Defining what teachers need to know to teach algebra successfully is important for informing teacher preparation and professional development efforts. Based on prior research, an analysis of video, interviews with teachers, and an analysis of textbooks, the authors define categories of knowledge and practices of teaching for understanding and assessing teachers' knowledge for teaching algebra. They argue that the combination of categories and practices must be covered in assessments of teacher knowledge, if the assessments are to be used in research that investigates the presumed links among teachers' content preparation, their knowledge, their practice, and student learning.

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In 2005, the International Association for the Evaluation of Educational Achievement (IEA), Michigan State University, and the Australian Council for Educational Research took an important step in advancing the field of education by partnering to develop and implement the first international and comparative study of mathematics teacher education. The study was made possible by the substantial funding received from the National Science Foundation, the IEA, and the collaboration of 17 participating countries. The purpose of this article is to illustrate the methodology used in this major cross-national study of teacher education-the IEA Teacher Education and Development Study in Mathematics, known as TEDS-M-and to share its main findings related to the mathematical preparation of future teachers.

In order to give insights into cross-national differences in schooling, this study analyzed the development of multiplication and division of fractions in two curricula: Everyday Mathematics (EM) from the USA and the 7th Korean mathematics curriculum (KM). Analyses of both the content and problems in the textbooks indicate that multiplication of fractions is developed in KM one semester earlier than in EM. However, the number of lessons devoted to the topic is similar in the two curricula. In contrast, division of fractions is developed at about the same time in both curricula, but due to different beliefs about the importance of the topic, KM contains five times as many lessons and about eight times as many problems about division of fractions as EM. Both curricula provide opportunities to develop conceptual understanding and procedural fluency. However, in EM, conceptual understanding is developed first followed by procedural fluency, whereas in KM, they are developed simultaneously. The majority of fraction multiplication and division problems in both curricula requires only procedural knowledge. However, multistep computational problems are more common in KM than in EM, and the response types are also more varied in KM.

Throughout the history of American education, learning to write proofs has been an important objective of the geometry curriculum for college-bound students. At the same time proof writing has also been perceived as one of the most difficult topics for students to learn. Until recently, the extent of students’ difficulties with writing proofs has been largely a matter of conjecture, for little research has been conducted in this area.