Marilyn Burns recently blogged about a 38 second video clip that I probably spent 380 seconds watching. (click on the image below to read the post — but then come back here!)

It’s a video of an upper elementary student, Adrian, explaining why he thinks $\frac{5}{6} > \frac{1}{4}$ orally. Please watch it.

#### Making Sense of Adrian’s Thinking

It’s a fascinating clip. After 38 seconds, I was left baffled by his reasoning. After 76 seconds, or two viewings, I felt like I had an emerging understanding of his strategy.

For the next 38 seconds, leading up to 114 seconds, I drew out a visual. Now I felt like I understood it, but this did not feel like an effective communication of it.

By the end of 5 viewings, or 190 seconds, I had mapped it out on polypad.

By 228 seconds, I had created a short screencast.

I returned to Marilyn’s blog post to see what she had written about Adrian’s thinking. Marilyn identified Adrian as having used common denominators to reason about this problem, but not in a way that I ever would have anticipated.

It was like Adrian was building up from unit fractions, but in order to do that with different denominators, he found common denominators. He started by comparing $\frac{1}{6}$ and $\frac{1}{4}$ by thinking of them as $\frac{2}{12}$ and $\frac{3}{12}$. Then he renamed yet another chunk of $\frac{1}{6}$ as another $\frac{2}{12}$, making for $\frac{4}{12}$ in all. That was already larger than the other fraction, and so he did not need to keep going. So instead of building from unit fractions — a strategy that we stress in 4th grade — he was building from renamed units. He preserved the chunk of the unit while renaming it.

All of my attempts to apply language feel somehow angular. Adrian’s explanation, though it took me repeated viewings to fully understand, was more elegant.

#### How might Adrian have communicated his thinking visually?

We frequently ask students to “show their reasoning.” So, I wondered, how would Adrian have done this? And would we have been able to understand this thinking from his work?

He might have used all numbers.

$\frac{4}{12} > \frac{3}{12}$

If I saw this on Adrian’s paper, I would have asked him to explain where the $\frac{4}{12}$ is in the problem. Without his explanation I would have been lost.

Or maybe he would have drawn something like this:

With a diagram like this, it’s easy to fixate on some of the mathematical errors — in size, or in how it’s labeled — and lose the sophistication that we hear in Adrian’s verbal explanation.

#### The Value of Clinical Interviews & Conferring

We make inferences all the time about student thinking, based on our experience with both the mathematics and other learners. If Adrian had drawn a diagram like this

I might have simply thought, “oh, he converted to common denominators,” and moved on without probing it. Maybe I would have been disappointed. “This would have been such a good problem to use other strategies. I wonder if he realized he could have compared quickly to a benchmark of a half.”

And I would have missed out on Adrian’s brilliance!

Time is a finite resource in the classroom, and we don’t have time to learn what every student thinks about every single problem. However, when we can make the time to listen — whether it’s conducting a formal clinical interview or conferring with a student during class — it’s transformative.

“The experiences I’ve had interviewing more than a thousand students have been professionally life changing. I realize that it’s quite a strong statement to say that something is life changing, but when I revisit the sentence, which I’ve done many times, it holds true.”

Marilyn Burns

When I ask students to explain their reasoning, I learn more about their thinking and sometimes even think more deeply about the math myself. Chunking is such an important strategy across mathematics — this is abundantly clear when I teach 7th and 8th grade — and here Adrian used chunking beautifully, in a way that I would not have anticipated.

This clip also reminds me of why I take notes when I interview students. Sometimes, it takes me time to process and ‘try on’ a student’s thinking. In another post, I’ll share a clip from a student named “Astrid” where note taking helped me understand her.

The reason that the Mathigon screenshot works better than my static initial drawing is because it models action. We can see the different unit pieces transforming into twelves as they are renamed. I think many students experience mathematical thinking as actions, and to perceive this action we need to hear it or see it play out. We need more communication about the movie in their mind.

If you take away nothing else from this, it’s that we can and should ask. This does not mean that students will always have an eloquent verbal explanation like Adrian did. I’ve had students who are new to the US draw pictures during interviews, or use gestures, or move around manipulatives. But when we give students a chance to communicate, we learn more about what they understand, and, when we are our best teacher selves, we leverage those understandings in our plans for future learning experiences.

## One more thing…

My session at Build Math Minds’ Virtual Math Summit 2023 is on using clinical interviews in the math classroom. It’s goes live on Sunday, February 26, at 9am PST/12pm EST in the US, but it’s available wherever you are! (Even the international space station? There’s only one way to find out!) Registration if free, and the recording will be hosted on the Virtual Math Summit site until March 6, 2023. I genuinely wrote this blog post in response to Marilyn’s, but, while we’re here…

1. Thanks for sharing your thinking here. I was fascinated by this clip, and now I feel like I got to talk to someone about it. My first watch I’m afraid I thought Adrian knew which was bigger because of course it is and couldn’t put a reason into words. By the time I got to his iteration argument, I was really humbled. Partly by that I would never have asked the question in the first place. 3/4 and 5/6, or 1/4 and 1/5… so much to think about. Also really looking forward to your VMS talk.

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1. Oh, for the first 20 seconds or so, I had a feeling Adrian was going to stumble through an explanation, but… not at all!

I tend to ask questions like the ones you gave, too: 3/4 and 5/6 or 1/4 and 1/5, which would push towards the strategies of comparing a distance from one or comparing the size of the unit. I might even listen FOR those strategies. Adrian’s humbled me, as well, because I was not listening for it at all. I had to listen closely to make sense of it.

Alos: iteration argument is a great way to describe it!

I’m a little nervous to watch my session. I don’t think I watched it back before submitting it! Here we go…

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