The psychology of science typically lacks integration between cognitive and social variables. We present a new framework of team innovation in multidisciplinary science and engineering groups that ties factors from both literatures together. We focus on the effects of a particularly challenging social factor, knowledge diversity, which has a history of mixed effects on creativity, most likely because those effects are mediated and moderated by cognitive and additional social variables. In addition, we highlight the distinction between team innovative processes that are primarily divergent versus convergent; we propose that the social and cognitive implications are different for each, providing a possible explanation for knowledge diversity’s mixed results on team outcomes. Social variables mapped out include formal roles, communication norms, sufficient participation and information sharing, and task conflict; cognitive variables include analogy, information search, and evaluation. This framework provides a roadmap for research that aims to harness the power of multidisciplinary teams.