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This study compares the views of teachers and professional development facilitators about effective professional development (PD). We analyzed interviews with 72 teacher participants and 23 PD facilitators involved in nine science and mathematics PD projects. The teachers' themes for characterizing effective PD included classroom application, teacher as learner, and teacher networking. Similarly, the PD facilitators discussed effective PD as having classroom application and experiences for teachers as learners. In addition, PD facilitators shared the need to develop collegial relationships with teachers and improve teacher knowledge. These views correspond to some of the standards and recommendations described in policy and research documents on effective PD. Criteria of effective PD in these documents that the participants did not mention included: (1) challenging teachers' content and pedagogical content knowledge with transformative learning experiences, (2) encouraging teacher leadership for sustained support, and (3) focusing on student learning by instructing teachers on how to use student data to inform their teaching practice. Our findings have implications for designing PD that reflects the criteria of standards-based reform.

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This study reports on the algebraic generalisation strategies used by two fifth grade students along with the factors that appeared to influence these strategies. These students were examined over 18 instructional sessions using a teaching experiment methodology. The results highlighted the complex factors that appeared to influence student strategy use, which included: (a) input value, (b) mathematical structure of the task, (c) prior strategies, (d) visual image of the situation, and (e) social interactions with the teacher and other student. Particular combinations of these factors appeared to increase the predictability of student strategy use. However, the complex nature of the factors influencing these strategies demonstrates the challenges that exist in encouraging students to move toward more sophisticated strategies.

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