The mathematical challenges coming from the social and behavioral sciences differ significantly from typical applied mathematical concerns. ""Change,"" for instance, is ubiquitous, but without knowing the fundamental driving force, standard differential and iterative methods are not appropriate. Although differing forms of aggregation are widely used, a general mathematical assessment of potential pitfalls is missing. These realities provide opportunities to create new mathematical approaches. These themes are described in an introductory, expository, and accessible manner by exploring new ways to handle dynamics and evolutionary game theory, to identify subtleties of decision and voting methods, to recognize unexpected modeling concerns, and to introduce new approaches with which to examine game theory. Applications range from avoiding undesired consequences when designing policy to identifying unanticipated voting (where the ""wrong"" person could win), nonparametric statistical, and economic ""supply and demand"" properties.