This descriptive study explored understandings of five learners in a third-grade mathematics class during a year in which their teacher attempted to change her teaching method. The study also examined the teacher's self-images of teaching while using the new discourse-based approach. Data was obtained from: (1) interviews with the teacher throughout the year; (2) interviews with each student in October and June; (3) small group problem-solving interviews in mid-June; (4) notes from weekly observations during the school year; (5) videotapes of all 20 days of instruction in a multiplication and division unit taught in April and May; (6) copies of students' mathematics notebooks; (7) students' grades on timed tests of number facts; and (8) students' California Test of Basic Skills scores from tests taken in the spring. The five students included an African-American boy, a Chinese boy, a Cameroonian boy, an Iranian-American girl, and a Malaysian girl. A narrative, interpretive presentation of data throughout the year comprises the majority of this document. Analyses revealed how different students had different experiences and constructed different understandings which related to the students' personalities and expectations of their learning in mathematics. Appendices include summaries of all data analyses; selected questions from student interview protocols; and explanations of categories used to code students' multiplication and division strategies. (MM)