Clip 6/9: Interpreting Lesson Part 2B
After Mia Buljan’s third-grade students work individually on a multiplication/division equal-grouping-scenario problem, Buljan brings them together on the carpet. She invites them to choose among several cards to find the one that describes in words “what’s happening in your problem.” Each student matches his or her math story/contextual word problem to an explanation in words.
MIA BULJAN: The second thing I notice is how I use this “match making” idea during math class. As predicted, we encountered some controversy on what is happening in the math stories. Esbin and Enmy both have the following problem: “Sam’s dad buys 24 hot dogs for Sam and his 3 friends. How many hot dogs can they each have?”
Again, in terms of reading comprehension, that tiny word “and” can throw a wrench into the works. Is it 24 divided among 4, or is it 24 divided among 3? Sure enough, Esbin is one of the first students we see, explaining his way of dividing 24 among 4. Enmy explains to me that she did it as 24 among 3. I am not surprised by her interpretation, but I know that at some point she is going to be talking to other kids who solved the same problem, and she would have to confront the issue of including Sam or not. Sure enough, Esbin comes by and shows me his work, and I immediately ask them to go to the carpet and talk to each other about what is happening.
I cannot overemphasize the power this teaching choice has had in my practice. Rather than group kids at the start of a session, when their ideas are still forming, I wait until they are deeper into their thinking, and then I curate groups that will have interesting conversations. Did these two have the same idea, but different strategies? They should talk. Did those two have different ideas and come up with totally different answers? They should talk. It takes some work, creating a class culture where students learn to listen to each other, where they focus on understanding each other versus “being right,” and where we have fostered the idea that it is a totally acceptable (and coveted) mathematical behavior to let yourself be convinced by mathematically persuasive arguments. Having said that, the work it takes to create that environment is one of the greatest investments I have made, as a teacher, in my classroom.
As I watch their conversation unfold here in the video, I’m reminded of how aware I felt that Enmy may be changing her mind because it was expected, rather than because she was legitimately changing her mind about whether the problem needs her to include Sam. She originally built the problem by laying down 3 “hundreds” flat place value blocks, as 3 children, and giving them 8 unit cubes (“hot dogs”) each. When Esbin explains that he included Sam, Enmy counters with it wouldn’t work to put down another flat. She knows that the problem works for three kids, so it doesn’t occur to her to dismantle her work and restart distributing the unit cubes among four. Since it works among three, she assumes that adding another flat means that the fourth flat (“child”) will have zero cubes. I encourage her to test it out, show us what would happen. When she sees that it does work, she immediately switches to saying that’s the right way.
But, again, I was asking myself, does she really think that’s the right way to do it now? Or does she just think, “oh it works, that must be what they want me to do”? I question her about the wording in the problem, and right when they both seem satisfied with 24 among 4, I can’t help myself, I ask one more question: How would the problem change if it were supposed to be 24 among 3? What words would be used? Enmy nails it with “Sam’s dad buys 24 hot dogs for Sam’s three friends” and, funny enough, when I ask Esbin the same question, he also gets there but sounds less confident, to my ear, than Enmy does.
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