How We Listen: Astrid’s Thinking about 5 x 6

“Clinical interviews are an opportunity to listen closely,” I caught myself telling a colleague recently. But… is that enough? How we frame the act of listening changes our entire experience.

Listening Stances

At NCTM Boston 2019, I attended a session on “cultivating listening” by Kassia Wedekind and Allison Hintz. There, Kassia and Allison pushed my thinking about our own listening stances. When are we listening to evaluate student thinking? When are we listening to interpret student thinking? How can we listen in a generative way, where the listening seeks to transform both individual and shared understandings.

This comes from research by Erna Yackel, Michelle Stephan, Chris Rasmussen, and Diana Underwood (2003), and it’s brilliantly built upon in a blog post by Amie Albrecht (“Teaching Through Listening,” January 9, 2022).

When we listen to evaluate student thinking, we’re thinking about a particular response to a question — a correct answer. We might funnel our questions in an attempt to elicit this correct answer. It’s like our brains have an on/off switch, waiting for those magic words to trigger the flip.

We do not want to carry that energy into clinical interviews.

Instead, I employ either interpretive listening, where I probe student thinking to make sense of it, and, from there, identify what to leverage as we move into new learning experiences. In my most beautiful moments during a clinical interview, I am listening in a generative way. Understanding how the student has thought about the problem transforms me, too, and we begin a dialogue. (This reminds me a little of listening to understand Adrian, although it was a one-way viewing rather than a conversation.)

Astrid’s Thinking

This clip comes from an interview with fourth grader Astrid*. The interview protocol is Interview 6: Foundations of Multiplication from Listening to Learn (by Marilyn Burns & Lynne Zolli, published by Heinemann). I asked Astrid to solve 6 x 5, and then explain her thinking.

Astrid looked at me nervously during the interview, and frequently bit her lip. Her speech was punctuated by plenty of filler “ummms” and “and then…,” with an occasional, “well, kind of.” She isn’t speaking with a tone of authority, but what she says is northing short of astounding to me.

You’re just going to count 6 five times, or 5 six times: 5, 10, 15, 20… and then two fives is already 10… and then… which means… and six divided by 2 is three. So you need to make 3… fives. Um, well, kind of… but, so, um, you get those and then you add the 10, 20, 30.

While listening, I was doing my best to interpret what she was saying. I was trying to picture it in my head: 5, 10, 15, 20… growing towards 30. But then Astrid took an abrupt left turn towards proportional thinking.

In the clip, you can hear me trying to make sense of it.

Two groups of five make that ten, and then you’re going to have three groups of two groups of five. Does that sound like what you were thinking?

I thought about different ways that I could represent Astrid’s thinking.

We might start with a 5 by 6 rectangular array.

And then you partition the rectangle to show that 2 columns of 5 makes 10, and that that can be iterated throughout the array.

Six has three groups of 2, so that 10 is iterated three times! 10, 20, 30.

There are 30 tiles in all.

This also has me thinking about how we work with prime factors and the associative property of multiplication.

Astrid may have thought of the 6 as 2 x 3.

5 \times (3 \times 2)

(5 \times 2) \times 3

10 \times 3 = 30

5 \times 6 = 30

…which is all to say that it was essential that in an interpretive or generative way. Sure, Astrid knew that 5 x 6 = 30. What was more enlightening was the window into Astrid’s proportional reasoning, and to think about how we can leverage that in class. Will she be able to apply this thinking to new and novel situations? Will it help if I show her one of the diagrams that I made? How can we continue to develop her thinking here? How can we use this thinking during the unit on multidigit multiplication?

It is also important to note that Astrid is fairly quiet during class. How can we help her share some of her brilliance with others, and affirm her as a math learner?

And I wouldn’t have any of these questions if I’d listened just for the answer.

Other resources:

Albrecht, Amie. “Teaching Through Listening,” January 9, 2022.

Burns, Marilyn. Listening to Learn. Heinemann.

Yackel, Erna, Michelle Stephan, Chris Rasmussen, and Diana Underwood. 2003. “Didactising: Continuing the Work of Leen Streefland.” Educational Studies in Mathematics 54 (1): 101–26.

*As always, students are given pseudonyms.

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