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As we near the turn of the century, there is considerable public discussion and frequent debate in the popular press and in scholarly journals about the quality of school mathematics programs. Many of the discussions center on how much mathematics American students know and are based on reports of studies of student achievement such as the National Assessment of Educational Progress (NAEP) and the Third International Mathematics and Science Study (TIMSS). The discussions have also focused on other issues including the latest set of recommendations from the National Council of Teachers of Mathematics (NCTM), Principles and Standards for School Mathematics: Discussion Draft. 1 The ongoing debates focus not only on how much mathematics students learn in school but also what mathematics they learn and how they learn it. The fervor of these debates is reminiscent of the rancorous discussions of the late 1950s and 1960s when the United States was thought to be behind in the space race and American students were thought to be ill prepared in mathematics and science.The current discussions and debates about mathematics education will likely lead to changes in mathematics education as they did in the 1960s. In the 1960s, for instance, elementary school mathematics textbooks became more formal in their treatment of many topics. This occurred in response to calls from reformers for more curricular attention to the structure and logic of the discipline of mathematics. Thus, students at the elementary school level began studying a curriculum that emphasized set ideas and the formal notation used to represent them. 209 Douglas A. Grouws is Professor of Mathematics Education at the University of Iowa. He was a Visiting Scholar at the Pacific Mathematics and Science Consortium at Pacific Resources for Education and Learning (PREL) when this chapter was written and he gratefully acknowledges their support. Kristin J. Cebulla is a Ph.D. student in mathematics education at the University of Iowa and a former teacher and research chemical engineer.