‘‘Once More About the Quadratic Trinomial…’’: On the Formation of Methodological Skills Alexander Karp Published online: 16 October 2007 Ó Springer Science+Business Media B.V. 2007 Abstract This article discusses certain aspects of mathematics teacher education in Russia. To be able successfully to teach through problems, a teacher must know how to select the necessary assignments, to construct mathematical tasks according to methodo- logical principles, and to organize work on these assignments in the classroom; in short, as they say in Russia, the teacher must possess certain well-defined methodological skills. This article analyzes examples of work with pre-service and in-service teachers leading to the formation of such skills. As the author tries to show, substantive and deep mathematical activity can take place in the context of tidying even the most routine school-level topics, such as the quadratic trinomial. Keywords Problem solving Á Methodological skills Á Quadratic trinomial Introduction This paper arises from teacher education both in the former Soviet Union and in present- day USA. My experience of working in each of these contexts has led me to understand that working on the construction of mathematical problems can be a critical aspect of teacher education. Although problem solving is today virtually universally recognized as the most important part of school-level mathematics (NCTM, 2000), in practice it is often perceived merely as the solving of certain special problems, for example, textbook exer- cises, or as something separate from the bulk of the teacher’s activity. Therefore, teacher education must aim at accustoming teachers to genuine problem solving, as a non-routine form of activity through which (and not alongside of which) the genuine study of math- ematics takes place. In this paper I shall address certain aspects of working with pre- service or in-service teachers on mathematical problems, relying on the experience of mathematics education in Russia. Note that such work in many respects resembles that A. Karp (&) Program in Mathematics, Teachers College, Columbia University, 525 West 120 Street, Box 210, New York, NY 10027, USA e-mail: apk16@columbia.edu 123 J Math Teacher Educ (2007) 10:405–414 DOI 10.1007/s10857-007-9037-9