The authors propose a framework for teacher preparation programs that aims to help prospective teachers learn how to teach from studying teaching. The framework is motivated by their interest in defining a set of competencies that provide a deliberate, systematic path to becoming an effective teacher over time. The framework is composed of four skills, rooted in the daily activity of teaching, that when deployed deliberately and systematically, constitute a process of creating and testing hypotheses about cause-effect relationships between teaching and learning during classroom lessons. In spite of the challenges of acquiring these skills, the authors argue that the framework outlines a more realistic and more promising set of beginning teacher competencies than those of traditional programs designed to produce graduates with expert teaching strategies.

Flexibility in the use of mathematics procedures consists of the ability to employ multiple solution methods across a set of problems, solve the same problem using multiple methods, and choose strategically from among methods so as to reduce computational demands. The purpose of this study was to characterize prospective elementary teachers' (n = 148) flexibility in the domain of proportional reasoning before formal instruction and to test the effects of two versions of an intervention that engaged prospective teachers in comparing different solutions to proportion problems. Results indicate that (a) participants exhibited limited flexibility before formal instruction, (b) the intervention led to significant gains in participants' flexibility that were retained six months after instruction, and (c) varying the source of the problem solutions that participants compared had no discernible effects on their flexibility. Implications for mathematics teacher preparation and for research on flexibility development are provided.

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We investigated whether the mathematics studied in 2 content courses of an elementary teacher preparation program was retained and used by graduates when completing tasks measuring knowledge for teaching mathematics. Using a longitudinal design, we followed 2 cohorts of prospective teachers for 3 to 4 years after graduation. We assessed participants' knowledge by asking them to identify mathematics concepts underlying standard procedures, generate multiple solution strategies, and evaluate students' mathematical work. We administered parallel tasks for 3 mathematics topics studied in the program and one mathematics topic not studied in the program. When significant differences were found, participants always performed better on mathematics topics developed in the program than on the topic not addressed in the program. We discuss implications of these findings for mathematics teacher preparation.