Observing Representation
"The ways in which mathematical ideas are represented is fundamental to how people can understand and use those ideas. When students gain access to mathematical representations and the ideas they represent, they have a set of tools that significantly expand their capacity to think mathematically."

## (NCTM, 2000, p. 67)

As we begin our work on this session, it's important to recall that even in high school, "representation" still encompasses the informal forms, such as diagrams or sketches, that students make up on their own. But at the same time, at this level, students are also introduced to a range of conventional and more complex representations, such as matrices, which have specific features that must be learned. The challenge for the teacher is to help students develop meaning for a new representation by connecting it to ideas and forms they already understand.

Rich problems provide a context for introducing new forms of representation. As students work, they can be encouraged to make sense of various aspects of the representation. This, in turn, can help students make decisions about when it is appropriate to use it, and help them recall how to use it in future situations.

Now let's observe how students worked on a rich problem. As you proceed, think about the importance of representation to the solution of the problems and the development of student understanding of the mathematical representations.

Mr. Karsky's 11th-grade pre-calculus class had worked with matrices earlier in the term. In this lesson, they were making their first use of matrix multiplication to model a real-world phenomenon, in this case probabilities relating to taxicab trips. This is a particularly elegant and powerful use of representation, for, as the students observe, the mathematics of the problem is nearly intractable when represented solely through tree diagrams. But the problem becomes mathematically and conceptually much clearer when represented in transition matrices.

Here is the problem:

 In yesterday's class, students were given the following table that represents the probabilities of ending a trip at each of three drop-off destinations for taxis traveling among three sections of town. For example, the probability of picking up a rider in Southside and dropping him off Downtown is 30%. The class challenge: Calculate the following. Show your method. 1. What is the probability of starting in Downtown and being there after two trips? 2. What is the probability of starting in Downtown and there after three trips?

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