## Wednesday, October 12, 2011

### Now what? Part III

Thus far we have considered ways middle school learners can extend their learning by generating their own problems based on young adult literature (here) and how preservice teachers  can extend their understanding by considering alternative solutions (here). Given the four different answers they usually come up with (1/15, 1/30, 1/21, and 1/36), the preservice teachers attempt to revise the original problem to match each answer.

For 1/15, the clearer question might be:
In the Community, you get two pets. There are six choices of pets to have: dog, cat, fish, snake, bird, and hamster. The Elders pick the pets for each family without any duplication (e.g. no cat-cat pairs).  What is the probability of getting a dog and a cat if the order doesn't matter (i.e. cat-dog is the same as dog-cat)?
Typically, this is how the preservice teachers read the original problem even though the original lacks many of the specifics. They see it as a combination problem and so add the necessary conditions in their head. It helps these future teachers to be aware of the implicit conditions hiding in many problems.

In the Community, you get two pets. There are six choices of pets to have: dog, cat, fish, snake, bird, and hamster. The Elders pick the pets for each family without any duplication (e.g. no cat-cat pairs).  What is the probability of getting a dog first and then a cat?
Because the preservice teachers are familiar with permutations this revision is fairly simple for them.

1/36 is usually the third answer they choose work to find the question for:
In the Community, you get two pets. There are six choices of pets to have: dog, cat, fish, snake, bird, and hamster. The Elders pick the pets for each family. Duplications, cat-cat pairs, are possible.  What is the probability of getting a dog first and then a cat?
This, too, comes quickly since they are comfortable with the multiplication principle.

In the Community, you get two pets. There are six choices of pets to have: dog, cat, fish, snake, bird, and hamster. The Elders pick the pets for each family. Duplication, cat-cat pairs, are possible.  What is the probability of getting a dog and a cat if the order doesn't matter (i.e. cat-dog is the same as dog-cat)?
While they come to this version of the problem easily based on the prior revisions, this is a new context for them and they are eager to explore it further.

Before we look at the math, I engage the preservice teachers in a discussion of the pedagogical worth of having "messy" problems with many possible interpretations. They are inclined to want to clean up problems before sharing them with students but they recognize this is often based on their own experiences in math class. Fortunately, there are usually some voices that identify how considering different points of view made the problem much richer. This ability to identify underlying conditions and considering the alternative problems supports the "Now what?" stance I hope to foster.

We explore the "Now what?" question regarding how to generalize the "1/21 case" in the next post.