How does a mathematician see the world?

And how does a math teacher help learners to see the world through a mathematician's eyes?

This was a major challenge during the past semester in my Intermediate Algebra sections. Many of the students came to class expecting me to show them a procedure that they would practice until test time - when they would reproduce the procedure and promptly forget it. Nearly all of the students had seen the Intermediate Algebra content in high school (linear, quadratic, and exponential functions) but it hadn't stuck.

This was the problem. They had seen the content. Now it was time for them to use it to see the world. So I shared pictures and videos and asked them to look at them through a mathematical lens.

At one point, a student said, "Just once, I wish I could see the world through your eyes." Exactly! Unfortunately, they were often so afraid of making a mistake, of breaking the mathematical glasses, that they were not willing to even put them on. They did not know how to be playful with the math we were exploring.

So I introduced them to Yes, And ...; this is a problem finding activity that I developed using a well-know improv game where participants accept and build on the ideas of their partners. Here are the instructions for my version:

• Provide a mathematical context (often pictures or videos) but without any identified problem;
• Pair up the students and find a fun way to identify Student A;
• Student A picks one of the contexts and finds a problem to solve;
• For one minute (this is usually enough time to get started without completely solving the problem), Student A talks through his or her thinking while Student B writes as much as possible down on a piece of paper;
• After one minute, the students switch roles. I say, "Yes, and ...," to signal the switch and to emphasize that Student B ought to build on the work that was already done.
• Student B thinks out loud for one minute while Student A records the thinking on the same paper.
• After one minute, I say, "Yes, and ...," and the roles reverse again.

Yes, And ... can go on as long as the teacher wants. I found six minutes (three rounds) to be about right the first time we played the game. One student submitted this to demonstrate her engagement with the task.

I cannot claim for certain that Yes, And ... changed my students' view of mathematics or helped them to see the world through a mathematical lens. What I know is that for six minutes they played with math. It's a start.

Where is the value in play?

This week, Kathy and I presented at the Michigan Council of Teachers of Mathematics Conference. We adapted our previous workshop on games to focus specifically on the Common Core Standards for Mathematical Practice. (Here is a PDF of the session PowerPoint.)

We used the grouping by one of the Common Core authors, William McCallum, to make the Practices more manageable for the participants. Then we concentrated our attention on Standards 7 and 8 (what McCallum refers to as "seeing structure and generalizing"). I shared how some preservice teachers had synthesized this pair into three key elements to look for while doing math:

• Noticing: recognizing patterns by breaking things down and identifying basic structures;
• Building: creating new knowledge by connecting ideas to what is already known; and
• Generalizing: identifying ways to create general methods/formulas

By intentionally narrowing our focus in this way, we hoped to model the importance of highlighting learning opportunities that occur during play.

Participants were given the opportunity to play three games. Two of the games, Race to 100 and Roll a Square, provide opportunities to examine the structure of our place value system and how the structure can be used to create methods for solving double-digit combining and separating problem. We asked participants to explore the games as teachers - keeping in mind scenarios that might be used to highlight Standards 7 and 8.

I provided the following as a model scenario:

While playing Race to 100, I saw Alyssa start on 14 and roll a 10. She ended on 24. What if she rolled 3 more tens in a row? What would the Rekenrek look like at the end of each roll?

While rolling four tens in a row is unlikely, we can use the shared experience of playing the game to provide learners with a chance to notice how our place value structure can be used to build a strategy for adding 10 to a number.

By playing the games before using them with learners, the teachers can be intentional about looking for opportunities to highlight "learnable moments." Then, teachers can use reflection time to talk about scenarios they observed during  game play (during planning, during the lesson, or even imagined). This can be as simple as sharing the scenario and adding one of the questions (from this PDF) associated with the Practice Standard(s) the teacher has decided is the focus of the lesson.

Too often we do not take the time to debrief around games and make explicit some of the mathematical practices that occurred. Is it any wonder that our students respond, "Nothing," when asked by their parents or guardians what they learned in math class today. Let's not leave learning to chance and assume learners will use the skills that will help them to improve their mathematical practice.