Connecting Outcomes and Counting
There is much documented evidence that students struggle when solving combinatorics problems. Some common errors are grounded in misunderstanding what is being counted and consequently applying inappropriate formulas, such as ignoring order when order matters, and using order when it does not. We suspect that the frequency of such errors can be reduced when students focus more explicitly on the objects they are trying to count. We discuss two types of interventions that direct attention to connections between sets of outcomes and the processes used to count them. We also discuss questions of the form "what is the math outcome," which have been particularly successful. We conclude by showing how questions of this type can be answered by using positional representations of integers.