Preface Welcome to the fourth volume of Research in Collegiate Mathematics Education (RCME IV). Student learning and calculus are major themes in this volume. As in previous volumes, these are examined in a variety of ways. Seven of the eleven articles that comprise RCME /Fconcern different aspects of calculus. The first two give overviews of calculus reform in France and in the United States. The next two are small- and large-scale longitudinal comparisons of students who were enrolled in first-year reform and traditional courses. Four detailed studies of students' un- derstandings of calculus and related topics follow. We then switch gears to focus more directly on relationships between Instruction and students' understandings for courses other than calculus—abstract algebra and number theory—and finish with a cross-sectional study of a cross-cutting concept—quantifiers. Calculus reform Michele Artigue gives an overview of calculus in France, including relevant re- search as well as a brief history of curricular change. Unlike the United States, France has a national program of study teaching calculus at the high school level became widespread at the beginning of the 20th Century. Alongside the national curriculum has been a coherent research enterprise which, for some decades, has explored mathematics teaching and learning at the upper high school (including cal- culus) and College levels. This research enterprise, much more homogeneous than in the U.S., is grounded in a particular sociocultural perspective, including the idea (called the "didactic contract") that students and teachers enter the classroom with a set of mutual, though often tacit, expectations, which play a strong role in shaping their classroom behavior. French cognitive studies have also focused on "epistemo- logical obstacles," conceptual issues that have proven difficult both historically for the field and for individual students when learning the material. Artigue describes some of the theoretical frameworks used in research and some of the different kinds of Student difficulties relevant to calculus which have been documented by research. This is followed by an account of the evolution of the teaching of calculus at the lycee level (grades 10-12).The syllabus changed in the 1960s and 1970s due to the influence of the Bourbaki. Another change occurred in 1982, this time influenced by the findings of mathematics education research, and the curriculum focused on approaches that were more intuitive than the formal approaches inspired by the Bourbaki. The Situation in France may be of particular interest to readers from the United States because about 70% of French students take at least two years of calculus in high school, hence the French have been concerned far longer with the problem of how to make calculus accessible to the majority of students. (The following statistics give a sense of how many high school students in the United vii
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