Six groups of students in an 8th grade class have rank ordered the eight criteria that need to be considered in purchasing a pair of sneakers. The class has aggregated the rankings by adding each group's rankings. In this snippet, the teacher raises a question about finding the average of the rankings.

*Natasha Speer, mathematics educator*: I was interested in the
knowledge of student thinking that a teacher might need to pull off
this type of instruction. Was the teacher surprised to hear students
say it could change the order? Should she answer the question herself
or pursue it further with them?

*Bill McCallum, mathematician*: What do students need to see
about the algorithm for computating averages, and its relation to the
computations so far, given that they don't know algebra? Is it clear
to the students that computing the average is matter of dividing
by 6 the sums they have already computed, or are they going to go
back to the original numbers and
compute from scratch? Understanding the structure of algorithms is a
preparation for algebra.

*Steve Paulsen, teacher*: She was trying to sum it up but also
pursue the moment further. It seemed an awkward moment: should I take
the time and pursue this to understand a key mathematical idea, or
will it come out in later examples? What's coming next once this is
over? Where does it lead from here? How much should these students
understand about average at 8th grade?

The process by which a group can resolve the different opinions of its members into a single choice has a mathematical basis in voting theory. Of the many different voting methods available, students working the Sneakers Problem used modified versions of two.

Some groups used a modified Plurality Method. Under the Plurality Method, the choice with the most first-place rankings wins. The groups used the Plurality Method to determine that cost should be ranked first, and then modified the method to find the criteria that should be ranked second, third, and so on.

Other groups used a modified Borda Count Method. They gave one point for each first-place ranking, two for each second-place ranking, and so on, and then summed the ranks and ranked the criteria based on the sums. (In some descriptions of this method, points are assigned in reverse order.)

For more information and references, see the Wikipedia article.

This case is taken from the project Developing Multimedia Case Studies for Preservice Teacher Education, directed by Janet Bowers, Helen M. Doerr, Joanna Masingila, and Kay McClain.

If you would like to learn more about it, contact Joanna Masingila.