A Mathematician’s Task (TEEN CIRCLE 1)
NOVEMBER 5, 2013
Suppose you have an opportunity to play a game that costs $1 to play. You have a 50% chance of winning. If you do win, you get $3, but if you lose, you get nothing. Should you play?
I asked this question after some brief discussion of Math Circle practices at the first session of our teen circle. Two of the six kids (ages 13-15) had never been to a Math Circle, so I needed to clarify the role of parents, students, and myself.* Then a few people called out answers to the question above. Conjectures stopped when someone asked the first big question: What if the dollar that you risk is your last dollar?
As the debate raged on, other questions came to light: How much money do you have? What is your spending style? How many times can you play the game? Do you believe that gambling is morally wrong?... Do these factors even affect your decision?
I kept three running lists on the board: Questions, Conjectures, and Assumptions. The students agreed that some of the above factors – particularly characteristics of the person making the decision – would affect the decision. The group decided that we would need to separate out the purely mathematical factors from the others. I ran into trouble writing this question on the board because I didn’t know what to call the non-mathematical factors. I asked the kids what to call them. Suggestions: economic, psychological, moral, bunny rabbits. The term “bunny rabbits” was controversial. Several students loved this term, while at least one vehemently opposed. We discussed whether we needed to reach consensus on this, or whether a majority would suffice. Then someone suggested that everyone could use different names, or that we could call them the “other” factors. Disagreement continued. I didn’t foresee quick resolution, so stepped out of my humble secretary role and announced that I had decided to call them “other” factors on the board, but that students could use whatever term they wanted. Some students were relieved; others grumbled. Since this is a new circle, the group’s collaborative style will have to emerge over time.
I then asked whether the conversation we were having so far was a mathematical conversation. One student said “definitely not.” One said “it definitely is.” The rest weren’t sure but had leanings one way or the other. What came out of the ensuing conversation is that there is more than one way to define the conversation as mathematical. We could (A) force math into it (i.e. “well, there is one yes vote and one no vote, etc.” – a counting approach), or we could (B) delve more deeply into whether the questions being asked involved mathematical thinking. We chose to discard option (A) and explore option (B). “If we’re going to do that,” said someone, “we’ll need to really define what math is.”
“Oh no,” said another, fearing that we would delve into circular reasoning. Several Math Circle veterans disagreed about whether circular reasoning is worth to get into. Others didn’t know what this is, so I gave a quick explanation. Then I made another executive (vs. secretarial) decision: we would not delve into it today. Instead, I explained that the tasks of clarifying the question and naming and defining terms are definitely part of a mathematician’s task.
In response to my jumping out of my appropriate role, one student said, “Then you should be called the SUP – the Secretary of Usurper Mathematics.” Not everyone understood what this meant, so we briefly discussed it so that everyone saw how I had taken some of their power with my pronouncements. Then I backed up into my role as asker of questions and recorder of answers. I hope that by carefully staying in my appropriate role, students can more readily own mathematics and enjoy its freedom.
“How exactly can we separate the mathematical factors from the other factors?” I asked. This was tricky. No one was sure at first. Finally one student suggested making an assumption. We talked about the role of assumptions in the mathematician’s task, then struggled with exactly what assumption to make. The students decided to discard the outliers when considering the population who is making the decision about playing the game. Three types of people were excluded: hobos, Howard Hughes** and his ilk and those morally opposed to gambling. So what do we call the people who are included? No one wanted to call them “rational.” No ideas came to light, so we left that decision open. I started a running list on the board of components of a mathematician’s task, and added “leaving things open when necessary.” Photos of our board work, BTW, are posted online.
We were 45 minutes into our session and finally ready to look at some numbers. I mentioned to the group that mathematicians can grapple with questions for years. “I can see why!” said one person. “Like Fermat’s last theorem,” commented another.
Everyone agreed that with our assumptions, the answer to the original question is “yes.” I asked for explanations, and put some on the board. Most people agreed that each explanation was basically saying that if you play multiple games, on average you will come out ahead. One student was not satisfied that this generalized arithmetic approach was sufficient to justify a “yes” answer. In the true spirit of mathematics, he repeatedly asked the others “Why? Why? Why?” (It’s worth noting that 45 minutes in, the students were now talking to each other and not to me.) Students debated whether stating that “on average, you will win 50% of the time” was a restatement of the given “you have a 50% chance of winning,” or an assumption. They decided collectively that it’s an assumption, and that the more times you play the game, the more likely it is to be true. Then they examined different specific win/lose scenarios to come up with justification for the generalization. Class ended as we continued to discuss the calculations involved.
Next time, the conversation will continue with our guest instructor Raissa Schnickel, who will talk about how the above questions relate to her work as an actuary, and to the students’ own lives. See you then!
*ROLES: Parents may be in the room with their children’s permission; parents may speak only if spoken to. Today, several kids felt that parental presence would have a chilling effect on their free speech while other parents were not interested in sitting in so we had no parents in the room. Students are not charged with the task of competitively solving a problem, so to the relief of some, pencils were not mandatory. I also made clear that the Math Circle leader’s role is simply that as secretary and that my job was to shut up. I charged the students with the task of telling me when I was not doing my job well.
NOTE: For this report I did not use my convention of using an initial letter to indicate who said what. In the next session, I will ask the kids how they want me to handle this in the reports: should I just keep calling everyone “someone” and “somebody,” or should each student be referred to distinctly somehow?
**”Who is Howard Hughes?” asked most of the very curious kids. I told them that he was an eccentric tightfisted millionaire, and to ask you for more info.