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What is a good mathematics problem? What is a good mathematics problem for students at a particular grade level? How can I make mathematics interesting to students and fun? Preservice teachers explored these questions as they participated in a pen pal project with fifth and sixth graders. The teachers presented the middle schoolers with custom-made, meaningful mathematics problems. The students honed their communication skills and received personalized feedback about their learning. Along the way, many became more motivated by mathematics than they had been, and some, stimulated by the interaction with college students, began thinking about the future and college.

Given a power series, the coefficients of the formal inverse may be expressed as polynomials in the coefficients of the original series. Further, these polynomials may be parameterized by certain ordered, labeled forests. There is a known formula for the formal inverse, which indirectly counts these classes of forests, developed in a non-direct manner. Here, we provide a constructive proof for this counting formula that explains why it gives the correct count. Specifically, we develop algorithms for building the forests, enabling us to count them in a direct manner.