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# Investigations Blog

I was watching one of those legal shows on TV the other night. The prosecutor was asking the defendant a version of the same question for the third time. The defendant’s lawyer, getting annoyed, objected: “Asked and answered!” I’ve heard this phrase a hundred times in the (made-up TV) legal context, but this was the first time it struck me how pervasive this idea was in my own mathematics education, and how powerful it still is: If a student has given a perfectly good answer to a math...

## The Lesson? Students Never Cease to Surprise Me

On a recent visit to a school in a small city in the Midwest, Karen and I joined a class of 5th graders as they learned a game in Unit 3 called Roll Around the Clock. In the previous session, students used a clock to find and name fractions and equivalent fractions. For example, if the minute or hour hand moves from the 12 to 3, it has rotated 3/12 or 1/4 or 15/60 around the clock. Students would use these ideas in this lesson. In this game, players take turns choosing which of two dice to...

## How Do We Support Students in Reflecting on Mathematics?

As I worked with teachers in classrooms this fall, the topic of how to help students reflect on their learning and the learning of others kept coming up. I’m still thinking about how to recognize, encourage, and promote student reflection about math ideas. What traits do reflective students possess? How can a teacher nurture a learning culture where reflection is a natural part? When students engage in math experiences that include time to reflect on their reasoning and the thinking of others...

## Incomplete, inarticulate, ill-formed, incorrect: Brilliant!

Over the last decade, much of my work has been focused on mathematical argument in the elementary classroom. Observing in our collaborating classrooms, I was struck again and again by how teachers supported students to build on each other’s incomplete ideas. Constructing a mathematical argument is difficult and challenging for elementary students and, therefore, necessarily collaborative. When students are learning what it means to make an argument, not just about the solution to a single...