As part of a 5-year project that examined teachers' knowledge and beliefs about mathematics and teaching mathematics, interviews were conducted with preservice elementary teachers during their certification year. The transcripts of five of these sets of pretest/posttest interviews were chosen as illustrative of the significant challenges that preservice teachers continue to face as they strive to deepen their subject-specific teaching knowledge and begin to address their beliefs about teaching and learning mathematics. (Contains 2 tables, 1 figure, and 2 notes.)