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
1-person Stone Game	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?
2-Person Penny Game I	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?
2-Person Penny Game II	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.

A Conversation with Jon Star

I am currently part of a cognitive science inquiry group with about ten other math teachers. The goals of the group are to learn more about what cognitive science research says about teaching and learning and also somehow apply this research to our classrooms through some sort of new or revised structure or routine. So far, we have read Make it Stick and Why Don’t Students Like School and  analyzed them through the framework of generating questions that are prompted by our reading and the relationship of the ideas to our own teaching experience.

I have all sorts of questions, but this week we posed three of our group’s bigger questions with Jon Star, who is an educational psychologist at Harvard who focuses on flexibility in problem-solving and acquisition of algebra. I came away with so much to think about because he was able to give specific suggestions for practice that were very obviously grounded in research.

Question 1: What can we do to support students, many of whom have identified disabilities, whose working memory has a smaller capacity or who have other challenges with working memory?

Answer: Here is a list of strategies, all of which are centered around the principle of modifications/accommodations that do not sacrifice the learning goals. These are strategies that can be considered “good teaching” because they can benefit all students. However, the negative consequences of not using these strategies hurt some students more than others.

Strategies:

• Reduce arithmetic complexity. Harder numbers make problems more complicated, but does not necessarily require a deeper understanding. He suggests making things conceptually harder, not computationally harder.
• Be aware of the amount and type of words in a problem.
• Be smart and organized about how you are using illustrations and board space. Processing everything verbally is extremely taxing on working memory. Board space and other visuals can be used a surrogate working memory for students.
• Reduce the need for dual processing (Ex: Expecting students to read and listen at the same time)

I really appreciated that he also clarified that awareness of when you are/are not doing any of these things is the more important than making sure to use all of these strategies all of the time. There are good reasons for not following each of these strategies when there is a specific purpose.

Question 2: How do we foster a culture where students are motivated to persevere through spaced, varied, and interleaved practice, even though it is harder and the results are not as immediately obvious?

Answer: It is important to complexify what cognitive scientists are saying about practice.

• Nuance #1: Practice should differ in different phases of learning—the first exposure to a concept/procedure vs. solidification after basic mastery. Some blocked practice may be necessary in the initial phase for motivation and initial formation of knowledge.
• Nuance #2: Elaborative recall is the main principle that drives recommendations about effective practice. The more opportunities to reconstruct/apply/elaborate/expand knowledge, the more solidified it becomes. With blocked practice, the concept/procedure gets too automatic and students don’t reap the benefits of the recall. Also, once students automaticity with a procedure, that knowledge is very stable and difficult to reexamine/extend/reconfigure. Thus if students gain automaticity before associating with concepts, it is extremely hard to do this later on. Ideally, we should be building automaticity and connections simultaneously.

Question 3: Ideally, we are building procedural fluency from conceptual understanding. Where does practice fit in this arc and should the practice look different depending on where in the arc you are?

Answer:

• Point #1: It’s a misconception that there is an order to how students should learn concepts and procedures. It is not true that conceptual understanding needs to precede procedural fluency. It is not true that procedural knowledge cannot develop conceptual knowledge. Order is arbitrary—instead it is more important that they are connected and developed iteratively.
• Point #2: There are the same best practices for developing procedural knowledge as conceptual knowledge. We can define conceptual practice as an opportunity recall and reconstruct concepts (just as we would define procedural practice as opportunity to recall and reconstruct procedures).
• Point #3: It’s hard to articulate what it looks like for conceptual understanding and procedural fluency to be intertwined. One possible way to “see” this is through flexibility: students should be able to solve a problem more than one way and identify which strategy is “better” (ex: more elegant) and what the criteria are that decides what makes one solution strategy different than others.

One final gem from Jon Star: 

When you are listening to a performance of a piano concerto, often the performance itself is evidence enough of mastery of the skills and concepts behind the music. You wouldn’t need to ask the soloist to explain why they made the decisions they made. The same should be true for math. A student does not always need to explain in order to demonstrate conceptual and procedural knowledge. Sometimes the work/thinking they do speaks for itself.

On my mind now:

My biggest questions coming out of this conversation are about Star’s points about the (as one of my coworkers put it) commutative nature of developing conceptual understanding and procedural fluency.  My whole teaching framework is built upon the idea of building procedural fluency from conceptual understanding. NCTM’s Principles to Actions articulates this as one of their math teaching practices: “Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems.” 

However, based on the challenges that I have experienced when trying to teach this way (When are they ready to move from concepts to procedures? How do I help students actually connect the procedures to what they have built conceptually? How do I encourage students to go back to the conceptual understanding in order to re-build procedures that they have forgotten?), I am definitely in a position where I want to read and think more about Star’s position. I think that a lot of the current emphasis on conceptual understanding is a direct response to a history of teaching procedures with no conceptual understanding. It makes sense to me that the most important part is that we are doing both and connecting them, not that there is an extremely delicate, perfect sequence to have procedural fluency build out of conceptual understanding. Star also stated that understanding a procedure is different than having conceptual knowledge of a procedure, which is something I would like more specifics on. What does understanding of a procedure look like then? So I’m waiting on some recommendations from Star for some further reading in order to help me incorporate (or not) this into my ever-changing math education philosophy.