The teacher opens the class by reminding students what they did the previous Friday: decomposing the number 7. From the outset, he gives students ownership of their learning by using the examples of two students’ work as the refresher. He calls on one of the students to come up and point out where her group of three was. 

It’s noteworthy that the teacher refers to “what we noticed” about decomposing 7, and asks students to think about the two decompositions of 10: “What do you notice? How are they the same…” This is important because every student is the authority in answering the question, “what did you notice?” This is not the case when a student is asked for an answer to a question for which the teacher is the authority, such as: what does 7 plus 3 equal? The teacher is displaying skill in asking discussable questions.

The teacher uses a common feature of APT: giving students a few seconds to think individually, then talking to a partner. With children this young, these times must be kept short if they are to be productive, and the teacher signals that they will be thinking only for 10 seconds. 

In this segment, the teacher is working on several things simultaneously. He is working on an important mathematical abstraction (moving from objects—hearts—that can be counted, to set size (a quantity of 8), to a number in an equation that represents the set size). But he is also working on the skill of listening—by asking students to repeat what their classmate just said. He supports the students as the authority by having the student who made the original statement confirm that it was correctly restated, and if it was not, to make the statement again. While this takes time, that time is an investment. The better students become at listening, and the more they see themselves as part of the conversation, the more they will learn. Derek demonstrates this: he asserts that the two should also be counted. The teacher agrees and Derek begins counting: one, two, three. The teacher stops at two, but Derek has the idea that the whole 10 should be counted, including the two. Though the teacher chooses against this additional use of time for counting, it is noteworthy that Derek is fully engaged in this task and has ideas about working with the number 10. 

In this segment, the teacher is grappling with the challenge that children this young sometimes have with articulating their thinking. Enrique has noticed something that is an important mathematical idea at this age: that the quantity of a group of two is the same, even if the specific two are different —located in different places on the card. He articulates this once clearly: “They’re not in the same place, but they have the same number.” But his subsequent efforts are less clear, “the two hearts are not in the same place, but the number is in the right spot.” When another student tries to repeat the point, she actually makes a different, valid point: “they’re the same numbers, but they're not in the same group.” There is quite a bit of back and forth to get this sorted out; neither Enrique nor Isabelle is able to be clear. But with the teacher’s help, the point gets made. 

In this segment, the student who is chosen with an equity stick is silent for what is a long time. The teacher thanks the other students for giving her think time, which contributes to the classroom culture he is creating: waiting patiently is something we give to our other community members. 

The teacher begins with asking Monse what she notices about the equations. She says they are in the same place. They are indeed in the same place, but this is not at all what the teacher is after. With some guiding questions, Monse demonstrates an important understanding: the numbers in the equation are representations of the groups on the cards. It is noteworthy that the class has covered some very important mathematics, but the teacher credits every one of the ideas to the student who first expressed it. 

In under 15 minutes, the class has covered a lot of complex mathematics. The teacher clarifies as the lesson comes to a close the main mathematical idea: “even though the groupings are in different places, they have the same equation.” But he adds: “is that what I’m hearing you say?” Even in this statement of the main mathematical idea of the lesson, he is giving ownership to the students. 

The teacher allows a student who has been anxious to say something to make his point before they close. The student says that the equation is the wrong way. Rather than the usual 2+8=10, it is written as 10=2+8. The teacher says they will talk about that next. What’s important about this exchange is that the student is invested in understanding. He is paying attention, and noticing that this is unexpected. The moment is revealing both because it is confirmation that the teachers’ effort to give students ownership creates interest and gives students a sense that they have the right to turn the conversation to the things they would like to understand. It is evidence of the value of APT for this student.