Overview:  In this summary the teacher displays the diagram from page 2 of the problem on the board at the front of the room.  The teacher asks students to consider the height of the trees in real life, not the height of the trees as they are drawn on the paper.  The teacher invites Khalil to the front of the room to share his computation for finding the height of the trees.  Khalil shows his work on the board, indicating that he found that the tree in the diagram was 5/6 the height of the house.  Based on that, Khalil determined that the height of the tree in real life should 5/6 times 15, which is the height of the house in real life.  Khalil determines the tree is 12.5 feet tall.  The teacher asks Khalil a question to clarify Khalil’s computations, and Paige raises her hand to say that she disagrees with Khalil’s solution.  The students have a disagreement about the unit of measurement, and the teacher resolves the disagreement by stating that Khalil’s answer is correct, modulo some error in measurement.  After Khalil sits down, the teacher sets up a proportion to indicate that the tree is 5/6 times the height of the house.

Prior knowledge:  The teacher uses Khalil’s solution to establish knowledge of the school mathematics concept of proportional relationships.  Although Khalil does not use the vocabulary of proportions, and he does not set up a proportion in his computations, the teachers seems to recognize that Khalil’s solution makes use of the proportional relationship between the height of the tree and the height of the house.  After Khalil sits down, the teacher attempts to formalize this relationship by writing the equation “T = 5/6H” on the board, where T represents the height of the tree and H represents the height of the house.  It is not obvious from this summary whether Khalil, or the other students in the class, see the connection between Khalil’s solution and the teachers’ formalization of that solution.

Other points of interest:  The teacher tries to facilitate a mathematical discussion with students, especially by having purposefully selected Khalil to present his solution and by attempting to connect Khalil’s solution with a bigger mathematical idea (Smith & Stein, 2011).  In addition, the teacher invites other students to respond or ask questions about Khalil’s solution, perhaps enabling students to control the direction of the mathematical discussion (Chapin, O’Connor, & Anderson, 2003).  When the teacher invites students to ask Khalil a question, initially no has a question.  It is unclear whether every student has understood Khalil’s solution or whether no student wants to raise a question.  When Paige raises an issue with Khalil’s units, the teacher does not seem interested to pursue this issue, perhaps because the teacher knows Khalil’s solution is correct and wants to pursue it further.  Although the teacher attempts to facilitate a discussion between students, by the end of this explore we do not see how students have come to understand the proportional relationship between the tree and the house.