Investigations Blog

What Happens When There Are More Than 10?

Imagine you are 6 years old. Or 7. You know you can use your fingers to model subtraction. For example, for a problem where there are 7 grapes and 2 get eaten, you can raise 7 fingers, put down 2, and count how many are left. But what do you do when the problem involves more than 10 things? Take a moment, before looking at the student work and video below. How would you use your fingers to solve a problem about having 12 scissors, and lending 5 to another class? A student I’ll call Miguel is a...

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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...

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Watch, Notice, and Learn

Classroom video is a powerful tool for studying and reflecting on mathematics teaching and learning. Unlike in-the-moment interactions with students, watching video enables us to slow down and more closely examine student-to-student exchanges. This affords us a unique opportunity to learn about students’ mathematical thinking. Several months ago, I began watching and discussing video footage of elementary mathematics classrooms with a group of colleagues. We were using the footage to study...

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“What Was Thomas Thinking?”

I often think about a lesson, captured on video some years ago. Liz, a fifth-grade teacher, gave her students two-digit multiplication problems—12 × 29 and 36 × 17— and asked them to come up with strategies other than the conventional algorithm to perform the calculation. For 12 × 29, Jemea thought about twelve 30s and then subtracted 1 for each of 12 groups. 360 – 12 = 348. For 36 × 17, Duane thought of 36 bowls, each holding 17 cotton balls. Ten bowls hold 170 cotton balls, and there are...

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