31The Mathematical Education of Prospective Teachers of SecondarySchool Mathematics: Old Assumptions, New Challenges1Joan Ferrini-Mundy, Michigan State UniversityBradford Findell, National Research CouncilWe address two questions: What mathematics do prospective secondary school mathematics teachers needto know? In what context should they come to know it? Consideration of both matters has implications forthe revision of the undergraduate program in mathematics.What mathematics do prospective secondaryschool mathematics teachers need to know?Teachers must know the mathematics they teach. Deciding exactly what this means, and then determiningwhat more mathematics they need, are not simple matters. Typically, two perspectives have influenced thedesign of programs for the preparation of secondary teachers, and both are relevant to mathematics depart-ments:1. Prospective high school teachers should study essentially whatever mathematics majors study—becausethis will best equip them with a coherent picture of the discipline of mathematics and the directions inwhich it is heading, which should influence the school curriculum.2. Prospective high school teachers should study mathematics education—methods of teaching mathemat-ics, pedagogical knowledge in mathematics, the 9–12 mathematics curriculum, etc.We argue in this paper that there is substantial knowledge that is necessary for effective teaching butwhich is neglected by this two-pronged approach. Furthermore, much of this knowledge is mathematical incharacter, and, as such, should be a responsibility of mathematics departments. Because this knowledge isparticular for the teaching of mathematics, it lies, in a sense, between mathematics education and traditionalundergraduate mathematics content. Keep in mind, however, that there is much outside of mathematics andmathematics education that all secondary school teachers need to know, about students, about learning,about teaching, about curriculum, and about the contexts of schooling.History of recommendations2The dominant approach to the mathematical preparation of secondary school teachers in the United States inrecent years is to require that they complete an undergraduate major (or a near-equivalent) in mathematics.——————1 The authors wish to thank Deborah Loewenberg Ball, Dick Stanley, Tom Rishel, Merle Heidemann, and Dawn Berk for their comments andassistance in preparing this paper.2 For more detail on the following history, see Gibb, Karnes, & Wren, 1970.32 CUPM Discussion Papers about Mathematics and the Mathematical Sciences in 2010Interestingly enough, a quick review of the recommendations in this century about the mathematical preparationof teachers reveals that this trend toward a near-major has generally grown stronger with each set of recommen-dations. For instance, the 1911 report of the American subcommittee of the International Commission on theTeaching of Mathematics recommended preparation in several areas of pure mathematics, applied mathematics(e.g., mechanics, astronomy, physics), surveying, a “strong course on the teaching of secondary mathematics,”other education, and “a course of an encyclopedic nature dealing critically with the field of elementary math-ematics from the higher standpoint” (International Commission on the Teaching of Mathematics, 1911, pp. 13–14). There is no explicit call for a major in mathematics. Likewise, the 1935 recommendations of the Mathemati-cal Association of America’s Commission on the Training and Utilization of Advanced Students of Mathematicscalls for “minimum training in mathematics that goes as far as 6 hours of calculus, Euclidean geometry, theory ofequations, and a history of mathematics course” (Commission on the Training and Utilization of AdvancedStudents in Mathematics, 1935). The courses that might have been more typical of a major at that time (advancedcalculus, mechanics, projective geometry, additional algebra) are described as “desirable additional training.”In reports from various groups in the late ‘50s and early ‘60s, the expectations for secondary teachersbegan to sound like a major, with calls for 24 semester hours of mathematics courses (National Council ofTeachers of Mathematics [NCTM], 1959), and 30 semester hours, including abstract algebra (American Asso-ciation for the Advancement of Science, 1959). It was the Committee on the Undergraduate Program in Math-ematics (CUPM) that first recommended that “prospective teachers of high school mathematics beyond theelements of algebra and geometry should complete a major in mathematics” (CUPM, 1961). Ten years later, thissentiment was still strongly held: “We regard it as a matter of great importance that a program for teachers shouldbe identical to the one offered to other mathematics majors, except for a few courses peculiarly appropriateto prospective high school teachers” (CUPM, 1971, p. 170).The 1983 CUPM recommendations do not explicitly call for a mathematics major, but instead list 13courses, including a 3-course calculus sequence, as the minimal preparation, with a call for additional workfor teachers of calculus (CUPM, 1983). It is worth noting that 13 courses is more than a major in someinstitutions. In 1991, the MAA’s Committee on the Mathematical Education of Teachers (COMET) assumedresponsibility for the preparation of teachers: “These recommendations assume that those preparing to teachmathematics at the 9–12 level will complete the equivalent of a major in mathematics, but one quite differentfrom that currently in place at most institutions” (Leitzel, 1991, p. 27). The recommendations list standards inseven content areas (e.g., geometry, continuous change, and mathematical structures) rather than specific courses.Since the first CUPM recommendations, most major sets of national committee recommendations of-fered by the mathematics community and most recommendations from the education community have rec-ommended the equivalent of a major in mathematics as the fundamental preparation for the secondaryteacher. Sometimes the recommendation is general and assumes that whatever is considered appropriate asa major is appropriate for future mathematics teachers. For instance, the new recommendations of the Na-tional Council of Accreditation of Teacher Education (NCATE), to go into effect next year, expect thatcandidates for teaching should “know the content of their field (a major or the