This article focuses on the subject matter knowledge of preservice elementary and secondary mathematics teachers. In order to examine what teacher candidates understand about mathematics as they enter formal teacher education, results from questionnaires and interviews with 252 prospective teachers participating in a large study of teacher education are discussed. The results reveal the mathematical understandings that these elementary and secondary teacher candidates brought with them to teacher education from their precollege and college mathematics experiences, understandings that tended to be rule-bound and thin. Based on these data, the article challenges 3 common assumptions about learning to teach elementary or secondary mathematics: (1) that traditional school mathematics content is not difficult, (2) that precollege education provides teachers with much of what they need to know about mathematics, and (3) that majoring in mathematics ensures subject matter knowledge. These assumptions underlie current teacher education practices as well as proposals to reform the preparation of teachers.
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