Research on the relationship between teacher characteristics and teacher effectiveness has been underway for over a century, yet little progress has been made in linking teacher quality with factors observable at the time of hire. To extend this literature, we administered an in-depth survey to new math teachers in New York City and collected information on a number of nontraditional predictors of effectiveness, including teaching-specific content knowledge, cognitive ability, personality traits, feelings of self-efficacy, and scores on a commercially available teacher selection instrument. We find that only a few of these predictors have statistically significant relationships with student and teacher outcomes. However, the individual variables load onto two factors, which measure what one might describe as teachers' cognitive and noncognitive skills. We find that both factors have a moderately large and statistically significant relationship with student and teacher outcomes, particularly with student test scores.