  Understanding degrees of freedom in classical mechanics is fundamental to
characterizing physical systems. Counting them is usually easy, especially if
we can assign them a clear meaning. However, the precise definition of a degree
of freedom is not usually presented in first-year physics courses since it
requires mathematical knowledge only learned in more advanced courses. In this
paper, we use a pedagogical approach motivated by simple but non-trivial
mechanical examples to define degrees of freedom and configuration spaces. We
highlight the role that topology plays in understanding these ideas.
