When teachers collaborate in meaningful, generative ways — digging into and questioning their own teaching practices to better support students — the benefits go both ways. Students learn more effectively, and teachers build a stronger, more supportive professional culture for themselves. How do coaches support this dialog to ensure that it is high-quality, which focuses on the relationships between students, the teacher, and the content? In other words, the instructional triangle (see part 1 of this three part post).
Some examples might be useful. Below we provide 2 examples, one in which a coach is planning a lesson with teachers. And the second is from a different coach reflecting with teachers after they have taught. In both these examples, you can see the different vertices of the instructional triangle coming into play.
Bringing the Instructional Triangle to Life: A Planning Conversation in Action
Along with our colleagues, Jim Hiebert and Kelly McKie, Lynsey analyzed the interactions of a highly skilled elementary mathematics coach who was supporting teachers who were enacting high-quality instructional materials. Below is a snapshot from one of our fifth-grade planning sessions that offers a clear example.
Teachers Carla, Tracy, and Diana, along with their coach, Nina, were deep in the process of identifying the lesson’s critical moment: the point where students would need to engage most directly with the day’s learning goal. The lesson centered on a fraction subtraction number string, and the team was deciding which expression would push students to choose between two possible models (a “clock” model and a “money” model) for solving it.
1/2 – 1/4
3/4 – 1/5
3/4 – 2/5
2/3 – 5/12
4/5 – 1/10
4/5 – 10/100
Carla offered her initial idea: the expression 2/3 − 5/12, because it’s the first problem where the money model simply won’t work, forcing students to reason about factors and denominators. Tracy agreed, adding that this expression is the first time students “can’t use the money model… [the] denominator really drives into a particular model.” Both teachers were reasoning about specific mathematical content—but also about how their own students typically default to certain strategies.
Carla then widened the lens, connecting the critical moment to what comes after it in the lesson:
“I really want to make sure I’m clarifying any…partial conceptions or ideas about the denominators to justify which model is the best or most efficient model for them to use. Because… I want them to walk away from this discussion feeling confident.”
In this snippet of discussion and beyond, Carla wove together all three vertices of the instructional triangle: the content (the relationship between the number string and the upcoming fraction task), her students (their potential partial conceptions and need for confidence), and her pedagogy (the discussion moves she’d need to make to get them there). Diana and Nina then added their own reasoning, each drawing on classroom experience to refine the group’s thinking.
What makes this conversation powerful is that the team arriving at that selection of the critical moment required teachers to move between thinking about the mathematics, their students, and their own instructional choices. This kind of integrated reasoning is what research suggests distinguishes planning conversations that meaningfully shape teaching practice from those that don’t.
Bringing the Instructional Triangle to Life: A Reflection Conversation in Action
Along with our colleague, Hannah Nieman, Lynsey analyzed the interactions of a different highly skilled elementary mathematics coach. Along with her principal, the coach had managed to create a productive learning environment for teachers in her school. They had regular opportunities to come together to analyze practice and engage in continuous improvement of their teaching. How did she do this? You can read more in Learning Together. But here, we want to talk about how she created the high-quality interactions with and among teachers that impacted their knowledge, skills and orientations toward teaching and learning—which ultimately impacted student learning opportunities and outcomes. You can read more about this study here.
This example comes from a second-grade team meeting in October, where teachers had just taught the same lesson: a game called Making 50, where students combine coins to reach 50 cents. The coach had visited each of the three classrooms briefly before the meeting and noticed that, even though all three teachers were teaching the same game, they had each set it up a little differently.
When the team gathered, the coach opened the meeting by naming what she’d seen the teachers launch the game in different ways: one teacher had a student demonstrate the game under a document camera, another played against the whole class in a circle, and a third had students play in pairs right away without playing as a class first. Another thing she’d seen: Two teachers gave students all the coins mixed together, while another teacher sorted coins into cups beforehand to make the game more accessible for kids who were still learning to tell coins apart.
What was striking about this conversation is how it moved between the three vertices: what students were doing and struggling with, the math they needed to understand, and the teaching decisions that shaped their opportunities to learn.
For example, one teacher shared that she wanted students to practice trading—swapping smaller coins for larger ones (like five pennies for a nickel)—but realized she’d had to keep prompting students to do this rather than them doing it independently. Another teacher described stopping his whole class mid-game to problem-solve together after some students ran into a conflict about not having enough pennies, which became a teachable moment about trading pennies for nickels or dimes.
Here’s the conversation that took place:
Anastasia: What if there is an emphasis on like the least amount of
coins or something?
Kris: I tried to have that. Well, I didn’t want to say, “least amount of
coins.” When we were playing in the circle, I would have different kids get the coins and then talk about which was the most efficient way to get them. … And then I count it in front of them. So it’s like 10, 20, 30.
Like, “Oh, that’s really efficient.” And then someone was like, “Well,
you can have a quarter and a nickel.” I’m like, “Let me count that. 25,
30.” I’m like, “Oh, which way was more efficient to count?”
Anastasia: Yeah, because when you say least, I notice there’s a
worksheet that they’re supposed to do, it said, “The least amount.”
They’re like, “Oh, well 25 pennies.” They’re thinking of value or
worth.
Tara: Mm-hmm (affirmative). Yeah, because pennies are the least.
Anastasia: Least. And that’s how I phrased it when I was like, “Okay,
with pennies, what’s the value, what’s the worth?”
Kris: So I’m hoping later on someone will make that connection of
like, “Oh, like the [fewer] coins there are, the more efficient it is to
count.”
Emily: Right. They kind of have to come to that on their own.
Kris: Yeah.
Tara: Because they’re going to have to understand what efficiency means here, so like- … emphasizing the efficient … that two quarters are more efficient than 50 pennies-
Emily: I like that.
This exchange brought together all vertices of the instructional triangle and supported teachers to consider how small instructional decisions have implications for students’ learning opportunities as they grappled with new ideas about what to emphasize with students moving forward.
By the end of the conversation, the team had a clearer, shared understanding of both the mathematical idea they wanted students to grasp and the specific instructional moves (e.g., language choices, materials, classroom setup) that could help students get there.
Summary
Across both examples, what stands out is the quality of the conversation itself. Whether teachers were planning a lesson before teaching it or reflecting on it afterward, the most productive moments happened when coaches created space for teachers to move fluidly between thinking about the mathematics, how their students respond to the mathematics, and their own pedagogical choices—rather than treating these as separate considerations. Carla’s planning conversation and Kris, Anastasia, Tara, and Emily’s reflection on “efficiency” both show how skilled coaches shaped what teachers talked about and how they connected it to the instructional triangle. This is the heart of high-quality coaching: cultivating conversations where students, content, and pedagogy stay in view together.
The ideas in this post are influenced by the following scholars: Jim Hiebert, Hannah Nieman, and Kelly McKie
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