Sunday, March 20, 2016

Math Games to Develop Experimental Thinking


In the past several months, I have had two “doing math” experiences that involve games—one at PCMI’s Boston teacher leadership weekend and the other at the Boston Math Teachers' Circle. Each game had a similar structure: there was a goal that you had to meet to win. There were the rules of the game, which gave the overall structure. Then there were the conditions in the game, which were particulars that could be changed without changing the structure of the game. And finally there was some sort of move that you could make.  Here are 3 of the games with each of their elements described:
 

Game
To Win
Rules
Conditions
Moves

All of the stones end up in one pile
- Take one stone each from two of the piles and put the stones in the remaining pile

- There are 3 piles of stones
- The piles have 6, 7, and 8 stones
Which piles do you take from?
Take the last penny
- You can take some number of pennies in your turn
- Then pennies are in a line
- Start with 11 pennies
- Take one or two pennies
- Who goes first
How many pennies do you take?
Take the last penny
- You can take some number of touching pennies in your turn
- The pennies are in a circle
- Start with 13 pennies
- Take one penny or two touching pennies
- Who goes first
How many pennies do you take and from where?

Mathematical Thinking Prompted by the Games

With all of these games, I felt like my experience went beyond playing (to win) a game and into deep mathematical thinking. I was doing math as I explored one or more of these questions:
  • Is it possible to win? Why or why not?
  • Under what conditions is it possible to win?
  • What moves do you need to make to ensure that you win?
  • These conditions are necessary to win, but are they sufficient?
  • Do these conditions set up all of the ways win or only a subset?
  • How does winning work if you modify the conditions of the game?
  • How does winning work if you generalize the conditions of the game?

If I replace the word “win” with “find a solution” in each of the above questions, I often consider the same questions when I am doing math that is not prompted by a game. For example, at the math teachers' circle last month, I spent about two hours working, starting with the prompt “Find two different-looking sets of three numbers that both have a mean of 5 and same standard deviation.” This problem still has structural, unalterable rules: how we calculate mean and standard deviation, and that the sets need to look “different.” It also has conditions that can be modified and experimented with: 2 sets of 3 numbers, and that the mean is 5.  However, what sets this apart from a game is that there aren’t moves.  I was still experimenting with different options, but it felt really different than moving stones or taking turns removing pennies.

Whether with a game or not, this type of experimenting, asking “what if” questions, and generalizing is at the heart of a lot of the mathematical thinking that I do when I take time to “do math” for myself. But my students very rarely, if ever, do this type of thinking. The closest that we come is me directing them to try out various options, look for patterns, and then generalize from that. But they don’t get to experience the experimenting that prompts them to ask these questions and then the joy of choosing what question(s) they will further explore based on what they think will be most interesting or worthwhile. And I think that is a problem. If this is the type of thinking that I find most fun, interesting, and fulfilling, then my students should be having that experience as well.

Implications for my Classroom

So what would it look like if I fostered a culture of doing this type of mathematical thinking in my classroom?

There would be two main goals:
  1. Developing students’ ability and desire to ask “what if” questions where they change and eventually generalize the conditions of a game/problem. From now on in this post, I will refer to this type of process as experimental thinking.
  2. Using experimental thinking to develop understanding of grade-level content standards. The first goal is less useful in my purposes as a teacher unless it is leveraged into this second goal.
 In order to fully develop these two goals, I could imagine the following trajectory:
  1. Play games that do not have a specific content focus, but would hook students and prompt experimental thinking. We would play the game for 10-15 minutes where students would get to know the game and figure out a winning strategy. We would then pause and generate questions about what the game made us wonder. Students would then choose one or more of those questions to investigate. Over several games, we would categorize our questions, and hopefully build a framework of questions similar to the ones that I mentioned above.
  2. Play games (or as one of the other people at the math teacher’s circle suggested “gametivities”) where there are moves that you can make, but the game lies in specific content. Playing the game would then prompt experimental thinking about this content
  3. Have content specific math challenges where there is still a goal, rules, and conditions, but there are not “moves” any more. Students ask the experimental thinking questions to deepen their own understanding and explore the concept.
Here’s what I am imagining for a “game” to introduce graphical solutions to systems of equations. I think as it is right now, this would lie in the third part of the trajectory because it is missing the experimentation through game moves.  On desmos, I would give students a graph of a line, say y = 3x – 5. The challenge would be to write equations for three more lines that do not intersect this one. Once they had time to experiment with this, I would imagine that the further exploratory questions that this could prompt could be:
  • what if we wanted the lines to have one intersection?
  • what  if we wanted the lines to have more than one intersection?
  • what if the line was something besides y = 3x – 5?
  • what if we started with any lines?
But I would really like to figure out how to gamify this more. I think the bridge of content-specific goals that are in the context of something that actually feels like a game is important. Having the game structure adds extra motivation to reach the goal and an easier framework in which to mess around and just try things.

This framework would need to happen over a longer period of time--at least a couple of months, if not the whole year. I haven't built the groundwork for this type of thinking in my classroom this year, but I could imagine trying out phase one in the last couple weeks of school. We wouldn't build content through the experimental thinking, but it would give me a chance not only to have a go-through at setting the groundwork for experimental thinking, but also decide if I think that this trajectory would be high-leverage enough in terms of development of both mathematical thinking and specific math content in order to dedicate significant time to it next year.

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