Now what? Part IV

So far in this series I have discussed the need to empower learners by getting them to ask and explore their own "Now what?" questions (here), considered possible answers to a messy learner-generated word problem (here), and identified implicit conditions associated with the different answers (here). In this final post of the series, I share my preservice teachers' efforts to extend our understanding of one of the possible answers to this word problem:

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?

Three of the possible responses represent combinatoric approaches that the preservice teachers are already familiar with: combinations, permutations, and the multiplication principle. The 1/21 answer (duplicates are allowed but order does not matter), however, represents a new approach.

Many of the learners decide to explore, "How could I generalize this result?" They quickly realize that the 21 comes from adding the combination, 6 choose 2, with the 6 pairs. Consequently, they hypothesize that the general case would be:

n choose 2 + n

In order to check this rule, they try out some simpler problems. 1 pet results in 1 possible pair. 2 pets result in 3 possible pairs. 3 pets result in 6 possible pairs. 4 pets result in 10 possible pairs. 5 pets result in 15 possible pairs.

While this satisfies many of them, a few embrace the idea of extending the problem and notice that the sequence 1, 3, 6, 10, 15, 21, ... looks familiar. In fact, it can be thought of as:

(n + 1) choose 2

This fascinates them and me. I knew this result going into the lesson but I resisted the urge to explore it further before the lesson. "Why does this work?" was a question I did not have an answer to ahead of time. I wanted to work with them instead of guiding them to the answer. (This is an important instructional approach that I wrote about here.)

We are able to show that the two approaches are equivalent fairly easily.

But this still does not explain why (n + 1) choose 2 works when selecting 2 pets from n animals allowing for duplicates.

Finally we begin listing possibilities which leads to us designing the following table. The first entry would be a pair of dogs.

The "+ 1" is the repeat column in the table. This satisfies the "Why?" question but leads us to consider what would happen if 3 pets were selected from 6 animals - allowing for duplicates. Time is up, however, meaning this will be something we can think about on the drive home. Enjoy!

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.

The next answer that my learners usually address is 1/30. The clarified question might read:

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.

Finally they address 1/21:

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.

Now what? Part II

In the prior post, I introduced a problem written by a seventh-grader as both an example of what middle school students could do when deciding what comes next and an opportunity for preservice teachers to develop and explore their own "Now what?" questions. This was the student-generated problem:

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...

Now what? Part I

Learners who lose the ability to make decisions are disempowered.

Brain Cambourne

This quote comes from Cambourne's theory of learning as it relates to responsibility. I see this "disempowerment" in the secondary math classes that I observe and the college courses that I teach. Students are constantly waiting for someone else, usually the teacher, to tell them what to do. The time when this is most evident is when a student finishes an assigned task and sits back waiting for the teacher to answer the question, "Now what?"

Disempowerment has tremendous consequences. It removes from the student the responsibility to be a self-directed learner. Once a task is complete, they fail to consider what might come next which results in a loss of cognitive momentum - disengagement. Their need to be directed by others to explore beyond the assigned task is unsustainable. What happens when a teacher is not available to tell them what comes next? They sit and wait.

This is why I spend a great deal of time in my classes, especially those populated with teachers-in-training, encouraging them to ask the "Now what?" question to themselves and not wait for me to tell them what comes next. Granted, this is not easy at first since it runs counter to years of training, but with time I have found that I can gradually release the responsibility of identifying extensions to my learners. It usually starts with me explicitly identifying the disempowerment issue and modeling what self-directed learners might do with extra time on their hands. After modeling this behavior multiple times, I share with my learners the responsibility of coming up with ways to maintain our momentum. My goal is that eventually learners will develop their own approach to extending their learning and thus empowering themselves.

An example of sharing this responsibility comes from a probability and statistics course I have taught for preservice K-8 math teachers. The activity is based on an article from Ann Lawrence, "From The Giver to Twenty-One Balloons: Explorations with Probability." I like this article because it models several possible extensions students might consider to further their learning. For example, after working on several probability problems related to The Giver, the middle school students in the article are asked to write their own problems given the story's context. Here are the results of this extension:

Novice teachers are sometimes hesitant to turn over this responsibility of generating problems to students because they feel unprepared to deal with what might be messy stories. Therefore, I ask my preservice teachers to look at Mark's story and consider possible solution methods and then what comes next. With practice, these future teachers can become more adept at dealing with unpredictable situations. Also, I explicitly state that their "Now what?" question can focus on either pedagogy (responding to Mark) or content (exploring the math) - it's up to them. Empowerment!

Before I share the typical results, I want to provide you the opportunity to explore Mark's problem. A chance to ask and answer your own "Now what?" question. I hope you will share your learning in the comments.