- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.
We focus on the practices for a few reasons. One is because in our work to support students in problem solving, through the PoWs, our Problem Solving and Communication Activity Series, and our Professional Development, we are very focused on students’ process. We want to help teachers and students think about how students are thinking and doing math, not just what they are doing. Whether you call it reasoning, sense-making, problem solving, mathematical habits of mind, or mathematical practices, it’s important to us and we’ve been focused since 1992 on helping students get better at it.
We also focus on the practices because they’ve gotten the short end of the stick, historically. The math Content Standards in the new Common Core documents are 93 pages. The Practices document is 3 pages. The Content Standards are broken out by grade level, and we can see concepts develop throughout students’ thirteen year journey through math education. The Practices are the same for all grade levels. We don’t yet know how we see the practices develop over thirteen years.
In looking at the practices, one thing that stands out about so many of them is that they require that students actively do mathematics. In order to get better at persevering, students need to do problems in which they get stuck. In order to construct arguments and critique others, students need to get into situations that are worth arguing about — situations in which they don’t know the answer or how to “mathematize” the situation best. In order to use tools strategically, students need to be in situations they’ve never encountered before, so they are forced to think hard about which of their many tools to use.
As teachers teaching with the PoWs, you clearly recognize the value of giving students challenging problems to work on and asking them to explain their reasoning and justify their thinking. You value letting students struggle productively, experience being stuck and then getting unstuck. Every time I read a PoW submission in which a student says, “I didn’t know what to do but then I had an aha! moment when….” I know that student is getting better at Mathematical Practice 1. They are learning to make sense of problems and persevere in solving them.
When I picture a classroom that is actively working at getting better at the Mathematical Practices, I picture one in which students encounter challenging mathematical situations, and teachers and students have lots of conversations about how they think about the situations, what’s working, and what they want to get better at. An observer might overhear questions like:
- What did you do when you weren’t sure how to get started?
- What does that [variable, number, etc.] mean in the story?
- When you said _________, can you tell me more about what you were thinking?
- How can you illustrate the relationship between those two things?
- What did you notice in the problem that made you think to use ________?
- Can you tell me what your answer means to the people in the story?
- What made this problem so hard?
- How did you decide to organize your work? What patterns did you see?
Did you notice there were eight questions above? One for each mathematical practice! They are the sort of questions we love to see students addressing in their PoW submissions, because we love having a glimpse of students’ reasoning, sense-making, problem solving, habits of mind, or, yes, mathematical practices.