We study the ideas about teaching and learning mathematics that undergraduate students generate when they encounter tasks designed to embed approximations of teaching practice in mathematics courses taken by a general population of students. These tasks attend to the dual goals of developing an understanding of mathematics content and an understanding of how teachers provide classroom experiences that foster mathematics learning. The study employs a qualitative, multiple-case study methodology, with four cases bounded by the content areas of abstract algebra, single variable calculus, discrete mathematics, and introductory statistics. The data for the study come from undergraduate students' written work on mathematical tasks, interviews with a subset of students from each course, and interviews with each instructor throughout the term during which they implemented the tasks. Our findings indicate that students identified the broad applicability of teaching skills (discussed by 32 of the 61 interviewed students), recognized the value of examining hypothetical learners' mathematical work (discussed by 59 of the 61 interviewed students), and reported empathy for hypothetical learners (discussed by 38 of the 61 interviewed students). These findings persisted across the course content and course levels we studied, leading us to conclude that our findings can transfer to additional mathematics courses in secondary mathematics teacher preparation.