I will discuss joint work with Alessandrini, in which we provide global coordinates for punctured surfaces and give explicit formulae for the action of each generator of the mapping class group in terms of these coordinates. If time permits, I will also explain, as an application, how these tools can be used to compute the stretch factor and the invariant foliations of a pseudo-Anosov mapping class in quadratic time, following joint work with Margalit, Strenner, and Taylor.s research area.