the ‘answer’ to a math problem? Do math problems have more ‘certain’ answers than other types of questions you see in school? What are the differences between the ways a mathematician might approach a question and, say, a scientist or an historian?” These questions can lead to a rich discussion, especially when done in conjunction with other game theory exercises. (A personal favorite is the so-called “Prisoner’s Dilemma,” a famous example of Cold War-era logic that pits an individual’s self-interest against the greater good of the group.) These exercises may take a bit more preparation and guidance from the teacher, but they are well worth the time spent.
Game theory has another added advantage in fostering classroom discourse: It is a brand-new branch of math that can nevertheless be presented in an accessible way. Most of the standard K-12 curriculum is ancient history, having been discovered and explicated long before the Common Era. On the other hand, many of the 20th century’s advances in mathematics are opaque to anyone without at least a graduate student’s background. Game theory, by contrast, had its genesis in the mid-20th century. It is still at the cutting-edge of much scientific research, and yet many of its fundamental ideas can be understood by a bright middle school student. I find that being in a “frontier” area of math excites many students, and John Nash and John von Neumann spark their curiosity as personalities in ways that Euclid doesn’t.
Student reaction to game theory tends to be overwhelmingly positive. I often have students asking me when the next game theory lesson will be or expressing a desire to learn more outside of class. They are excited to be playing games in math class and eager to solve math problems that aren’t the typical step-by-step fare.
Many math teachers have not studied game theory. Fortunately, there are a wide variety of resources for the math teacher interested in the subject. Books such as Rock, Paper, Scissors, by Len Fisher, provide easy introductions with plenty of examples that can be adapted to the classroom.1 Yale University offers a free online course on game theory, taught by economics professor Ben Polak (http://oyc.yale.edu/economics/econ-159). 2
Sparking discourse in a math classroom may be difficult, but it certainly isn’t impossible. With game theory, we can bring our students a new, exciting branch of mathematics that allows them to have great discussions, engage in metacognition, and relate math to other areas of study. No doubt, there are other branches of math that could serve our purpose here, and the important thing is to be willing to explore and to keep talking.
Notes
1. Len Fisher, Rock, Paper, Scissors: Game Theory in Everyday Life (New York: Basic Books, 2008).
2. Ben Polak, ECON 159: Game Theory (an introduction to game theory and strategic thinking), Open Yale Course; online at http://oyc.yale.edu/economics/econ-159.