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This report considers what it means for one teacher, Elaine Hugo, to teach mathematics for understanding. By looking closely at the goals Hugo has for her students and how these goals are reflected in how she talks about her teaching and what she does in the classroom, the authors explore what teaching for understanding means for her. The authors also look closely at a two-week unit on decimal fractions. Prior to and after this unit six target students were interviewed about their knowledge of decimals. In spite of Hugo's extensive knowledge of mathematics and concern that her students understand the mathematics they study, only two of the six target students showed different ways of thinking about decimals after the decimals unit. The authors conclude that teaching for understanding is a complex process. This complexity arises from many questions teachers trying to teach for understanding must address. These questions include what content should be taught in elementary mathematics classes, what it means to understand something, and what instructional techniques should teachers use. (Author)