From the perspective of Polya's problem solving steps, you can see that Marilyn Burns spent a significant amount of set-up time to having students come up with some examples of dividing the square into halves in different ways. This time spent on examples is part of helping children really understand the problem. The challenge in this problem isn't quantitative (big numbers) it's qualitative (lots of different ways to do things that you thought had only one or maybe two ways to do), so the examples go a long way to explaining what the problem really is.
This ties description in the first standard that students "start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals." In this lesson, we see the teacher guiding this process with the students' participation. Because the teacher is guiding the prcess, we can say that this is deliberately included in the lesson, but we can't say that students are taking ownership of the process in this lesson.
Marilyn describes different approaches taken by different children (counting first or drawing first), and shares a variety of solutions that children came up with. Children are clearly choosing different strategies to solve the problem, though it's not clear if the children are really aware of the strategy-development process. The strategies she describes all involve a step where the equality of the two parts is checked, so the children are reflecting sufficiently to verify that their solutions are correct.
Solving and checking solutions to a problem like this that is different from what the children have solved before involves some amount of making "conjectures about the form and meaning of the solution" (std 1) and "check their[ing] answers to problems using a different method, and ... continually ask[ing] themselves, “Does this make sense?”
In going back and forth between numerical and geometric reasoning, children were developing skills for "explain correspondences between" different representations of the problem situation.
At the end of the lesson, Marilyn had several children share their solutions and solution strategies, which gives children practice with "understanding the approaches of others to solving complex problems." (std 1)
One strategy that the class used was to count squares and find areas to determine if a shape was indeed half of the square. Moving back and forth between the counted areas and the drawn shapes involves contextualizing (interpreting as a geometric half) and decontextualizing (using numbers to describe or analyze why and whether it is half). (std 2)
This lesson involved students learning more about area as a model for understanding and representing fractions, but students weren't creating new models or using math to solve a real world problem, so standard 4 doesn't apply to this lesson.