execution of paper-and-pencil skills through demonstrations of procedures and repeated practice.

Teacher Preparation, Certification, and Professional Development

A bachelor’s degree and a teaching certificate are required to teach in most public schools in the United States. Teaching certificates are granted by states, usually based on the completion of specific undergraduate coursework and field experience in schools. Some states also require that candidates pass an examination. A teaching certificate from one state is occasionally honored across state lines; states without reciprocity of certification commonly offer a provisional certificate to out-of-state teachers until they have met all the requirements. Some states also offer alternative routes to certification for prospective teachers with a bachelor’s degree but lacking some of the requisite coursework or field experience.

Programs of teacher education have traditionally separated knowledge of mathematics from knowledge of pedagogy by offering separate courses in each.78 A common practice in university-based programs has been for prospective teachers to take courses in mathematics from the mathematics department and courses in pedagogy from the college or department of education, which is where they also get field experience and do supervised teaching practice. The standards for both types of courses have, in recent years, been influenced by reports such as A Call for Change,79 which listed expectations for the mathematics courses required in teacher preparation, and the Professional Standards for Teaching Mathematics,80 which concentrated more on issues of pedagogy.

Nationally, two-year colleges have been urged to play a larger role in recruiting future elementary and middle school teachers and providing college-level mathematics courses for them.81 At the same time, universities are exploring different ways of connecting courses on mathematics content and pedagogy and on giving students earlier and more intensive experience in school mathematics classes. Some recent programs have attempted to bring content and pedagogy together in both teacher preparation and professional development by considering the actual mathematical work of teaching.82

Although states have long set such requirements for teachers seeking certification, some have recently begun to impose higher standards for the knowledge teachers should have to teach children at a given age or grade level, requiring teachers to take specified courses and to pass assessments of their subject matter knowledge.83 There is considerable variation across states

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