Profound has three related meanings -deep, vast, and thorough-and profound understanding reflects all three. It is primary because it contains advanced mathematical topics in rudimentary form, and it is elementary because it is at the beginning of students' learning. ![]() Fundamental mathematics is a foundation for later learning. To analyze the responses of both groups of teachers, she developed the notion of profound understanding of fundamental mathematics (PUFM). She investigated this suspicion in her dissertation research, asking Chinese elementary teachers the same questions that had been asked of U.S. Her memories of elementary teaching in China suggested that Chinese teachers would react quite differently. As a graduate student, the first author of this paper worked on this project and was struck by the teachers ' responses. ![]() Teachers were asked how they would respond to classroom scenarios in which mathematical ideas played crucial roles. What mathematical knowledge do elementary teachers use in teaching? About 10 years ago, Deborah Ball and her colleagues at Michigan State University carried out a research project to address this question (Ball, 1988 Kennedy, Ball, & McDiarmid, 1993). Knowledge of Fundamental Mathematics far Teaching 1
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