The pursuit of multiple educational outcomes makes teaching a complex craft subject to potential conflicts and competing commitments. Using a dataset in which teachers were randomly assigned to students paired with videotapes of instruction, we both document and unpack such a tradeoff. Upper-elementary teachers who excel at raising students' math test scores often are less successful at improving student-reported engagement in class (and vice versa). Further, the teaching practices that improve math test scores (e.g., cognitively demanding content) can simultaneously decrease engagement. At the same time, paired quantitative and qualitative analyses reveal two areas of practice that support both outcomes: active mathematics with opportunities for hands-on participation; and established routines and procedures to proactively organize the classroom environment. In addition to guiding practice-based teacher education, our mixed-methods analysis can serve as a model for rigorously studying and identifying dimensions of "good" teaching that promote multidimensional student development.