In the Community, you get two pets. The Elders pick the pets for each family. There were six choices of pets to have: dog, cat, fish, snake, bird, and hamster. What was the probability of getting a dog and a cat?Typically, the preservice teachers came up with an answer fairly quickly. After all, since it comes after a unit on combinations the solution method seems obvious. Still, I ask them to explore the problem further by using one of the extension questions we collect over the course of the semester.
It is sometimes difficult for the preservice teachers to consider alternative answers, however, because of their own experiences with math problems having a single correct answer and the fact they think this problem is so cut-and-dry. Fortunately, I have examples of alternatives to their expected answer of 1/15 that were identified in previous classes. If no one comes up with these alternative answers in the current class, I offer them as other possibilities we ought to consider. I say, "A group came up with an answer of 1/30. Another was pretty sure that it was 1/21, although they also considered 1/36 after they hear the 1/30 rationale."
The preservice teachers' initial reaction is, "Those answers are wrong." I remind them that as educators we must consider that learners are not wrong but they may have answered a question different than what we expected. (I wrote about this here.) Therefore, the natural "Now what?" question that a teacher can consider is, "What question does this answer?"
To be continued...