Investigations Blog

The Bad Rep of Word Problems: “Two trains leave the station…”

When people want to make a joke about how difficult, convoluted, or inaccessible word problems are, they often cite some version of the “two trains” problem. You can see an example of this problem here: Maybe you want to try solving this problem yourself before reading on. (Maybe not!) The “two trains” problem has become an emblem in popular culture. Saying the opening phrase, “Two trains leave different stations at the same time …,” invariably results in uncomfortable laughter. It surfaces...

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Developing Mathematical Language is Hard Work

Using language to effectively communicate one’s mathematical thinking is an important skill—one that is a focus of Math Practice 6: Attend to Precision. Many of us know firsthand that clearly articulating mathematical ideas is challenging work, and that when students use ambiguous, imprecise terms in their explanations, their language can actually get in the way of understanding. Developing precise language is key if we want to students to engage in rich, collaborative discussions in which...

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“That Seems Way Too Big”

On a recent visit to a small district in the Midwest, I got the chance to visit a third grade class that was working on division (3U5, Session 3.4). When I joined Nicole, she was in the middle of working on the following problem: Gil loves toy cars. He saved enough money to buy 32 toy cars. How many 4-packs of toy cars did he get? (SAB p. 333) Below the problem, Nicole had written I asked her to tell me about her thinking. She said she wanted to start with something she knew, which I agreed...

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Asked and Answered: Why Ask the Same Question When You’ve Already Gotten a Perfectly Good Answer?

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...

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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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Using Tools in Art and Math

Recently, I was chatting with a 7-year-old I know pretty well. I asked her about school, and she quickly started telling me about her current math work, complete with eye rolls and boredom. I decided to change the subject a little. “I know you like to tell me about math because I love math, but what’s your favorite subject?” I asked. She thought for a moment and then said, “Art.” Art was never my favorite subject. I’m not even sure I’ve passed stick figure drawing yet. “Why?” I asked. She...

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