We discuss the current methods for determining the dynamic critical index z for the dynamic universality class n=1, d=2 where the nonconserved order parameter is the only slow mode (model A). We conclude that essentially all known methods (expansions, high-temperature expansions, Monte Carlo calculations, Monte Carlo renormalization-group calculations, and the real-space dynamic renormalization method) are, at their present level of development, inconclusive. We show, in particular, that if we analyze the available high-temperature expansion data using methods similar to those used in carrying out the expansions, the resulting series is too short to extract any nonconventional value of z. At this level of expansion, the series is compatible with a conventional value of z. We show that these difficulties appear to be associated with the existence of an asymptotic dynamic critical region much narrower than the asymptotic static critical region.